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1","1":"On the distinction between the first motion and the second or proper motions; and in the proper motions, between the first and the second inequality.","2":"The testimony of the ages confirms that the motions of the planets are orbicular. Reason, having borrowed from experience, immediately presumes this: that their gyrations are perfect circles. For among figures it is circles, and among bodies the heavens, that are considered the most perfect. However, when experience seems to teach something different to those who pay careful attention, namely, that the planets deviate from a simple circular path, it gives rise to a powerful sense of wonder, which at length drives people to look into causes.","3":"It is just this from which astronomy arose among humans. Astronomy's aim is considered to be to show why the stars' motions appear to be irregular on earth, despite their being exceedingly well ordered in heaven, and to investigate the specific circles whereby the stars may be moved, so that by their aid the positions and appearances of those stars at any given time may be predicted. ","4":"Before the distinction between the first motion(1) and the second motions(2) was established, people noted (in contemplating the sun, moon and stars) that their diurnal paths were visually very nearly equivalent to circles. These were, however, entwined one upon another like yarn on a ball, and the circles were for the most part smaller(3) circles of the sphere, rarely the greatest(4) (such ","5":"as here ABCE, FMNG cutting the equator AB in CN), part of them north and part south. They also saw that the stars have different speeds in this diurnal and apparent motion. The fixed stars are fastest of all, since those that are in conjunction with any of the planets on the preceding day (such as H with A and F) come to their setting first (such as H, moving along LK back to I). The sun (on ABE) is slower, as on the following day it stands at E and so its setting follows that of the fixed stars at I with which on the previous day it was conjoined on HA. Slower still than this, slowest of all the stars, is the moon, since after setting with the sun today (at A, the moon being at F), it lags by an appreciable interval (EG) tomorrow when the sun sets (at E, the whole heaven and the moon along with it having made a circuit around the earth along FMNOG). Hence the Pythagoreans, when they shared out musical tones among the stars, gave the lowest (the hypate among the strings of the lyre)1 to the moon, because the motions of both were slowest. Hence have originated the words \u03c0\u03c1\u03bf\u03b7\u03b3\u03bf\u1f7b\u03bc\u03b5\u03bd\u03bf\u03c2 and \u1f51\u03c0\u03bf\u03bb\u03b5\u03b9\u03c0\u03c4\u03b9\u03ba\u1f79\u03c2. The former of these terms originally corresponded to a star which, on the next day, comes to its setting earlier (the sun E is said to be \u03c0\u03c1\u03bf\u03b7\u03b3\u03bf\u1f7b\u03bc\u03b5\u03bd\u03bf\u03c2 with respect to the moon G). The latter term corresponded to a star that is slower in the first motion (such as the moon here), which is, as it were, abandoned and left behind (at G) by the swifter ones (E and I). For more on this subject see our Optics, Chapter 10. 3 ","6":"This first adumbration of astronomy consists, not of the unfolding of a cause, but solely of the experience of the eyes, extremely slowly acquired. It cannot be explained in figures or numbers, nor can it be extrapolated into the future, since it is always different from itself, to the extent that no spiral is equal to any other in increments of time, and none carries over into the next with a curvature of the same quantity. Nevertheless, there are some people today who, riding roughshod over two thousand years' work, care, erudition, and knowledge, are trying to revive this, obtruding admiration of themselves far and wide (an attempt which has not been fruitless among the ignorant). Those with more experience consider them with good reason to be incompetent, or (if, like that man Patricius, they want to be known as philosophers) to act mad with reasoning.","7":"For it was helpful to astronomers to understand that two simple motions, the first one and the second ones, the common and the proper, are intermingled, and that from this mingling that continuous sequence of conglomerated motions follows. And so, when that common and extrinsically derived diurnal revolving effect is removed, the fixed stars are suddenly no longer the swiftest and the moon slowest, but quite the opposite, the latter being the swiftest in itself and in its proper motion FG while the former are clearly very slow or immobile. When a planet (such as the moon at G) is \"left behind\" (by the sun at E or the fixed stars at I), it is carried in consequence* through FG more swiftly than the sun (through AE) or the fixed stars (through HI). It, however, it appears to be 'leading' among the fixed stars, it goes along with a retrograde motion. For example, if the sun at A along with a fixed star at H had been released from the same starting line AH on the previous day, so as to traverse BCDE and arrive at P while the fixed star traversed HLK and arrived at I, the sun, in the space of one day, would have retrogressed through the interval AP.","8":"This turned out to be of great profit in astronomy in grasping the simplicity of the motions. Instead of unending spirals, a new one always being added to the end of the earlier one at E or G, there remained little but the solitary circles FG and AE, and a single common motion, either of all the planets and the whole world as well in a direction opposite to the proper motions, or (with the world standing still, according to Aristarchus) of the earth's globe T around the axis QR in the same direction as the proper motions.","9":"Now that the first and diurnal motion had thus been set aside, and only those motions that are apprehended by comparison over a period of days, and that belong to the planets individually, were considered, there appeared in these motions a much more complicated mingling than before, when the diurnal and common motion was still mixed in with them. For although this residual mingling was there before, it was less observed\u2014less striking to the eyes\u2014because the diurnal motion was very swift. And so this residual motion was divided into minute parts and spread out over several days and several diurnal spirals. But now, that minute division and distribution of the star's proper motions over so many days was removed (that is, by the removal of the diurnal motion), and so all the proper motions of the stars, as many as they are, and all the confusion of this multitude shone forth more obviously. First, it was apparent that the three superior planets, Saturn, Jupiter, and Mars, attune their motions to their proximity to the sun. For if the sun would approach them they moved forward and were swifter than usual; where the sun would come to the signs opposite the planets they retraced with crablike steps the road they had just covered; between these two times they became stationary; and these things always used to occur, no matter what the signs of the zodiac in which the planets might have been seen. At the same time, it was clear to the eye that the planets appeared large when retrograde, and small when anticipating the coming of the sun with a direct and swift motion. From this, the conclusion was easily reached that when the sun approaches they are raised up and recede from the lands, and when the sun departs towards the opposite signs they descend again towards the lands. And finally, it was observed that these phenomena of retrogressions and increase of luminosity, just described, was moved through the signs of the zodiac in the order that tended from west through the meridian eastward, so that whatever has happened at one time in Pisces soon would come to pass similarly in Aries, then in Taurus, and so on in consequence.","10":"If one were to bundle all this together, and were at the same time to believe that the sun really moves through the zodiac in the space of a year, as Ptolemy and Tycho Brahe believed, he would then have to grant that the circuits of the three superior planets through the aethereal space, composed as they are of several motions, are real spirals, not (as before) in the manner of balled up yarn, with spirals set side by side, but more like the shape of a pretzel, as in the following diagram. ","11":" This is the accurate depiction of the motions of the star Mars, which it traversed through the aethereal air from the year 1580 until the year 1596, on the assumption that the earth stands still, as Ptolemy and Brahe would have it. These motions, continued farther, would become unintelligibly intricate, for the continuation is boundless, never returning to its previous path. Take note, too, that since the orb of Mars requires such a vast space, the spheres of the sun, Venus, Mercury, the moon, fire, air, water, and earth, have to be included in the tiny little circle around the earth A, and in its little area B. In addition, the greatest part even of this little space is given to Venus alone, much greater in proportion than is given to Mars here out of the whole area of the diagram. Moreover, we are forced to ascribe similar spirals to the remaining four planets, and much more complicated ones to Venus, if the earth stands still. Ptolemy and Brahe offer explanations of the causes, order, permanence, and regularity of these spirals, the former using individual epicycles carried around on the eccentrics of the individual planets, in imitation of the sun's motion, and the latter by having all the eccentrics carried around upon the single orb of the sun. Nevertheless, both leave the spirals themselves in the heavens. Copernicus, by attributing a single annual motion to the earth, entirely rids all the planets of these extremely intricate spirals, leading the individual planets into their respective orbits7 quite bare and very nearly circular. In the period of time shown in the diagram, Mars traverses one and the same orbit as many times as the 'garlands' you see looped towards the center, with one extra, making, say, nine times, while at the same time the earth repeats its circle sixteen times.","12":"Again, however, it was noticed that these loops in each planet's spirals are unequal in different signs of the zodiac, so that in some places the planet would retrogress through a longer arc of the zodiac, at others through a shorter, and now for a longer, now for a shorter time. Nor is the increment of brightness of a retrograde planet always the same. Because, if one were to compute the times and distances between the midpoints of the retrogressions, neither times nor arcs would be equal, nor would any of the times answer to its arc in the same proportion. Nevertheless, for each planet there was a certain sign of the zodiac from which, through the semicircle to the opposite sign in either direction, all those things successively increased.","13":"From these observations it came to be understood that for any planet there are two inequalities mixed together into one, the first of which completes its cycle with the planet's return to the same sign of the zodiac, the other with the sun's return to the planet. ","14":"Now the causes and measures of these inequalities could not be investigated without separating the mixed inequalities and looking into each one by itself. They therefore thought they should begin with the first inequality, it being more nearly constant and simple, since they saw an example of it in the sun's motion, without the interference of the other inequality. But in order to separate the second inequality from this first one, they could not do anything but consider the planets on those nights at whose beginning they rise while the sun is setting, which thence were called \u1f00\u03ba\u03c1\u03bf\u03bd\u03c5\u03c7\u03af\u03bf\u03c5\u03c2 [acronychal, or \"night rising\"]. For since the presence and conjunction of the sun makes them go faster than usual, and the opposition of the sun has the opposite effect, before and after these points they are surely much removed from the positions they were going to occupy through the action of the first inequality. Therefore, at the very moments of conjunction with and opposition to the sun they are traversing those very positions that are their own. But since they cannot be seen when in conjunction with the sun, only the opposition to the sun remains as suitable for this purpose.","15":"But since the sun's mean and apparent motions* are two different things, for the sun, too, is subject to the first inequality, the question is raised which of these releases the planets from the second inequality, and whether the planets should be considered when at opposition to the sun's apparent position or its mean position. Ptolemy chose the mean motion, thinking that the difference (if any) between taking the mean sun and the apparent sun could not be perceived in the observations, but that the form of computation and of the proofs would become free from difficulty if the sun's mean motion were taken. Copernicus and Tycho followed Ptolemy, carrying over his assumptions. I, as you see in Chapter 15 of my Mysterium cosmographicum, instead establish the apparent position, the true body of the sun, as my reference point, and will vindicate that position with proofs in Parts 4 and 5 of this work.","16":"But before that, I shall prove in this first part that one who substitutes the sun's apparent for its mean motion establishes a completely different orbit for the planet in the aether, whichever of the more celebrated opinions of the world he follows. Since this proof depends upon the equivalence of hypotheses, we shall begin with this equivalence.","17":"Chapter 2","18":"On the first and simple equivalence, that of the eccentric and the concentric with an epicycle, and their physical causes.","19":"And now, to begin, I take up the equivalence of hypotheses adopted to save [the appearances of] the first inequality, which were demonstrated by Ptolemy in Book III and by Copernicus in Book III Chapter 15.","20":"There, an eccentric is shown to square accounts with an epicycle on a concentric, provided, that is, that the line of apsides in the eccentric and the line through the center of the epicycle and the planet on the concentric always remain parallel, and that the semidiameter of the epicycle in the latter is equal to the eccentricity in the former, while the semidiameters of eccentric and concentric are equal. And also provided that, in the former, the planet is moved uniformly on its eccentric, so as to traverse equal arcs in equal times. ","21":"First, let A be the position of the observer and the center of the concentric BB on which is the epicycle BC, BE. Let the arcs between two B's, or the angles BAB, be equal, and the planet be first at C, then at E and G, with the lines BE, BG parallel to BC. Next, let \u03b2 be the center of the eccentric \u03b3\u03b6, with \u03b2\u03b3, \u03b2\u03b5 equal to AB, and let \u03b1 be the point at which the observer is, and \u03b2\u03b1 (the eccentricity) be equal to the semidiameter of the epicycle BC, BE, and parallel to them. Also, let the arcs \u03b3\u03b5, \u03b3\u03b6, that is, the angles \u03b3\u03b2\u03b5 \u03b3\u03b2\u03b6, be equal both among themselves and to the former angles BAB. I say that the distances AC, \u03b1\u03b3, are equal, and likewise AE to \u03b1\u03b5, AG to \u03b1\u03b7, AD to \u03b1\u03b4, AH to \u03b1\u03b8, and AF to \u03b1\u03b6; and again, that the angles EAC, \u03b5\u03b1\u03b3 are equal; and that in each instance the planet, although its motion is uniform, will appear from A, \u03b1 slow at C, \u03b3, and swift at D, \u03b4. As I said, Ptolemy demonstrated this in Book III, so there is no need for further discussion. To geometers, the diagram speaks for itself, and others may go to Ptolemy. ","22":"As for the physical accounts of these models, there is a greater difference between the two. That this may be very clear, it must be researched at somewhat greater depth, in one way from Peurbach2 using Aristotle's principles, and in another way from Tycho.","23":"Ptolemy has described these circles to us in their bare form, such as geometry applied to the observations shows. Peurbach set up a way for them to be traversed that follows Aristotle, who attempted this same thing upon the geometrical suppositions of Eudoxus and Calippus, by which they had treated their astronomy.","24":"And while these authors used 25 orbs to demonstrate all the inequalities of the planets, Aristotle (since he believed the heavens to be filled with solid orbs)3 thought that 24 other revolving orbs had to be interposed in order to free each lower orb from the revolving effect which, on account of the contiguity of surfaces, it was going to receive from the orb above it. So, having thus accumulated 49 orbs in all (or 53 or 55, following Calippus), he attributed to each its own mover. Any one of these would be responsible for the perfectly uniform motion of its own orb and all inferior ones which it encompassed. This motion would take place inside the closest surrounding superior orb, as if in a sort of place, and from the mover would proceed a constant determination [ratio] of both the direction in which the motion was to occur, and the swiftness with which the orb was to return to its starting position. Moreover, since that philosopher held the motion to be eternal, he also stipulated that the movers were eternal. Since they created motion for an infinite time, and since Aristotle knew that nothing material could receive the form of infinity, he maintained that they are also immaterial, and separable principles, and consequently immobile. Also, since he had constructed the world's eternity from the eternity of motion, and this duration of essence, the goodness and perfection of the whole world, was opposed to destruction, which was bad, he therefore attributed to those principles the highest perfection and the understanding thereof, and from good understanding the will to see it accomplished, lest the good not be done well. In this way, he introduced to us separate minds which, it turned out, were gods, as the perpetual administrators of the heavens' motions. They also bestowed a moving soul [anima motrix], more closely attached to the orbs and giving them form, so that the mind would only have to give assistance. This was either because it seemed necessary that the mover and the moving thing have some common ground, or because the potentiality, considered in relation to the distance to be traversed, should not be infinite, just as there exists no infinite motion, but only motion through a certain space in a certain time. They therefore transferred this potentiality for creating motion to a soul, for the meantime becoming subject to matter for the purpose of inhering in the orbs of the heavens.","25":"Now this coupling of mind and soul is indeed quite in agreement with the detailed considerations of the astronomers, even though the philosophers' mode of argument is chiefly metaphysical. For it is the same in humans: the moving faculty is one thing; that which makes use of the moving faculty according to the indications of the senses\u2014the Will\u2014is another. The senses differ from the moving faculty both in the means they use and in the excellence of their structure, which in the organs of sense is more admirable than in the seats of the motive faculty. Similarly, if we should propose these Aristotelian orbs as objects of contemplation, two things will present themselves to us: 1) the motive force, sufficient for the round orb, from whose activity and constant strength the time of revolution arises; 2) the direction in which it acts. The former is more correctly ascribed to the animate faculty, and the latter to its intelligent or remembering nature. Now although through this solidity of orbs all motions or celestial appearances are so provided for that nothing is left to the providence of the presiding movers, while the whole variety of motions is a consequence of the number and disposition of the orbs; an although nothing else is required but that the moving souls receive and retain their activity and be set going from the first moment of creation in whatever direction is theirs, sent forth from their prisons, as it were, into space;5 nevertheless, it must be kept in mind that there is need of the supreme mind to launch any of the planets in its own direction, as if into its fixed and proper province. Aristotle, who knew nothing of the world's beginning and did not","26":"believe in it, of necessity ascribed this function instead to the governors of the motions. The followers of Aristotle, and even Scaliger, who professes to be a Christian, openly contend that this motion of the orbs is voluntary, and that the principle of volition for them is intellectual intuition and desire.","27":"So, to return to Peurbach, certain others along with him (chiefly authors of books on the sphere), explain the first model by imagining for themselves one solid concentric orb of the thickness of the whole epicycle, with an epicycle in it, and in the epicycle a planet. Then they attributed two moving souls to these two orbs (if they should carry through with their physical considerations), both with the same amount of power, proportionally, so that they would complete their periods in the same time, although moving in opposite directions.","28":"The other model requires two deferents (which remain motionless so long as we abide by the simplicity of the motions, mentally removing the progression of the apogees), and one orb with the thickness of the planetary body. In this orb is a soul which drives it around with a uniform effort in that direction in which it was projected in the beginning. Thus, if this solidity of the orbs and the other assumptions be granted, in the first model BC and BE will remain parallel, and in the second, the orb \u03b3\u03b5 will go around the center \u03b2, even though the movers in the former pay no attention to AC, nor in the latter, to \u03b2. For they are governed by material necessity or by the arrangement and contiguity of the orbs.","29":"But, with arguments of the greatest certainty, Tycho Brahe has demolished the solidity of the orbs, which hitherto was able to serve these moving souls, blind as they were, as walking sticks for finding their appointed road; and hence the planets complete their courses in the pure aether, just like birds in the air. Therefore, we shall have to philosophize differently about: these models.","30":"Let it then be assumed among the first: principles, that every force by which motions of this sort: are administered dwells in the body of the planet itself, and is not: to be sought: outside it","31":"Now the planet: must execute a perfect circle in the pure aether by its inherent: force, an epicycle in the first model and an eccentric in the second. It is therefore clear that the mover will to have two jobs: first, it must have a faculty strong enough to move its body about, and second, it: must have sufficient knowledge to find a circular boundary in the pure aethereal air, which in itself is not divided into such regions. This is the function of mind. Please don't tell me that the motive faculty itself, a scion of a simple and brute soul, has a native altitude for circular motion, exactly like a stone's nature to descend in a straight line. For I deny that God has created any perpetual non-rectilinear motion that is not ruled by a mind. Even in the human body, all the muscles move according to tire principles of rectilinear motions. They either swell by contacting info themselves, or stretch out, the ends moving apart: in the former case, the member approaches the muscle, while in the latter, the member recedes. This same thing takes place in its own way in the circular muscles that are set up as guards for orifices. When they are extended by the circular filaments, they relax and open the passage, and constrict it when the filaments return into the form of a smaller circle. There is no member whatever that rotates uniformly and without impediment. On the contrary, the bending of the head, feet, arms, and tongue is expressed in certain mechanical devices by many straight muscles carried across or stretched out from one place to another. In this way it is brought about that the motive faculty, which by its own nature tends in a straight line, swings its member in a gyre. Likewise, certain machines raise water to great heights, not because the nature of the body, which conveys the motion, tends to an exalted position, but because, by an arrangement of channels, it is brought about that the water necessarily gives way upwards when a greater weight tends downwards. And even if the motion of certain members were perfectly circular, they7 nonetheless could not be perpetual. There should be no great wonder at this, since in the human body mind presides over the animate faculty. Surely, then, if there had been any way of so constructing some moving faculty that it might be able to rotate some body, it would not have been neglected in the human body.","32":"Besides, it is quite impossible for any mind to manifest a circular path without recourse to the guidepost either of a center or of some body which might appear under a greater or smaller angle according to its approach or recession. For a circle is both defined and brought to perfection by the same criterion, namely, equality of distance from the middle. No matter how many of these motive faculties you set up, a circle, even for God, is nothing other than what was just said. Geometers do, of course, show how, given three points on a circumference, to form a continuous circle, but in this very proof it is presupposed that some portion of the circumference (that which passes through the three points) is already constructed. Who, then, will show the planet this starting place, in conformity with which it will make the rest of its path? This is possible in no other way than if the planet's mover (as in Avicenna's opinion) imagine for itself the center of its orb and its distance from it, or if it be assisted by providing some other property of a circle in order to lay out its own circle.","33":" We will therefore now form the phys","34":"ical hypothesis of these two models in another way In the latter, simpler one, if our supposition is valid that the mover driving the planet around the path \u03b3\u03b5\u03b4 is in the planet itself, there will have to accrue to the planet's mover some sort of awareness of the apparent magnitude of the body at \u03b1 seen (or as if seen) from \u03b3, \u03b5, \u03b7, \u03b4. The planet will therefore have to strive both to move forward uniformly (this the undivided","35":"and unimpeded forces of the moving soul provide), and to exhibit all the distances \u03b1\u03b3, \u03b1\u03b5, \u03b1\u03b7, \u03b1\u03b4, in such order that they follow by a geometrical law from the eccentric \u03b2\u03b3. To this end, the mover should also know how much longer \u03b1\u03b3 is than \u03b1\u03b4; that is, by how much the path which it will traverse is eccentric from the body at \u03b1 around which it is to go. The planet's mover will thus be occupied with many things at once. To escape this conclusion, one must assert that the planet pays attention to the point \u03b2, entirely empty of any body or","36":" real quality, and maintains equal distances from that point.","37":"The prior model is explained physically thus. Let a motive power be","38":"conceived which, seated on the concentric B and itself without body, moves around the body at A with a uniform exertion of forces, maintaining equal distances from that point. Let there be another power in the body of the planet C, capable of holding its attention on the incorporeal power at B, estimating and maintaining its distance from that power, and moving uniformly around it. Thus, as before, this power again will have numerous tasks. But it is also incredible in itself that an immaterial power reside in a non-body, be moved in space and time, but not have a substrate to move itself (as I said) from place to place. And I am making these absurd assumptions in order to establish in the end the impossibility that every cause of a planet's motions inhere in its body or somewhere else in its orb, and to build a path for other less difficult forms of motions that are more readily persuasive. ","39":"I have presented these models hypothetically, the hypothesis being astronomy's testimony that the planet's path is a perfect eccentric circle such as was described. If astronomy will discover something different, the physical theories will also change.","40":"In this equivalence of hypotheses, not only the equality of the apparent angles at A and \u03b1, but also the actual paths themselves of the planets through the surrounding aether, each remain the same. For in both shape and size, the planet traverses an arc from C to E through angle CAE that is also the same as it traverses from \u03b3 to \u03b5 through the equal angle \u03b3\u03b1\u03b5.","41":".","42":"CHAPTER 3","43":"On the equivalence and unanimity of different points of observation and of hypotheses, for laying out one and the same planetary path.","44":"Next, I must show how one and the same planetary motion, while in itself remaining the same, can display one or another appearance, and how the pair of forms are equivalent here. ","45":"About centers A and \u03b3, with equal radii AC, \u03b3\u03b5, let the circles CD, \u03b5\u03b6 be described with CA, \u03b5\u03b3 drawn through the centers parallel to one another, and other lines AB, \u03b3\u03b4, and AD, \u03b3\u03b6, through the centers inclined to the former, both pairs likewise parallel. Also, about B let an epicycle be described, with radius BE, and likewise about D with radius DG equal to BE. Let the planet be placed at E and G, so that DG and AB are parallel. On the line \u03b4\u03b3 let a segment equal to BE be set out on the side opposite \u03b4, and let it be \u03b3\u03b2. Let G be joined with A and \u03b6 with \u03b2. The hypotheses will therefore be equivalent, by the preceding chapter, and to an observer placed at A and \u03b2, EAG and \u03b4\u03b2\u03b6 will be equal. EA and \u03b4\u03b2 will also be equal, as well as GA and \u03b6\u03b2. And finally, the arcs EG and \u03b4\u03b6 will be equal.","46":"Now let a smaller epicycle be described upon BCD with radii BI, CF, DH, and let AC be extended to F, and BI and DH be parallel to CF. And let the planet be on IFH. Again, by ch. 2, the circle IFH will be equal to the circle \u03b4\u03b6.","47":"Next, extend the arc IF from the point \u03b4, so as to end at \u03b5, and from \u03b5 through \u03b3 draw \u03b5\u03b3, such that \u03b5\u03b3 is parallel to CA. And let a magnitude equal to CF be set out on the line \u03b5\u03b3, and let it be \u03b3\u03b1, on the side opposite \u03b5. Let I and H be joined with A, and also \u03b4 and \u03b6 with \u03b1. Again, therefore, the hypotheses will be equivalent by the preceding chapter, and to an observer placed at A and \u03b1, FAH and \u03b5\u03b1\u03b6 will be equal, as well as FAI and \u03b5\u03b1\u03b4. FA and \u03b5\u03b1 will also be equal, as well as HA and \u03b6\u03b1, and IA and \u03b4\u03b1. And finally, the arcs FH and \u03b5\u03b6 will be equal and similar, as are also FI and \u03b5\u03b4, by construction. ","48":"Therefore, if the path of the planet remain the same while the observer is moved from \u03b2 to \u03b1, different appearances will be produced at the same moments of time. For the same places, \u03b4 and \u03b6, are viewed in different ways from \u03b2 and \u03b1. On the other hand, if the observer remain at A, and the quantity of the planet's path EG, IH remain the same while changing its place, the planet will again appear in different places, even when at the same place on its path, because the entire path has been shifted. Accordingly, since the planet, whether viewed from \u03b1 or \u03b2, is at \u03b4, or at \u03b6, at the same moment in each observation, and the hypotheses are entirely equivalent, it must also be said that I and E, which are positions of different epicycles, are occupied by the planet at the same moment. The same is true of G and H. The only difference is that in the first diagram the planet's path is shifted by altering the epicycle in its position, the observer remaining in the same place, while in the second diagram, for the planet's path the position also stays the same while the observer's position is changed by the same amount in the opposite direction. If required, however, it is possible to keep the path fixed in the former and the observer fixed in the latter fixed by shifting what is now fixed, in accordance with the ","49":"demonstrations of the preceding chapter.","50":"This demonstration will be put to use below. For surely, if the first inequality of the superior planets could be accounted for by the simple hypothesis of the second chapter, no difficulty would arise as to whether one should examine the first inequality at mean opposition to the sun, or at apparent opposition. For the actual path would in fact remain the same, and in both models the planet would be at the same points of the path at any given moment. Only the position of this path, in the first model, would be altered through the space of the sun's eccentricity, while in the second, although the path would stay fixed, the point whence the eccentricity is reckoned would also be shifted by the same amount.","51":"In the physical account, the above characteristics remain unchanged. Only their quantities change as the motive powers are intensified.","52":"CHAPTER 4","53":"On the imperfect equivalence between a double epicycle on a concentric, or eccentric-epicycle, and equant on an eccentric.","54":"That is how it would be if there were scope for the simple hypothesis of ch. 3 to account for the first inequality of the superior planets. However, for demonstrating the first and simple inequality of the planets, Ptolemy makes use of a more elaborate hypothesis.","55":"About center B let an eccentric DE be described, with eccentricity BA, so that A is the place of the observer. The line drawn through AB will indicate the apogee at D and the perigee at F. Upon this line, above B, let another segment BC be extended, equal to BA. C will be the point of the equant, that is, the point about which the planet completes equal angles in equal times, even though it lays out the circle around B rather than around C.","56":" In Book V ch. 4, as well as Book IV ch. 71,","57":"Copernicus marks this hypothesis among other things in this respect, that it offends against physical principles by instituting irregular celestial motions. For let a point E be chosen on the circle which the planet is bodily traversing, and let it be connected with C, B, and A. Now let DCE be a right angle, as well as ECF. Now since these angles are equal (for they are set up in equal times), and the exterior angle DCE is equal to the interior angles CBE and CEB, therefore, when the part CEB is subtracted, the remainder CBE or DBE will be less than DCE. Consequently, FBE will be greater than DCE or FCE. But the arc DE measures the angle DBE, and the arc EF measures the angle EBF. Therefore, DE is smaller than EF, and the planet passes over them in equal times. Therefore, the same solid orb (Copernicus believed in them)2 in which the planet inheres is slow when the planet borne by the orb proceeds from D to E, and fast when the planet goes from E to F. Therefore, the entire solid orb is now fast, now slow. This Copernicus rejects as absurd.","58":" Now I, too, for good reasons, would reject as absurd the notion that the moving power should preside over a solid orb, everywhere uniform, rather than over the bare planet. But because there are no solid orbs, consider now the physical coherence of this hypothesis when very slight changes are made, as described below. This hypothesis, it should be added, posits two motive powers to move the planet (Ptolemy was unaware of this). It places one of these in the body A (which, in the reformed astronomy will be the very sun itself), and says that this power endeavors to drive the planet around itself, but possesses an infinite number of degrees corresponding to the infinite number of points of the distance from A. Thus, just as AD is the longest, and AF the shortest, so the planet is slowest at D and fastest at F, and in general, as AD is to AE, so is the slowness at D to the slowness at E3, as will be demonstrated at great length in Part III below. The hypothesis attributes another motive power to the planet itself, which has the capacity to adjust its approach to and recession from the sun, either by the strength of the angles or by inspection of the increase or decrease of the solar diameter, and to make the difference between the mean distance and the longest and shortest equal to AB. Therefore, the point of the equant C is nothing but a geometrical short cut for computing the equations from a hypothesis that is clearly physical. But if the planet's path is a perfect circle, as Ptolemy certainly thought, the planet also has to have some additional perception of the swiftness and slowness by which it is carried along by the other external power, in order to adjust its own approach and recession in such accord with the power's precepts that the path DE itself is made to be a circle. It therefore needs both a comprehension of the circle and a desire to effect it. Also, the ratios of its own slowness and swiftness must differ from the gradations of the external power. However, if the demonstrations of astronomy, founded upon observations, should testify that the path of the planet is not quite circular, contrary to what this hypothesis asserts, then this physical account too will be constructed differently and the planet's power will be freed from these rather troublesome requirements.\n But let me return to Copernicus. In avoiding the absurdity explained just above from his own opinions, he substituted another epicycle for the equant, in the following way. About center \u03b1 with radius \u03b1\u03b2 equal to BD let the concentric \u03b2\u03b4 be described, so that the observer is at \u03b1; let \u03b1\u03b2, parallel to BD, be extended in both directions; and let the angle \u03b2\u03b1\u03b4 be set up equal to DCE. Now let BC be bisected at I, and about the centers \u03b2 and \u03b4 with radii \u03b2\u03b3 and \u03b4\u03b6 equal to AI let the first or greater epicycle be described, and let \u03b4\u03b6 be parallel to \u03b1\u03b2. Next, about centers \u03b3 and \u03b6, but with radii \u03b3\u03b5, \u03b6\u03b7 equal to IC, let the second epicycle be described, and let its motion be eastward, with twice the speed of the motion of the first. And let the westward motion of the first epicycle be equal to the motion of the eccentric. And since \u03b3 is on \u03b1\u03b2, let the planet be at \u03b5, the point nearest \u03b2. And since \u03b2\u03b1\u03b4 is right, let the planet be at \u03b7, the point farthest from the center of the greater epicycle \u03b4. This particular hypothesis of Copernicus is also followed by Tycho Brahe religiously, in all particulars.","59":"Physically considered, this hypothesis is in any event valid if you grant solid orbs. However, if you remove these, as Brahe does with good reason, it says something practically impossible. For it attaches three movement- producing minds to a single planet; and besides, the other two [minds] will be thrown into confusion by the motion and impulse of one of them towards the body at \u03b1. For that any of them should pay heed to its own center, which is not distinguished by any body and is mobile besides, cannot be represented even in thought. Further, while Copernicus strives to outdo Ptolemy in the uniformity of motions, he is in turn outdone by him in the perfection of the planetary path. For, in Ptolemy, the planet bodily traces out a perfect circle in the aethereal air. Copernicus, on the other hand, says in Book V ch. 4 that for him the path of a planet is not circular, but goes outside the circular path at the sides. This is easily demonstrated in the present diagram.","60":"If from \u03b5, the planet's position at apogee, you extend the distance \u03b1\u03b2, the semidiameter of the orb, to \u03b8, and from \u03b8 draw \u03b8\u03ba parallel to \u03b1\u03b4, the circle \u03b5\u03ba described about \u03b8 will indeed go through \u03b5 and the perigeal point opposite, but since it touches the straight line \u03b4\u03b7 only at k, while the planet goes through \u03b7, it does not stay on the circle \u03b5\u03ba, but strays outside this track. To this excursion of the planetary path from the perfection of the circle Ptolemy might well have objected against Copernicus, but I do not. For below, in Part IV, it will be demonstrated that by the agency of two physical powers, simple in capability, acting in concert to move the planet, it necessarily happens that the planet turns aside from the circle for a short time, though not by running outside of it, as in this Copernican hypothesis, but in the opposite direction, towards the center; that is, by making an incursion. ","61":"Besides, should Copernicus retain that liberty he had of setting up the ratios of the epicycles, it can happen that the planet's path would come out twisted, higher before and after apogee than at apogee itself, and lower before and after perigee than at perigee itself. This happened to Tycho in his lunar theory, inasmuch as he followed Copernicus.","62":"That these two forms of hypothesis are not simply equivalent, I shall demonstrate numerically.","63":"In the Ptolemaic form it can be computed more simply than Ptolemy did in the following manner. First, in triangle CBE, given the mean anomaly ECB or DCE, the side CB \u2014 the eccentricity of the equant\u2014is also given, as well as the radius of the orb, BE. Therefore, as the radius of the orb is to the sine of ECB, so is CB to the sine of CEB. And since ECD is equal to the two opposite interior angles CEB and CBE taken together, therefore CEB subtracted from DCE leaves CBE. Therefore, in triangle EBA, the angle at B is given, together with the sides about it. For BA is the eccentricity of the eccentric, while EB is the radius of the orb. Therefore, following the rule for this form of triangle, the angle BEA is given. But CEB was given before. Therefore, the whole equation CEA will be given. ","64":"We shall now make use of numbers belonging to the motion of Mars. Although Ptolemy made CB and BA equal, Copernicus, freed from this rule, nonetheless also adopted other ratios, which Tycho Brahe undertook to imitate. Let CB be 7560, BA 12,600, where BE is 100,000; and, first, let DCE be 45\u00b0, whose sine is 70,711. Therefore, as 100,000 is to 70,711, so is 7560 to 5346, the sine of the arc of 3\u00b0 4' 52\", which is CEB. Subtracting this from 45\u00b0 leaves CBE, 41\u00b0 55' 8\", whose half is 20\u00b0 57' 34\", the tangent to which arc is 38,304. And since EB is 100,000, while BA is 12,600, the difference, 87,400, multiplied by the radius and divided by the sum, 112,600, gives 77,620. Multiply this by the tangent found above (38,304). The product, that is, 29,732, is the tangent of the arc 16\u00b0 33' 30\", which, subtracted from the half of CBE, found above, leaves 4\u00b0 24' 4\", which is the angle BEA. Therefore, the whole, CEA, is 7\u00b0 28' 56\", in the Ptolemaic form. In the Copernican form, although the ordinary means of finding the equation is clearly presented in Tycho's lunar tables in Vol. I of the Progymnasmata, and in Copernicus himself, let me nonetheless follow a different, less usual procedure, which is adapted to an anomaly of 45\u00b0. Let \u03b2\u03b1\u03bb be 45\u00b0, and \u03bb\u03bd or \u03b2\u03b3 be 16,380, \u03b3\u03b5 or \u03bd\u03bf be 3780, and \u03bf\u03bd\u03bb be right, that is, twice \u03b2\u03b1\u03bb. Now let \u03bd\u03bb be parallel to \u03b2\u03b1, and let \u03bd\u03bb and \u03b4\u03b1 be extended, so as to meet at \u03bc. From \u03bf let \u03bf\u03be be dropped parallel to \u03bd\u03bc. Therefore, \u03bb\u03b1\u03bc is 45\u00b0, and consequently \u03b1\u03bc, and also \u03bc\u03bb, are 70,711. Add \u03bb\u03bd, 16,380, and \u03bc\u03bd or \u03bf\u03be will be 87,091. And because \u03b3\u03b5, \u03bd\u03bf, and \u03be\u03bc are equal, subtract \u03be\u03bc from \u03b1\u03bc. The remainder, \u03b1\u03be, is 66,931. Therefore, as \u03bf\u03be is to \u03be\u03b1, so is the whole sine4 to the tangent of \u03b1\u03bf\u03be or \u03bf\u03b1\u03b2, 76,852, giving an angle of 37\u00b0 32' 37\", which differs from the arc 45\u00b0 by 7\u00b0 27' 23\". Therefore, the difference between the Copernican and the Ptolemaic equations at this position is 1' 33\", a very small difference indeed.","65":" Again in the Ptolemaic form, let DCE be 90\u00b0. Therefore, since ECB is right, and EB is 100,000, BC will be the sine of the angle CEB, or 4\u00b0 20' 8\". Therefore, EBC is 85\u00b0 39' 52\", and EC is 99,713. Now, as EC is to CA, so is the radius to the tangent of CEA, 20,218. Hence, the equation, CEA, is 11\u00b0 25' 48\". But in the Copernican form, the whole magnitude \u03b7\u03b4, equal to CA, becomes the tangent, because \u03b7\u03b4\u03b1 is right and \u03b4\u03b1 is the radius. Therefore, \u03b7\u03b1\u03b4 is 11\u00b0 23' 53\". The difference is 1\u2019 55\". ","66":"Thus you see that, as far as the eccentric equation is concerned, there is something very slight lacking that prevents the two forms of hypothesis from being equivalent.","67":"They are different, however, in the distances of the planet from the observer at \u03b1, and as a consequence, in the annual equations of the center as well. For, in the Ptolemaic form, as the sine of the angle AEC is to AC, the whole sine is to AE, which becomes 101,766 when DCE is 90\u00b0. But in the Copernican, \u03b7\u03b1 is the secant of angle \u03b7\u03b1\u03b4, that is, 102,012. The difference is 246 parts, and this can have a somewhat greater effect upon the equation of the center for the annual orb, as will be clear below in Part IV. We can eliminate even this extremely slight difference in the equations by positing 20,103 as Mars's eccentricity in the Ptolemaic form where Brahe, in the Copernican form, found it to be 20,160. However, the distances in the Copernican form cannot be made equal to those in the Ptolemaic unless the equation be altered by 43'. In a certain equivalence I tried out in Tycho's hypothesis of the lunar tables, I transposed those two Copernican epicycles into such a Ptolemaic eccentric with an equant point. Nevertheless, I added yet another epicycle on account of another inequality, peculiar to the moon.","68":"Finally, in accord with ch. 2, the greater epicycle with its concentric in the Copernican form can, by virtue of its complete equivalence, be transformed into an eccentric whose eccentricity is equal to the semidiameter of the greater epicycle. Therefore, when a smaller epicycle is added to this Copernican eccentric, an eccentric with an epicycle will be created which matches the double epicycle on a concentric to a hair, and which differs from the Ptolemaic eccentric with an equant by no more than does this double epicycle.","69":"Chapter 5","70":"The extent to which this arrangement of orbs, using either an equant or a second epicycle, while remaining entirely one and the same (or very nearly one and the same), can present different phenomena at one and the same instant, according to whether the planets are observed at mean or at apparent opposition to the sun.","71":" This is done in two ways: one, in which the Ptolemaic and Copernican forms are equivalent, and another which is peculiar to the Copernican form. This latter, as it is further from our enterprise, we shall explain first, for it remains more allied to itself than the other.","72":" About center \u03b3, with radius \u03b3\u03b4, let an eccentric be described, upon which, in the first instance, let \u03b1\u03b3 be the line of apsides, with \u03b1 the observer, and let this line be extended to \u03b5. And let \u03b3\u03b1 be the magnitude of the eccentricity, or the radius of the greater Copernican epicycle (the equivalence of the two was discussed at the end of Chapter 4, preceding). Next, about center \u03b5, with radius \u03b5\u03b7, let the smaller epicycle be described, and since its center is at \u03b5, let the planet be at \u03b7, falling on the line \u03b5\u03b3, so that it is not the star but the center of the epicycle bearing the star that runs along the eccentric \u03b5\u03b4. Thus, by Chapter 4, it is the Copernican form that is expressed here. By Chapter 3, we shall set up another that is equivalent to it in reality or in the indication of the exact planetary path, but is different in appearance. This","73":"we shall do by moving the observer from \u03b1. By what was said at the end of Chapter 3, we could do the same thing even if the observer were to remain at \u03b1, by moving the eccentric while keeping lines parallel, in such a way that the size of the eccentric remains constant and only its position changes. But we shall carry it to completion as we have begun it. Adopting a position for the observer not on the prior line of apsides (let it be \u03b2), such that \u03b2\u03b3 has a magnitude different from that of \u03b1\u03b3 (that is, a new eccentricity or new radius of the greater epicycle), let us draw a new line of apsides \u03b2\u03b4 through \u03b2\u03b3, and at \u03b4 let us describe an epicycle equal to the former. Here, although the center of the epicycle is at the apsis \u03b4, nonetheless we shall not place the planet at the point nearest \u03b3, as before, but, taking the measure of the angle \u03b5\u03b3\u03b4, we shall set out the angle \u03b8\u03b4\u03b3 twice that size, in the direction of \u03b5, and shall place the planet at \u03b8, when the epicycle is at the apsis \u03b4. For this is where the planet would be, were the observer at \u03b1 and the epicycle at \u03b4. In this way, the true compound planetary path remains the same to a hair\u2019s breadth, while the appearance is altered. For when the lines of sight are inclined to one another, as \u03b2\u03b8 and \u03b1\u03b8 here, or \u03b2\u03b7 and \u03b1\u03b7, they fall upon different positions beneath the fixed stars.\nYou may object that even when the lines of sight are parallel, they fall upon different positions beneath the fixed stars, and it is therefore not essential that they be inclined to one another. I answer, this is indeed true, but in that case the space of the sphere of the stars intercepted between the two lines is not perceptible to the power of sight unless the distance between the parallels is perceptible in relation to the radius of the sphere of the stars.","74":"In the physical account, this must be posited in addition to what was said in Chapter 3, in order to establish the identity of the path while the appearances are altered: the mind to which the smaller epicycle is committed pays attention to a point on the circuit different from the one regarded by the mind of the greater epicycle. For in the second position, the greater epicycle or the eccentricity returns to its starting point on the line \u03b2\u03b4, while the smaller epicycle does so on the line \u03b1\u03b5 which does not pass through the place of the observer. This is because, in the second position, the observer is located at \u03b2, while in the first position, where the observer is located at \u03b1, both epicycles return to their starting points on the same line \u03b5\u03b1. The form of the hypothesis thus does not remain simply the same physically in such a way that that the planet keeps the same path. But suppose you were going to simulate the same path in the second position by having both epicycles return to their starting point at the same line of apsides \u03b2\u03b4. If so, while the eccentric as well as the epicycle stay the same in both instances, the position of the planet on the epicycle will be different at every single moment. Thus, although in the second instance the form of Ptolemaic hypothesis presented is the same to a hair\u2019s breadth, the actual path of the planet will be altered. Hence, it will be inferred below, even though the first inequality of the planets may be entirely accounted for by the compound hypothesis of Chapter 4, it cannot happen that the first inequality have the same measure at the planet\u2019s mean opposition to the sun as at apparent opposition, unless the planet\u2019s orbit be moved from its own location at the same time (unlike the circles in the theory of the sun), or the Ptolemaic form of Chapter 4 be changed.","75":"Maestlin made use of this form of transposition in constructing the table in Chapter 15 of my Mysterium cosmographicum. For when Copernicus transformed the Ptolemaic [models] into his own general form of hypotheses, he supposed the observer to be stationed at some nearly motionless point near the sun, distant from the sun\u2019s own body by the entire eccentricity of the solar orb. I, however, in adapting Copernicus to the subject matter of that book, needed a different fiction. The observer was to be imagined as transported from that point to the very center of the solar body, and from there (that is, from the body of the sun) the departures of the planetary bodies were to be computed, moving on the same path which the suppositions of Copernicus fashioned. But, as has just now been shown, by reason of the particular times my translation of the line of apsides did not effect exactly the same path. The difference, however, was very slight, and was clearly of no importance in that little book. For there the question concerned only the location of the path, which this procedure did not affect.\nAs for the rest, in order to avoid confusion in what follows, I shall no longer make use of this Copernican eccentric, described by the center of the epicycle rather than the star. It differs from the planet's true path, which is higher at perigee and lower at apogee. The term \"eccentric\" from now on we will use only to designate the actual path of the planet, or of the point to whose motion the first inequality belongs. In proceeding in this way, it is appropriate that we imagine only the Ptolemaic eccentric, or something like it. For it was shown in the fourth chapter that our computation of the equation, based upon the Ptolemaic form, will differ from the Copernican by only two minutes at most. Then, too, the procedure for computing the first inequality is easier in the Ptolemaic form than in the Copernican. Finally, as was said, this Ptolemaic form of the first inequality is better accommodated to nature herself and to our speculations that follow in the third and fourth parts. However, because of the equivalence, anyone who so chooses will always be able to supply in thought the Copernican eccentric-cum-epicycle considered so far in this fifth chapter.","76":"I now proceed to the prior procedure for setting up the two sorts of equivalence that I have proposed\u2014the one common to the particular hypotheses of the authorities. I shall demonstrate this first in the Ptolemaic form.","77":"About center \u03b2 let the Ptolemaic eccentric \u03b9\u03b6\u03b7 be described, with \u03b9\u03b2 the line of apsides, \u03b1 the observer, and \u03b3 the equalizing point. ","78":"Now when I say that the observer is at \u03b1, I mean it either as a fiction or as truth. Physically speaking, it is not so much the observer which is to be placed at \u03b1 as the power which renders the planetary circuit around itself slow or swift according to the ratios of its proximity to \u03b1, as was said above. Let some point on the circumference not on the line of apsides (say, \u03b7) be connected with \u03b3, \u03b2, and \u03b1. Let it be that about as many angles \u03b9\u03b1\u03b7 may be computed by this hypothesis throughout the entire circuit as are observed from \u03b1, and after certain periods of time, which the angle \u03b7\u03b3\u03b9 measures uniformly. Later, in the second part, it will be shown how one can find, through astronomical observations, how great the angle \u03b7\u03b1\u03b9 should be for any given \u03b7\u03b3\u03b9. Again, let the observer or moving power be at some point not on the line \u03b9\u03b1, and let this be \u03b4. Also, let it be granted us that the apparent angles about \u03b4 be apprehended by astronomical observations at certain times; that is, how much the planet appears to move forward in sidereal position in a given time when seen from \u03b4. Let this be granted as well: that these appearances at \u03b4 square with a hypothesis in conformity with the previous one, with only the magnitude of the eccentricity altered. But since it is certain that at one and the same time the planet traverses one and the same path in the heavens, not one seen to an observer at \u03b4 and another to an observer at \u03b1, it is also certain, as a consequence, that the planet cannot appear to both observers (both the one at \u03b1 and the one at \u03b4) to be equally moved in the same time. For let \u03b9\u03b7 be a portion of the planet\u2019s true path, and let the planet traverse this in a given time, say twenty days. Now since \u03b1 is nearer \u03b9\u03b7 than \u03b4 is, \u03b9\u03b7 will appear greater at \u03b1 than at \u03b4, by what is demonstrated in optics. Therefore, during the same twenty days the planet will appear to make greater progress to one who is at \u03b1 than to one who is at \u03b4. And since any planet maintains a fixed and constant number of days in which it is returned to the same sidereal position, the slowness has to be compensated by a contrary speed. Therefore, since in the portion \u03b9\u03b7 the planet appears slower to one at \u03b4, it will in some other portion appear swifter to the one at \u03b4 than to one at \u03b1. Hence it happens that it appears slowest to the one at \u03b4 in one place, and to one at \u03b1 in another. Nevertheless, the planet itself can be truly slowest in but one place on its orbit. \nWith these as preliminaries, the question is raised whether one and the same true path of the planet in the heavens (this is presupposed) can present two sets of appearances each proper to an observer, one at \u03b4 and the other at \u03b1, and each such as comply with and admit the Ptolemaic form of computation.","79":"If the planet were of equal speed at all parts of its orbit, the answer, according to what was said in Chapter 3, is yes. But since in terms of real and true increment of time2 the planet is slowest at one point on the eccentric and fastest at the opposite point, the answer must therefore be, clearly not.","80":"The reason for this is that the two slowings are intermingled, one real and physical, occurring at one place on the eccentric, and the other optical and apparent and not occurring at a single place, but in the place most distant from whatever position is chosen for the observer. Now, when the observer \u03b1 lies upon a line drawn through the center of the eccentric \u03b2 and the center of the equant \u03b3 on the side of \u03b2 opposite the center of the equant \u03b3, then both slowings verge toward the same sidereal position \u03b9. But when the observer departs from this line, as at \u03b4, then a straight line drawn from \u03b4 through the center of the circle \u03b2 marks the place of the optical slowing, \u03b7, while the true and physical slowing is at \u03b9. Furthermore, each of these inequalities or slowings diminishes the other, and they are accumulated at an intermediate point between \u03b9 and \u03b7, as would be if a line were to be drawn from \u03b4 through \u03b3 to the point \u03b6. Consequently, were one to adopt a form of computation in which \u03b4\u03b2 were the line of apsides of the eccentric and \u03b2\u03b3 the line of the equant\u2019s eccentricity, then even though the planet\u2019s true path \u03b9\u03b7 remained the same, it would be represented differently at \u03b4 than at \u03b1. For to the observer at \u03b4 the planet would be slowest at \u03b6, and to the one at \u03b1 it would be slowest at \u03b9. But no such thing would be represented at \u03b4 that ought to have been represented by a hypothesis of the same form as the previous one, according to our presuppositions above. For the forms of the hypotheses differ in that in the former, \u03b2 is the midpoint between \u03b1 and \u03b3 (as physical considerations require, if the moving power is in \u03b1), while in the latter the center of the eccentric \u03b2 would not be the midpoint between \u03b4 and \u03b3, nor would the line of the equant\u2019s eccentricity pass through the observer \u03b4, as before. Even if it did pass through \u03b4, as \u03b4\u03b3, it would not cut the eccentric into two equal parts, because it would not cut it at the center \u03b2, and it would not allow the planet to appear on one side fastest and on the other slowest, at opposite places. ","81":"It is thus established that when a planet's path in the heavens remains in all respects unchanged, the form of hypothesis cannot persist entirely unchanged. This raises the further question, how much the path of a planet is changed from its prior position if the same form of hypothesis is set up about \u03b4, and how much this newly established hypothesis at \u03b4 will be at variance with the appearances as seen at \u03b1. First, if the center of the equant be transferred from \u03b3 to the line \u03b4\u03b2, and \u03b2\u03bc be made equal to \u03b2\u03b3, the position of the planetary path is quite unchanged, but the planet is slowest, in physical terms, at \u03b7 rather than \u03b9. This changes what cannot be changed in the planet\u2019s path, because, unlike the optical slowing, the physical slowing is independent of the observer\u2019s viewpoint. Even though the planet would traverse the same path \u03b9\u03b7 in twenty days (which path appears greater at \u03b1 and smaller at \u03b4), nevertheless, if you consider the parts of this time, their ratio in comparison with the parts of this path will be violently perturbed, and much more so at other places, not between \u03b9 and \u03b7. In particular, for the observer at \u03b1 the quantity of his equations will be changed noticeably if, for the observer at \u03b4, you will prevent the planet\u2019s being slowest at \u03b9; that is, if you transfer the equant from \u03b3 to \u03bc. For if you draw a straight line through \u03b3\u03bc to the point \u03bd on the circumference and connect \u03b1\u03bd, this equation \u03b1\u03bd\u03bc alone will be equal to the prior, \u03b1\u03bd\u03b3. Above \u03bd the equations about \u03bc will be smaller, and below \u03bd, greater. For example, at \u03b7 the angle \u03bc\u03bd\u03b1 is much less than \u03b3\u03b7\u03b1. But then what we proposed to do has not been done, for the prior form of hypothesis has not yet been quite established. For \u03b1\u03b2 is not to \u03b2\u03b3 as \u03b4\u03b2 is to \u03b2\u03bc, since \u03b2\u03bc is equal to \u03b2\u03b3, while \u03b4\u03b2 is greater than \u03b1\u03b2. But if, on the other hand, you make \u03b4\u03b2 be to \u03b2\u03bc as \u03b1\u03b2 is to \u03b2\u03b3, \u03b2\u03bc will become greater than \u03b2\u03b3. Whence it follows that for the observer at \u03b1, his equation will be more in error, even when it is at a maximum, on account of the increased eccentricity. Not only will the planet be slowest in a different place than before, but in also by a different and in fact greater measure of its true slowness. It is clear, therefore, that the equivalence we have been seeking cannot be established by drawing the line of apsides from \u03b4 through the center of the eccentric \u03b2. And since it has at the same time become clear how important it is to keep the same equant point \u03b3, a breakthrough must in all events be made here, or nowhere.","82":"But what will happen if a new line of apsides be drawn from \u03b4 through the old equant point \u03b3, and a new hypothesis of the same form as the old be set up? That is, if the center of the eccentric be transferred from \u03b2 to the line \u03b4\u03b3, and \u03b4\u03b8 be made to \u03b8\u03b3 as \u03b1\u03b2 is to \u03b2\u03b3, \u03b8 thus being the center of the eccentric? Obviously, the result will be this: the path of the planet in the heavens does not remain entirely the same. About \u03b8 let the eccentric \u03b5\u03ba\u03bb be described, equal to the previous one, and through \u03b8\u03b2 let a straight line be extended to the circumference, on one side to \u03be\u03bf and on the other to \u03c1\u03c0. Therefore, \u03be\u03bf and \u03c1\u03c0 are both of the same magnitude as \u03b8\u03b2, and the planet is this much closer to \u03b2 at \u03bf, and more remote at \u03c1, than it would have been had it traversed the previous eccentric. However, the planet is slowest in a different region, for previously the apsis was at \u03b9, and now it is at k. Through this mutual tempering it is brought about that the observer previously stationed at \u03b1 has his observations pretty much unchanged, which is the only thing sought for here. But now we shall prove it with numbers belonging to Mars\u2019s motion, although Brahe recorded somewhat different ones. This should prove no impediment to us, since we are only performing a preliminary exercise3 here.","83":"Let the magnitudes on the line \u03b4\u03b3\u03b1 be taken as follows: let \u03b4\u03b1 be the quantity of the sun\u2019s eccentricity, 3584; \u03b4\u03b3 the eccentricity of Mars, 30,138 of the same parts, and the angle \u03b1\u03b4\u03b3 47\u00b0 59\u00bc\u2019, which is the angular difference between the sun\u2019s and Mars\u2019s apogees. Now, from these three given quantities, \u03b3\u03b1, Mars\u2019s new eccentricity, will also be given, and will be 27,971, while the angle \u03b4\u03b3\u03b1 is 5\u00b0 27\u2019 47\". On the supposition that the old apogee of Mars \u03b4\u03b3 is positioned at 23\u00b0 32' 16\" Leo, Mars\u2019s new apogee \u03b1\u03b3 will fall at 29\u00b0 0\u2019 16\" Leo. ","84":"Now let \u03b2\u03be be 100,000 and \u03b1\u03b3 be 18,034 of the same parts. Before, it was 27,971, in units of which \u03b4\u03b3 was 30,138. Therefore, in these units, \u03b4\u03b3 will be 19,763. Next, let both be divided by the letters \u03b8 and \u03b2 in such a ratio that \u03b4\u03b8 is to \u03b8\u03b3, and \u03b1\u03b2 to \u03b2\u03b3, as 1260 is to 756: \u03b4\u03b8 will be 12,352, \u03b8\u03b3 7411; and \u03b1\u03b2 11,271, \u03b2\u03b3 6763. In this way, a Ptolemaic hypothesis for the first inequality may be set up about both \u03b4 and \u03b1. Then, in the former units, of which \u03b4\u03b1 is 3584, \u03b8\u03b2 or \u03bf\u03be will be 1344, but in units of which \u03b2\u03be is 100,000, \u03b8\u03b2 or \u03bf\u03be will be 880. These should be kept in mind.","85":"To find the basis of a computation whereby we may investigate, for the observer at \u03b4, how much his appearances are changed by transposing the eccentric from \u03c1\u03b8\u03bf to \u03c0\u03b2\u03be, we must proceed as follows. Since \u03b3 is the common center upon whose circle the times are indicated, let the line \u03b3\u03b5\u03b9 indicate the same moment in both hypotheses. Therefore, if the planet is traversing the eccentric \u03b5\u03bf, it will at that moment be at \u03b5 with equation \u03b4\u03b5\u03b3, but if it is traversing \u03b9\u03be, it will be at \u03b9 with no equation, since the line of apparent motion \u03b1\u03b9 coincides with the line of mean motion \u03b3\u03b9. Again, after a certain amount of time, whose measure shall be \u03b9\u03b3\u03b6 or \u03b5\u03b3\u03ba, whose vertical angle is \u03b4\u03b3\u03b1, just found to be 5\u00b0 27\u2019 47\", let a common moment be taken, represented by \u03b3\u03ba\u03b6. At that time, the planet traversing the eccentric \u03b5o will be at k with no equation, while the planet traversing \u03b9\u03be will be at \u03b6 with equation \u03b3\u03b6\u03b1. Thus, in both instances, the planet is always on a line drawn from \u03b3, at the point where that line cuts one or another eccentric. If the observer were at \u03b3, there would be no difference in the appearances, whether the planet were at k or at \u03b6. But since in the present model the observer is placed at \u03b4 by the practitioners4 and at \u03b1 by myself, the question arises at what point on the circumference the distance between the eccentrics is perceptibly a maximum for the observer at \u03b4. In order that this difference become perceptible, three factors must concur. First, the distance itself must be large (just as it is a maximum around \u03bf\u03be and \u03c1\u03c0). Second, as nearly as possible it should be presented directly to an observer at \u03b4 (just as it vanishes at \u03b6\u03ba and the point opposite) according to optical principles. Thus, it appears greatest at the intermediate regions, below \u03be and above \u03c1. Third, it must be close to \u03b4 (just as it is closer above \u03c1 than below \u03be, because the center of the other eccentric \u03b2 lies off to the right of \u03b4). If we construct a right angle to point \u03b3 of the line \u03b3\u03b4, this perpendicular from \u03b3 to the circumference will bring us as close as possible to the place where the apparent magnitude is greatest. Let \u03c3\u03c6 be drawn through \u03b3 perpendicular to \u03b4\u03b3, intersecting the eccentric about \u03b8 at \u03c3 and \u03c5, and the other at \u03c4 and \u03c6, and to this let the perpendicular \u03b2\u03c7 be dropped. Therefore, at the moment \u03b3\u03c3 the planet will be at \u03c3 and \u03c4, and at the moment \u03b3\u03c6, at \u03c5 and \u03c6. First of all, the quantity \u03c5\u03c6 must be found. Let \u03b8\u03c5 and \u03b2\u03c6 be joined. In \u03b8\u03c5\u03b3, \u03b8\u03c5 is given as 100,000, because \u03b8 is the center of the eccentric \u03c5. The magnitude \u03b8\u03b3 is 7411, and \u03b8\u03b3\u03c5 is right. Therefore, \u03b3\u03c5 is 99,725. The same is to be done in \u03b2\u03b3\u03c6. But first, \u03b2\u03c75 has to be found. This will be revealed by the triangle \u03b2\u03b3\u03c7, in which \u03b2\u03c7 is parallel to \u03b8\u03b3, the angle at \u03c7 is right, and \u03b3\u03b2\u03c7 is equal to \u03b8\u03b3\u03b2 (5\u00b0 27\u2019 47\") and \u03b2\u03b3 is 6763. Hence, the sides are found to be: \u03b3\u03c7 644, \u03b2\u03c7 6732. Therefore, in the right triangle \u03b2\u03c7\u03c6, since \u03b2\u03c6 is 100,000 (\u03b2 being the center of the eccentric \u03c6) and \u03c7\u03b2 is 6732, \u03c7\u03c6 will be 99,773. Add \u03c7\u03b3, 644, and \u03b3\u03c6 is given, 100,417. But \u03b3\u03c5 was 99,725. Therefore, the quantity sought, \u03c5\u03c6, is 692.","86":"With \u03c5 and \u03c6 joined to the position of the observer \u03b4, the magnitude of angle \u03c5\u03c6\u03b4 is found as follows. Above, \u03b4\u03b3 was 19,763 in the new units, and the angle at \u03b3 right. Therefore, as \u03b4\u03b3 is to \u03b3\u03c6 and \u03b3\u03c5, so is the whole sine to the tangents of the angles \u03b3\u03b4\u03c6 and \u03b3\u03b4\u03c5. These come out to be 78\u00b0 51\u2019 54\" and 78\u00b0 47\u2019 30\". The difference of these angles, the angle \u03c5\u03b4\u03c6, is 4\u2019 24\". The angle \u03c3\u03b4\u03c4 will be much less, because \u03c3\u03c4, as it is closer to the intersection of the eccentrics, is smaller than \u03c5\u03c6.","87":"You see, then, how nearly the appearances remain unchanged for the observer at \u03b4, despite the substitution of a new planetary path in the heavens by transposition of the observer and change of hypothesis. Moreover, it is still within the practitioner's power to vary somewhat the mean motion and the ratio of the eccentricities both among themselves and in relation to the radius of the orb, should this become useful, for the purpose of obliterating this discrepancy of some five minutes.","88":"This equivalence pertains chiefly to the first inequality\u2014that is, to the appearances at \u03b4, close to the center of the eccentric. However, in the second inequality (the equation for the annual orb), it makes a big difference whether the planet goes around on \u03be\u03c0 or o\u03c1, as was also noted above in the other equivalence. And there we could not ignore the 246 units (the difference between the Ptolemaic and the Copernican hypotheses). Much less can we overlook these 880 units, which are 1344 of the old units. In the next chapter we shall see how much of a difference in Mars\u2019s apparent position this would occasion.\n Hitherto, we have transposed the observer from \u03b4 to \u03b1. Let it now be demonstrated that very nearly the same things happen if the observer remains fixed while the point of the equant is transposed. This we do to make it clear that the same thing can be done in this chapter with an eccentric that has an equant as could be done on a simple eccentric (at the end of Chapter 3 above). In the earlier case, the result was the same whether the observer or the center of the eccentric were transposed, while here, likewise, the result is almost the same whether the observer or the center of the equant be transposed. However, it necessary to adapt this demonstration to this divergence, on account of the great dissimilarity of the opinions followed by the practitioners in demonstrating the second inequality of the planets, which opinions will be keeping us in court in the next chapter.\n Let the points \u03b1 and \u03b4 merge into one, so that the observer remain in the same place. Let \u03b4, \u03b8, and \u03b3 remain the same, but let the line \u03b3\u03b2\u03b1 of the previous diagram be deleted, and replaced by \u0391\u0392\u0393 parallel to it and passing through the point \u03b4 or \u0391. Let the segments \u0391\u0392, \u0391\u0393 be equal to the previous segments \u03b1\u03b2, \u03b1\u03b3. Therefore, \u0393\u03b3 will be the transposition of the equalizing point \u03b3, equal to the previous transposition of the observer \u03b1\u03b4. Once more, two eccentrics or planetary paths through the aethereal air will be described, about \u0392 and \u03b8. All the letters on each circle will be carried over along with them, and the magnitudes of the lines will remain precisely the same. The only difference is that the two points on the two eccentrics at which the planet is to be placed at a given moment are no longer determined by a single line, but by parallel lines drawn from the two equant points \u0393, \u03b3, each extended to its own eccentric. For example, when the eccentric \u03b8\u03ba has its planet at \u03ba, the eccentric \u0392\u0399 will have it at \u0396, where \u03b3\u03ba and \u0393\u0396 are parallel. And when the former has the planet at \u03b5, the latter will have it at \u0399, where \u03b3\u03b5 and \u0393\u0399 are again parallel. The rest is clear from the diagram without demonstration.","89":" Now, suppose it is not permissible to shift the observer (and it is not permitted by those who make the earth the center of the world, as will be remarked in the next chapter), and that the planet has been observed in several positions on the zodiac, always at opposition to the sun's mean position, and that the practitioner will use the positions and the intervals of time between them to construct this sort of hypothesis, with the observer at \u03b4, \u03b4\u03b8 the eccentricity of the eccentric \u03b8\u03ba, \u03b8\u03b3 the eccentricity of the equant, and \u03ba the apogee. Comes now Kepler, who would change the observed positions and times; that is, he would observe the moments and points at which the planet is at opposition to the apparent position of the sun rather than its mean position. From these positions and times he will have come up with another hypothesis, in which the observer would be left unmoved at \u03b4 or \u0391, but where the eccentricity would come out to be \u0391\u0392 in a new eccentric \u0392\u0399, and the eccentricity \u0391\u0393 of a new equant \u0393, and there would be a new apogee \u0399. The question now is whether, if the prior practitioner combines the new eccentric \u0392\u0399 with his original equalizing point \u03b3, the computed equation and sidereal position of the planet will turn out much different from what he had formerly found using his eccentric \u03b3\u03ba. (It is the first inequality that is in question; this discussion is not concerned with the second inequality, and the nature and magnitude of the changes which this procedure would effect therein.) The answer, arising from the equivalence of transpositions, is that the discrepancy will be extremely small. Its maximum, reached in the neighborhood of the points \u03c5 and \u03a6, will not exceed five minutes, exactly as before when the observer was transposed, except that now the line \u03c5\u03a6 is closer to the observer \u03b4 than is its endpoint \u03c5. Consequently, the angle \u03c5\u03b4\u03a6, which previously was 4\u2019 24\", is now 4\u2019 43\". The opposite happens at \u03c3\u03a4.","90":"It has thus been demonstrated in a Ptolemaic eccentric what sort of disturbances would arise if one should transpose either the observer or the orb and construct a new eccentric, making use of the planet\u2019s oppositions to the sun\u2019s apparent position.","91":" I do not think there is any need to repeat the arguments and demonstrate the same equivalence in the Copernican or Tychonic form, which makes use of two epicycles. I shall only show, by what was established at the end of Chapter 3, how to delineate both the eccentric-cum-equant that suits the planet, and its transformation into different magnitudes and different positions for the observer, in terms of the Copernican double epicycle. This is done in such a manner that, while the observer is transposed, the path of the planet through the aethereal air is invariant, as nearly as possible, in accord with what has been said in this fifth chapter. (This is exactly the possibility adumbrated in Chapter 3).","92":"Let the triangle \u03b4\u03b3\u03b1 be constructed equal to the previous one, with corresponding lines parallel, and through \u03b1 let \u03b1\u03b2 be drawn parallel to \u03b4\u03b3, and through \u03b4, \u03b4\u03b8 parallel to \u03b1\u03b3. And about centers \u03b1, \u03b4 let two concentrics be described, equal to the previous eccentrics \u03b4\u03b8, \u03b1\u03b2. Let \u03b4\u03b3 be extended to \u03b6 and \u03bb, and \u03b1\u03b3 to \u03b5 and k, and let \u03b4\u03b6 and \u03b1\u03b5 be semidiameters, as before, and lines of apsides (since both go through the same point \u03b3). Now let \u03b4\u03b3 and \u03b1\u03b3 be cut at \u03b7 and \u03be in the same ratio as before, and let \u03b7\u03b3 and \u03be\u03b3 be bisected at \u03c8 and \u03c9. Then, with radius \u03b4\u03c8 and upon centers \u03b8 and \u03b6 let the epicycles \u03b9 and \u03bb be described, and let \u03b6\u03bb be parallel to \u03b8\u03b9. Then, about centers \u03b9 and \u03bb, with radius \u03c8\u03b3, let epicyclets be described through \u03c0\u03bc and \u03c1\u03c4.","93":"Again, let the epicycles \u03ba and o be described about centers \u03b5 and \u03b2 with radius \u03b1\u03c9, and with \u03b2\u03bf parallel to \u03b5\u03ba. Next, about centers \u03ba and o with radius \u03c9\u03b3, let epicyclets be described through \u03c0\u03c5 and \u03c1\u03bd, and let \u03b8\u03b9\u03bc and \u03b2\u03bf\u03bd be double \u03b4\u03b3\u03b1. Let the planet on epicyclet \u03ba\u03c0 be at \u03c5 nearest \u03b5, and the planet on epicyclet \u03bb\u03c1 be at \u03bd nearest \u03b6. Therefore, according to the hypothesis deriving from \u03b4 the planet falls upon \u03c4\u03bc, while according to the hypothesis deriving from \u03b1 the planet falls upon \u03bd\u03c5. Here you see that the points \u03bc and \u03c5, as well at \u03c4 and \u03bd, hardly differ when seen from \u03b4 and \u03b1, respectively, when the planet is near the apsides. But around the middle intervals these points will be separated from each other only by the amount that separated \u03c5 and \u03a6 in the previous diagram. The magnitudes will all be nearly equal and the demonstrations entirely the same. For if \u03b8\u03b9 and \u03b5\u03ba be extended to meet at \u03c0, and \u03b6\u03bb, \u03b2\u03bf to meet at \u03c1, the triangles \u03b8\u03c0\u03b5 and \u03b6\u03c1\u03b2 will be congruent to triangle \u03b4\u03b3\u03b1, and corresponding sides will be parallel.","94":"These demonstrations will be perplexing enough in themselves, so it is not really advisable to make them more involved by a heaping up of Copernican or Brahean epicycles. Therefore, in what follows, we shall decree that the Copernican or Brahean form count as belonging to the first inequality. For the procedure for treating the hypotheses of the second inequality, since it is always going to concern each of the three, will furnish us with a great abundance of things to do.","95":"I now immediately state it as a postulate that whatever we shall demonstrate using the Ptolemaic equant-cum-eccentric also be taken as demonstrated in the Copernican or Brahean concentric-cum-double-epicycle, or eccentrepicycle. For in Chapter 4 above, the difference was found to be very small.","96":"Chapter 6","97":"On the equivalence of the hypotheses of Ptolemy, Copernicus, and Brahe, by which they demonstrated the second inequality of the planets, and how each one varies when accommodated to the sun\u2019s apparent position and when accommodated to its mean position.","98":"The discussion so far has concerned the hypotheses of the planets\u2019 first inequality, which completes its cycle each time the planet returns to the same sign of the zodiac. Now we pass on to the other inequality, which completes its cycle not at a single constant sign of the zodiac but with the sun\u2019s opposition to, or conjunction with, the planet. People have wondered exceedingly at this, different ones proposing different reasons why a planet in conjunction with the sun is made swift, direct, high, and small; and opposite the sun, retrograde, low, and large; while in between it becomes stationary and of a medium size.","99":"The Latin authors considered that in the sun\u2019s aspects and rays there is a force by which the other planets are in fact attracted. Their opinion cannot be shown numerically, because it is not astronomical. But it is also improbable, now that the true causes have been found, and manifestly false, since Saturn begins to retrogress at quadrature with the sun, or beyond; Jupiter, at trine; Mars, at biquintile or before sequiquadrate, and all at variable distances. ","100":"Ptolemy said that at a determined point on the planetary circle that serves for the first inequality, there is fixed, not the planet itself, but the center of an epicycle bearing the planet fixed upon its circumference, which is in turn borne by the planet\u2019s chief circle. The motion has the following form: if the center of the epicycle be in conjunction with the sun, the planet is also at the highest point of the epicycle and is moved along with the sun in the same direction, [and] when the sun, which is faster, departs from the center of this epicycle, the planet simultaneously descends on the epicycle. But since the epicycle\u2019s motion about its center is faster than the motion of its center about the earth, it hence happens that when the planet traverses the lower parts of the epicycle while the epicycle\u2019s center is at opposition to the sun, the compounding of motions makes it actually retrograde. Thus Ptolemy made his opinions correspond to the data and to geometry, and has failed to sustain our admiration. For we still seek the cause that connects all the epicycles of the planets to the sun, so that they always complete their periods when their centers are in conjunction with the sun.","101":"Copernicus, with the most ancient Pythagoreans and Aristarchus, and I along with them, say that this second inequality does not belong to the planet\u2019s own motion, but only appears to do so, and is really a by-product of the earth\u2019s annual wheeling around the motionless sun. In this way, just as in Chapter 1 the diurnal motion was separated from the motion proper to the planets, the second inequality of the planets is likewise now separated from the first by Copernicus, and in quite the same way. For some practitioners admit that the first motion is extrinsic to the planets, but still think it is in fact in the planets, inserted into them, so that the planets, too, are moved with the same motion. Copernicus holds that it is neither intrinsic to the planets nor inserted, but only attached to them through an optical illusion. For while the earth rotates upon its axis from west to east, it appears to our eyes that the rest of the world rotates from east to west. It is, I claim, in just the same way that Copernicus asserts that the planets do not really become stationary and retrograde, but only appear so. For he says that since the earth is in addition carried along by another motion, which is annual, in a very large circle (which he calls the orbis magnus), those who believe that the earth is at rest think that the planets and the sun are carried in the opposite direction; and he says that when the sun is between the planet and the earth, in the appearance the motions of the earth and the planet in are added, whence the planet appears to be swift; and when, on the other hand, the earth is between the sun and the planet, the planet is apparently left behind and thus retrogresses, owing to the earth\u2019s being swifter than the planet.","102":"Tycho Brahe holds something in common with the Latins: although the sun does indeed not attract the planets through the aspect, the planets do fawn upon the sun. For he says that they strive to keep the sun (although it is moving) nearly in the middle of their circuits, and indeed, that they arrange their real paths around the sun as if it were motionless. Thus any given planet, besides its own path, traverses the sun\u2019s path in the aethereal air, and out of these motions compounded with one another there is produced exactly what Ptolemy had (that is, a spiral), as described in Chapter 1. In astronomical terms, Ptolemy put epicycles on eccentrics, while Brahe put eccentrics on a single epicycle, which is the sun\u2019s orb itself.","103":"I, in the demonstrations that follow, shall link together all three authors\u2019 forms. For Tycho, too, whenever I suggested this, answered that he was about to do this on his own initiative even if I had kept silent (and he would have done it had he survived), and on his death bed asked me, whom he knew to be of the Copernican persuasion, that I demonstrate everything in his hypothesis.","104":"Furthermore, we shall demonstrate, both right here and through the entire book (though while doing other things), that these three forms are absolutely, perfectly, geometrically equivalent. For the present we must carry out what we set out for ourselves and what is to be demonstrated, namely, that there is a very great error indeed in the second inequality if the apparent position of the sun is replaced by its mean position, to which the planet is at opposition at the beginning of this second inequality.","105":"I shall begin with the Copernican opinion. About center \u03b2 let the earth\u2019s eccentric \u03b3\u03c5 be described, such as Copernicus, putting his trust in Ptolemy, imagined. Let \u03b3\u03b2 be its line of apsides, \u03ba the position of the motionless sun, and \u03b2 the point about which the earth\u2019s motion is uniform.","106":"Through \u03b2 and perpendicular to \u03b2\u03b3 let \u03c5\u03b2\u03c3 be drawn, intersecting the circumference at points \u03c5 and \u03c3, and let \u03c5 and \u03c3 be joined to \u03ba.","107":"Copernicus, intending to transfer the Ptolemaic numbers into his own form of hypothesis, reckoned the eccentricities of the planets from the conjectural center \u03b2 of the earth\u2019s uniform motion rather than from the sun \u03ba. For if lines be drawn from \u03b2 (as \u03b2\u03b3, \u03b2\u03c5, \u03b2\u03c3), whenever a planet and the earth lie upon one of these, the planet was supposed to shed the second inequality, to which it was subject on account of the earth\u2019s motion, as, for example, if the earth were at \u03c5 and the planet were found on \u03b2\u03c5 extended. ","108":"In effect, by adopting this procedure, Copernicus established a fictitious observer at the point \u03b2. For provided that the planet is on the line \u03b2\u03c5, it makes no difference for the purpose of designating its sidereal position whether it be viewed from \u03c3 or from \u03b2. The same may be truly said of the lines \u03b2\u03b3, \u03b2\u03c3, and all the other infinite lines intersecting at \u03b2. Therefore, \u03b2 is the point of intersection of all the lines of vision, and is thus the fictitious common point for all observers. In fact, however, the point of vision is the earth, our home, which is found at one or another point on the circle \u03c3\u03b3\u03c5 at various times.","109":"So Copernicus believed the planet was freed from the second inequality whenever the planet and the earth were found on any one line passing through \u03b2. Consequently, he endeavored to find the apparent sidereal position of a planet at those moments of opposition to the mean position of the sun through mathematical instruments. For if the planet\u2019s position was found on one of the nights near the planet\u2019s opposition to the sun, and if at that time the mean position of the sun was found by calculation to be at the point precisely opposite, then that was the moment of time desired. However, if there was still a little distance between them on that night, then he tracked down this moment of time, and the point or position held by the planet at this moment, by a comparison of two or more nights and of the diurnal motions of Mars and the earth over the interval. When he had done this as many times and in as many places on the zodiac as he thought necessary (suppose, for example, \u03b2\u03b3, \u03b2\u03c5, \u03b2\u03c3), the practitioner now began to use these known sidereal or zodiacal positions of the planet, \u03b2\u03b3, \u03b2\u03c5, \u03b2\u03c3, to investigate the hypothesis of the first inequality. This involved finding the magnitude of the planetary circle's eccentricity from the selected point \u03b2, and the parts of the zodiac into which the apogee points, by comparing the angles which the observed positions set up about the point of observation \u03b2, with the time intervals between them. The method of this undertaking, however, I shall present clearly in its proper place, below.","110":"Suppose these things already done, giving a line of apsides \u03b2\u03b4, eccentricity of the equant point \u03b2\u03b4, and the center of the eccentric on this line at the point \u03bb, and let this hypothesis correspond to all positions observed at the moment of opposition to the sun\u2019s mean position.","111":"Now, Kepler, what more could you ask of Copernicus? Are you denying that this hypothesis corresponds entirely to observations or to astronomers\u2019 experience?3 That is indeed not in question at present. Nor was I, when I first entered upon this undertaking, tempted by the observations to take up a different opinion. But it is this that I have been wanting: Let \u03b2\u03b4 be extended so as to intersect the eccentric at \u03c7 and \u03be, and near \u03c7 let some point \u03c4 on the eccentric be chosen, and lines be drawn from \u03c4 to \u03b4 and \u03bb. Now \u03c7\u03c4 is the measure of the angle \u03c7\u03bb\u03c4, while the angle \u03c7\u03b4\u03c4 is greater than the angle \u03c7\u03bb\u03c4 by the amount \u03b4\u03c4\u03bb, and \u03b4 is the point of temporal uniformity. Therefore, the time designated by \u03c7\u03b4\u03c4 is greater, with respect to the whole periodic time designated by four right angles, than is the arc \u03c7\u03c4 with respect to the whole circumference. The planet, then, is actually slower over the arc \u03c4\u03c7 (this is not just an optical illusion), and fast over the opposite arc; and at \u03c7 it is slowest, and at \u03be it is fastest. Nevertheless, it is not farthest from the sun \u03ba when it is at \u03c7, nor is it nearest \u03ba at \u03b6. But by all the arguments, even the testimony of the very hypothesis set up upon \u03b2 which I am refuting, it is fitting that this real slowing down of the planet arises from its moving away from the body of the sun, and the speeding up from its approach to the sun itself, seated at \u03ba. On the other hand, it is impossible even to conceive of how a force could inhere in point \u03b2, which has no body, rather than in k, quite nearby, in which there is the sun, the heart of the world, which force would drive the planet around more swiftly or slowly according to its approach and recess. Furthermore, even if one who is not prepared to admit that the slowings and quickenings arise physically from the close interconnection of the eccentrics, should consequently assert that these affects of motion are naturally under the control of the motive faculties residing in the body of the planet, we will again maintain the same probable conclusion. For what would be the reason why those ","112":"minds would bypass the point \u03ba (which has a geometrical affinity for motion, being invested with a body of no small magnitude) and pay attention to the point \u03b2, only four semidiameters (diameters, according to the authorities)5 of the solar body distant from the sun, empty, and propped up by nothing but imagination alone? Even Copernicus himself admits, in Book 5 Chapter 16, that the sun is really fixed at \u03ba, wherefore the eccentricity \u03b4\u03ba is constant, while he holds that the point \u03b2, which he takes as the center of the annual orb, is displaced over the ages, thus making \u03b2\u03b4 shorter. Thus either \u03b2 is no longer in the center of the world today, or it was previously not there. But it is probable either that the motion originates from the center of the world, or that the moving minds pay attention to the center of the world, and thus not to \u03b2 but to \u03ba, which Copernicus says is fixed, as the center of the world should be.","113":"Led by these probable arguments, I concluded that the line of apsides taken in order to produce the planet\u2019s first inequality, ought to go right through \u03ba itself rather than \u03b2. But we shall obtain this result when we make use of the sidereal positions which the planet has at the moment when the sun\u2019s apparent position and its own are opposite. ","114":"And when the points \u03ba and \u03b2 are collinear with the earth \u03b3, and when the planet itself falls on the same line at the same time, as at \u03c4, it is at the same moment at opposition to both the sun\u2019s mean position and its apparent position. Its position is the same whether it be designated by \u03b2\u03c4 or by \u03ba\u03c4 extended out to the fixed stars, and the planet is truly stripped of its second inequality, whether it depends upon the earth\u2019s apparent position or its mean position. But when the earth comes to the side of its eccentric, or rather the middle intervals, a fairly large difference arises between them. For let the earth have traveled from \u03b3 to \u03c5 (that is, let the sun, directly opposite, have moved from perigee in Capricorn to Aries), and let the line of the sun\u2019s mean position \u03c5\u03b2 be found in Aries, while the line upon which the planet is observed in Libra be exactly opposite, namely, \u03c5\u03c9. Now since \u03c5\u03ba is farther eastward beyond \u03c5\u03b2, the sun\u2019s apparent position is beyond opposition to the planet. And since \u03c5 is the observer\u2019s home, the earth, and \u03c9 is the planet, and both are going down towards \u03be, the earth \u03c5 more swiftly, the line \u03c5\u03c9 will at a later time be still more inclined to the line of the sun\u2019s visible position \u03c5\u03ba. Therefore the apparent opposition precedes the mean. So, at a time preceding the moment designated by \u03b2\u03c5 (call it \u03b2\u03b8), the planet will fall upon a line drawn from \u03ba through \u03b8, that is, at \u03b6. And then the planet\u2019s line of vision \u03b8\u03b6 (as the less experienced should diligently note) lies farther eastward beneath the fixed stars than the line \u03c5\u03c9 of the later time. This is because, although \u03b8\u03b6 precedes the line \u03c5\u03c9 in being farther west, it is nonetheless exactly as if \u03b8, \u03c5, and absolutely all points on the earth\u2019s circle were a single point and were the center of the sphere of the fixed stars. Therefore, it is not the distance of the endpoints \u03b8 and \u03c5 but the inclination of the lines \u03b8\u03b6 and \u03c5\u03c9 that causes the lines to strike upon different zodiacal positions, since they would be perceived as coinciding if they were parallel. But that \u03b6 is inclined towards \u03c9 is clear from the supposition that the planet is moved from \u03b6 to \u03c9 in the same time that the earth is moved from \u03b8 to \u03c5. For the earth is swifter than the planet. Therefore, the earth traverses a greater space \u03b8\u03c5 than does the planet along \u03b6\u03c9.","115":"But it can be shown even more easily that at an earlier time the planet is farther east, since at opposition it is retrograde, as is clear to everyone. It is therefore clear what is altered in the positions stripped of the second inequality, in this transition from the mean to the apparent positions of the sun.\nFor, at \u03c4 and the point opposite, the original positions remain; \u03b1t \u03b6 \u03bfr \u03c9 an addition is made to the observed position, since \u03b8\u03b6 (as was said) is farther eastward than \u03c5\u03c9. A subtraction is made in the intervening time, because \u03b8\u03b6 is the line of vision at an earlier time than is \u03c5\u03c9. At the opposite position the outcome is the contrary, that is, an addition is made to the time and a subtraction from the position. Accordingly, these positions of the planet differ considerably from the original ones. And therefore, the operation set up in this new way produces quite different results. That is, since we have transferred the fictitious point of observation to the sun \u03ba (by virtue of our having viewed the planet when it was at \u03c4 and \u03b6 and the earth was on the lines \u03ba\u03c4 and \u03ba\u03b6, namely, at the points \u03b3 and \u03b8), the eccentricity will now originate at the point \u03ba. But in Chapter 5 above, it was shown that when the observer is shifted from \u03b2 to \u03ba and a line is drawn from \u03ba through the original point of uniformity \u03b4, although this new hypothesis does result in a new eccentric, it is one that, for the observer at \u03b2, leaves nearly all his appearances undisturbed. So, by joining \u03b4\u03ba, dividing it at \u03bc so that \u03b4\u03bb is to \u03b4\u03bc as \u03b4\u03b2 is to \u03b4\u03ba, tracing out a new eccentric \u03b7\u03b5 about \u03bc equal to the previous one \u03be\u03c7, and drawing a new line of apsides through \u03ba\u03b4, a new hypothesis will be formed, whose apsis is at \u03b7. Previously, however, we had improperly called \u03c7 the apogee, because the Copernican center \u03b2 on the line \u03c7\u03b2 was the successor to the Ptolemaic position of the earth. Now, following my own notions, we shall call \u03b7 the aphelion (since we are in the Copernican hypothesis), and the point opposite it perihelion, because the sun\u2019s distance from \u03b7 is a maximum. \n It has been said how this pair of opinions, mine and those of the authorities, differ as regards physics. It has also been shown how each is to be constructed geometrically in the Copernican form. Third, it has been driven home that in astronomical terms those opinions8 do not differ in any important way at the moments of conjunction and opposition9. The next thing for me to do is to demonstrate what remained unexplained in Chapter 5 above, that there is a considerable difference between the two hypotheses if you are required to use them to compute the planet\u2019s position outside the acronychal location. ","116":"If a line be drawn through the centers of the eccentrics \u03bb, \u03bc, parallel to \u03b2\u03ba, and extended to intersect each eccentric in two points, above and below, there will be set up, below, the maximum space \u03b5\u03c1 between the two, equal to \u03bb\u03bc. But because it is lines from \u03b4, not those from \u03bb, that designate certain unchanged moments of time, which is what we need here, let \u03b4\u03c1 be drawn intersecting the eccentrics at \u03b5 and \u03c1, so that at one and the same moment the planet on one might be at \u03b5 and on the other at \u03c1. When the earth is on the line \u03b4\u03c1, at \u03c0, the planet, whether it is at \u03b5 or \u03c1, will in either case be seen at the same place on the zodiac. For, optically considered, the line \u03b5\u03c1 appears as a point. But when the earth departs towards either side of this line, the magnitude of the line \u03b5\u03c1 appears greater and greater, since it is viewed obliquely.\nIt is required to find the point on the earth\u2019s orb from which the lines of vision passing through \u03b5 and through \u03c1 are at their greatest distance from one another and form the greatest angle of vision, and at which the error would be greatest if the planet were placed at \u03c1 when it should have been placed at \u03b5.","117":"First, the angle will be greater down at \u03b5 than up near \u03c4 because the earth\u2019s orbit, described about \u03b2, moves the observer nearer to \u03b5\u03c1 than to \u03c4. Next, since \u03b4\u03c1 is beyond \u03c4\u03b2, \u03b5\u03c1 is seen more obliquely from the left side than from the right. It will consequently appear less at the former place than at the latter even when the distances of the earth from the line \u03b4\u03c1 are equal. Therefore, the point we are to find is on the right side. I say that \u03b5\u03c1 subtends a maximum angle of vision when the observer is stationed at the point where a circle drawn through \u03b5 and \u03c1 is tangent to the earth\u2019s circle. For let such a circle be described through \u03b5\u03c1 tangent to the circle \u03c5\u03c3 on the side towards \u03c3, and let the point of tangency be \u03bd. From \u03b5 and \u03c1 let lines extend both to the tangent point \u03bd and to several other points of circle \u03c5\u03c3 before and after the point of contact. Now since circles touch one another in one and only one point, the sides of all angles extending from \u03b5 and \u03c1 and meeting with points on circle \u03c5\u03c3 will therefore be cut by the circle through \u03b5\u03c1, with the exception of those that terminate at the tangent point of the circles, \u03bd. But the sides from \u03b5 and \u03c1 that are cut by the circle \u03b5\u03c1 before they intersect would form a greater angle had they intersected at either of the points of section, by Euclid\u2019s Elements I. 21. And by Euclid III. 21, all angles at the circumference set up on segment \u03b5\u03c1 are equal. Therefore, the one at \u03bd (the point of contact) is greater than all the others. q.e.d.","118":"Next, to investigate its magnitude in appropriate numbers, we need to find \u03b5\u03c1 and also the perpendicular from \u03b2 to \u03b4\u03c1.","119":"We shall find both by solution of the triangles \u03b4\u03bb\u03c1 and \u03b4\u03bc\u03b5. Now we said above12 that in \u03b4\u03bb\u03c1, \u03b4\u03bb is 741113 where \u03bb\u03c1 is 100,000, and \u03c1\u03bb\u03b2 is 47\u00b0 59\u2019 16\". This gives \u03c1\u03b4\u03bb a value of 44\u00b0 59\u2019 10\", and \u03b4\u03c1 105,123. Therefore, in \u03b5\u03b4\u03bc, since \u03b5\u03b4\u03bb is 44\u00b0 59\u2019 10\", and \u03bb\u03b4\u03bc earlier came out to be 5\u00b0 27\u2019 47\", the whole \u03b5\u03b4\u03bc is 50\u00b0 26\u2019 57\", and \u03b4\u03bc above was 6763 where \u03bc\u03b5 is 100,000. Therefore, in \u03b5\u03b4\u03bc, with three magnitudes given the rest are given: \u03b5\u03bc\u03ba is 53\u00b0 26\u2019 17\", and through this, \u03b4\u03b5 is 104,170. But earlier, \u03b4\u03c1 was 105,123. Therefore, the remainder, \u03b5\u03c1, is 953. Above, \u03bb\u03bc was 880, to which \u03b5\u03c1 would be equal were \u03b5 and p on the line \u03bc\u03c1. But because here \u03b5 is on the line \u03b4\u03c1 which is inclined to \u03bc\u03c1, you should not be surprised that \u03b5\u03c1 is longer than \u03bc\u03bb14. Now let \u03b2\u03b9 be drawn from \u03b2 perpendicular to \u03b4\u03c1. In triangle \u03b4\u03b2\u03b9, the angle at \u03b9 is right, and \u03b2\u03b4\u03b9 is 44\u00b0 59\u2019 10\", and \u03b2\u03b4 was found to be 19,763, above. Therefore, the required perpendicular, \u03b2\u03b9, is 13,971, and \u03b4\u03b9 is 13,978. Consequently, \u03b9\u03c1 is 91,145. It is also necessary to figure out the quantity of the radius \u03b2\u03c5 in the same units. Above, when the line corresponding to the present \u03b2\u03ba was taken to be 3584 parts, \u03b2\u03c5 was assumed as 100,000. Now, however, \u03bb\u03c1 is assumed as 100,000, and \u03bb\u03c1 is to \u03b2\u03c5 (taken above) as approximately 61 to 40, whence the other ratios are extrapolated. Thus 60 is to 41 as 100,000 is to 65,656\u00bd, the appropriate magnitude for \u03b2\u03c5. ","120":"Next, let a circle passing through \u03b5 and \u03c1 touch the circle \u03b2\u03c5 at point \u03bd, and, \u03b5\u03c1 being bisected at \u03bf, let \u03c8\u03bf be set up perpendicular to \u03b9\u03c1, and \u03b2\u03bd be extended so as to intersect \u03bf\u03c8 at \u03c8. The center of the circle will be \u03c8. For the center of the circle is on the line passing through the center of one of the tangent circles and the point of tangency, by Euclid III. 11: hence, it is on the line \u03b2\u03c8. Again, by Euclid III. 3, the center of the circle is on the perpendicular bisector of the chord \u03b5\u03c1, which connects the points of intersection \u03b5 and \u03c1. Therefore, the center is on the line \u03bf\u03c8, and hence is at the point \u03c8 common to the two lines. Let \u03b5\u03c8 be joined, and from \u03b2 parallel to \u03b9\u03c1 let \u03b2\u03b1 be drawn intersecting o\u03c8 at \u03b1. Therefore, \u03b2\u03b1 is equal to \u03b9\u03bf, and \u03b1\u03bf is equal to \u03b2\u03b9. But \u03b2\u03b9 was just found to be 13,971, while \u03b9\u03bf is known through \u03b9\u03c1 and \u03b5\u03c1. And \u03b9\u03c1, above, was 91,145, and \u03b5\u03c1 was 953. But \u03bf\u03c1 is half of \u03b5\u03c1, and therefore, \u03bf\u03c1 is 476\u00bd. So, when \u03bf\u03c1 is subtracted from \u03b9\u03c1, the remainder \u03b9\u03bf or \u03b2\u03b1 is 90,668. Now since \u03b1 is a right angle, \u03b2\u03c8 is the power15 of the two, \u03b2\u03b1, \u03b1\u03c8. However, \u03b2\u03c8 is composed of \u03b2\u03bd, which is known (65,656), and \u03bd\u03c8. But because \u03bf is a right angle, \u03bd\u03c8, that is, \u03b5\u03c8, is the power of the known \u03bf\u03b5 (476\u00bd) and o\u03c8 which is composed of the known \u03bf\u03b1 and \u03b1\u03c8, which is unknown but was also noted previously. Therefore, \u03bf\u03c8 must be made long enough that when you add the powers of \u03c8\u03bf and \u03bf\u03b5, the side \u03b5\u03c8 or \u03c8\u03bd will be just so long that when the power of the sum of \u03b2\u03bd and \u03bd\u03c8 is diminished by the power of \u03b2\u03b1, it leaves the power of \u03c8\u03b1 of such a magnitude that when it is compounded with \u03b1\u03bf the sum is equal to \u03bf\u03c8 that was taken at the beginning. ","121":"I take \u03c8\u03bf as a figured unit [x]. Its square will also be a figured unit [x2]. Add the square on \u03b5\u03bf, 227,052, and the sum of the two will be the square of \u03c8\u03b5 or \u03c8\u03bd. But the square of \u03b2\u03bd is 4,310,747,475. If you add this to the square of \u03c8\u03bd and complete the rectangle, the result will be the square of the whole \u03c8\u03b2. Then each rectangle is the root of","122":"4,310,747,475x2 + 978,763,835,536,363.","123":"And thus the square on \u03b2\u03c8 is obtained for the first time. 18","124":"Now since \u03b1\u03bf is 13,971, \u03c8\u03b1 will be represented by the figured unit diminished by 13,971. Its square will be","125":"x2 - 27,942x + 195,188,841.","126":"Add to this the square of \u03b2\u03b1, 8,220,686,224, so that the square on \u03b2\u03c8 may be established for a second time:","127":"[\u03b2\u03c82 =]x2-27,942x+8,415,875,065.","128":"Previously it was","129":"x2+4,310,974,527 augmented by double the root of","130":"4,310,747,475x2+978,763,835,536,363.","131":"Subtract x2 from both, and also 4,310,974,529. In the former, the remainder will be\n\u201427,942x+4,104,900,538,","132":" and in the latter,","133":"twice the root of 4,310,747,475x2 + 978,763,835,536,363,\nand these are equal. Therefore, the simple root in the former is equal to","134":"\u201413,971x + 2,052,450,269.","135":"And since this is equal to the latter\u2019s root, its square will be equal to the quantity itself. This square is","136":"+195,188,841x2-57,349,565,416,398x+4,212,552,106,718,172,361.\nSubtract from each","137":"195,188,841x2 and 978,763,835,536,363,","138":"and add to each","139":"57,349,565,416,398x.","140":"The two will remain equal, the former being","141":"4,115,558,634x2+57,349,565,416,398x,","142":"while the latter,","143":"4,211,573,342,882,635,998.","144":"In least terms,","145":"x2+13,934x is equal to 1,023,329,690.","146":"Solving the equation gives the figured unit \u03bf\u03c8 the value of 25,772. ","147":"Now that the semidiameter of the circle is known, the angles are easily obtained. From \u03c8\u03bf subtract \u03bf\u03b1, 13,971. The remainder, \u03c8\u03b1, will be 11,801. And \u03b2\u03b1 is 90,668\u00bd, and \u03b2\u03b1\u03c8 is right. Therefore, \u03b1\u03b2\u03c8 is 7\u00b0 30\u2019 10\". But above, \u03b1\u03b2 or \u03c1\u03b4 were inclined to \u03c1\u03bb or \u03b2\u03ba by 3\u00b0 0\u2019 6\", which latter lies at 5\u00bd\u02da Cancer. Therefore, \u03c1\u03b9 or \u03b1\u03b2 will be at 8\u00bd\u00b0 Cancer, and \u03c8\u03b2 is consequently at 16\u00b0 Cancer. Therefore (assuming these numbers), when the sun is passing through 16\u00b0 Cancer while the planet is at 8\u00bd\u00b0 Capricorn in its mean and uniform motion, and in the neighborhood of 27\u00b0 Scorpio in its apparent motion, \u03b5\u03c1 appears maximum. If the planet is beyond 8\u00bd\u02da Capricorn, that is, beyond \u03c1\u03b5, even though \u03c1\u03b5 is then diminished, its apparent size can increase when viewed from a point beyond \u03bd owing to the greater nearness of the orbs. This apparent size is now immediately obtained. For since \u03bf\u03c8 was found to be 25,772, and \u03bf\u03c1 476\u00bd, \u03bf\u03c8\u03b5 will be 1\u00b0 3\u2019 32\" But \u03c1\u03bd\u03b5 (which is what we have been seeking) is equal to \u03bf\u03c8\u03b5, by Euclid III. 20. This is because the whole angle at the center \u03c1\u03c8\u03b5 is twice the angle at the circumference \u03c1\u03bd\u03b5, and at the same time \u03bf\u03c8\u03b5 is half of \u03c1\u03c8\u03b5. But if \u03b2\u03b4 and \u03ba\u03b4 be bisected [at \u03bb and \u03bc], and \u03bb\u03bc were to be taken as half \u03b2\u03ba (on which more will be said below), then \u03c1\u03b5, and consequently its angle at \u03bd as well, could possibly become greater by one fourth. So, at last, you see how much my transposition of the hypothesis from the mean to the apparent position of the sun creates a disturbance in the annual parallax.","148":"Therefore, by the observations as well, a door has opened for us to determine what I had deduced a priori by consideration of moving causes: that the planet\u2019s line of apsides, which is the only line bisecting the planet\u2019s path into two semicircles equal both in size and in vigor\u2014this line, I say, does not (as the practitioners have it) pass through some point other than the sun, but right through the center of the solar body. In the conclusion of the work I shall demonstrate this by means of observations, in Parts IV and V.","149":"Now, to the extent that this is possible, I shall deduce the same things in the Ptolemaic hypothesis. ","150":"About center \u03a8 let the eccentric of the sun \u0393 be described, upon which let \u03a8\u0393 be the line of apsides, with the motionless earth at point \u039a of the line \u03a8\u0393 on the side nearest \u0393, and \u03a8 be the point about which the sun\u2019s motion is reckoned as being uniform. From \u03a8 and \u039a let the perpendiculars \u03a8\u03a3 and \u039a\u03a5 be set up, and let \u03a3\u039a be joined. Let \u039a\u03a3 be the line of the sun\u2019s apparent motion, and \u039a\u03a5 the line of the sun\u2019s uniform motion.","151":"Now Ptolemy measured out the courses of the planets, not on the lines \u039a\u03a3, but on the lines \u039a\u03a5 drawn from \u039a parallel to the lines \u03a8\u03a3 passing through the sun\u2019s body. For whenever the planet fell upon these lines \u039a\u03a5 on the side opposite the sun, it was supposed to have shed its second inequality, which fell to it (in Ptolemy\u2019s opinion) because of the epicycle. Then the planet\u2019s sidereal position was sought out by means of instruments, and the center of the epicycle was supposed to be found at that time on the same line. This was done a number of times and at various places on the zodiac: say, \u039a\u0393, \u039a\u03a5, and the points opposite. From three such positions of the planet (or of the center of the epicycle, which, according to Ptolemy, accounts for the second inequality), the practitioner sets out to investigate the hypothesis for the first inequality by comparing these angles which the observed positions form about \u039a, the center of the earth and of observation, with the intervening times. The method for this enterprise is found in Ptolemy Book IX. Now suppose the treatment done, and let the eccentric's line of apsides come out to be \u039a\u039b\u0394\u03a7, with \u0394 the equalizing point, the center of the eccentric upon this line and at point \u039b, and the eccentric \u03a7\u0396. And let this hypothesis correspond to all positions observed at the times of the planet\u2019s opposition to the sun\u2019s mean position.\nHere, the objections I raised against Copernicus concerning the ordering of the physical motion are also not clearly in agreement with Ptolemy. For the center of the epicycle which accounts for the second inequality, here just as the planet itself previously, is borne slowly or swiftly according to its approach to or recession from the earth \u039a, performed on the circle \u03a7\u0396. Furthermore, to stipulate that there is in the earth \u039a (just as there was before in Copernicus\u2019s sun, the heart of the world) a motive force that drives the centers of this kind of epicycle around, is absurd and monstrous. The hypothesis can also be discredited by another physical argument. For intrinsic to this form of hypothesis, in one way or another, is solidity of the orbs; and since this has been destroyed (by Tycho Brahe\u2019s observations of comets), the hypothesis appears, so to speak, to collapse under its own weight. For a moving force would be declared to reside in the center of the epicycle (not in a body, but in a mathematical point), and to rouse itself to move from place to place, by an unequal amount in equal times; but at the same time would draw the planet along with itself at the distance of the epicycle\u2019s diameter; all the while causing the planet to gyrate around itself, by equal amounts in equal times. That this enormous variety cannot fall to a single moving mind (unless it be God), Aristotle gives supporting arguments in the Metaphysics Book XII Chapter 8: he holds that the individual minds preside over individual, perfectly uniform and simple circular motions. Besides, how will some power sit in a non-body, and flow out of the non-body into the planet? Even if you divide up the tasks, and locate one motive intelligence at the center of the epicycle and another in the body of the planet, the one at the center will use the earth (a body, of course) as its reference, and will move around the earth nonuniformly in a circle, while the one at a point on the circumference (that is, in the planet's body) will move around an incorporeal center and do so uniformly. The question, then, is (as before), by what aids would it move around the incorporeal point? Not by geometrical imagination, for it does not admit of geometrical imagination of itself. Nor can a mobile point subsist in a non-body even in imagination. We humans, in imagining points of this sort, have recourse to the aids of tablets or paper, which we draw upon with our hands or at least remember ourselves to have drawn once. But it likewise cannot happen through a physical flowing of power (which is in the center of the epicycle) out to the circumference and the planet\u2019s body. For we have submitted to this outflow of power on the supposition that the tasks of the compound motions are divided between two minds. Why not also question whether in the first, eccentric motion, some natural force constituted to produce motion might subsist in some point entirely devoid of any body? And even more, whether this kind of incorporeal power can move itself around the earth, and go from place to place? And most of all, whether it can communicate or transfer another's motion through an efflux from itself, although it is not supported by any body that could serve as its nest? Those sublime considerations of the essence, motion, place, and operations of the blessed angels and of separate minds which some people will want to raise in opposition to me, are irrelevant. For we are arguing about natural things which are far inferior in dignity, about powers that exercise no choice of how to vary their action, and about minds which are not in the least separate, since they are yoked and bound to the celestial bodies which they are to bear. These, then, are the general objections that can be raised against Ptolemy. \nBut something further may be said to Ptolemy, on account of which in particular he would wish to abandon his mean position of the sun and embrace the apparent with us. For if the power moving the planet (whether single or double) acts with respect to the sun, so that it places the planet at the lowest point of the epicycle whenever the center of the epicycle is opposite the sun, I ask (as above) why it looks to the imaginary point \u03a5 (which sometimes precedes and sometimes follows the sun, here marked \u03a3, and is sometimes above and sometimes below it), rather than the sun's body itself? Or how it is that that power can have any perception whatever of the motion of \u03a5 around the earth K, since there is no body at \u03a8? Is it not more probable that the epicycle is restored to the lines \u039a\u03a3 of the sun\u2019s apparent position, when these pass through the center of the epicycle?\n Let us then see what is changed on the eccentric by substitution of the sun\u2019s apparent position. Again (as before) when the sun \u0393, the center of the sun\u2019s eccentric \u03a8, and the earth \u039a are on the same line, so that the line of the sun\u2019s apparent position \u03a8\u0393 and the line of its mean position \u039a\u0393 coincide, then for the center of the epicycle \u03a4 this place remains the same, whether its sidereal position is designated by \u039a\u03a4 or by \u03a8\u03a4, and the planet is really on the line \u039a\u03a4 or \u03a8\u03a4 and at the lowest point on its epicycle \u03a6, since here it is nearest to both \u03a8 and \u039a. The planet is thereby truly divested of its second inequality. But when the sun comes to the side of its eccentric (that is, to the middle distances), a great enough difference arises. For let the sun have gone from \u0393 to \u03a3, and the line of the sun\u2019s mean position \u039a\u03a5 be found in Aries, and the planet\u2019s line of vision \u039a\u03a9 precisely opposite in Libra, so that \u03a5\u039a\u03a9 is a single line. Now since Ptolemy declared that the planet \u03a9 on this line of vision \u039a\u03a9 is divested of its second inequality, he places the center of the epicycle \u0396 on the line \u039a\u03a9. But since \u039a\u03a3 has gone beyond \u039a\u03a5, the sun\u2019s apparent position is beyond opposition to the planet. And in the last part of the time, \u039a\u03a9 is not descending, so as to be opposite \u039a\u03a3, but ascends towards \u039a\u03a6, because the lowest parts of the epicycle \u03a9 are retrograde and swifter than the center \u0396, and it is there, of course, that the planet is at opposition to the sun. Therefore, this apparent opposition precedes the mean one.\nTherefore, at some time preceding the moment designated by \u039a\u03a5 (let it be \u039a\u0398) when the sun is seen on the line \u039a\u039e, the planet will be seen opposite it (suppose it to be at \u0399) on \u039a\u0399 which forms a straight line with \u039a\u039e. Also, because it is now laid down that it sheds the second inequality at this true opposition, the center of the epicycle will also be seen on this line \u039e\u039a, say at \u039f. And because the planet is retrograde, at the time \u039a\u0398 prior to \u039a\u03a5 the planet is on the line \u039a\u0399 later than at \u039a\u03a9. But \u039a\u0399 and \u039a\u03a9 are parts of the lines \u039a\u039f and \u039a\u0396. Therefore, \u039a\u039f too is more to the east than \u039a\u0396.\nThus it is clear what would be changed on the line of the epicycle\u2019s center by this restoration from the sun\u2019s mean position to its apparent position. For at \u03a4 and the opposite point on the original line the motions of the epicycle\u2019s center remain unchanged. At \u0396 this line, and the center of the epicycle upon it, is moved forward and a subtraction is made from the intervening time. At the opposite place the contrary happens: an addition is made to the time, and the line of motion of the center of the epicycle is moved back westward. And thus these lines of the center of the epicycle differ much from the original lines. For this reason also, when we investigate the causes and measure of the first inequality by a new and repeated operation using these several observed positions of the center of the epicycle (that is, using the observed positions of the planet, after which we suppose the center of the epicycle to lie upon the same line), the outcome of this operation differs much from that of the preceding one. That is, since the time in the semicircle containing the apogee will have been diminished, so that the planet is correspondingly faster, the eccentricity of the equant will come out smaller. And since in the greater quadrant \u0392\u0396 of the semicircle containing the apogee the time will have been diminished by an amount equal to the diminution in the other, smaller part, the planet has been made proportionally much faster in the remaining part of the semicircle. Therefore, the perigee has come closer to the latter, and the apogee has descended from \u03a7 towards \u0396.\nThe quantification of the new hypothesis will be made clear thus. It is presupposed that the planet \u03a9 falls on the line drawn from the center of the epicycle \u0396 through the earth \u039a, only when \u039a\u0396 is collinear with \u039a\u03a3, the sun\u2019s apparent position. Therefore, \u039a\u03a3 and the line drawn from \u0396 through the body of the planet always proceed in parallel. Moreover, we have just taken it from Ptolemy that at the time when the line of the sun\u2019s mean position was \u039a\u03a5, drawn through \u03a9, the planet was observed on the line \u039a\u03a9, and nonetheless we do not grant him that the center of the epicycle \u0396 is on \u039a\u03a9 at the same time. Therefore, in accord with our position, let a line be drawn from the planet\u2019s position \u03a9 parallel to \u039a\u03a3, and let this be \u03a9\u039f. We are supposing that the center of the epicycle is at this moment on the line \u03a9\u039f, or some other line nearby and parallel, according as \u03a9 (representing the planet) is closer to or farther from \u039a on the line \u039a\u0396. Let \u03a9\u039f be drawn from whatever point on the line \u039a\u0396 (now, it is \u03a9), equal to \u03a9\u0396. From \u039f let a line be drawn to \u0396\u039a parallel to \u039a\u03a8, and let this be \u039f\u0396. Now since \u0396\u03a9\u039f is equal to \u039a\u03a3\u03a8, and \u039a\u03a3 is imperceptibly greater than \u03a8\u03a3 or \u03a9\u039f (because \u039a\u03a8\u03a3 is right, and the angle at \u03a3 is not greater than 2\u00b0 3\u2019, so that if \u03a8\u03a3 is 100,000, \u039a\u03a3 is 100,064), \u039f\u0396 is also imperceptibly less than \u039a\u03a8. Let \u0396\u0394 be joined, and let a line be drawn at \u039f parallel to \u0396\u0394. Now the moment of time at which the center of the epicycle is placed at \u0396 by Ptolemy and at \u039f by me, is the same, and is designated by both of us by the line \u039a\u03a5 in the theory of the sun. In the theory of Mars, that moment should be designated in the former theory by \u0396\u0394, because \u0394 is the point of equality, while in the new theory it will be designated by a line parallel to \u0396\u0394. Therefore, there will be a new point of equality, about which the time is counted, on this parallel through \u039f. ","152":"Further, the same things occur when the center of the epicycle (in Ptolemy's account) is at the other end of the line of the sun\u2019s mean position in the neighborhood of \u039a\u03a5 (which, for the sake of brevity, I shall not prove). Therefore, if some line is again drawn parallel to the Ptolemaic line of the mean position of the center of the epicycle, a line \u0394\u0394 drawn from \u0394 to where the two new parallels intersect will be parallel to \u0396\u039f or \u03a8\u039a, equal to \u0396\u039f, and very nearly equal to \u03a8\u039a, and the new \u0394 will be the common point of equality in the new hypothesis. ","153":"But at the end of Chapter 5 above, it was shown that if through \u0394 a line \u0394\u0394 be drawn parallel to \u039a\u03a8, \u039a\u03a8 being equal to \u0394\u0394, and if the new \u0394 be joined to \u039a, and the new line \u039a\u0394 be cut at \u039c in the same ratio in which the previous \u039a\u0394 was cut at \u0394, then by this new hypothesis a new eccentric is constructed, that is, one having a different position from the former, which even if applied in the prior hypothesis would still leave all the appearances to an observer at \u039a very nearly undisturbed. Let such a new eccentric be described about \u039c, equal to the former, and let \u039a\u039c be extended in both directions. The new apogee will be \u0397, and the center of the epicycle will be on the points \u0392, \u039f, of the new eccentric, with the planet nearer at \u0391 and farther at \u0399 than before. However, in positions in which the second inequality is involved, the former observations are thoroughly and vehemently perturbed by this new eccentric introduced into their hypothesis. (This is clearly so if an epicycle equal to the sun\u2019s eccentric is attributed to the planet, as it must be if we wish to carry over the force of Copernicus\u2019s and Tycho\u2019s discoveries into the Ptolemaic form.) The reason is not that the point of equality \u0394 does not remain the same, but that near the positions of the sun\u2019s apsides the centers of the two eccentrics, the Ptolemaic and ours, are distant by an interval \u039b\u039c. Also, from this distance of the centers there necessarily follows an equal distance of the positions of the planetary body. Furthermore, this discrepancy is not greatest when the center of the epicycle is about the sun\u2019s middle distances. For it has been said that at those places the position of the center of the epicycle on either eccentric is nearly the same, even though they stand apart by parallels from \u0394\u0394. It is therefore greatest near the sun\u2019s apsides, and greater near perigee on \u039c\u039b extended so as to intersect the eccentrics at \u03a1 and \u0395. For \u03a1\u0395 is of the same magnitude as \u039c\u039b. But this one line \u039c\u039b does not designate the same moment, since it is not \u039c or \u039b that is the point of equality, but \u0394. Therefore, let parallels be drawn from \u0394\u0394 towards \u03a1 and \u0395, which shall designate the same moment, and let them be \u0394\u03a1 and \u0394\u0395. Also, let the epicycles \u039d and \u03a0 be described about \u03a1 and \u0395.","154":"The question now is, where would this discrepancy appear greatest, in terms of the circumference of the epicycle? It is certain that this does not occur at the parts of the epicycle nearest \u039a, the earth, because these parts would be in the same direction as \u039a, nor at the highest parts of the epicycle, because they would be too far away. Therefore, this will occur at the parts of the epicycle near perigee, therefore when the sun, and the planet with it, are not exactly at perigee but nearby: namely (to put it briefly) at the points \u039d and \u03a0 corresponding to the same moment of time, such that the small circle through them and \u039a is a minimum. The center of this small circle is, moreover, on the line through \u039a which, extended upward and meeting \u03a1\u0394 likewise extended, makes an angle of 7\u00bd degrees.","155":"Let anyone who disagrees with this adapt the previous demonstration to the present conditions. The numbers are just the same, except for Ptolemy \u039c\u039b is greater than \u03bc\u03bb, in the numbers taken above. For this reason, the difference in apparent position, \u039d\u039a\u03a0, is also greater.","156":"For previously, \u03b2\u03ba was to \u03bb\u03bc as \u03b4\u03b2 was to \u03b4\u03bb, which is less than half \u03b4\u03b2. To Ptolemy, \u0394\u0394\n(equal to \u03b2\u03ba) would be to \u039c\u039b as the whole \u039a\u0394 is to the half, \u039a\u039b. ","157":"Finally, I shall deduce the same things in the Tychonic hypothesis. ","158":" About center B let the solar eccentric GS be described, with line of apsides BG, C the position of the immobile earth, and B the point of equality, following the opinion of the authorities. For it will be shown in due course that the point of equality and the center of the eccentric are not the same in the theory of the sun. Upon BC let the perpendiculars BS and CV be erected, and let SC be joined, so that CV may be the line of the sun\u2019s mean position, and CS, of its apparent position.","159":"Now, although Tycho Brahe had never finally decided whether to refer the planets to the lines CV, or to CS instead, in his initial conception he used the lines CV, according to the explanation he left in the Progymnasmata, vol. I p. 477 and vol. II p. 188. This is the way shown him by the footprints of Ptolemy and Copernicus. Of this path trodden by Tycho (if we proceed on it with Ptolemy\u2019s views), it should be said that whenever a planet is on one of the lines of the sun\u2019s mean position CV, on the side opposite the sun, the planet is divested of its second inequality, which, in Brahe\u2019s opinion, is applied to it on account of the motion of the center of the eccentric around the earth in the same time as the sun does so.","160":"This common point, with respect to which all the planets are said to perform their eccentric motion, and at which the whole planetary system is conceived as being attached to the sun\u2019s orbit\u2014this point, I say, is always on the line of the sun\u2019s mean position, distant from the earth C by a constant distance BS, and describing a concentric V equal to the eccentric GS. This was Tycho Brahe\u2019s opinion\u2014except that he denied solid orbs. So what we said about the attachment of the whole system to the sun\u2019s orbit, we said to win over those who believe in solid orbs. Let VC be extended, and let the planet be on that line beyond C. In this configuration, Brahe will place the point at which the planetary system is attached at V. The planet will be observed on the line VC. So, even though the observer is on the earth, C, it is exactly as if he were at V, the point upon which the first inequality depends. Next, let the planet\u2019s sidereal position be taken with instruments whenever it shall be seen at some point on the line CV opposite V beyond C (let this be on the lines CV, CG, and points opposite). Thus the center of the planetary system will be on the circle VP, the sun at S and G, and the body of the planet opposite it at O and F. (Nevertheless, in the theory of Mars, the planet\u2019s eccentric is so small in proportion to the sun\u2019s eccentric that Mars\u2019s eccentric and the points O and F are nearer to the earth C than is the sun S, which was one of the reasons why Brahe denied the solidity of the orbs.) From many positions of this sort, as many, in fact, as he could obtain, Tycho Brahe used to investigate the hypothesis of the first inequality, by removing the magnitude of the orb VP and treating it as a single point, as though in the meanwhile the center VP of the planetary system, the point at which the system is attached, had remained at rest. So he set up a comparison of the elapsed times with the angles which VO and PF, drawn from a single point (V and P coinciding), would form. These are in fact identical to the angles OCF and VCP. \nSuppose this treatment done, and let the line of apsides of the eccentric come out to be VLD or PLD, D the equalizing point, L the center of the eccentric on this line, and the eccentric HO and FH. Let this hypothesis correspond to all positions of the planet observed at the moment of the planet's opposition to the sun\u2019s mean position.","161":"For the present, I am postponing a more careful examination of whether this sort of hypothesis is in accord with physical principles\u2014a hypothesis, that is, in which the sun, through its motive mind paying attention to the earth, moves around the earth, and of itself (since it lacks an orb) drives itself forward nonuniformly according to its approach to or recession from the earth (unless you would make the earth more important than the sun and ascribe to the earth the sun's motive force); while this same sun (as in Copernicus) sends out a motive force to all the planets, sweeping them around itself with a degree of speed corresponding to their degree of nearness to the sun; the planets meanwhile striving to accomplish their approach to and recession from the sun on a small epicycle, and at the same time to follow the sun, wherever it is supposed to be, in those same tracks not proper to them; and thus any planet, and most of all the sun, attends to many things at once, and the actual trajectories of the planets through the aethereal air (as with Ptolemy) make spirals, such as were depicted in Chapter 1. Whether, I say, these are fitting, we shall consider elsewhere when the occasion arises. Here, let this form of hypothesis be supposed true in its general features. The question is, whether it is also fitting, in particular, that the planets follow the sun\u2019s body S, G, or whether they instead follow the point V, P, void of any body, four solar semidiameters (no more) from the center of the sun, which is now above the sun and now below, now before and now behind. And further, whether it is more fitting that the force that drives the planets in an orb around the sun make its nest in the body of the sun S, G, or in some other point such as VP void of body. In brief: if the axle [axis] (to derive the sense of the word roughly from a wagon) of the planetary system, to which the orbs of the planets are fastened as with a nail\u2014if this, I say, is near the sun, why not in the sun itself? If this axis or point of attachment travels around the earth both close to the sun and in the very same period, why does it describe its own peculiar path? Why does it not hold to the very same path as the sun?","162":"I therefore wholeheartedly conclude that if, perchance, Tycho Brahe\u2019s opinion on the world system be universally true, it must be accepted in such form that the center of the planetary system lies upon SG, exactly on the sun\u2019s path, not upon VP; and finally, that it is in the sun itself, and that for liberating the first or eccentric inequality from the second, one should use the planet\u2019s oppositions to the apparent position of the sun, not the mean. Brahe himself in his final days embraced this procedure unreservedly. Let us, then, see what is changed on the eccentric. Once again, as before, when the sun is on the line BC, as at G, and the planet is at F opposite point P, the planet F will be opposite the sun itself, G. Consequently, the sidereal position of the planet will appear to be the same, along the line GF, whether the continuation of the line be CP or CG, because both have been made into one line. Therefore, according to either procedure, the planet is divested of the second inequality. But when the sun comes to the side of its eccentric (that is, to the middle distances), an appreciable difference arises. For let the sun have gone from G to S, let the line of the sun\u2019s mean position CV be in Aries, and the planet\u2019s line of vision CO be exactly opposite in Libra, so that VCO is one line. Since CS is beyond CV, the sun's apparent position is beyond opposition to the planet. But, because of my alteration, the center of the planetary system is not at V but at S when the planet is seen along CO. Therefore, SO being connected, the earth C will lie outside the line SO, and hence, the planet\u2019s apparent position on the line CO still is intermingled with the second inequality. Nor will CO be farther east at a later time, so as to be at opposition to CS. Instead, it will ascend toward CF, because the sun\u2019s motion, and with it the motion of the center of the planetary system and of all its parts (and thus those of the planet O and of the center of the eccentric L), is from the line CO upward towards F, and is much faster than the motion of the eccentric or planet at O about L from the point H towards the bottom. Consequently, O is appreciably drawn back westward by a motion extraneous, not proper, to the eccentric; and indeed, it is well known that the planets are retrograde at opposition to the sun. Therefore, at a time preceding the moment designated by CV (let it be CT), when the sun appears on line CQ, the planet will be seen at the point opposite its apparent position, at I. And now, since in the present case it is supposed to have shed the second inequality, QCI will be collinear; that is, the point whence the eccentricity arises will be on the line CQ. But CI, the retrograde planet\u2019s line of vision, earlier in time, is beyond the later (and hence more westerly) line of vision CO. Therefore, CQ will also be beyond CV, and Q will be the system\u2019s new center, beyond the old, V. Moreover, the line IQ was made to be more distant from the line \u039fV to the east by the angle OCI, while the line of apsides VD or PD (from which the motion begins) remains parallel to itself in the entire circuit. As a result, it appears to be established that the planet goes farther in a shorter period of time around the center Q of the system than it previously did in a greater period of time about the center V of the system. ","163":"It thereby becomes apparent what is changed in the apparent eccentric motion by this restoration from the sun\u2019s mean position to its apparent position. For when the center of the system is at G and the point opposite, the line of the apparent eccentric position remains unchanged, at Q it is moved forward, and opposite Q it is moved back, for at Q the time is diminished, and opposite Q it is increased. Also, these lines differ greatly from the original ones. Accordingly, when we repeatedly use a new operation, based upon these several apparent positions of the planet (opposite which we are supposing the center of the planetary system to be found; that is, in the sun itself), to investigate the causes and measure of the first inequality, the result of the operation differs greatly from the previous result.","164":"For we have just transposed the point of attachment from the circle VP, upon which Brahe had it move around, to the circle GS, that is, to the body of the sun. This new center always stands on a line parallel to CB, at a distance CB from the original Brahean point; for example, above V and P at S and G. Therefore with the point of equality D fixed (so that CV represents the same moment), in order for it to be possible both for the planet to be at O and for the point of attachment to be at S, a new line of apsides needs to be drawn through D and S or G. Consequently, according to the demonstrations of Chapter 5 (which we have brought forward above, in explaining the Copernican form), with DS or DG drawn and divided at M in the ratio in which DP or DV are divided at L, let there be described about center M, with the same radius as before, a new eccentric. This will not only account for the later observations, upon which it was constructed, but, introduced into the prior hypothesis, it will also save the observations previously cited, with a precision of within five minutes.","165":"However, computations which are carried out upon the previous eccentric and this new one at positions other than night rising will in some places (particularly near the sun\u2019s perigee) be able to differ by more than one degree, if we use numbers fitting and proper for Mars, which have been furnished by Brahe. ","166":" There is no need to repeat the demonstration. The drawing is easiest in the Copernican diagram, if from the earth \u03bd you set up a line parallel to \u03b2\u03ba, measure off upon it, at an interval \u03b2\u03ba above \u03bd, the center of the sun's eccentric, and upon this center set up the Brahean eccentric of the sun through k, and delete the Copernican center of the earth\u2019s eccentric.\n The differences of the hypotheses, and their equivalence in the first inequalities but discrepancy in the second, have now been expounded. Let us then conclude the first part, which, as I see it, is the most difficult of the entire work, because of the almost inescapable labyrinths of opinion and the perpetual ambiguities of words and extremely tiresome circumlocutions. What necessity it was, however, that forced me to prefix this body of instruction, will now directly appear in Chapter 7. Anyone who is less energetic can defer the whole part until he understands the easier parts. ","167":"Chapter 7 ","168":"The circumstances under which I happened upon the theory of Mars.","169":"It is true that a divine voice, which enjoins humans to study astronomy, is expressed in the world itself, not in words or syllables, but in things themselves and in the conformity of the human intellect and senses with the sequence of celestial bodies and of their dispositions. Nevertheless, a kind of fate also invisibly drives different individuals to take up different arts, and makes them certain that, just as they are a part of the work of creation, they likewise also partake to some extent in divine providence.","170":"When, in my early years, I was able to taste the sweetness of philosophy, I embraced the whole of it with an overwhelming desire, and with practically no special concern about astronomy. I certainly had enough intelligence, nor did I have any difficulty understanding the geometrical and astronomical topics included in the normal curriculum, aided as I was by figures, numbers, and proportions. These were, however, required courses, nothing that would bespeak an exceptional inclination towards astronomy. And since I was supported at the expense of the Duke of W\u00fcrttemberg, and saw my comrades, whom the Prince, upon request, kept trying to send to foreign countries, stalling in various ways out of love for their country, I, being hardier, quite maturely agreed with myself that whithersoever I was destined I would promptly go.","171":"The first to offer itself was an astronomical position; however, to tell the truth, I was driven forth to take it by the authority of my teachers. I was not frightened by the distance of the place, for (as I have just said) I had condemned this fear in others, but by the low opinion and contempt in which this kind of function is held, and the sparsity of erudition in this part of philosophy. I therefore entered upon this better furnished with wits than with knowledge, protesting loudly that I by no means gave over my right to follow another kind of life which seemed more splendid. What came of those first two years of study may be seen in my Mysterium cosmographicum. The additional goads which my teacher Maestlin gave me towards embracing the rest of astronomy, you will read of in the same book, and in that man\u2019s prefatory letter to Rheticus\u2019s Narratio. I had the very highest opinion of what I discovered there, and all the more so when I saw that Maestlin, too, held it in similar esteem. It was not so much his untimely promise to the readers of what he called my \u201cuniversal uranic opus\" that spurred me on, as it was my","172":"own ardor to seek, through a reworking of astronomy, whether my discovery would support observations of complete accuracy. For it had then been demonstrated in that book that the discovery was consistent with the observations within the limits of accuracy of ordinary astronomy.","173":" So from that time I began to think seriously of comparing observations. In 1597, I wrote Tycho Brahe asking his opinion of my little book, and when he, in reply, mentioned (among other things) his own observations, he ignited in me an overwhelming desire to see them. Moreover, Tycho, who was indeed himself a large part of my destiny, did not cease from then on to invite me to come to him. And since I was frightened off by the distance of the place, I again ascribe it to divine arrangement that he came to Bohemia. I thereupon came to visit him at the beginning of 1600 in hopes of learning the correct eccentricities of the planets. But when I found out during the first week that, along with Ptolemy and Copernicus, he made use of the sun\u2019s mean motion, while the apparent motion would be more in accord with my little book (as is clear from the book itself), I begged the master to allow me to make use of the observations in my own manner. At that time, the work which his aide Christian Severinus4 had in hand was the theory of Mars. The occasion had placed this in his hands, in that they were busy with the observation of the acronychal position or opposition of Mars to the sun in 9\u00b0 Leo. Had Christian been treating a different planet, I would have entered upon it as well.","174":"I therefore once again think it to have happened by divine arrangement, that I arrived at the same time in which he was intent upon Mars, from whose motions alone we will have to come to a knowledge of the hidden secrets of astronomy, or remain forever ignorant of them. ","175":"A table of mean oppositions was worked out, starting with the year 1580. A hypothesis was invented which was proclaimed to represent all these oppositions within a distance of two minutes in longitude. (The numbers I used in Chapter 5 were taken from among these, or differed from them only slightly.) The apogee at the beginning of the year 1585 was placed at 23\u00b0 45\u2019 Leo. The maximum eccentricity, made up of the semidiameters of the two small circles, was 20,160, in units of which the semidiameter of the greater epicycle was 16,380. Therefore, in the Ptolemaic form of the first inequality, the eccentricity of the equalizing point was 20,160 or a little less.","176":"From this hypothesis, a table of eccentric equations for individual degrees was constructed, as well as a table of the corrected mean motion, made by adding 1\u00be\u2019 to the mean motion of the Prutenic Tables. These mean motions, apogees, and likewise the nodes, were extended over a period of forty years, exactly as was done for the solar and lunar motions in Book I of the Progymnasmata. It was only in the latitude at acronychal positions and also the parallax","177":"of the annual orb that Christian got stuck. There was, actually, a hypothesis and table for the latitudes, but they failed to elicit the observed latitude. This result was a problem for him, as he was about to brood over the lunar motions.","178":"Now since I suspected what proved to be true\u2014that the hypothesis was inadequate\u2014I girded myself for the work following opinions that had been preconceived and had been expressed in my Mysterium cosmographicum. At the beginning there was great controversy between us as to whether it were possible to set up another sort of hypothesis which would express to a hair\u2019s breadth so many positions of the planet, and whether it were possible for the former hypothesis to be false despite its having accomplished this so far over the entire circuit of the zodiac.","179":"I consequently showed, using the arguments presented already in Part I, that an eccentric can be false, yet answer for the appearances within five minutes or better, provided that the equant point be true. As for the parallax of the annual orb, and the latitudes, that prize was still unclaimed, and had not yet been won by the hypothesis of those others. What remains, then, is to find out whether they, with their computation, might not somewhere differ from the observations by five minutes.","180":"I therefore began to investigate the certitude of their operation. What success came of that labor, it would be boring and pointless to recount. I shall describe only so much of that labor of four years as will pertain to our methodical enquiry.","181":"Chapter 8","182":"Tycho Brahe\u2019s table of observed and computed observations of Mars\u2019s oppositions to the line of the sun\u2019s mean motion, and an examination thereof.","183":"Now the table, mentioned above, was the following:","184":"An accurate rendering of the motion of the planet Mars on its eccentric from trustworthy acronychal observations in a variety of positions, carried out sedulously of twenty years (1580 to 1600). 1","185":" The amended mean longitudinal motion of Mars at the beginning of 1585 was found to exceed the numbers provided by the Prutenic calculation2 by at least a minute and a half, or at most 1\u00be\u2019, which by all indications appears more nearly correct. However, the position of its apogee then fell short of the Prutenic calculation at the same time, by 5\u00b0 2\u2019, both being compared with the first star of Aries in the Copernican manner. Hence, owing to our removal of the vernal equinox westward from that star, which was then 28\u00b0 2\u00bd\u2019, it is concluded that Mars\u2019s apogee was at 23\u00b0 25\u2019 Leo. For the first date it was set at 23\u00b0 20\u2019 Leo, and for the last, at 23\u00b0 45\u2019 Leo.","186":"Also, the maximum eccentricity, composed of the semidiameters of the two small circles, was found to be 20, 160 parts, of which the semidiameter of the greater epicycle, or distance between centers as given by Copernicus, is 16,380. In both of these, however, he differs both from himself and from Ptolemy. Care was taken, where appropriate, concerning refraction, using the solar parallax.","187":"THIS, THEN, IS BRAHE\u2019S TABLE","188":" We shall begin the examination of the sun\u2019s mean motion with the listed instants of equal time, as many as the table presents. For indeed, it is the sun\u2019s mean position in opposition to which the table says the star Mars was found, referred to the ecliptic. ","189":"You see here that the sun\u2019s mean position differs from opposition to Mars\u2019s apparent position on the ecliptic by 13\u00bd\u2019 in some cases, nearly thrice the error which could arise through a change of hypothesis. Therefore, the exactness of their hypothesis did not prevent my seeking another.","190":"But they permitted this discrepancy advisedly. This is clear from the following: since the nodes are about 17\u00b0 Taurus and Scorpio, and the limits about 17\u00b0 Leo and Aquarius, as will be said below, the additions and subtractions are made chiefly at 17\u00b0 Cancer, 25\u00b0 Virgo, 4\u00b0 Scorpio, 27\u00b0 Sagittarius, and 13\u00b0 Pisces, intermediate points, and none at 21\u00b0 Leo and 18\u00b0 Scorpio, the nodes and the limit. They therefore had reason to believe that a planet is not divested of its second inequality unless the sun\u2019s departure from the node is as great as the planet\u2019s on its own orbit. Their intentions, moreover, were not consistent. For, in their way of thinking, the difference ought to be greatest at 3\u00b0 Cancer, because Cancer is closest to the 45th degree, where this difference is generally greatest. But at 17\u00b0 Cancer they subtracted 5 minutes, while at 3\u00b0 Cancer, one minute only. Because of this, the following table, comparing the positions (referred to the orbit of Mars) with the mean positions of the sun at these moments, is presented. ","191":"Clearly, they did not compensate the whole difference in this way.","192":"We shall discuss this plan of theirs once more a little later.","193":"Now, we shall also examine Mars\u2019s mean motion, for the sake of which, see the following table. 6 7","194":"I am therefore missing something small in the mean longitude. For, that nearly everywhere there is half a minute too much, may be so because of my having computed the mean motions from the most recent table, in which something might possibly have been deliberately altered.\nThere follows a table of Mars\u2019s eccentric positions. ","195":"All positions are tolerably accurate except 27\u00b0 Sagittarius. Here, for various reasons, a small but appreciable quantity has been accumulated. First, the sun\u2019s position is 26\u00b0 45\u2019 24\" Gemini. Now the computed position on Mars\u2019s orbit is 26\u00b0 24\u2019 43\" Sagittarius. In the opinion of the table, 10\u2019 20\" are to be subtracted from the latter to refer it to the ecliptic. Therefore, the computed position on the ecliptic would be 26\u00b0 24\u2019 13\" Sagittarius, a difference from opposition to the sun of 21\u2019 11\".","196":"Chapter 9","197":"On referring the ecliptic position to the circle of Mars.","198":"But it is now time for us to discuss in detail this adjusting to the ecliptic or to the planet's orbit, which serves as a foundation.","199":"First, the table provides us the following information from the observations: the northern latitude takes its rise from 18\u00b0 Taurus, at which it was five minutes, reaches its observed maximum at 21\u00b0 Leo, decreased thereafter and became only 1\u2154\u00b0 at 3\u00b0 Scorpio, but right away at 27\u00b0 Sagittarius it was south and a rather large 4\u00b0, and still","200":"greater at 13\u00b0 Pisces. From this one concludes roughly that the ascending node is a little before 18\u00b0 Taurus, and the descending node far beyond 3\u00b0 Scorpio. The nodes will therefore be around 17\u00b0 Taurus and 17\u00b0 Scorpio, and the limits around 17\u00b0 Leo and Aquarius. Since the plane of Mars\u2019s eccentric is inclined to the plane of the ecliptic, nearly the same thing that happens with the right ascensions of parts of the ecliptic will happen here: the observed arcs of one circle do not correspond to the same observed arcs on the other, except the ones beginning at a node and ending at a limit. I use the term, \u201cobserved arcs\", because here one must mentally separate out the planet\u2019s eccentricity, and proceed as though Mars\u2019s path were in the orb of the fixed stars, exactly as is the ecliptic, and as though it really intersected the latter. And indeed, when asked what is the ecliptic position of a planet, astronomers define it thus: it is that point on the ecliptic at which the circle of latitude (at right angles to the ecliptic) passing through the sidereal position of the planet\u2019s body intersects the ecliptic.","201":"It is therefore clear from the demonstrations in Theodosius\u2019s On the Sphere that unless this circle passes through the poles of both circles (the ecliptic and the planet\u2019s path), its points of intersection will always cut off unequal arcs as measured from the point at which the two circles intersect. And since the circle of latitude is at right angles to the ecliptic, it will always be oblique to the planetary orbit if it does not pass through the poles of the orbit. Consequently, the arc between the planet\u2019s position on its orbit and the nearest node is always greater than that between its ecliptic position and the same node.","202":"Now when we observe the planets, we do not feel convinced that we have defined their exact positions until we have referred them to the ecliptic. We do this by indicating the point on the ecliptic at which the circle of latitude passing through the planet is found. The ecliptic position is used, therefore, to aid our memory and comprehension. But when, on the other hand, we compute the planet in its own hypothesis, we are not concerned with the ecliptic to which it is inclined, but with the exact path of the planet. Therefore, to be able to","203":"compare the observed position with the computed position, we must either extend the arc between the ecliptic position and the nearer node, or abridge the arc between the body of the planet and the same node, so that from the former operation the position on the orbit might be given, and from the latter, the ecliptic position. This is actually accomplished by adding or subtracting, according as the node precedes or follows the planet\u2019s position.","204":"Such care concerning the planets Ptolemy considered unnecessary. Copernicus did not forego it in treating the moon, and Tycho Brahe diligently embraced it for the sake of precision.","205":"To continue: in the referring process which we have been considering, there are two things I would like to know, both of which I can seek using the same procedure and diagram. \n ","206":"Let A be the sidereal position of the node, AB the arc of the ecliptic, and upon it let AC be set equal to AB, and let the planet be observed beneath C. Further, from C let an arc be drawn perpendicular to the ecliptic, and let it be CE. ","207":"Now in the first place the ancients thought that since E is the position on the ecliptic and C the position of the orbit of the planet I, the planet is at the point opposite the sun when the sun is at E, the planet being observed at C. However, as was said above, those who constructed the tables thought that the planet is not exactly at opposition to the sun unless arc AB,","208":"the elongation of the place opposite the sun from the same node, is made equal to AC (the observed distance of the planet from the node).","209":"Now the truth of the matter is quite different. The planet is, indeed, seen in opposition to the sun at that time, but it really is not, and the advantage we seek from the planet\u2019s opposition to the sun is more harmed by making AC and AB equal, than they themselves were hoping it would be corrected. For why are the planets observed at opposition to the sun? In order, of course, that they then lack the second inequality of longitude. And when the point opposite the sun is at B and the planet is at C, both being between the nodes and the limits, the planet is more wrapped up in the second inequality of longitude than if the point opposite the sun were at E, the planet remaining at C. For let G be the sun, the center of the planetary system, at which all the orbs intersect the ecliptic, whether in the Copernican or the Brahean form. Let G be joined to A and E, points on the ecliptic, and let the earth be on the line EG, at H. Let HC be joined, and from H let the sun G be observed opposite E, while from the same point H let the planet be seen at C, its sidereal position along the line HC. Therefore, in this sighting, the planet is really on the line HC. It is, however, far below the fixed stars. Let it be at the point I on the line HC, and let a straight line be drawn from G through I, which will intersect the arc CE. For the whole plane CEHG is beneath the arc EC. Let its point of intersection be F, and let a third arc AF be drawn from A through F to BC, cutting BC at D. It is obvious that the plane of the planet\u2019s eccentric viewed from H towards C is set beneath AF, not AC, and that when the point opposite the sun is at E, the planet will really be beneath F, while when the point opposite the sun is located at B the planet is really going to be beneath D, although both do appear beneath C. But AD is shorter than the legs of the isosceles triangle BAC. Therefore, the point B opposite the sun is farther from A than is D, the position beneath which the planet is at the moment they have chosen. Therefore, the sun really stands beyond the point opposite the planet's true position. This is contrary to what they proposed to do.","210":"But it is likewise false that if the orbit of the planet were beneath AC, AB should on that account be taken equal to AC. For the orbit\u2019s really being beneath AD is likewise not sufficient reason for taking AD equal to AB. For since the planet is observed at the point opposite the sun in order that it might shed the second inequality of longitude, while the longitude is to be reckoned on the planet\u2019s true orbit, or on AD which stands above it, surely, unless the point opposite the sun falls upon the arc drawn through the planet at right angles to the orbit (that is, unless ADB is a right angle), the point B opposite the sun will not coincide in longitude with D. But if ADB is a right angle, AB is longer than AD. They are therefore not equal. Clearly, therefore, the equality between arcs AC and AB that the table aspires to is undermined. ","211":"However, for practical purposes, these differences are smaller than can be perceived. I therefore do not hesitate to allow the place opposite the sun to be at E, with AEF right and AFE consequently acute, even though it has just been demonstrated that AFE should be right instead. But it was also necessary to proceed against a new claim of accuracy, by means of accurate arguments. What follows here is the harm arising from this accuracy.","212":"In the second place, then, I wish to establish this: that in the table of adjustment they followed a procedure that is unsound. For, given Mars\u2019s ecliptic position E and apparent latitude EC, they computed the length of AC, and stated that the planet on its orbit was then distant from the node by their quantity AC. Now the orbit of the planet (whose first inequality we are investigating) is not beneath AC, but beneath AD, as was just shown. Therefore, the arc AC has nothing to do with the first inequality, but adulterates the planet\u2019s true elongations from A. And furthermore, the apparent latitude is EHC, while the true latitude of the point F, the inclination of the line GF to the ecliptic, is EGF. Thus, although the second inequality of longitude is swallowed up at opposition to the sun, the second inequality of latitude is nonetheless near its maximum there, and its measure is the angle HIG. Therefore, just as the whole latitude EC causes AC to be longer than AE by the arc EB, similarly, the part FC or HIG of this apparent latitude, which is a result of the second inequality, makes AC longer than AF. So it is longer than it should be. And this error cannot be ignored, as it can be as much as 9 minutes.","213":"This error could also have been perceived in the inconstancy of the angle BAC, which they attributed to the inclination of the planes of the ecliptic and of Mars's orbit. This is clear from the result obtained if you suppose the arc AC to be increased by the amount of the addition expressed in the table, and use this and AC to compute the angle EAC. For the angles come out as in the accompanying table, from which it is clear that in the northern semicircle they suppose an angle of maximum northern latitude of 4\u00b0 33\u2019, and in the southern, of 6\u00b0 26\u2019 south. According to this, at the subtending line connecting the nodes, which passes through the sun or earth, the plane of the eccentric would somehow be bent, since the upper part is less inclined than the lower. Or rather, the whole path or plane of the planet\u2019s eccentric would be full of twists and turns, just as is the path described beneath the fixed stars by the observed latitudes of Mars, which is no circle.\nHowever, all this is in conflict with the simplicity of the celestial motions, as many examples from experience will attest.","214":" Therefore, the true procedure for referring the inclinations to the orbit is this: from the planet\u2019s position E on the ecliptic, known from the observations, to find the angle of inclination EGF for that position, using the method that will follow below. Then, since the angle E is right, from AE and EF (the measure of the angle EGF) AF is found by trigonometry, or instead of EF the constant angle EAF may be used. And since, from arguments which I shall present below, it is clear that for the star Mars the angle EAF is not greater than about 1\u00b0 50\u2019, the adjustment about 45\u00b0 from the node (where it is maximum) accordingly does not exceed one minute, for which the table nonetheless has one add 8 or 10 minutes in certain places. So for this reason, too, the hypothesis can be in error by as much as 7 and 9 minutes, since the observations upon which it was founded suffered no little loss through this adjustment. I was consequently subject to much less restraint than before in seeking out a new hypothesis.","215":"Chapter 10","216":"Consideration of the observations themselves, with which Tycho Brahe hunted for the moments of opposition to the mean sun.","217":"In an enquiry of such precision, I could not have foregone a deeper inspection of the foundations themselves. And Brahe had given me the opportunity to make use of his observations. This is what I found.","218":"On 1580 November 12 at 10h 50m, they set Mars down at 8\u00b0 36\u2019 50\" Gemini without mentioning the horizontal variations, by which term I wish the diurnal parallaxes and the refractions to be understood in what follows. Now this observation is distant and isolated. It was reduced to the moment of opposition using the diurnal motion from the Prutenic Tables. ","219":"For in Maestlin, on the twelfth at noon, Mars is put at 8\u00b0 20\u2019 Gemini, and on the seventeenth, again at noon, it is at 6\u00b0 25\u2019 Gemini. Therefore, the motion over five whole days would be 1\u00b0 55\u2019. In Stadius, it is 1\u00b0 52\u2019. Therefore, on the seventeenth at the same hour of 10h 50m, Mars ought to have been seen at either 6\u00b0 41\u2019 50\" Gemini, or 6\u00b0 44\u2019 50\". At 9h 40m (which Tycho gives as the moment of opposition), it is 1\u2019 4\" farther forward, at either 6\u00b0 42\u2019 54\" or 6\u00b0 45\u2019 54\". They put it at 6\u00b0 46\u2019 10\" Gemini.","220":"You see that this opposition (as regards exactness) is a little more uncertain because it makes use of a diurnal motion which is not observed but imported from elsewhere, and about which the different authors differ from one another by three minutes over these five days.","221":"On 1582 December 28 at 11h 30m, they set Mars down at 16\u00b0 47\u2019 Cancer by observation. The moment of opposition assigned by Tycho comes 46 minutes later, during which the planet retrogressed less than one minute. Tycho therefore puts it at 16\u00b0 46\u2019 16\" Cancer. On an inserted sheet here, an attempt was made to correct for a refraction of two minutes. This was, I think, first trial of the theory of refraction then being developed. Nevertheless, he followed the observed value unchanged, thus declining to consider the planet as something which could alter its position. Nor was there any need for correction, since it was in Cancer, beyond the reach of refraction, and was in mid-sky where, in Cancer, there is no longitudinal parallax.","222":"On 1585 January 31 at 12h 0m, Mars was placed at 21\u00b0 18\u2019 11\" Leo. The diurnal motion, by comparison of observations, was 24\u2019 15\". The moment of opposition followed at 19h 35m, 7 hours and 35 minutes later. To this period belongs 7\u2019 41\" of diurnal motion westward. Therefore, at the designated moment, it would have been at 21\u00b0 10\u2019 30\" Leo, which is what was accepted. There is no mention of parallax. Nothing had to be done about refraction, because Mars was high and at mid-sky. I therefore find the bit of advice in the table about refraction (properly) ignored.","223":"On 1587 March 7 at 19h 10m they deduced the position of Mars from the observations, which was 25\u00b0 10\u2019 20\" Virgo. This they kept in the table, but changed the time to 17h 22m. The difference of 1h 48m multiplied by a diurnal motion of 24\u2019 gives the same number of minutes and seconds (that is, 1\u2019 48\"), no more. It therefore should have been 25\u00b0 8\u2019 32\" Virgo, which also approaches nearer the point opposite the sun. The difference is of practically no importance. ","224":"On 1589 April 15 at 12h 5m they established the position of Mars very carefully at 3\u00b0 58' 21\" Scorpio, and corrected for longitudinal parallax so as to make it 3\u00b0 57\u2019 11\". There remain 1h 30m until the designated moment of opposition, which, for a diurnal motion of 22', bring the planet back 1\u2019 22\", so as to be at 3\u00b0 55\u2019 49\". They took the value of 3\u00b0 58\u2019 10\". The former is closer to the sun\u2019s mean position.","225":"On 1591 June 6 at 12h 20m, Mars is placed at 27\u00b0 15\u2019 Sagittarius. There remained 2 days 4 hours and 5 minutes until the designated moment. In four days it was found to be moved forward 1\u00b0 12\u2019 47\". Therefore, to 2d 4h 5m correspond 39\u2019 29\". Consequently, at that moment Mars was at 26\u00b0 35\u2019 31\" Sagittarius. There is no need to consider any horizontal variations in longitude, since Mars is at mid-sky and at the beginning of Capricorn. The table has 26\u00b0 32\u2019 Sagittarius.","226":"On 1593 24 August at 10h 30m they report Mars as being at 12\u00b0 38\u2019 Pisces with an observed diurnal motion of 16\u2019 45\", and this near the nonagesimal13 where there is no longitudinal parallax. The moment designated for the opposition preceded this by 8h 17m (for it was at 2h 13m), to which corresponds a motion of 5\u2019 48\" eastward. Therefore, the planet falls at 12\u00b0 43\u2019 48\" Pisces. And the table has 12\u00b0 43\u2019 45\".","227":"On 1595 October 30 at 8h 20m, they found Mars at 17\u00b0 48\u2019 Taurus, with a diurnal motion of 22\u2019 54\". The designated moment preceded by 11h 48m, for which is required a motion of Mars of 11\u2019 7\" eastward, so that it would be at 17\u00b0 59\u2019 7\" Taurus. But it was projected eastward on account of parallax. Therefore, possibly using another observation on the meridian, they put down 17\u00b0 56\u2019 15\" Taurus in the table.","228":"On 1597 December 10 at 8h 30m, they first placed Mars at 3\u00b0 30\u2019 Cancer, and again at 4\u00b0 1\u2019 Cancer, the mean being 3\u00b0 45\u00bd\u2019 Cancer. The moment of opposition came 3 days 5h 5m later, to which, from Magini, corresponds 1\u00b0 15\u2019 westward. Therefore, Mars would have been at 2\u00b0 30\u00bd\u2019 Cancer. In the table, it was put at 2\u00b0 28\u2019. The reason for the rough measurement, carried out with a measuring staff, is clear from the date. Tycho had left the island, leaving all instruments but the staff behind. Nevertheless, he did not wish to ignore this opposition completely. But I wish he had still remained on the island, for this opposition was a marvelous opportunity, not often recurring within a man\u2019s lifetime, for examining Mars\u2019s parallax.","229":"On 1600 January 13\/2317 at 11h 50m, the right ascension of Mars was: 18 19","230":"Hence, Mars is at 10\u00b0 38\u2019 46\" Leo, at an adjusted time of 11h 40m reduced to the meridian of Uraniborg. But on January 24\/February 3 at the same time it was at 6\u00b0 18\u2019 Leo. This gives a diurnal motion of 23\u2019 44\", and a position on January 19\/29 at 9h 40m of 8\u00b0 18\u2019 45\" Leo, just as they put it.","231":"I have presented these discrepant values for the right ascension in order to show that even in the observations themselves there is an uncertainty of several minutes unless extreme care be exercised and all possible aids used. At that time the instruments (except the largest) had arrived in Bohemia, but they were still not well enough positioned, and were affected by the journey besides. However, even in observations at the island it too often happened that right ascensions measured from two different stars differed by three minutes. When I asked Christian [Longomontanus], on this subject, whether I should consider this an effect of the limitations of observation or vision, he replied, \u201cThis is not unusual.\"","232":"Finally, the reader should be advised that Tycho, in his table, claimed to have made use of the solar parallax in correcting the positions of Mars. But it will now shortly be made clear that the parallax of Mars is a slippery and imperceptible business. However, this does not much affect the certainty of the positions in this table, as Mars can almost always be observed in mid-sky where it has no longitudinal parallax.","233":"Chapter 11","234":"On the diurnal parallaxes of the star Mars.","235":"The starting point of my new elaboration and renewal of the motions was where I have just stopped. For it is clear from Part I that the positions of Mars should be taken at the moments of true opposition to the sun. However, it is also clear that not every trace of the second inequality is thus removed, it being also necessary to refer the arc measured on the ecliptic to the planet\u2019s orbit. But the planet\u2019s orbit must first be investigated by the inclination of the planes and by finding out the nodes. Again, the inclination and the nodes cannot be found without knowing the diurnal parallax, at least if this should turn out to be relatively large. One must therefore start with the parallax. I shall present two ways to find it.","236":"The first way (and the one more familiar to others) will be examined using the Brahean observations.","237":"In 1582, when Mars was opposite the sun in Cancer, I found the observation done with incredible care, with Tycho\u2019s manuscript title, \u201cFor investigating the parallaxes of Mars\", from which you will, however, deduce either no parallax at all, or one exceedingly small. I pass over without saying, that (as is customary) they compared the star Mars with nearby stars on the ecliptic, and frequently with ones at a great distance. Now it is usual to find the parallax of a mobile star (for Mars moves, with a retrograde motion when opposite the sun) by comparing morning and evening observations. As a result, it has happened that almost all the stars by which Mars was observed in the morning are different from those by which it was observed in the evening. For a fixed star which is at hand in the morning (and higher than Mars), if it be near the ecliptic, has either set by evening (when Mars is in the west) or is rendered useless by refraction for this delicate procedure. Another star therefore had to be substituted. But if the fixed stars are substituted for one another, there is always less trust in the procedure than if the same star should be retained.","238":"Brahe, however, announced to the learned world in many places that from the observations of that year, the parallax of Mars was found to be considerably greater than that of the sun. I therefore very carefully scrutinized the whole book in order to examine his operation or computation more deeply. I did indeed find a chapter which professed to offer a procedure for investigating the parallax of Mars from the observations of that year. But here was something really surprising: they fitted the position of Mars found by observation into a Copernican diagram drawn very laboriously and carefully. In this diagram, they took up the immense labor of solving all triangles created by the double epicycle on the concentric, in numbers replete with a great many digits. Finally, this was the goal of the calculation: to issue a pronouncement that the parallax of Mars is indeed greater than the solar parallax. Brahe had thus asked one thing, but his assistants in calculation carried out something else. He wanted them to find out the parallax of Mars by comparing morning and evening observations with one another, but they had in fact found out how much parallax the Copernican diagram would bring about. Whether Brahe\u2019s pronouncement on parallax was founded solely upon his trust in his assistants, is unknown to me. ","239":"As for us, let us consult the actual observations, insofar as they pertain to our undertaking.","240":"In 1582 on the night between 23 and 24 November, the distances from the fixed stars remained the same at different times. This, then, was a station point. ","241":"On the following two days, the motion was 11\u2019 and 15\u2019.","242":"On the night of 26 December, it passed between the second and seventh stars of Gemini, its distance (measured with the staff) from the head of the lower of the Twins (the second star) being 2\u00b0 25\u2019 or 2\u00b0 26\u2019, but from the seventh, ","243":"1\u00b0 6\u2019 or 1\u00b0 7\u2019, making the latitude about 4\u00b0 9\u2019. Then, at 8h 28m, it was 44\u00b0 41\u2019 from","244":"the eye of Taurus, whose latitude is 5\u00b0 31\u2019 south, longitude 4\u00b0 12\u00bd\u2019 Gemini, in 1600. Hence, Mars\u2019s longitude as if the year were 1600 is 17\u00b0 53\u2153\u2019 Cancer, or, at the end of 1582, 17\u00b0 38\u2019 Cancer, at an altitude of 40\u00b0 50\u2019. It is thus beyond the effects of refraction.","245":"Again, at 7h 15m on the morning of December 27, it was 36\u00b0 43\u2019 from Cor Leonis, whose latitude is 0\u00b0 26\u00bd\u2019; hence, its longitude at the end of 1582 is 17\u00b0 28\u2153\u2019 Cancer, altitude 14\u00b0 4\u2019, and thus affected by refraction. Therefore, from 8h 28\u00bdm in the evening to 19h 15m, an interval of 10h 46\u00bdm, it was observed to retrogress 9\u2154\u2019.","246":"For the diurnal motion, on the 29th at 7h 47m, the distance of Mars from the southern foot of Erichthonius8 was 29\u00b0 38\u00bd\u2019 But on the 30th at 8h 8m the distance from the same star was 29\u00b0 13\u00bd. Therefore, over 24h 21m it moved 25\u2019. And this diurnal motion remained the same on the 27th. Therefore, to 10h 46\u00bdm there should have corresponded 11\u00bd\u2019 of arc, but we found only 9\u2154\u2019. Let us consider this.","247":"On the previous evening, when Mars was rising farther to the east (because it was retrograde), parallax moved it eastward, and in the morning, when it was setting and was farther to the west, parallax moved it westward. So, just as the moon\u2019s diurnal parallax apparently slows its motion, the same parallax in turn quickens Mars\u2019s retrograde motion. Therefore, if parallax is perceived, it is perceived through an excessively enlarged diurnal motion. But this motion is diminished. Therefore, there is no parallax. Again, refraction is perceived as contrary to parallax. Now the refraction at altitude 13\u00b0 is 4\u2019 from the table of fixed stars, and 8\u2019 from the table of the sun, and only a very small part of this affects the longitude, as Cancer descends quite obliquely. So the refraction in longitude comes to three minutes at most, which, added to 9\u2154\u2019, makes the refraction-free motion 12\u2154\u2019 over 10\u00be hours, which motion, if it also were free from parallax, ought to have been 11\u00bd\u2019. Therefore, the excess of 1\u2153\u2019 is the longitudinal parallax for the two observations\u2014clearly minimal, untrustworthy, and entirely contemptible.","248":"On 1583 January 16 at 7h 30m in the evening, Mars was 23\u00b0 29\u2019 from the bright star in the foot of Erichthonius at an altitude of 51\u00b0. The next morning at 5h 0m, it was 43\u00b0 58\u2019 from Cor Leonis at an altitude of 15\u00b0. And, measured by the straight edge, Mars was perfectly collinear with the two stars. And so, since Mars\u2019s motion is carried out along this line, Brahe made a note that the longitudinal parallax is given from the diurnal motion of Mars. This is obtained here as follows. On January 16 at 10\u00bdh it was 23\u00b0 27\u2019 from the bright star in the foot of Erichthonius. On January 17 at 10\u2157h 12 it was 23\u00b0 12\u00bd\u2019 from the same star. Therefore, the diurnal motion would be 14\u00bd\u2019. Now in order to comply with Brahe\u2019s advice, we must set out the distance between the foot of Erichthonius and Cor Leonis, which is found to be 67\u00b0 21\u2019. Subtracting Mars's distance from the bright star in the foot of Erichthonius, 23\u00b0 29\u2019, leaves Mars's distance from Cor Leonis, 43\u00b0 52\u2019, at 7\u00bdh in the evening, which at 5h in the morning was 43\u00b0 58\u2019, 6 minutes greater. The time interval is 9\u00bd hours, to which ought to correspond 5\u215d\u2019 of the diurnal motion. Here, therefore, the sum of the two parallaxes is no more than \u215c\u2019 except that the amount of Mars\u2019s longitudinal refraction at 15\u00b0 is added to it. But this is quite small. For Cancer and Leo are setting extremely obliquely, and Mars's large northern latitude put it at nearly the same altitude as Cor Leonis.","249":"On January 17 at 5h 20m in the evening, Mars was 23\u00b0 16\u2019 from the foot of Erichthonius. On the following day, the 18th, at 3h 0m in the morning, this distance was 23\u00b0 9\u2019, and at 5h 5m in the evening it was 23\u00b0 1\u00bd\u2019. So the motion over 23h 45m is 14\u00bd\u2019, but over 9h 40m it is 7\u2019. This should have been 6\u2019. We are left with a longitudinal parallax of no more than 1\u2019. Refraction does not affect anything, since in both instances the altitude of Mars was about 30\u00b0.","250":"Likewise, at 7h 34m its distance from the seventh star of Gemini was 7\u00b0 51\u2019. At 4h 52m in the morning, it was 7\u00b0 59\u2019 from the same star. Therefore, over 9h 18m it moved 8\u2019. We thus have one minute more than before. Of this star (at the shoulder of Gemini), Brahe wrote, \u201cNote that I am taking Mars\u2019s distance from this star because its course passes through it, as it were, so that the morning and evening distances compared might show the parallax.\" I have reproduced this here so that the reader may rest assured that Brahe did not proceed without a purpose.","251":"On January 18 at 8h 52m in the evening15 there was 44\u00b0 22\u2019 between Mars and Cor Leonis. At 4\u00beh in the morning the same distance was 44\u00b0 27\u2153\u2019. Therefore, the motion over 7h 53m was 5\u2153\u2019. On January 19 following, at 7h 3m, this distance was 44\u00b0 32\u00bd\u2019. Therefore, for 22h 11m the motion is 10\u00bd\u2019. And for 8 hours there would be less than 4\u2019 of motion. Our profit is about 1\u00bd\u2019 of parallax.","252":"But now let us calculate, for January 17, how much of an increase in the hourly motion should result from a parallax greater than the solar parallax as usually accepted. Since we consider the sun\u2019s parallax to be three minutes, let Mars have four. 16","253":" It follows that the motion of mars over those hours should have appeared 4\u2019 greater than what follows proportionally from the diurnal motion. Since this is repudiated by the observations, Mars\u2019s parallax is found \u2013 often none at all. There was even an occasional note in Brahe\u2019s hand saying, \u201cIt strayed over to the wrong side.\" So this is the first way of investigating Mars\u2019s parallax.","254":"I shall now add the other way, because of its beauty; I cannot use the Brahean observations in it. Therefore, in using my own observations, I am going to give you a clown show, and will show by example why Brahe needed such diligence, precision of instruments, assistants, and other equipment.","255":"I have two instruments, which I use through the generosity of Baron Friedrich Hoffmann, L.B. an iron sextant and a brass azimuthal quadrant. The latter is two and a half feet in diameter and the former three and a half, and both are calibrated in one-minute divisions.","256":"Now at this very time, 1604, at which I am considering parallax (whether that of the sun more than Mars\u2019s is hard to say, for my Hipparchus18 requires Mars\u2019s aid even in the lunar eclipses), a very suitable occasion for observation arose, if only the climate zone had been different and Mars had moved a little higher. For about 19\/29 February19 of this year 1604, Mars was stationary both in longitude and in latitude at the same time. This occurred in Libra, and therefore from Mars\u2019s rising to the sun\u2019s rising the angle of the horizon with the ecliptic continually decreased. Consequently, according to Chapter 9 of the Astronomiae Optica20 the latitudinal parallax, if any, continually increases. But from the increment (found through the columns of the parallactic table21 opposite the initial and final angles of the ecliptic with the horizon), the whole horizontal parallax (at the front of the column) is known.","257":"There follows the series of my observations. ","258":"On the night between Thursday and Friday, which was February 17\/27, while Corvus was on the meridian, there was 9\u00b0 44\u2019 between Mars and Spica, and between it and the Northern Pan, 17\u00b0 41\u2019; and between Mars and Arcturus, 29\u00b0 13\u2019. Also, to test the sextant, we measured the interval between Arcturus and Spica as 32\u00b0 57\u2019, which should have been 33\u00b0 1\u2019 45\", as is clear if you calculate it using the right ascensions and declinations, or latitudes and longitudes, which Tycho assigned to these stars in Book I of the Progymnasmata. Therefore, my distances are smaller than the true distances by 4\u00be minutes. I applied this correction to the distances of Mars from the fixed stars, so as to make it 9\u00b0 48\u2019 45\" from Spica, 17\u00b0 45\u2019 45\" from the Pan, and 29\u00b0 17\u2019 45\" from Arcturus. ","259":"I then used the quadrant to obtain the meridian altitude of Mars, 32\u00b0 4\u2019, and of Spica, 30\u00b0 50\u2019. Since the latter has a declination of 9\u00b0 2\u2019, Mars is left with a declination of 7\u00b0 48\". However, the altitude of Spica showed that my perpendicular was not well enough set up. For the altitude of the equator, at my location, is 39\u00b0 54\u2019. Accordingly, the meridian altitude of Spica is 30\u00b0 52\u2019, and of Mars, 32\u00b0 6\u2019. Now, from the declination of Mars and its distance from the fixed star, its right ascension comes out: 26","260":"I am not certain whether, when (as happened a few times) the clamps holding the arm loosened and it (being a heavy piece of iron) fell precipitously and hit hard, it might not have changed the position of the sights, since they are removable and subject to dislocation. But from this right ascension, I first select, from Tycho\u2019s table of right ascensions, the degree of the right sphere that was riding at the same time, 28\u00b0 1\u2019 0\" Libra, whose declination, from another of that author\u2019s tables, is 10\u00b0 48\u2019 30\" [South]. But that of Mars is 7\u00b0 48\u2019 [South]. Therefore, Mars is distant from the ecliptic, the oblique path, by 3\u00b0 0\u2019 30\" on the circle of declination. But the angle which the circle of declination makes with the ecliptic is 68\u00b0 59\u2019 from the appropriate table. Its complement is 21\u00b0 1\u2019. And in my table of parallax, under the heading of 60\u2019, I find, opposite 68\u00b0 59\u2019, the entry 56\u2019 1\". Under 30\", however, I find 28\". But since I have thrice 60\u2019 in this distance of Mars from the ecliptic (which I call the base of the latitude ), I multiply what I extracted under 60\u2019 by 3. This gives me a latitude of 2\u00b0 48\u2019 31\". The same operation opposite 21\u00b0 1\u2019 shows me what has to be subtracted from the place rising at the same time, namely, 1\u00b0 5\u2019 4\". Accordingly, Mars\u2019s position will be 26\u00b0 56\u2019 Libra. I come within a minute of this using a computation whose fundamentals I am going to be presenting in this work.","261":"To test the latitude of Mars, I also consulted the distance from Arcturus, through the star\u2019s latitude and longitude provided from Tycho, and Mars\u2019s longitudinal position just found, and it replied to me that Mars was at a latitude of 2\u00b0 47\u2019 48\". Before, it was at 2\u00b0 48\u2019 31\".","262":"On February 19\/29 we had moved the sight, and began to observe Mars rising. Its distances from Arcturus were noted, and were these: ","263":"I think we are ten minutes too high. For the wind was blowing so hard that it was only by a glowing coal that we could cast light upon the scale so as to read it. And the altitude of Mars was then 11\u00b0. Later, the back of Leo culminated at altitude 62\u00b0 37\u2019, according to a corrected plumb line. Thus the altitude of the equator was shown to be 39\u00b0 55\u2019: almost correct. At that moment, Mars\u2019s altitude was 23\u00b0. We next reinvestigated the former distance [to Arcturus], which came out to be: Therefore without doubt the previous distance was ","264":"For first, when Mars was near the horizon, refraction moved it toward Arcturus, later letting it drops down when Mars had acquired some altitude. It was, however, the cold and the extremely biting winds that occasioned so much variety in observations made at the same time. For it was impossible to handle the iron and close the clamps with bare hands, and with gloves the arm was not securely enough clamped to be read to the minutes. Vindemiatrix32 showed a meridian altitude of 53\u00b0 5\u2019, a little greater than it should have had. But Spica\u2019s 30\u00b0 54\u2019 was correct within one minute. The altitude of Mars at culmination was 32\u00b0 6\u2019, the same as it had been two days before, and of Arcturus, a correct 61\u00b0 13\u2019. Hence, by computation, it is concluded that the distance of Mars and Arcturus was 29\u00b0 18\u2153\u2019. Now since, according to the Prutenic Tables and to my computations, Mars was stationary in longitude at this time, there could be no change in meridian altitude resulting from its wandering about on the ecliptic. For this reason, since the meridian altitude remained quite the same (for my instrument allows an uncertainty of one minute), neither did any change in the latitude occur during this time.","265":"On February 22 or March 3 we tested the sextant, just as we had done above, and found 26\u00b0 2\u2019 between Canis Minor and Orion\u2019s higher shoulder, which the calculation shows to be 26\u00b0 2\u2019 15\". So also, between Canis minor and Palilicium34 was found 46\u00b0 22\u00bd\u2019, which Tycho, in his Epistolae, indicates is 46\u00b0 22\u2019. Therefore, when the fifth star of Leo culminated, with the arm of the instrument fixed above 29\u00b0 13\u00bd\u2019 the distance was now more, while no error could be found at 29\u00b0 15\u2019. Then the whole sky unexpectedly clouded over. It became clear again on the morning of March 4, and now, when Antares culminated, with the arm fixed at 29\u00b0 19\u2019, both stars were seen evenly [with the sights]. However, it seemed that something needed to be added, but at 29\u00b0 20\u2019 too much had been added. When the observation was finished, Saturn preceded the meridian by less than Jupiter preceded Saturn.","266":"On the night following February 29 or March 10, the instrument having meanwhile been displaced, this distance was, first, between 29\u00b0 9\u2019 and 29\u00b0 10\u2019, half an hour before the culmination of the Heart of Hydra. Later, it appeared to the investigators to be between 29\u00b0 12\u2019 and 29\u00b0 13\u2019, as it was now higher and free from refraction. For at the end of this observation it had an altitude of 19\u2159\u2019. But a little afterward \u2013 I do not know whether the sights were disturbed \u2013 it would not allow that much, for it appeared to be 29\u00b0 9\u00bd\u2019. The Tail of Leo was about half a degree from the meridian. The altitude of Mars was then 24\u00be\u00b0. The Tail of Leo, when culminating, had the correct altitude of 56\u00b0 44\u2019, within one minute. When one third of the distance between Mars and Spica had crossed the meridian, it [i.e., the distance to Arcturus] first appeared to us to be 29\u00b0 9\u00bd\u2019, but the cylinder was not well enough applied, as it was too long. Thus, a little later, this could not be accepted, but 29\u00b0 10\u00bc\u2019 or a little less seemed to be required. And Mars was see on both sides of the cylinder.","267":"Then, between Mars and Spica, there was 9\u00b0 26\u2019, and less than 9\u00b0 27\u2019.","268":"Mars culminated at an altitude of 32\u00b0 19\u00bd\u2019. 39","269":"There was then 18\u00b0 25\u2019 between Mars and the Norther Pan [of Libra]. ","270":"For the investigation of the sextant, the distance between Spica and the Pan was taken as 27\u00b0 39\u2019, although it should have been 27\u00b0 34\u2019. Also, the distance between Spica and the northern star in the head of Scorpius41 was 39\u00b0 32\u00bd\u2019, which should have been 39\u00b0 26\u00bd\u2019. So the sextant read five minutes high. Furthermore, calculation of Mars\u2019s position also provides evidence of this. For unless you diminish Mars\u2019s distance from the fixed stars by 5 minutes, the right ascension measured through Spica and through the Pan will be discrepant by 10 minutes, but if (as required by the examination of the sextant) the five minutes be subtracted, they will coincide exactly, and will be 205\u00b0 27\u2019 10\", with declination 7\u00b0 35\u00bd\u2019. Therefore, the position is 26\u00b0 18\u2019 48\" Libra, with latitude 2\u00b0 47\u2019 20\". You see that the latitude is manifestly the same, though meanwhile the planet had retrogressed 38\u2019 in longitude. If, from the position of Mars found thus, you find its distance from Arcturus, it will come out to be 29\u00b0 9\u2159\u2019, while using the faulty instrument it was 29\u00b0 14\u2019.","271":"Now, with the Heart of Scorpius42 culminating, our distance [to Arcturus] was 29\u00b0 13\u00bd\u2019, the instrument in the meantime having been displaced and later repositioned. We next tested the sextant again, which showed 44\u00b0 45\u2019 between Polaris and the Tail of Cygnus. This should have been 44\u00b0 39\u00bd\u2019. Therefore, the instrument was in its pristine state. But when Saturn had passed the meridian by one degree, the distance could not admit 29\u00b0 13\u00bd\u2019; nevertheless, it was greater than 29\u00b0 12\u00bd\u2019, about 29\u00b0 13\u2019.","272":"So this is the series of observations. I would be crazy if I tried to use them to build something of great precision. Therefore, I am presenting an example to another more diligent and successful observer, rather than an argument. I also hope that the nausea evoked by these uncertain observations will lead readers to desire all the more fervently the extremely certain Tychonic ones. Now, on with the example.","273":"The first and second days, agree only in showing the station in latitudinal motion. In both, Mars was 29\u00b0 18\u2019 from Arcturus, and in both, its meridian altitude was 32\u00b0 7\u2019 or 6\u2019. Those days were busying me in preparation for meeting the following days properly, should the instruments be needed.","274":"But on March 3, with the Mouth of Leo culminating, the distance was 29\u00b0 15\u2019, and with the Heart of Scorpius45 culminating, 29\u00b0 19\u2019 plus. Therefore the distance changed about 4\u00bc\u2019 over the interval. And since Arcturus and Mars have very nearly the same longitude, this change of distance bespeaks a variation of latitudinal parallax. I am not unaware that 29\u00b0 19\u2019 is hardly different from 29\u00b0 18\u2019, and that by analogy with the previous day the latter ought to be the distance at about the same time if Mars is standing still. I also know that when the Mouth of Leo is on the meridian Mars is 12\u00bd\u00b0 high and is somewhat affected by refraction. But of these we shall speak afterwards. For now, let us entirely ignore them, so as not to complicate our example. Now the altitude of the nonagesimal was 57\u2153\u00b0 (about) when the Mouth of Leo was culminating, but finally, after the Heart of Scorpius had culminated, it was 20\u2153\u00b0. Let me therefore look through the parallactic table to find the column in which, from a distance from the zenith of 32\u2154\u00b0 to a distance of 69\u2154\u00b0, the entry in the table would change 4\u00bc\u2019. I find that this happens in the column whose heading is 9\u2019. Therefore, the maximum parallax of Mars would be 9\u2019. And since, on that day, the distance of Mars and earth was to the distance of Mars and the sun as 28 is to 60 (given approximately by an anticipatory acquaintance with the hypotheses of Tycho and Copernicus), the ratio of the parallaxes will be the inverse of this, and the maximum solar parallax will be about 4\u2019 24\". It is set at 3\u2019 0\".","275":"But now let us consider that at an altitude of 12\u00bd\u00b0 Mars would be subject to refraction, if the table of fixed star refractions constructed at Hven is valid for Prague. At this altitude, it was 4\u2019 20\", 2\u2019 18\" of which is owed to the latitude, by which Mars is caused to be closer to Arcturus. If, however, we apply the sun\u2019s refraction to Mars (as it often appears we should), it is 8\u2019 45\" at this altitude, twice as great. Therefore, the latitudinal parallax would also be twice as great, 4\u2019 36\". In this way, the whole difference which observation imposes upon itself, at these two different moments, would be due entirely to refraction. Reckoning it in the former way would leave a latitudinal parallax of 2\u2019 \u2013 the difference in parallax in the column whose heading is 5\u2019. This would give the sun only 2\u2019 25\" of maximum parallax. So refraction makes our third day, too, suspect and doubtful, and ultimately quite worthless. I know that since Arcturus and Mars are 9 degrees apart, which is one third of the amount by which Arcturus\u2019s latitude exceeds Mars\u2019s, it comes out that not all the latitudinal refraction is subtracted from the distances from Mars, and that the parallax changes Mars\u2019s latitude more than it changes this distance from Arcturus. But since this is very small, I have, with greater fear, considered it as something to be left buried. Let him observe it who is versed in more subtle matters.","276":"Now on the fourth day, what seems to have been accomplished is nothing other than the total destruction of Mars\u2019s parallax. The meridian distance ought to have been 29\u00b0 9\u00bd\u2019 with an accurate instrument, and consequently 29\u00b0 14\u2019 with the faulty one. But at the end it was found to be 29\u00b0 13\u00bd\u2019, when the latitudinal parallax (had there been any) ought to have been greater, and hence the distance from Arcturus greater. Hence, from that time when Mars attained an altitude of 19\u00b0, the distance was found to be 29\u00b0 12\u00bd\u2019, and one minute greater at the end. This would be a very small parallax. And what is this ratio? When its altitude was 9 degrees (when Hydra was culminating), the distance was 29\u00b0 9\u2019 with a faulty instrument, and still subject to refraction. Later, at an altitude of 25\u00b0 and near the meridian, it was again 29\u00b0 9\u2019, measured twice at different moments. Could the refraction have been zero initially, that the arc thus remain constant? Or should it rather be said that I (though to myself I might have seemed most diligent) erred in observing? Especially on account of the length of the cylinder.","277":"Still, it is at least established by these observations, whatever their quality, that Mars\u2019s parallaxes of latitude are not greater than 4\u2019, which is the amount of uncertainty in the instrument. It is more credible that the parallax is very small. In Chapter 64 below, you will find further discussion of this point.","278":"That the parallaxes of Mars are greater than the sun\u2019s, on the other hand, is argued by the proportion in the Tychonic and Copernican hypothesis; and from this proportion Mars\u2019s parallax could easily be computed if we were certain of the sun\u2019s parallax. Is, then, the procedure for finding the sun\u2019s altitude and parallax from eclipses uncertain? It is indeed relatively uncertain quantitatively, but as concerns the thing itself it is perfectly certain. The sun is not nearer than 230 semidiameters of the earth, but it is not an infinite number of semidiameters away. But between 700 and 2000 semidiameters (of which the first quantity is from my Mysterium cosmographicum, while the other two are given as the upper and lower limits in observations of eclipses) it does not appear that any indisputable number has yet been demonstrated, as I shall prove in my Hipparchus.","279":"Chapter 12","280":"Investigation of Mars\u2019s Nodes.","281":"Means are not wanting of investigating the planets\u2019 first inequality through observations, even when these are entangled in the second inequality. Nevertheless, in this second part, I prefer to follow the footsteps of the authorities and make use of acronychal observations, in order to establish my credibility. For I want to be sure that later, when I bring forth something contrary to accepted opinion, no one may complain that I am hiding behind the briar-path of my own method.","282":"Furthermore, it is now clear that nothing of any importance is wanting in Mars\u2019s diurnal parallax as taken by Tycho. I shall therefore move on, little by little, towards the reduction of the observed positions of Mars to the point opposite the sun\u2019s apparent position.","283":"As a first step, we must find the nodes. Tycho Brahe used to investigate them thus:","284":" In the diagram of Chapter 9, let A be the position of the node, E the planet\u2019s position on the ecliptic in 1595, C the observed sidereal position of the planet, 17\u00b0 56\u2019 5\" Taurus, EC the observed latitude, 0\u00b0 5\u2019 15\" north. Now it is presupposed that the angle EAC is very nearly 4\u00b0 34\u00bd\u2019, the same as the maximum northern latitude observed in the same way in 1585. In the right triangle CEA (or the isosceles triangle CBA; the difference is unimportant in this procedure), from side CE and angle EAC he sought the length EA, the distance of the ecliptic position from the node. There is nothing wrong with this operation, since EC is small and near","285":"the node. However, the need for accuracy1 in this demonstration commends another. For it was said in Chapter 9 that the angle EAC is not constant; hence, through different latitudes of different oppositions they will also show different positions for the node. Nor is EAC as great as the maximum apparent latitude, because AC is a curved arc; and neither is it AC, but some path inside (AF, say), that is the planet\u2019s path as it would be seen from the center of the sun. Therefore, A will not necessarily be the node, at least as found through this operation.\nI, therefore, investigated the nodes differently, using observations on the day on which they were at the node. Even though this method depends upon some preconceptions, and the subject will be treated more accurately below in Part 5, we ought to have a first taste of it here, if only to give confirmation.","286":"My presupposition was that when the planet, in its eccentric position, is truly at the node, it can by no disposition of the earth or sun be made to appear elsewhere than at the node. For in the Copernican hypothesis, this in itself is in agreement with the nature of things: that the moving faculty of any star is not bound to observe a star foreign to it (including the earth), but has its own laws governing its circuit. In the Ptolemaic hypothesis, this would be exactly as if one were to say that the epicycle pays heed, not to a line from the sun through its own center, but to certain positions beneath the fixed stars, beneath which it places the planet in the plane of the ecliptic. In the Tychonic system, the same will be said of the eccentric.","287":"Further, that which I presupposed, I have found to be true using the following observations:","288":"On 1590 March 4 at 7h 10m in the evening, Mars's declination was 9\u00b0 26\u2019","289":"N, and the right ascension was 22\u00b0 35\u2019 10\". Hence, its position comes out as 24\u00b0 22\u2019 56\" Aries, and its latitude as 3\u2019 12\" south. Parallax and refraction were opposite and approximately equal, and are therefore ignored.","290":"On 1592 January 23 at 10h 15m in the evening, Mars was at 11\u00b0 34\u2019 30\" Aries with latitude 0\u00b0 2\u2019 south. The altitude of Mars was 25\u00b0, and therefore (from the table for the fixed stars), there was no refraction. The parallax was about as much as the sun's, because Mars and the sun were sextile, and were therefore about equally distant from earth. Nearly all of this was latitudinal. Therefore, about two minutes of latitude must be added northward to Mars in order to free it from parallax, and it thus will fall upon the ecliptic. For on February 6 it was already at about 7\u2019 northern latitude. ","291":"On the evening of 1593 December 10, Mars was observed at the ascending node. For after correction of the horizontal variations it retained no more than 0\u00b0 0\u2019 45\" north latitude. ","292":"On 1595 October 27 at 12h 20m, Mars's true latitude after the removal of parallax was 0\u00b0 2\u2019 20\" south. On the 28th, when the parallax had similarly been removed, the latitude was 0\u00b0 0\u2019 25\" north. Therefore,* in the meanwhile, it was at the ascending node.","293":"Counting backwards 687 days, the number of days in Mars\u2019s revolution on its eccentric, starting from noon on 28 October, the end will fall on 1593 December 10, while on the preceding night Mars was observed near the node. Count back another 687. This will end up at 1592 January 23, when the planet was observed right at the node. If you do the same a third time, you will come out at 1590 March 7, while on the fourth day preceding, Mars had some southern latitude, which it would make up over the four days, so as to fall at the node about the seventh.","294":"From this it is known that it makes no difference where the earth is, either in sidereal position or with respect to Mars. In the Ptolemaic system, it makes no difference where the sun is with respect to the center of Mars\u2019s epicycle and where Mars is on the epicycle, and in the Tychonic system, it makes no difference where the center of the eccentric or the sun is located with respect to the line from Mars through the earth, so drawn that Mars lies in the plane of the ecliptic. For the diameter of the nodes is always the same in Copernicus and Ptolemy, and in Tycho it is always parallel to itself, except that over the ages the nodes move slightly. This motion was not perceived over these six years.","295":"Now let us find the opposite node.","296":"I. On the morning of 1595 January 4, when Mars was observed at 7h 10m at altitude 8\u00b0, with reference to Spica Virginis and Cor Scorpii, it was observed at latitude 0\u00b0 3\u2019 46\" N, and it was itself at 13\u00b0 36\u2019 40\" Sagittarius. The parallax is small, because Mars was more than twice as far from the earth as is the sun. Refraction, on the other hand, is large: 6\u2019 45\" from the table for the fixed stars, or 11 minutes from the table for the sun. This is nearly all latitudinal, owing to the low altitude of the nonagesimal. Therefore, Mars (reckoning by the refraction displayed by the sun) was really at a few minutes south latitude\u2014about 2 or 3, or even more.","297":"II. On the night of 1589 April 15 Mars's observed latitude was 1\u00b0 7\u2019 north. The parallax of the annual orb was drastically increased, owing to the approach of Mars and earth. After 21 days, the latitude decreased to a paltry 6\u2154\u2019 north. And thus, although on May 6 it decreased somewhat more slowly, since the star was moving away from the earth, we shall still not be far wrong if we extrapolate proportionally: we make 21 days be to the number of days after which Mars falls upon the ecliptic, as 60 minutes of diminution is to the 6\u2154 minutes remaining. This rule shows it to be two and one third days, so that Mars was at the node on May 9. Once again, counting thrice 687 days thence, we will come out on 1594 December 30. On this day, Mars ought to have been at the node, and for the next 5 days\u2014up to the morning of January 4\u2014it should have been moving off southward. And indeed, from observations of it on January 4, we have given it a few minutes of south latitude. Mars was not observed more frequently at this eccentric position. It is enough that we have this 1595 observation, provided that it does not disagree with us. In 1589 there is nothing that we can bring into question. Nor should it disturb you that in 1589 we assigned a latitudinal motion of 6\u2154\u2019 to 2\u2153 days, while we do not allow so much over a 5-day period around 1595 January 4. For, as will appear in the course of this work, the latitude is most attenuated at conjunction with the sun (as in 1595), owing to the parallaxes of the annual orb, and at opposition (as in 1589) it is augmented. It is therefore fitting that the diurnal motion in latitude appear less in 1595, and greater in 1589.","298":"Now, how are the sidereal positions of the two nodes found? Thus: one finds an approximate value for the mean position of Mars at each place, using tables for Mars (which, accordingly, we presuppose for this purpose). Whether you do this with the help of the Prutenic or Tychonic tables, taking into account the true precession of the equinox, you will find that on the morning of 1594 December 30, the mean position of Mars is 27\u00b0 14\u00bd\u2019 Scorpio, and on the morning of 1595 October 28 it was at 5\u00b0 31\u2019 Taurus. It therefore appears that the diameter of the nodes does not pass through the center of equable motion, but far beneath it. For from 5\u00b0 31\u2019 Taurus to 27\u00b0 14\u00bd\u2019 Scorpio is more than from the latter to the former.","299":"If, on the other hand, you make use of the Tychonic equations, 11\u00b0 17\u2019 must be added to the latter figure and 11\u00b0 30\u2019 subtracted from the former. Accordingly, the one comes out to be 15\u00b0 44\u00bd\u2019 Scorpio, and the other, 16\u00b0 48\u2019 Taurus, which are Mars\u2019s equated eccentric positions. As you see, the nodes are nearly opposite one another at about 16\u2153\u02da Taurus and Scorpio, when viewed from the center of the planetary system, which Ptolemy described as a point very near the earth, and Copernicus and Tycho as a point very near the sun.","300":"Further, it will be seen in Part 5 below how much we are going to change these positions of the nodes when we change the equations by transposing the theory of the sun from the sun's mean position to its apparent position.","301":" Chapter 13","302":"Investigation of the inclination of the planes of the ecliptic and of the orbit of Mars.","303":" In the previous chapter, the nodes and limits have been found quite accurately according to the opinions of Tycho Brahe and myself. Now it is to be inquired what exactly the inclination of the plane of Mars\u2019s orbit is to the plane of the ecliptic.","304":"It is not so evident how to deduce this from the observations themselves. For the angle of this inclination is set up around the center of the planetary system, which for Copernicus and Tycho is the sun.","305":"But the eye cannot be placed at the sun so that this inclination might thence appear beneath the fixed stars and be measured, and the maximum distance of the limit from the ecliptic, seen from another place, will also be seen under another angle. In the Ptolemaic form there might appear to be a more direct procedure, but this is not so. For it will be demonstrated that the plane of the epicycle always remains parallel to the plane of the ecliptic. Therefore, place the center of the plane of the epicycle at either limit, and let the planet lie on the same line of longitude passing from the center of vision through the center of the epicycle. The planet will then either be more distant than the center of the epicycle from the observer, and thus its distance from the ecliptic will appear less than the distance of the center of the epicycle from the same ecliptic, or it will be nearer to the observer, and will thus appear greater than what we seek.","306":"In this difficulty we may take consolation from this one circumstance: that the purpose for which we seek to know the inclination as one of our principles is not such as to require the highest accuracy. It will consequently permit us to use those means which furnish indirect evidence of the quantity of the inclination: we shall offer three of these.","307":"Now it is apparent from what has just been said that it will be most directly helpful to us if we find an observation of the star Mars at that moment at which Mars is reported to be equidistant from the sun and the earth and on the line drawn from the sun to 16\u00b0 or 17\u00b0 Leo or Aquarius (the positions of the limits). In the Ptolemaic form, this is where the center of the epicycle is at 16\u00b0 or 17\u00b0 Leo or Aquarius and Mars is as far from the earth as is the center of the epicycle. In Mercury alone this problem does not occur.","308":"Let B be the sun, A the earth, and upon AB let the isosceles triangle ABC be set up, with C the planet\u2019s position in the plane of the ecliptic; and, CE being drawn perpendicular to the orbit of Mars, let the body of Mars be at E. EC will therefore appear the same whether seen from B the sun or A the earth \u2014this is immediately evident.","309":"But in order that it be known at what position Mars is equidistant from the sun and the earth, observe that when the lines from Mars at C and the earth at A falling upon the sun at B make the angle CBA right, then CB is shorter than CA. Consequently, BA, the position opposite the sun\u2019s, should make with BC, the eccentric position of Mars, an angle less than 90\u00b0, in order that CAB and CBA be equal. Therefore, if BC is directed toward 17 Leo, the sun ought to be beyond 17 Taurus and before 17 Scorpio. Or if on the contrary BC is directed toward 17 Aquarius, the sun should be beyond 17 Scorpio and before 17 Taurus. We use these circumlocutions to denote morning risings or evening settings, with Mars and the sun sextile or quintile. ","310":" In the Ptolemaic form, if C be the earth, A the center of the epicycle, and B Mars, CAB will not be able to be right, as CA and CB are to be made equal. So the anomaly of commutation2 ought to be more than 90\u00b0 or less than 270\u00b0.","311":"If you wish to work a little more precisely, take from Copernicus or from an anticipation of Tycho\u2019s reformation the approximate ratio 1525:1000 as (in Copernicus) the ratio of the orbits of Mars and earth, (in Tycho) of Mars and the sun, or (in Ptolemy) of the eccentric and the epicycle. At 16\u00b0 or 17\u00b0 Leo, this is about 5:3, and at 16\u00b0 or 17\u00b0 Aquarius, about 11:8.","312":"So the triangle ACB is isosceles with sides AC, CB equal, and (with AB=1000), BC is 1666\u2154 when directed toward 17\u00b0 Leo. Therefore (CD being dropped perpendicular to AB), where AD, which is half AB, is 1000, AC will be 3333\u2153. Looking this up in a table of secants, we find the angles CAD and CBD to be 72\u00b0 33\u2019. So also, at 16\u00b0 or 17\u00b0 Aquarius, with AB=1000, AC is 1375, so if AD=1000, AC is 2750, showing 68\u00b0 40\u2019 in the table of secants.","313":"Therefore, with BC directed towards 16\u00b0 or 17\u00b0 Leo or thereabouts, the apparent position of Mars, AC, ought to be 72\u00bd\u00b0 from the apparent position of the sun AB. And with BC at 16\u00b0 or 17\u00b0 Aquarius, these ought to be 68\u2154 from one another. And since the sum of the two (CAB, CBA) at 17\u00b0 Leo is 145\u00b0, ACB will be 35\u00b0 at 17\u00b0 Leo. Wherefore, Mars, which lies on the line AC, ought to be seen at 22\u00b0 Virgo (the sun being on AB at 5\u00b0 Sagittarius), or at 12\u00b0 Cancer (the sun being at 30\u00b0 Aries).","314":"Similarly, at 17\u00b0 Aquarius, since the sum (CAB, CBA) is 137\u2153, ACB will be 42\u2154\u00b0, wherefore Mars, lying on the line AC, ought to be seen at 24\u2153\u02da Sagittarius (the sun being on AB at 16\u00b0 Libra) or at 0\u00b0 Aries (the sun being at 9\u00b0 Gemini).\nSomething approximating this could have happened, first, in November of 1586 or 1588; again, in April of 1581, 1583, 1596, and 1598; third, in September or October of 1587 and 1589; and fourth, in May or June of 1580, 1582, 1595, and 1597. In the last instance, suitable observations are lacking, since Mars in Aries, on account of its small ascension (where the sun in Gemini makes the nights bright) can hardly be observed, or even be seen at all.","315":"Accordingly, on 1588 November 10 at 6h 30m in the morning the planet Mars was seen at 25\u00b0 31\u2019 Virgo, with a latitude of 1\u00b0 36\u2019 45\" north, the sun being at 21 Scorpio. The sun is thus only 62\u00bd\u02da from Mars, although it should have been 72\u00b0 from Mars in order to make the triangle isosceles, as the problem requires. Therefore, Mars is then farther from the earth than from the sun. Consequently, its latitude at that place appeared less than was the true inclination.","316":"On December 5 following, at 6h in the morning, Mars was seen at 9\u00b0 19\u2156\u2019 Libra, with a latitude of 1\u00b0 53\u00bd\u2019 N., the sun being at 23\u00b0 Sagittarius. Therefore, since the sun was 73\u00bd\u00b0 from Mars, the digression of the point which Mars then occupied on its orbit was a little less than 1\u00b0 53\u00bd\u2019 (since there should have been 72\u00b0 between Mars and the sun). Since the present angle is greater, the distance of Mars from the earth turns out to be less than the distance of Mars from the sun. The apparent magnitude of the inclination\u2014at least, of this point from the plane of the ecliptic\u2014is consequently greater. But since on December 5 the planet in its eccentric motion was already several degrees beyond the limit, again diminishing its true digression from the ecliptic, this was consequently greater right at the limit. And since these two effects cancel one another, the maximum inclination of the planes will be about 1\u00b0 50\u2019.","317":"Similarly, on 1586 October 22 at 6h in the morning, about dawn, there was about 6\u00b0 9\u2019 eastward between Mars and Cor Leonis. The declination of Mars from the equator was 13\u00b0 0\u2019 40\" north. Hence, its apparent longitude is found to be 0\u00b0 7\u2019 Virgo, latitude 1\u00b0 36\u2019 6\" N. The sun stood at 8\u00b0 Scorpio, 68\u00b0 from Mars. It should have been farther. Consequently, the line between Mars and the earth was longer than that between Mars and the sun. And so the apparent latitude was less than the true digression of the planet from the ecliptic, and this was, in fact, long before it reached the limit.","318":"But on November 2 at 4\u2154h in the morning, with the sun at 19\u2156\u00b0 Scorpio, Mars was seen at 5\u00b0 52\u2019 Virgo, with latitude 1\u00b0 47\u2019 N. The sun was 73\u00bd\u00b0 from Mars, by a nearly exact measure. But Mars was a few degrees before its northern limit which is at about 16\u00b0 17\u2019. Therefore the latitude at this position appeared about right, although exactly at the limit it is reckoned to be greater than 1\u00b0 47\u2019, namely, about 1\u00b0 50\u2019.","319":"On December 1 following, at 7\u00bdh in the morning, the equatorial distance10 between Cor Leonis and Mars was 25\u00b0 12\u00bc\u2019, and Mars\u2019s declination was 6\u00b0 2\u00bc. Hence is found its longitude, 20\u00b0 4\u2019 30\" Virgo, and latitude, 2\u00b0 16\u2019 30\", with the sun at 18\u00b0 Sagittarius, which is 88\u00b0 from Mars. It should have been only 72\u00bd.\u00b0 Therefore the line between Mars and the earth is made less than that between Mars and the sun, and because of the lesser distance, the digression appeared greater than it really was. At this point, therefore, the digression from the ecliptic was less than 2\u00b0 16\u00bd\u2019. Indeed, it was much less, but not thereby much greater than 1\u00b0 47\u2019. So here the magnitude of the maximum inclination is indirectly confirmed to be 1\u00b0 50\u2019.","320":"On the other hand, on 1583 April 22, at 9\u00beh in the night, an interval of 20\u00b0 58\u2019 was observed between Mars and the Dog12 and 22\u00b0 47\u00bd\u2019 between Mars and Cor Leonis. Hence, the position of Mars is found to be 1\u00b0 17\u2019 Leo, with latitude 1\u00b0 50\u2154\u2019 north. The sun was at 11\u00b0 Taurus, 80\u00b0 distant from Mars. This should have been 72\u00bd\u00b0. Accordingly, Mars is closer than it should be. Therefore, its observed latitude is greater than its true digression from the ecliptic. But Mars is more than twenty-one degrees beyond the northern limit. So at the limit, is digression will again be greater. Therefore, the opposite causes again cancel one another, and the maximum inclination is 1\u00b0 50\u2019.","321":"Likewise, at 8h in the evening on 1596 March 9, it was observed at 15\u00b0 49\u2019 Gemini, with latitude 1\u00b0 49\u2154\u2019 north. The sun was at 30\u00b0 Pisces, 76 degrees from Mars. It should have been a little closer. Therefore, Mars\u2019s true digression from the ecliptic was a little less than the observed latitude. However, this digression was not at its maximum, since Mars had not yet approached within about 25 degrees of the limit. So once again, indirect support is provided for a maximum digression of about 1\u00b0 50\u2019 at the limit.","322":"Now, at the other limit, at 17\u00b0 Aquarius, although observations are rarer, there is one available.","323":"On 1589 September 15 at 7\u00bch in the evening, Mars was observed at 16\u00b0 47\u2153\u2019 Sagittarius with 1\u00b0 41\u2154\u2019 southern latitude. But when the correction for refraction of light which it underwent at this low altitude is applied, its position was 16\u00b0 45\u2154\u2019, with latitude 1\u00b0 52\u2153\u2019 south. The sun was at 2\u00b0 Libra, 74\u2153\u00b0 distant from Mars. It ought to have been only 68\u2154\u00b0. Therefore, the observed latitude is a little greater than the digression of its position from the ecliptic. However, that is not the most distant point, as it is several degrees before the limit. Therefore here, too, the effects cancel.","324":"On November 1 following, at 6\u2159h, it was seen at 20\u00b0 59\u00bc Capricorn, with latitude 1\u00b0 36\u2019 south, the sun being at 19\u00b0 Scorpio. While it was then no more than 62\u00b0 from Mars, it should have been 68\u2154. Therefore, the apparent latitude is less than the true digression from the ecliptic. But at the same time, the digression at this point is less than the digression at the limit, because this point is beyond the limit. Therefore, the maximum inclination is much greater than 1\u00b0 36\u2019, and by all indications is about as great as the apparent latitude on September 15, namely, 1\u00b0 50\u2019, approximately.","325":"I have carried through one method, in which a knowledge of the approximate proportion of the orbs is presupposed. The observations followed this method within the limits of calculation, indicating readily enough the maximum inclination of the planes.\nI shall now present another method, for which rarer, more select observations are required. If these are to be had, what we are seeking is found without prior knowledge of the ratio of the orbs and this without the encumbrance of a laborious computation.\nWhen two planes cut one another, any two lines drawn in the respective planes to the same point on the line of intersection and at right angles to that line always include one and the same angle.","326":" Let the plane of the ecliptic be ACDB, the plane of Mars\u2019s orbit AEFB, and let them intersect one another in AB. Let the sun be at A, the earth at B, and from A and B, perpendicular to AB, let AC and BD be set up in the plane of the ecliptic, and AE and BF in the orbit of Mars. Let the planet be at F. The inclination EAC of the limit E will be equal to the apparent latitude of the planet at F, namely, FBD. You will therefore note that if ever there is a perfect quadrature of the sun and Mars, with the line BA, that is, the sun, at 16\u00b0 or 17\u00b0 Taurus or 16\u00b0 or 17\u00b0 Scorpio\u2014where between the line BA from earth to sun (which in this instance is also the line of intersection of the planes), and the line BF from earth to Mars, there is 90\u00b0 or one quadrant intervening\u2014then, whatever the apparent latitude of Mars FBD will be there, that will also be the maximum inclination of the planes EAC, although at that place F, Mars is not as far from the ecliptic as at E.","327":"The first such day fell on 1583 April 22, which only just now I had under consideration. The sun was at 11 Taurus, five or six degrees below the node. The earth, therefore, was above the line of intersection towards Mars. On this account, the apparent latitude will be greater than the truth, since it is seen from nearer. On the other hand, since there are not 90\u00b0 between the sun and Mars, the apparent latitude will on this account be less than the truth. On the supposition that these opposite deviations cancel one another, the inclination of the planes will approximately equal the observed latitude. The observed latitude was 1\u00b0 50\u2154\u2019. Therefore, the inclination of the planes is approximately that much.","328":"On 1584 October 30, there was a select occasion, but no observation is available. However, on November 12 following, at 1\u00bdh in the night, when the sun had already fallen about 14\u00b0 or 15\u00b0 below the diameter of intersection, the earth having risen that much (for Copernicus), or the diameter of intersection having fallen that much toward the earth (for Tycho), Mars was seen at 23\u00b0 14\u2019 Leo with latitude 2\u00b0 12\u2156\u2019 north, while the sun was at 1\u00b0 Sagittarius. This angle is somewhat diminished owing to the inclination of Mars\u2019s line of vision to the line of intersection. But it is greatly augmented by its approach toward the earth. Therefore, the inclination is much less than 2\u00b0 12\u2019, namely, 1\u00b0 50\u2019.","329":"On 1585 April 26 at 9h 42m, Mars was seen at 21\u00b0 26\u2019 Leo with latitude 1\u00b0 49\u00be\u2019 north. The sun was at 16\u00b0 Taurus, right near the node. Mars\u2019s line of vision was a little inclined, since Mars was beyond 16\u00b0 Leo. Therefore, the angle of maximum inclination of the planes was only a little greater than 1\u00b0 49\u00be\u2019; that is, 1\u00b0 50\u2019 or a little greater.","330":"Similarly, near the other limit, on 1591 October 16 at 6h 30m in the evening, Mars was seen at 1\u00b0 27\u2153\u2019 Aquarius with latitude 2\u00b0 10\u215a\u2019 south, decreasing. (For on October 10 preceding, the latitude was 2\u00b0 18\u2154\u2019, and on October 2 it was 2\u00b0 38\u00bd\u2018.) The sun was at 2\u00bd\u00b0 Scorpio, above the node. The earth was therefore below the node towards Mars. So because of this proximity, the observed latitude was greater than the inclination of the plane of the ecliptic. Fourteen days later, when the sun was at the node, if it were again to have decreased 28 minutes (the amount of decrease in the previous 14 days), there would remain 1\u00b0 45\u2019. But the ratio of decrease does not remain the same when the earth departs from a star, or vice versa. For at a greater distance the decrease is always less. Therefore nothing can be adduced here against a maximum inclination of 1\u00b0 50\u2019. On the contrary: it is indirectly confirmed.","331":"The demonstration can be extended farther. Let BA be a line drawn from the earth through the body of the sun at the place of the node, 17\u00b0 Scorpio or Taurus, and let the planet be observed at any point whatever on the zodiac. Now the latitude which it appears to have measures the inclination of a point on the plane truly removed from the limit by an amount equal to Mars\u2019s apparent removal from the limit. Let Mars be observed on BG. Draw AH parallel to it. The apparent latitude of point G seen from B will be the same as the inclination of point H. And BG and AH are directed towards the same sidereal degree, because they are parallel. For example, in the observation of 1585 April 26, the sun was at 16\u00b0 Taurus, and Mars was observed at 21\u00b0 26\u2019 Leo, with latitude 1\u00b0 49\u00be\u2019. Therefore, the inclination at the eccentric motion of 21\u00b0 26\u2019 Leo is 1\u00b0 49\u00be\u2019. And since 21\u00b0 26\u2019 Leo is 5\u00b0 from the limit, and the sine of 85\u00b0 is 1\/250 less than the whole sine, the maximum inclination here will be greater by 1\/250 of it, that is, about 1\u00b0 50\u00bd\u2018.","332":"In the Ptolemaic hypothesis, the demonstration of this theorem proceeds on the following basis.","333":" Let A be the earth, AB the line through the sun and the point opposite, at 17\u00b0 Taurus or Scorpio, AD Mars\u2019s line of vision, D Mars, and BAD a right angle. AD will accordingly be at 17\u00b0 Leo or Aquarius. And because D is Mars, a line drawn from D parallel to BA will pass through the center of the epicycle C (since the motion of Mars on its epicycle follows the motion of the sun in its orb). Take a point E on AD such that AE equals AC. Therefore, since AC will not be at 17\u00b0 Leo or Aquarius, it will not be so far from the ecliptic as the northern limit E. D will likewise not stand so far from the ecliptic as E, because CD and all the points of the epicycle are equally distant from the ecliptic, since the plane of the epicycle, in order to make the hypothesis equivalent, is supposed always to remain parallel to the plane of the ecliptic. But proportionally as D or C is less distant from the ecliptic than E, D is closer to A than E, with the result that the distance D may be seen from A as proportionally greater, and both may be seen under the same angle from A. Now, according to spherical trigonometry, as the distance of C from the ecliptic is to the distance of E from the ecliptic, so is the sine of arc CB (that is, AD) to the whole sine AE, because ECB is a circle inclined above AB. But C and D are equally distant from the ecliptic, as was just said. Therefore, AD is to AE as the distance of D (or the perpendicular drawn from D to the ecliptic) is to the perpendicular from E. Therefore, the triangles ADD and AEE will be similar, since they have right angles at points D and E on the ecliptic, and the sides are proportional. Also, they will be concurrent, since the sides (AD, AE) are drawn in the plane of the ecliptic through the same point A, and are directed toward the same point of longitude, 17\u00b0 Leo or Aquarius. Therefore, the lines AD, AE in the orbit are also concurrent; that is, the line extended from the earth A through Mars D at this position will hit upon E, the center of the epicycle, when it is at the limit. Thus the angles of maximum inclination and of observed latitude of Mars will be the same at this place.\n A third way depends upon computation and upon prior knowledge of the ratio of the orbs. We shall have a taste of this one only for the sake of confirmation. A true and accurate treatment is reserved for Part V and ch. 63, and is not required here.","334":"In Tycho\u2019s table of oppositions, the apparent latitude at 21\u00b0 16\u2019 Leo was 4\u00b0 32\u2159\u201922","335":"Let A be the sun, B the earth, C Mars on the eccentric. Thus the line AE running out through the earth B among the fixed stars will intersect the ecliptic, and AC will intersect the orbit of Mars. And since Mars is at 21\u00b0 Leo, near the limit, the angle EAC is near its maximum. I track this down as follows. Let BA be 1000, AC 1664, and EBC 4\u00b0 32\u2159\u2019. Therefore, as AC is to EBC, so is BA to BCA, 2\u00b0 43\u2019 27\u201c. This, subtracted from EBC, leaves the required angle BAC, 1\u00b0 48\u2019 43\u201c; hence, right at the limit, it would be about 1\u00b0 49\u2019 and is somewhat altered if the ratio BA to AC is altered (more on this below). In this manner, from any given acronychal observation with a comparatively great latitude, the inclination is found, first, of that point on the orbit, and then at its maximum, by considering the distance from the node or limit. For example, on 1593 August 24, the apparent latitude at opposition to the sun comes out to be 6\u00b0 3\u2019 south. Mars was at 12\u00bd\u00b0 Pisces. Now let BA be 1000 and AC 1389, from our prior knowledge. As CA is to the sine of CBE, BA is to the sine of BCA, 4\u00b0 21\u2019 10\u201c. Subtracting from CBE, this leaves the required angle BAC, 1\u00b0 41\u2019 50\u201c. However, this position is about 26\u00b0 from the limit, 64\u00b0 from the node. So, as the sine of 64\u00b0 is to this digression from the ecliptic of 1\u00b0 42\u2019, so is the whole sine to the maximum inclination of the planes, which comes out to be 1\u00b0 53\u2019. We need not be concerned about the three minutes\u2019 excess, for they arise from the assumed ratio, for which see Part IV below.","336":"In the Ptolemaic form, A will be the earth, C the center of Mars\u2019s epicycle, and D the lowest point on the epicycle, since Mars is situated at opposition to the sun. And because the sun\u2019s line EA is on the ecliptic, while the plane of the epicycle is set up parallel to the plane of the ecliptic, CD will be parallel to EA. Therefore, BAC and ACD, the inclinations of the eccentric and the epicycle, are equal. But, owing to the full equivalence of the hypotheses, CD is equal to BA; that is, as AB is to AC in Copernicus, so is the semidiameter of the Ptolemaic epicycle DC to CA, the line from the earth to the center of the epicycle. Therefore, CDA and CBA are equal, and EBC and BAD are equal: the apparent latitude.","337":"Chapter 14","338":"The planes of the eccentrics are non-liberating. ","339":"The convolutions of Ptolemy\u2019s hypothesis forced him to accumulate many monstrosities in the theory of latitudes. For when he decided to make the plane of the epicycle tip every which way (it not being immediately clear, through the mists of his hypothesis, that the plane of the epicycle is parallel to the plane of the ecliptic), he contrived the latitude from three components, and in order that contraries counterbalance one another,* utterly wrenched his epicycle from its parallel position. He did not choose to find average values, whether on the grounds that his observations were not closely spaced, or that he distrusted those that were so, and hence accepted extreme values which were in error.","340":"As a consequence, you may see that in the usual computation (e.g., in Magini\u2019s Ephemerides) there is no conjunction whatever of Mars and the sun that is not, as they say, \u201cthrough the body\u201c. If this were true, it would been in vain that nature devised latitudinal temperings, which prevent the excessive arousal of the sublunar powers that often repeated physical conjunctions would cause.","341":"Copernicus, ignorant of his own riches, ever took it upon himself to express Ptolemy, not the nature of things, to which, nonetheless, he of all men came closest. (In this regard, see Rheticus\u2019s Narratio prima.) For although he rejoiced to find that when the earth approaches a celestial body, the apparent latitudes are increased, he still did not dare to reject the remaining Ptolemaic increases in the latitudes (which this approach of the earth would not bring about). Instead, in order to reproduce these as well, he fabricated librations of the planes of the eccentrics, in which the angle of inclination (which for Ptolemy was fixed) would be varied. Moreover, in a manner close to being monstrous, this happens not according to the laws of motion2 of its own eccentric, but to those of the earth\u2019s orb, clearly foreign. See Copernicus Book VI Chapter 1.","342":"Armed by my own skepticism, I always opposed this gratuitous connection of diverse orbs as a cause of motion, even before seeing Tycho\u2019s observations. In this I more greatly congratulate myself that the observations were found to stand in agreement with me, as has happened with many other preconceived opinions.","343":"But lest anyone deem me untrustworthy on this very account, claiming that I would treat the observations with prejudice, let him now witness that I","344":"have most solidly demonstrated that there are no liberations in the inclination of the eccentric. For three ways of investigating the maximum inclination were proposed. In the first, the sun was near Mars\u2019s sextiles and quintiles, that is, about as close to Mars\u2019s conjunction as it could be and still allow Mars to be conveniently seen and observed. In the second, it was near quadrature with Mars, and in the third it was right opposite Mars. But when the sun was at all three places, Mars exhibited one and the same maximum inclination (1\u00b0 50\u2019, about), northwards, at the same place on its eccentric, and the same amount southwards at the opposite position. Likewise, in Chapter 12, it appeared that when Mars was near the nodes of its motion on the eccentric, no matter what position the sun occupied on its orb (whether near Mars or far from it), Mars was never seen to have any latitude. And in the fifth part below it will be proved in many ways that the declination of Mars\u2019s orbit from the ecliptic is constant at any particular position on its orbit.","345":"And so let us conclude with great certainty that the inclination of the planes of the eccentrics to the ecliptic does not vary at all. (For why should I not form a general conclusion, seeing that there is no reason why it should exist in only a single planet? Even so, I have demonstrated the same for both Venus and Mercury from the observations.) And a follower of Ptolemy may learn from this that the plane of the epicycle is always parallel to the plane of the ecliptic. For this is already demonstrated where the center is near the limits, and it was proved above in Chapter 12 that when the center is near the nodes, the epicycle lies entirely on the ecliptic.","346":"Now who will set me up a fountain of tears, by which, for his deserts, I might bewail the pathetic industry of Apianus, who in his Opus Caesarium, following his trust in Ptolemy, spent so many hours and wasted so many ingenious meditations trying to express, by means of spirals and corollae and helices and volutes and a vast labyrinth of the most intricate curves, a human figment which the nature of things clearly disowns? But that man shows us that he was easily capable of equaling nature by the divine talents of his most perspicacious wits. Apart from this, he entertained his mind with these tricks (in which he rivaled nature herself), which were thoroughly mastered and assembled in his models, and he has consequently won the prize of undying fame, whatever diminishment fortune herself might have in store for the works themselves. But what are we to say of the empty artistry6 of those who made the devices?7 For they make six hundred, nay rather twelve hundred little wheels, so they can triumph in the presentation of the latitudes (that is, human figments thereof) in their works, and claim the consequent reward.","347":"Chapter 15","348":"Reduction of observed positions at either end of the night to the fine of the sun\u2019s apparent motion1","349":"Now, with that investigation carried through to its conclusion, and the positions of the nodes, the inclination of the planes, and its constancy all demonstrated, all of which were necessary for the coming reduction, we shall now define the positions which a planet may occupy on its orbit when the sun itself is diametrically opposite it. The years 1580 and 1597 may be omitted from the argument, as they present no suitable evidence owing to uncertainty of the observations. ","350":"Suppose, however, that on 1580 November 12 at 10h 50m Mars was observed at 8\u00b0 37\u2019 Gemini, and the motion over five days was 1\u00b0 55\u2019. Since at the given time the sun stood at 0\u00b0 45\u2019 36\u201c Sagittarius, and its motion over five days is 5\u00b0 5\u2019, the sum of the two motions will come to 7\u00b0 0\u2019. But the sun is 7\u00b0 51\u2019 24\u201c removed from opposition to Mars. Of this, seven degrees exactly are traversed in 5 days, or 120 hours. So, according to the same ratio, the remaining 51\u2019 24\u201c will be traversed in 14 hours 41 minutes. Therefore, the moment of opposition was November 18 at 1h 31m. Its position was 6\u00b0 28\u2019 Gemini on the ecliptic. Now this is 20\u00b0 away from 16\u00bd\u00b0 Taurus. I want to know how much longer this makes the arc on the orbit extended from the node to the arc of latitude through 6\u00b0 28\u2019 Gemini. So I turn to Philip Lansberg\u2019s trigonometry. I mention him out of honor and gratitude, for he has supplied me in abundance with the finest axes, best adapted for building astronomical foundations, from nearby, and at little expense of time. Without him, these would have had to be sought from afar with great deal of trouble and toil, and the handles would not have been so well fitted. From Lansberge, then, the tangent of the 20\u00b0 side multiplied by the secant of the angle of inclination, 1\u00b0 50\u2019, the last five digits being dropped, gives an increase of only 18\u00bd of the smallest units, to which corresponds about 35 seconds. Mars therefore, standing opposite 6\u00b0 28\u2019 Gemini, is 35\u201c further along on its orbit. It should therefore be placed at 6\u00b0 28\u2019 35\u201c Gemini, a tiny and quite unnecessary correction. The latitude is 1\u00b0 40\u2019 north.","351":"On the night following 1582 December 28, at 11h 30m, Mars was observed at 16\u00b0 47\u2019 Cancer, while the true position of the sun was 17\u00b0 13\u2019 45\u201c Capricorn. The moment of opposition had therefore passed. Now the sun\u2019s diurnal motion was 61\u2019 18\u201c, that of Mars 24\u2019, and their sum, 85\u2019 18\u201c. At this moment, the distance between the stars was 26\u2019 45\u201c. Therefore, as 1\u00b0 25\u2019 18\u201c is to 24 hours, so is 26\u2019 45\u201c to 7 hours 32 minutes. Subtracting this from 11 hours 30 minutes gives December 28 at 3h 58m after noon as the moment of true opposition. Its position on the ecliptic was 16\u00b0 54\u2019 32\u201c Cancer, and by reduction to the orbit (a 50\u201c correction), 16\u00b0 55\u00bd\u2019 Cancer. The latitude was 4\u00b0 6\u2019 north, as given by Brahe\u2019s table of oppositions. For among the observations I find various latitudes: on the night following December 26, 4\u00b0 6\u2019 or 4\u00b0 2\u2019, while on the night following December 29, 4\u00b0 8\u2019 or 4\u00b0 6\u00bd\u2019.","352":"On 1585 January 31 at 12h 0m, Mars was observed at 21\u00b0 18\u2019 11\u201c Leo. The sun was at 22\u00b0 21\u2019 31\u201cAquarius. The true opposition had therefore passed. The distance was 1\u00b0 3\u2019 20\u201c. The sun\u2019s diurnal motion was 61\u2019 16\u201c, that of Mars 24\u2019 15\u201c, and their sum 85\u2019 31\u201c. Now as 1\u00b0 25\u2019 31\u201c is to 24 hours, so is 1\u00b0 3\u2019 20\u201c to 17 hours 46 minutes, to which correspond about 18\u2019 of Mars\u2019s motion. Therefore the time was January 30 at 19h 14m, and Mars\u2019s ecliptic position was 21\u00b0 36\u2019 10\u201c Leo. For reduction, some very small quantity is subtracted, because Mars is then beyond the limit. Therefore, the extension of the arc on the orbit from the following node is directed westward. But because Mars was only 4 or 5 degrees from the node, the subtraction is rendered quite imperceptible. The latitude, on the authority of the Tychonic table, was 4\u00b0 32\u2019 10\u201c north. For the observation on January 31 at 12h gave 4\u00b0 31\u2019. They added the remainder to the Tychonic figure, on account of diurnal parallax.","353":"On the night following 1587 March 4, at 1h 16m past midnight, the position of Mars was found to be 26\u00b0 26\u2019 17\u201c Virgo, from Cor Leonis and Spica Virginis, with an observed latitude of 3\u00b0 38\u2019 16\u201c north. But because Mars was elevated 37\u00bd\u00b0 above the horizon, diurnal parallax comes into the reckoning, and subtracts some small quantity from the longitude, thus making it 26\u00b0 26\u2019 Virgo, with a slightly greater latitude. For since the sun is nearly twice as far away from earth as is Mars, Mars\u2019s parallax will consequently be nearly twice the sun\u2019s. On the supposition that the sun\u2019s is 3\u2019, Mars\u2019s will be about 5\u2019. Now when 9\u00b0 Sagittarius is rising, the nonagesimal10 is 55\u00b0 from the zenith. Opposite this number in our parallactic table, under the column headed 5\u2019, the latitudinal parallax 4\u2019 is shown. Therefore, the latitude observed from the center of the earth would be 3\u00b0 42\u2019 22\u201c north. In Part V below, this will be useful to us in a more accurate examination of Mars\u2019s parallaxes, where a determination will be made of the precise inclination and of the absolutely certain distance of Mars for this position. The sun\u2019s true position was 23\u00b0 59\u2019 11\u201c Pisces. The true opposition was therefore still to come. The stars were 2\u00b0 26\u2019 49\u201c apart. The sun\u2019s diurnal motion was 59\u2019 35\u201c, that of Mars 24\u2019, and their sum 1\u00b0 23\u2019 35\u201c. As this is to 24 hours, so is 2\u00b0 26\u2019 49\u201c to 1 day 18 hours 7 minutes, to which corresponds 42\u2019 7\u201c of Mars\u2019s motion. Therefore, the time of true opposition was March 6 at 7h 23m. Mars\u2019s position on the ecliptic was 25\u00b0 43\u2019 53\u201c Virgo. For the reduction to the orbit, 55\u201c must be subtracted. Therefore, on the orbit it was at 25\u00b0 43\u2019 Virgo. The latitude was decreasing. It was therefore somewhat less than 3\u00b0 38\u2019 N., or 3\u00b0 42\u2019 corrected for parallax.","354":"On the night following 1589 April 15 at 12h 5m, the planet was found at 3\u00b0 58\u2019 20\u201c Scorpio, with latitude 1\u00b0 4\u2019 20\u201c north, decreasing. Mars\u2019s altitude was 22\u2155\u00b0, where refraction from the table for the fixed stars was zero, and from the table for the sun, 3\u00bd\u2019. But the parallax was about twice as great as the sun\u2019s, that is, 6 minutes at the horizon. The degree rising was 24\u00b0 Sagittarius. Therefore, the nonagesimal was 64\u00b0 from the zenith, giving a diurnal latitudinal parallax of 5\u2019 24\u201c. Whether it really was that much will become apparent below, through a careful consideration of latitudes. For there, the northern latitude, free from diurnal parallax (and if there is no refraction), would come out to be 1\u00b0 9\u2019 45\u201c north. And because the altitude of the nonagesimal is 26\u00b0, the longitudinal parallax15 at the horizon is 2\u2019 38\u201c. But Mars is 40\u00b0 from the nonagesimal, counting from 4\u00b0 Scorpio to 24\u00b0 Virgo, which, under the column headed 2\u2019 38\u201c shows a true longitudinal parallax of 1\u2019 42\u201c. That hastens Mars further forward than when viewed from the center of the earth, and this is so on the assumption that it underwent no refraction. But to me it is more probable that it undergoes the same refraction as the sun, greater, that is, than that of the fixed stars, because the opposition of the sun and Mars stirs up the air, while the fixed stars are observed when the air is as calm as possible. Still, let there be no refraction at all, and let Mars be placed at 3\u00b0 57\u2019 Scorpio. At that moment the sun was at 5\u00b0 36\u2019 20\u201c Taurus. At this time, therefore, Mars was 1\u00b0 39\u2019 20\u201c past opposition to the sun. Mars\u2019s diurnal motion, as is clear from comparison with April 13, is 22\u2019 8\u201c; the sun\u2019s, 58\u2019 10\u201c; the sum, 1\u00b0 20\u2019 18\u201c. As this is to 24 hours, so is 1\u00b0 39\u2019 20\u201c to 1 day 5 hours 42 minutes. Therefore, the moment of opposition was April 14 at 6h 23m PM. Its position was 4\u00b0 24\u2019 30\u201c Scorpio, or a little past, if refraction were applied or if the previously assumed value for the diurnal parallax was too great. For reduction to the orbit, some imperceptible quantity must be subtracted, since it is barely 12 degrees from the node. This would be about 24 seconds, which are of no importance, and Mars would be at 4\u00b0 24\u2019 Scorpio, with latitude three minutes greater than before. For from that position, the latitude was decreasing from the eighth of March; nor was it a maximum at opposition.\nVI. On the night following 1591 June 6 at 12h 20m, Mars was found at 27\u00b0 14\u2019 42\u201c Sagittarius, with latitude 3\u00b0 55\u00bd\u2019 south. Refraction was of course provided for (from the table for the fixed stars), since it was large, in that Mars had no more than 6\u00b0 altitude on the meridian. There was, however, no mention of parallax. But at that time Mars was distant from the earth by half the solar distance. Therefore, the horizontal parallax is greater than 6 minutes, on the supposition that the sun\u2019s parallax is 3\u2019. I omit it nonetheless, partly because the refraction is supplied from the table for the sun (which, as I said, is the more probable), exceeding that which Brahe took here by 4\u00bd\u2019, which almost completely cancels out the parallax; and partly because Mars was on the meridian and near the winter solstice point, and thus had no longitudinal parallax. Of the latitude, on the other hand, it will have to be seen below in Part IV whether it might not be a few minutes less, since the parallax projects the planet too far south.\nThe sun was at 24\u00b0 58\u2019 10\u201c Gemini. The difference in position between the stars was 2\u00b0 16\u2019 10\u201c. The sun\u2019s diurnal motion was 57\u2019 8\u201c, and Mars\u2019s (for four days) was 1\u00b0 12\u2019 24\u201c, since on June 10 at 11h 50m it was at 26\u00b0 2\u2019 18\u201c Sagittarius. For one day, therefore, 18\u2019 12\u201c. The sum of the diurnal motions is 1\u00b0 15\u2019 20\u201c. This corresponds to 1 day 19 hours 24 minutes, which, added to the 6th at 12h 20m (because opposition was yet to come), shows [the moment of opposition to be] the 8th at 7h 43m. Mars\u2019s position was 26\u00b0 41\u2019 48\u201c Sagittarius, to which are added 52\u201c for reduction to the orbit, so as to make it about 26\u00b0 43\u2019 Sagittarius. The latitude was six minutes greater than on 6 June, because, by the observations, the latitude here was increasing until the fortieth day after opposition, and increased nearly thirteen minutes between the 6th and the 10th of June. Therefore, ignoring parallax and keeping the same refraction, it would be 4\u00b0 1\u00bd\u2019. VII. On 1593 August 24 at 10h 30m, the ecliptic position of Mars was found to be 12\u00b0 38\u2019 Pisces, with latitude 6\u00b0 5\u2019 30\u201c south. The altitude was great enough that horizontal variations canceled one another. On the August 29th following, at 10h 20m, Mars was observed at 11\u00b0 15\u2019 24\u201c Pisces, with latitude 5\u00b0 52\u2019 15\u201c south. It was decreasing precipitously. For before August 10 it was maximum, fourteen days before opposition. The motion for the five days was 1\u00b0 22\u2019 36\u201c, and for one day, 16\u2019 31\u201c. The sun\u2019s position on August 24 at 10\u00bdh was 11\u00b0 2\u2019 31\u201c Virgo. The stars were 1\u00b0 35\u2019 30\u201c apart. The sun\u2019s diurnal motion was 58\u2019 20\u201c. The sum of the diurnal motions was 1\u00b0 14\u2019 51\u201c. This requires 1 day 6 hours 57 min. until opposition, so that this will be August 26 at 5h 27m in the morning. Mars\u2019s position was 12\u00b0 16\u2019 Pisces. Its latitude was 6\u00b0 2\u2019 south, approximately, if the horizontal variations do indeed cancel.","355":"VIII. On 1595 October 30 at 8h 20m, the planet was found at 17\u00b0 47\u2019 15\u201c Taurus, not far from the nonagesimal. We may thus be sure of the parallax, although we must take it into account. The latitude was 0\u00b0 5\u2019 10\u201c north. The sun\u2019s position was 16\u00b0 50\u2019 30\u201c Scorpio. The distance between the stars was 56\u2019 45\u201c. The sun\u2019s diurnal motion was 1\u00b0 0\u2019 35\u201c; that of Mars, 22\u2019 54\u201c, as appears by comparing the nearby observations. The sum of the diurnal motions was 1\u00b0 23\u2019 29\u201c. If the distance between the stars be divided by this, it comes out to 40\u2019 47\u201c of a day, or 16 hours 19 min. Therefore, the true opposition was 0h 39m PM on October 31. Mars\u2019s position was 17\u00b0 31\u2019 40\u201c Taurus. This needs no reduction to the orbit, as it is nearly at the node. The latitude was about 0\u00b0 8\u2019 north. But comparison with the preceding and following days shows a latitude of about 5\u2019 north.\nIX. Let, at any rate, Mars\u2019s position on 1597 December 10 at 8h 30m be 3\u00b0 45\u00bd\u2019 Cancer (as above). The sun\u2019s position was 29\u00b0 4\u2019 53\u201c Sagittarius. The distance between the stars was 4\u00b0 40\u2019 37\u201c. The sun\u2019s diurnal motion was 61\u2019 20\u201c, that of Mars 23\u2019 40\u201c (for in 1580 the diurnal motion in Gemini was 23\u2019, and in 1582, at 17\u00b0 Cancer, it was 24\u2019). Therefore, the sum of the diurnal motions was 1\u00b0 25\u2019 0\u201c. These data show that the time of true opposition followed 3 days 7 hours 14 min. later, on December 14 at 3h 44m in the morning. Mars\u2019s position was 2\u00b0 27\u2153\u2019 Cancer. The reduction to the orbit (quite ridiculous here, since the observation itself has an uncertainty of several minutes) requires the addition of about 52\u201c. Therefore, the corrected position was 2\u00b0 28\u2019 Cancer. The latitude, from the table, was 3\u00b0 33\u2019 north.\nOn the same night, the one following December 10, at 12\u2159h, Fabricius in East Frisia found Mars\u2019s position to be 3\u00b0 40\u00bc\u2019 Cancer with latitude 3\u00b0 23\u2019 N. In this observation the longitude comes out nearly the same. For the motion over 3h 40m is 3\u00bd\u2019, so that in the Brahean observation too, at 12\u2159h Mars could have been at 3\u00b0 42\u2019 Cancer\u2014two minutes beyond the Fabrician position.\nX. On 1600 January 13\/23 at 11h 40m, the time being adjusted to Uraniborg time, the planet was observed at 10\u00b0 38\u2019 46\u201c Leo. The sun\u2019s position was 3\u00b0 26\u2019 30\u201c Aquarius. The stars were 7\u00b0 12\u2019 16\u201c apart. The sun\u2019s diurnal motion for the next few days was 1\u00b0 1\u2019 3\u201c; that of Mars, 23\u2019 44\u201c. The sum: 1\u00b0 24\u2019 47\u201c. The opposition therefore followed 5 days 2h 22m after; that is, on January 19\/29 at 2h 2m in the morning, before dawn. Mars was at 8\u00b0 38\u2019 Leo. There is no need for reduction, since it is near the limit. The latitude, from the table, was 4\u00b0 30\u2019 50\u201c N.\nXI. On the evening of 1602 Febr. 18\/28 at 10h 30m, using the Tychonic instruments (with the help of the learned Matthias Seiffard, bequeathed us by Tycho), I took the distance of Mars from the middle star of the tail of Ursa Major 23 to be 52\u00b0 22\u2019. And since the distance between Cor Leonis and Procyon was 37\u00b0 22\u2019 20\u201c, which should have been 37\u00b0 19\u2019 50\u201c, we know that the sextant reads 2\u00bd minutes high. Therefore, the corrected distance of Mars from the tail of Ursa was 52\u00b0 19\u00bd\u2019. And since the latitude of the fixed star is 56\u00b0 22\u2019, the remainder, by subtraction, is 4\u00b0 2\u00bd\u2019\u2014supposing that Mars was at precisely the same longitude as the fixed star. But because there was a difference of 3\u00be degrees between them (as is clear from the following observations), a slight correction is required.\n For let AB be 3\u00b0 43\u2019 30\u201c on a parallel close to the ecliptic, B Mars, C the fixed star, and BC 52\u00b0 19\u2019 30\u201c. Dividing the secant of BC by the secant of AB gives the secant of CA, 52\u00b0 14\u2019, which, subtracted from 56\u00b0 22\u2019 (the latitude of the fixed star) leaves 4\u00b0 8\u2019 north as Mars\u2019s observed latitude. At the same time, we found 19\u00b0 23\u2019 between Mars and Cor Leonis (19\u00b0 20\u00bd corrected), and 21\u00b0 20\u2019 between Mars and the bright star in the wing of Virgo (21\u00b0 17\u00bd\u2019 corrected). From these two distances (using the latitudes of the stars and Mars), Mars\u2019s longitude is found to be 13\u00b0 19\u2019 6\u201c Virgo, by consensus of all measurements.\nAlternatively, the meridian altitude of Mars was found at 12h 40m, using two quadrants, to be 50\u00b0 19\u2019, while the tail of Leo was 56\u00b0 45\u2019. Therefore, from the declinations and right ascensions of the fixed stars and our distances, Mars\u2019s position is determined as 13\u00b0 19\u2019 30\u201c Virgo, latitude 4\u00b0 7\u2019 55\u201c. This is the Tychonic procedure. I have included the other for the sake of showing a consensus, and also that it might be evident that despite the lack of absolute perfection in the demonstration, short cuts either in computation or in our understanding can under certain circumstances be applied. For in that previous procedure there is less in the actual work than in the reporting of it. At Prague, 5\u00b0 Scorpio was rising. Therefore, the nonagesimal was about 32\u00bd\u00b0 from the zenith. And since Mars\u2019s distance from the earth was rather more than half the sun\u2019s distance, the resulting parallax of about 5\u2019 opposite 32\u00bd\u00b0 in our parallactic table, shows a latitudinal parallax of 2\u2019 41\u201c. Thus the latitude as seen from the middle of the earth would be 4\u00b0 10\u2154\u2019 north. And because the altitude of the nonagesimal was 57\u00bd\u00b0, the longitudinal parallax at the horizon was 4\u2019 13\u201c. But since Mars was 38\u00b0 from the nonagesimal, the longitudinal parallax corresponding to this position is 2\u2019 36\u201c, and if this be eliminated, Mars would be placed at about 13\u00b0 18\u2019 Virgo. At that moment, the sun\u2019s position was 10\u00b0 16\u2019 42\u201c Pisces. The distance between the bodies was 3\u00b0 1\u2019 18\u201c. The sun\u2019s diurnal motion was 1\u00b0 0\u2019 4\u201c, and that of Mars, 24\u2019 5\u201c (for it was 24\u2019 18\u201c at 21\u00b0 Leo in 1585, and 24\u2019 at 26 Virgo in 1587). The sum of the diurnal motions was 1\u00b0 24\u2019 9\u201c. The true opposition therefore followed 2 days 3 hours 43 minutes later, on 21 February\/3 March before dawn at 2h 13m, Mars being at 12\u00b0 27\u2019 35\u201c Virgo. Forty seconds must be subtracted to reduce the position to the orbit, putting Mars at 12\u00b0 27\u2019 Virgo, with slightly less latitude than before, since the latitude was then decreasing. It was therefore about 4\u00b0 10\u2019, or 4\u00b0 7\u2153\u2019 if the parallax be neglected.\n But because, since Tycho\u2019s death, we have not made frequent observations nor continued them over several days, it would be best, for certainty\u2019s sake, also to make use of those observations which David Fabricius of East Frisia, a sedulous practitioner of astronomy, has communicated to me.\nOn February 16, old style, at 5h in the morning, he took the planet\u2019s distances from the tail of Leo (for the latitude), from the neck of Leo, and, on the other hand, from the bright star in the southern wing of Virgo, so as to check its longitude by working it out twice.\nI could make use of Tycho\u2019s line of reasoning, which he routinely adopted in volume I of the Progymnasmata when (as here) the declination of the planet was not known. But because that method extends to ten operations, I prefer for the sake of brevity to proceed as I did before with my own observations. Here there are no hidden dangers.\nFirst, the star in the wing of Virgo, adjusted to our time, is at 4\u00b0 36\u2019 30\u201c Libra with latitude 2\u00b0 50\u2019 north. Fabricius found that Mars is 20\u00b0 18\u2019 westward from it on the zodiac, which puts Mars at about 14\u00b0 18\u2019 30\u201c Virgo. This is a preliminary, approximate figure: the longitude will shortly be corrected. Now the tail of Leo is at 16\u00b0 4\u2019 Virgo, with latitude 12\u00b0 18\u2019 North, and Mars was found to be 8\u00b0 17\u2019 distant from the Tail. What is sought is the distance of its parallel from the Tail, since the difference in longitude is 1\u00b0 45\u2019. Dividing the secant of 8\u00b0 17\u2019 by the secant of 1\u00b0 45\u2019 gives the secant of 8\u00b0 6\u2019, the arc sought. This subtracted from the fixed star\u2019s northern latitude of 12\u00b0 18\u2019 leaves Mars\u2019s latitude, 4\u00b0 12\u2019 north. This I now take as determined, and compare it with the latitudes of the fixed stars according to the laws of trigonometry. From the Wing of Virgo, I find Mars\u2019s longitude to be 14\u00b0 19\u2019 Virgo; from the Neck of Leo, 14\u00b0 23\u2019 36\u201c. The mean of these two is 14\u00b0 21\u2019 18\u201c Virgo. And since the sextant gave distances larger than the truth, the latitude would come out at 4\u00b0 14\u2019 north.\nOn the night following February 23, at 12h, he observed Mars in relation to five fixed stars: the Tail of Leo and Arcturus for latitude, and for the longitude, in the first instance, Spica (which followed it), and, in the second, the Neck of Leo and Cor Leonis (which preceded it).\nBy a rough estimate, I foresee that Mars is going to fall at 11\u00bc\u00b0 Virgo. It was found to be 9\u00b0 24\u2019 from the Tail of Leo, and hence, its latitude comes out at 4\u00b0 6\u2019. And now, through this and the latitudes of the fixed stars, together with their distances (17\u00b0 26\u2019 from Regulus, 17\u00b0 51\u2019 from the Neck of Leo, 37\u00b0 28\u2019 from Spica, 44\u00b0 15\u2019 from Arcturus), Mars\u2019s position comes out at 11\u00b0 21\u2019 23\u201c Virgo (from Regulus), 11\u00b0 20\u2019 52\u201c Virgo (from the Neck of Leo), and 11\u00b0 17\u2019 40\u201c (from Spica). Again (as you see) the distances err in being too great. For from the Heart and the Neck [of Leo], Mars is pushed a little eastward, and from Spica and Arcturus a little westward, and more from Arcturus, owing to its greater northern latitude. The mean (ignoring Arcturus), 11\u00b0 19\u2019 20\u201c Virgo, is very nearly true. Also, the latitude is greater, namely, 4\u00b0 7\u2019 40\u201c north. Now from February 15 at 17h to 23 February at 12h, a period of 7 days 19 hours, Mars moved 3\u00b0 0\u2019: 180 minutes in 187 hours. That is, about one minute per hour. And if you wish to take it into account, the parallax (if any) is subtracted from the longitude on February 16 and a little is added on February 23.\n Because the last observation follows the time of observation found by me by 2 days 21 hours 47 min., add the motion corresponding to this time, 1\u00b0 7\u2019: the position will come out 12\u00b0 26\u2019 Virgo. The agreement is thus very good\u2014it could not be better\u2014given that both of us [that is, Kepler and Fabricius] work independently and do not rely on the facilities used by Tycho Brahe.","356":"As for the latitude, on the 16th it was 4\u00b0 12\u2019, and on the 23rd it was 4\u00b0 7\u2154\u2019. It is therefore fitting to set it at 4\u00b0 9\u2019 on the 21st, which comes between the other two dates. Correction for parallax makes it somewhat greater. And I, too, was putting it at a little less than 4\u00b0 10\u2154\u2019, or 4\u00b0 10\u2019.","357":"XII. Finally, in 1604, when I had published my previously written Ephemerides, in which on the night between March 29 and 30\/April 8 and 9 the planet was placed on a line from Arcturus to Spica, that very thing appeared. For on the evening of April 8 it was slightly east of that line, but already on April 9 it was to the west. At that time, with the help of Johann Schuler and using Hofmann\u2019s sextant, I found 33\u00b0 4\u2019 between Arcturus and Spica, which ought to have been 33\u00b0 1\u00bd\u2019. Therefore, the reading was 2\u00bd\u2019 too great. Immediately after, between Arcturus and Mars there was 29\u00b0 43\u00bd\u2019; therefore, correctly, 29\u00b0 41\u2019. And since the latitude of Arcturus is 31\u00b0 2\u00bd\u2019 north, this left 2\u00b0 21\u00bd\u2019 as the latitude of Mars. There was then 54\u00b0 8\u00bd\u2019 between Cor Leonis and Mars, and at the same time, the same amount between Cor Leonis and Spica. This should, however, have been 54\u00b0 2\u2019. There were therefore 6\u00bd minutes too much, while before the excess was only 2\u00bd\u2019. The origin of this uncertainty of four minutes could not be ascribed to the intervention of obstacles, as we were unable to eliminate it while observing. But let us suppose that, as before, the excess was 2\u00bd\u2019, making the distance between Mars and Cor Leonis 54\u00b0 6\u2019. The error could then be in Spica\u2019s position, possibly because Mars was mistaken for Spica, since they were close to one another. Hence, Mars\u2019s latitude comes out to 2\u00b0 21\u00bd\u2019, longitude 18\u00b0 25\u2019 Libra. The hour is known since, at the time of observation, the Back of Leo was culminating, whose right ascension is 163\u00b0 13\u2019. Now the sun\u2019s position at noon is 18\u00b0 56\u2019 24\u201c Aries, whose right ascension is 17\u00b0 27\u2019 55\u201c. Hence the difference in ascensions is 145\u00b0 45\u2019, which resolves into 9 hours 43 min. The rising point was 22\u00bd\u00b0 Scorpio. Therefore, the nonagesimal was 39\u00b0 from the zenith, and the distance of Mars from the earth was a little greater than half that of the sun from the earth. So the parallax was about 5\u00bd\u2019, and its latitudinal part, 3\u2019 28\u201c. Therefore the latitude without parallax was 2\u00b0 25\u2019 (whether this correction was rightly made, we shall consider below). And because the altitude of the nonagesimal was 51\u00b0, and Mars\u2019s distance from the nonagesimal was 56\u00b0, the longitudinal parallax was 3\u2019 32\u201c. Therefore, Mars would be at 18\u00b0 21\u00bd\u2019 Libra. At our chosen moment, the sun\u2019s position is 19\u00b0 20\u2019 8\u201c Aries. The two celestial bodies are 58\u00bd\u2019 apart. The sun\u2019s diurnal motion is 58\u2019 38\u201c, and that of Mars, 22\u2019 36\u201c. (For in 1587 in Virgo it is 24\u2019, and in 1589 at 4\u00b0 Scorpio it is 22\u2019 8\u201c). The sum of the diurnal motions is 1\u00b0 21\u2019 14\u201c. From all these beginnings, it follows that the true opposition preceded the observation by 17 hours 20 min., namely, on 29 March\/8 April at 4h 23m in the morning. Mars\u2019s position was 18\u00b0 37\u2019 50\u201c Libra. For reduction to the orbit, subtract about 39 sec., making Mars\u2019s position 18\u00b0 37\u2019 10\u201c Libra. The latitude was slightly greater than 2\u00b0 25\u2019, but, when parallax is ignored, it is 2\u00b0 22\u2019 north.","358":"Now these twelve eccentric positions of Mars (so called because the longitudes are freed from the effects of the second inequality) have been established with all possible care. If in this prickly business something has escaped me somewhere (and it did escape sometimes for a period of as much as eighteen months: I relied upon a false foundation\u2014false, that is, for the applied observation\u2014and all that work was in vain), I am entirely at a loss to imagine what it could be.","359":"I shall therefore set out all the positions in the following table, with the addition of the mean longitudes from Tycho. I could have gotten these from the Prutenic tables, or from the computation upon which Ptolemy based his demonstrations and which he designed for that purpose, but this would be unnecessary. For if the mean motion needed correction, it will be corrected later. For the present, it will serve to measure the time intervals without any appreciable error. 36","360":"Chapter 16","361":"A method of finding a hypothesis to account for the first\ninequality.","362":"Ptolemy, in Book 9 chapter 41 of the Great Work, where he is about to take up the first inequality, made by way of preface a somewhat cursory declaration of the suppositions of which he wished to make use. It is, in summary, as follows: We see that a planet spends unequal times on opposite semicircles. As, although from 2\u2154\u00b0 Cancer through Leo to 26\u00be\u00b0 Sagittarius is less than a semicircle, and from 26\u00b0 Sagittarius through Aquarius to Cancer is more than a semicircle, nonetheless the planet is found to spend longer on the former than on the latter, although a law of uniformity2 would require the contrary. For from a mean longitude of 2s 23\u00b0 18\u2019 to 9s 5\u00b0 44\u2019 is 6s 12\u00b0 26\u2019, more than a semicircle, that is, more than half of the planet\u2019s periodic time. So from 12\u00b0 16\u2019 Pisces through Leo to 12\u00b0 27\u2019 Virgo is about a semicircle and 11 minutes. But if the mean longitude of the former position (11s 9\u00b0 55\u2019) be subtracted from the longitude of the latter (5s 14\u00b0 59\u2019), the difference is seen to be 6s 5\u00b0 5\u2019, which is 5\u00b0 5\u2019 more than half. The planet consequently takes a proportionally shorter time from Virgo through Aquarius to Pisces. Now if you examine adjacent positions one at a time and compare the intervening arcs with the times or with the arcs of mean longitude, you will see that the planet is slowest at one fixed point on the zodiac, and swiftest at the opposite point, and that at the intermediate points its motion gradually increases or decreases, according to its proximity to one or the other.","363":"These things reveal first of all that the motion of a planet (however irregular it may appear) is governed according to cycles, and that the present cycle is the successive modification of motion and a return to its same state. For if the planet moved in straight lines joined by angles (such as if it should move around a pentangle\u2014I was once engaged in such ideas), its motion would sometimes suddenly change from swifter to slower in an evident manner, according to the relationship of the lines, and this would happen not in one but in many places on the zodiac, according to the number of lines. However, since so great an inequality still remains in the planet\u2019s motion, after the removal of the inequality that depends upon the sun, it therefore will be incapable of being either governed or demonstrated by the supposition of a simple circle (one set up at the center of observation). This can, however, be done by composition of several circles, or the equivalent (as Ptolemy said in his preliminaries to Book 3). The simplest ways of doing this are two: by using either an eccentric circle or a concentric with an epicycle.\nThus Ptolemy chooses an eccentric for the first inequality, for the sake of distinguishing between the two and providing an aid to comprehension, since an epicycle would be required for the second inequality. Then, thinking over this general description, he denies that a mere eccentric suffices the planets. For he first considered closely what would duly follow from the simultaneous revolution of an epicycle (to account for the second inequality) and an eccentric (for the first inequality), and it was then evident, by comparing observations, that the center of the epicycle approaches much nearer to the earth at apogee, and flees farther from it at perigee, than the simple eccentric that accounts for the first inequality allows. From this discovery, by a continuous train of thought, he alights on the measure of this approach, and relates that he discovered that the center of the eccentric that carries the center of the epicycle is exactly at the midpoint between the center of observation, the earth, and the center of uniformity or of the eccentric accounting for the first inequality. And, without a single demonstration, he nevertheless relies upon this principle for the three superior planets.","364":"Copernicus, as he frequently did on other occasions, here too followed his master religiously, his form of hypothesis being accommodated to this measure.","365":"Not without reason, astronomers have wondered about this, and I among them (using Maestlin\u2019s voice), as you see in the Mysterium cosmographicum Chapter 22 p. 79. Despite my having opined, in that passage of the said book, that Ptolemy used blind guesswork to establish this, the truth is the opposite. For he was able to prove it with a perfectly good demonstration given a suitable observation, as I shall demonstrate below. One finds fault with the theorist only in that he did not transmit to posterity those observations along with the demonstration.","366":"And so, since I thought then that this was altogether too much to assume, and also saw it pointedly called into question by Copernicus when he argued for a change in Mars\u2019s eccentricity on the basis of his figures which were not in accord with this bisection of the eccentricity, I envisaged a method which would lead me to a knowledge of the ratio of the two eccentricities (because, as I said, it is not indubitably 2:1). And since Ptolemy used three acronychal5 observations and this preconceived opinion of the ratio of the eccentricities to find the position of the apogee, the correction of the mean longitude, and finally, the magnitude of the eccentricity, I saw that in order to weaken the sinews of the problem (once the axiom of the ratio of the eccentricity is taken away), it will be indeterminate and not having a single case, and thus I would need in addition the support of a fourth acronychal observation. And so, in the year 1600, having acquired knowledge of this art, I came to Tycho, and was happy to learn that he too did not assume this ratio, but made an investigation, as his figures indicate. For he makes the center of the (Copernican*) eccentric distant from the center of vision by 13,680 units, while the point of equality in turn is another 3,780 of these units beyond that. In the Ptolemaic form, this would be as if he were to make the distance of the centers of vision and of the eccentric 9,900, and the remaining distance between the center of the eccentric and the point of equality 7,560.","367":"Now I myself could also have taken the bisection of the eccentricity as certainly established, and with better reason than Ptolemy, because in Chapter 22 of my Mysterium I had brought forward a physical cause for the bisection. Indeed, it was for this very reason that I had come to Tycho, that I might use his observations to inquire further into my opinions expressed in that book. I of course did this without prejudice, and continue to do so. And if I survive to see astronomy achieve its purity and perfection, so that a verdict can be given in the case which I have brought before her tribunal in that book, I promise the reader that I shall retract that book and, upon confirming what is seen as true, will faithfully reveal the remainder that has turned out not to be so. ","368":" But back to the argument. About center B let the eccentric FG be described, and on it, through B, the diameter of the apsides HI, taken as if immutable over any number of years. If there be danger of error in this assumption, we are not lacking ways of taking it into account. On this line below B let A be the observer, and above B let C be that center about which angles are made proportional to amounts of time, since (as was just said above) these are not proportional about A. Now let F, G, D, and E be four observations distributed about the circumference of the circle, so situated that the planet, stripped of the second inequality, would appear there as if viewed from point A. For indeed, according to Ptolemy, A is the true center of vision or the center of the earth, while according to Tycho and Copernicus vision takes place along the lines FA, GA, DA, and EA, and A is the sun. But it was said above that in either way the planet is likewise shorn of its second inequality. Now let each point be connected with each of the others, and let AF be at 25\u00b0 43\u2019 Virgo, AG at 26\u00b0 43\u2019 Sagittarius, AD at 12\u00b0 16\u2019 Pisces, and AE at 17\u00b0 31\u2154\u2019 Taurus. Hence, the four angles about A are given: FAG is 91\u00b0 0\u2019, GAD is 75\u00b0 33\u2019, DAE is 65\u00b0 15\u2154\u2019, and EAF is 128\u00b0 11\u2153\u2019. These must be corrected somewhat on account of the precession of the equinoxes. For in relation to the fixed stars the planet is not so far forward at E (the last observation) as is indicated by these numbers. Wherefore FAE is a little greater, and the others smaller by the same amount. In the same way, by subtraction of the [mean] longitudes, the angles about C are also obtained.","369":"Proposition. It is now required to select values for angles FAH and FCH such that, once these are supposed, the points F, G, D, E stand on one circle, and that center B of that circle lie between the points C and A on the line CA.","370":"The solution is not geometrical, at least if algebra is not geometrical, but proceeds by a double iteration. For algebra, too, forsakes us here, because terms communicated by strictly straight lines through straight line do not extend to angles, unless perchance one would wish to cram the entire theory of sines into this one operation.","371":"But behold what we are required to do. If we were to assume a value for the angle FAH, then since the line AF has a certain sidereal position, the other leg AH would also be assumed to have a certain sidereal position. But let AH be the line of the apogee, or the line of the aphelion in the Copernican and Tychonic notion. We are thus required to assume and posit that which is sought. For it was in order to learn the position of this aphelion that we embarked upon this path. In the same way, since the sidereal position of AH (that is, CH) was arrived at through this assumption of ours, and it passes through C the center of our equant circle (and therefore also through the starting point from which the numbering of its parts has its beginning, namely, the apsis which is conceived as being above H); and since we are also required to assume the angle FCH, the line CF therefore also will acquire its position on the circumference of the equant. And indeed, this is the mean longitude corresponding to the observed position of the planet at F. And we were seeking to know what this mean longitude is. Therefore, in addition to the apogee, we are assuming yet another thing among those which we were seeking.","372":"At the same time, however, it is not unusual, whether in geometry, or arithmetic, or dialectic, to use a form of argument which leads to an impossibility, so that if something absurd is seen to follow from the assumptions, they are rejected as false; and this is carried out until the consequent removal of excesses and defects unveils the exact truth (which in the mathematical disciplines lies hidden in the middle between the two). In the present case this comes about in the following manner.","373":"Let the line CA be taken as the nominal standard (and thus, let it be given). Since the angles FCH and FAH are assumed, and hence also the inclinations of the remaining lines to HCA, and AC is the common side of the four triangles CFA, CGA, CDA, CEA, whose angle are given, therefore the four lines AF, AG, AD, and AE will be given in relation to the length AC. And since in the four new triangles FAG, GAD, DAE, and EAF, the sides are already given with the angles at A between two sides, the individual angles at the bases of the several triangles (that is, AFG, ADG, ADE, and AFE), will not be a matter of ignorance. But AFG and AFE are parts of the angle GFE. And in the quadrilateral DEFG (if, indeed, it is inscribed in a circle, which is one of the hypotheses here), it is a consequence that two opposite angles (as GFE, GDE) taken together are equal to the sum of two right angles. Therefore, when the four angles which we have just found are combined, if their sum differs from the measure of two right angles, we shall pronounce the assumptions false, whether the falsehood be in one or the other of the assumed values, or both.","374":" Then, one angle, FCH, being kept the same, and the other, FAH, being changed, a return will be made to the beginning, and the sum of the four angles will once more sought. If this sum differs more than the previous sum from two right angles, it suggests that FAH was altered in the wrong direction. Therefore, the opposite must be done: if you had added something to it, you now would diminish it, or vice versa. But if, on the other hand, you have come closer to the correct measure, then you will be sure that you are on the way. And then by comparing the original discrepancy with that which still remains, you will carry on in that same proportion by increasing or decreasing the angle FAH.","375":"But it is still not certain that this second correction will directly reconcile your four angles with the exact measure. For the rate of increase of circular variables is not the same as that of straight ones. Your labor will have to be repeated again and again, until your sum for the angles in question is 180\u00b0 or very nearly as much (you may safely ignore very small discrepancies).","376":"When you have carried this out until the angles F and D (and therefore the remaining angles G and E) truly stand upon the same circumference, now, in turn, an enquiry must be made into the other matter that it is fitting to pursue, and that is, whether the center of that circle B lies between C and A on the same line. For on this point it was said above that Ptolemy assumed it outright, and physical considerations demand that the slowest motion occur where the star is at its greatest distance from A, the sun, as at H. This can happen in no other way than if A, B, and C are on the same line. To find this out, let the known angles GAD, DAE, be taken as one, so that the angle GAE may be known, and in GAE from this angle and sides GA, AE, let the side GE be sought. Now in the triangle GFE the angle GFE stands upon the circumference. Therefore, GBE, the angle at the center, is its double. But the angle GFE was previously found through its parts GFA, AFE. So, again, in the isosceles triangle GBE the angle GBE and the side GE are given. Consequently, the angles at the base will not be unknown, as well as GB the radius of the circle, in proportion to AC, the eccentricity taken at the beginning. And because BG and BGE are now had, and AG and AGE were had before, therefore, by subtracting AGE from BGE (or vice versa, as the case requires), the remainder will be AGB. Next, in triangle AGB, AG, BG, and the included angle AGB are given. [Hence, the other sides and angles will not be unknown, and thus the angle BAG will be given]. If this differs from CAG, which was taken at the beginning, it shows that B itself, contrary to what should happen, does falls outside the line CA. So again we shall pronounce the assumed magnitudes of angles FCH and FAH to be false. But if we keep FCH fixed and change FAH we fall into yet another absurdity, namely, that the positions D, E, F, G, do not fall upon a circle (just as they did not above, before we had finally established the magnitude of FAH). Therefore, it is obvious that FCH, too, has to be changed. Let it be changed, then; that is, let another quantity be taken at will for the angle FCH, and keeping that constant, let the angle FAH be adjusted four, five, or six times, until once again the four angles at F and D add up to two right angles. Then let an attempt be made at a second enquiry, using the triangles GAE, GFE, GBE, and BGA, to find BAG, in comparison with CAG as it has now most recently been established. Here you will again see whether you have departed farther from the truth, or have in truth come closer, and according to the qualities of excess or defect and ratios of the additions you will thence return to the beginning until you find BAG equal to the value you had assumed during that trial for CAG or HAG. When you have arrived at this point, then finally, in triangle BGA, you will assign a round number (100,000) to BG as a standard, and, in the same ratio (through the mediation of the angles), you will seek out BA the eccentricity of the eccentric and CA the eccentricity of the equant. Whence, by subtracting BA, CB remains. Then you will issue the pronouncement, concerning both the position of the apogee and the correction of the mean motion (which you had assumed in the final operation), that they are well established, at least as far as pertains to this form of hypothesis.","377":" If this wearisome method has filled you with loathing, it should more properly fill you with compassion for me, as I have gone through it at least seventy times at the expense of a great deal of time, and you will cease to wonder that the fifth year has now gone by since I took up Mars, although the year 1603 was nearly all given over to optical investigations.","378":"There will arise subtle geometers such as Vieta who will think it something great to show up the contrived nature13 of this method. For in this matter, Vieta did object to Ptolemy, Copernicus, and Regiomontanus. Let them therefore go forth themselves and solve the figure geometrically, and they will be to me great Apolloes. For me it is enough to draw four or five conclusions from a single argument (which includes four observations and two hypotheses); that is, in getting from the labyrinth back to the highway, to show, instead of a geometrical light, a contrived thread, which nonetheless will lead you to the exit. If this method is difficult to grasp, the subject is much more difficult to investigate with no method at all.","379":"There now follows an example of this instruction based upon the four proposed observations.","380":"Because of precession, all positions are reduced to [the time of] the first observation. Here, the apparent longitude was 25\u00b0 43\u2019 Virgo, the mean longitude 6s 0\u00b0 47\u2019 40\u201c, and the annual motion of the fixed stars is 51 seconds, as Brahe has demonstrated in the Progymnasmata. Therefore, from 1587 March 6 to 1591 June 8 is 4 years 3 months, to which corresponds a motion of precession of 3\u2019 37\u201c. Therefore, we must set the apparent position in 1591 at 26\u00b0 39\u2019 23\u201c Sagittarius, and the mean longitude at 9s 5\u00b0 40\u2019 18\u201c. Similarly, from 1587 March 6 to 1593 August 25 is 6 years 5\u00bd months, to which corresponds a motion of precession of 5\u2019 30\u201c. And so Mars is to be placed at 12\u00b0 10\u2019 30\u201c Pisces, with mean longitude 11s 9\u00b0 49\u2019 34\u201c. Finally, from 1587 March 6 to 1595 October 31 is nearly 8 years 7 months, to which corresponds a motion of 7\u2019 18\u201c. And so Mars is to be placed at 17\u00b0 24\u2019 22\u201c Taurus, with mean longitude 1s 7\u00b0 6\u2019 51\u201c.","381":"Now first, we shall assume that the apogee or aphelion in 1587 is at 28\u00b0 44\u2019 0\u201c Leo, and second, we shall assume that the mean longitudes should be increased by 3\u2019 16\u201c, so that the mean longitudes are 6s 0\u00b0 50\u2019 56\u201c, 9s 5\u00b0 43\u2019 34\u201c, 11s 9\u00b0 52\u2019 50\u201c, and 1s 7\u00b0 10\u2019 7\u201c. 18","382":"For the angles of the equations. ","383":"For the lines from A.","384":"Let AC be taken as the standard, with magnitude 10,000. Now as the [sines of the] angles of the equations are to AC, so are the [sines of the] angles at C to the lines from A. Therefore, the sines of the angles at C, multiplied by 10,000, are to be divided by the sines of the angles of the equations. ","385":" For the angles at A. ","386":"For the angles at F and D","387":" The angles AFG, AFE, ADG, ADE, are approximately the halves of the supplements of the angles at A. Those at F, however, are smaller, because the lines AG (50,703) and AE (52,302) have been found to be smaller than AF (59,433); and those at D are greater, because those lines AG and AE are longer than AD (48,052). And since those four angles about A are equal to four right angles, the sum of their supplements consequently will also equal four right angles, because four semicircles are equal to eight right angles. Therefore, half of the sum of the supplements is equal to two right angles, which is what we wish the sum of GFE and GDE to be. Consequently, the angles at D ought to exceed [the halves of] their supplements by the same amount that those at F fall short of theirs. But the tangents of the differences of the angles at the bases in this kind of triangle are found if you divide the differences of the sides by the sums of the sides, and multiply the quotient by the tangents of the halves of the supplements. Therefore, if the differences of the two angles at F together equal the sum [of the two differences] at D, angle at F plus the angle at D will equal two right angles. ","388":"From this it is clear, therefore, that the sum of F and D is less than two right angles, because the difference to be subtracted exceeds that to be added.","389":"The amount wanting is 24\u2019 15\u201c. Now I know from many repetitions of this task that by adding 3\u2019 20\u201c to the aphelion, the sums come together. This I shall prove.","390":"The angles of the equations and their sines will stay the same, as well as the tangents of the halves of the supplements of the angles at A. ","391":" Here the sums differ by no more than 1\u2019 48\u201c. So we have now moved the apogee too far forward, and must move it back by another 12\u201c. But not much care is needed with such a small difference. We shall make it up \u201cfrom the equal and the good\u201c [that is, by interpolation], in order to be able to carry on with our method. Before, when we had erred in defect by 29\u2019 15\u201c, the sum of the differences at F and D was 12\u00b0 1\u2019 44\u201c. Now, when we have erred in excess by 1\u2019 48\u201c, this sum comes out to be 12\u00b0 19\u2019 46\u201c. And so, since 31\u2019 will produce 18\u2019 in the sum of the differences, therefore, 1\u2158\u2019 produces about 1\u2019, so that the exact sum is 12\u00b0 18\u2019 44\u201c, whose half, 6\u00b0 9\u2019 22\u201c, is the sum at either F or D.\nFor the Triangles GFE, GBE. \nNext, GE is sought, from the sides GA, AE, and the angle GAE. \nAs the sine of AGE is to AE, so is the sine of GAE to GE. \nTherefore, in GBE, as [sine] GBE is to GE, let [sine] BGE be to BE.  \n Therefore B is a little off the line CA on the side of G, since CAG is greater than BAG by 30' 38\". But I know from many trials that by the addition of one half minute to the mean longitude, B is made to lie upon the line CA. But at the same time, to keep the quadrangle on the circle, the aphelion has to be moved forward 2'. If is worthwhile seeing this through, at the same time demonstrating the eccentricity. So since 30\" are added to CF and its counterparts, and 2' to CH, HCF will be diminished by 1' 30\". Therefore, ","392":"HCF is 32\u00b0 2' 6\"; GCI 53\u00b0 8' 32\"; DCI 11\u00b0 0' 44\"; and ECI 68\u00b0 18' 1\".","393":"But the angles of the equations are increased and decreased by 30\". ","394":"We have six minutes left over, which are removed by moving the aphelion back 38\". So, since it was at 28\u00b0 49' 8\" Leo, it will now be at 28\u00b0 48' 30\" Leo. ","395":"Test ","396":" And so, with the quadrangle contained in the circle, let it again be enquired whether B lies upon the line CA. And from the sum of 70\u00b0 22' 29\" established above, subtract the difference of 6\u00b0 2\u2019 20\" just found. The remainder is ","397":"B still lies 7\u2019 20\" from the line CA towards G. ","398":"Whence we see that because before, by adding 30\" to the mean motion and 82\" to the aphelion, we moved forward 23' 18\", we will take up the remaining 7' 20\" by adding 9\" to the mean motion and 25\" to the aphelion. Therefore, the total addition to Tycho's longitude is 3' 55\", and the aphelion is placed at 28\u00b0 48\u2019 55\" Leo.","399":"And with such a small error, one incurs no disadvantage by finding BA from the given angles and sides of triangle CAG as if B lay exactly on the line CA.  Therefore BA is 11283 where BG is 100000.","400":"But as BG, 53,866, is to 100,000, so is 100,000 to AC. But to exclude all error, let us interpolate. 24 25 So the whole eccentricity remains 18,564 but that of the eccentric 11,332 and of the equant 7,232.","401":"In the Copernican and Tychonic form, the diameter of the small epicycle would be 3,616, and of the greater, 14,948. Or, following what was said at the end of the fourth chapter, the tangent may be used instead of the sine, in this manner.","402":"Let the maximum equation be investigated at the ninetieth degree. Let HCG be 90. BC will be the sine of the angle BGC, 4\u00b0 8' 51\". And GBC will be 85\u00b0 51' 9\". And GC 99,738. But in the Copernican form, with C at the center of the concentric, GC will be 100,000. Therefore, in order that CGA, the angle of the equation, remain unchanged, the same 18,564 is to be increased in the same ratio for Tycho and Copernicus: \nThe composite Copernican-Tychonic eccentricity. And in the tangents, this shows 10\u00b0 32' 38\" as the common angle of the equation at 90\u00b0 of anomaly.\nTherefore the corrected diameter of the smaller epicycle is 3,628. of the greater, 14,988.","403":"Compare all this with Chapter 5, where I transposed the Tychonic rendition from the mean to the apparent motion of the sun, and see how slight the difference is.","404":" So this is the method by which the hypothesis of the first inequality was investigated using four acronychal26 positions of Mars. In this, with Ptolemy, I have supposed that all positions of the planet throughout the heavens are arranged on the circumference of one circle; also that the planet moves most slowly where it is at its greatest distance from the center of the earth (according to Ptolemy) or of the sun (according to Tycho and Copernicus); and that the point about which this slowing is measured is fixed. Everything else I have demonstrated, although the demonstration is a \"reduction to the impossible.\" But whether the things I assumed in the demonstration are in fact so, or the opposite, will become clear in what follows.","405":"I shall now test the remaining eight positions against this hypothesis, for the sake of consensus. But in order that the test be universal and legitimate, I shall also throw in the motion of the apogee. So I shall take this up first.","406":"Chapter 17","407":"A superficial1 investigation of the motion of the apogee and nodes.","408":"This investigation will be of the same degree of certainty as the observations (or rather, those things handed down by Ptolemy). If that practitioner had not existed, we would know even less today about those very slow motions. For besides him there is no one at all to be found, from the earliest records of civilization to the present, who could help us here.","409":"We lay down here those suppositions found in Ptolemy, which are not in all respects perfectly certain. First, that the fixed stars have remained exactly in the zodiacal positions in which Ptolemy placed them (Ptolemy book 7). Second, that Ptolemy's figure for the sun's eccentricity was correct: 4153, where the semidiameter of the orbit is 100,000 (Ptolemy book 3 Chapter 4). Third, that the sun's apogee was fixed at 5\u00bd\u00b0 Gemini (in the same chapter). Fourth, that Mars's apogee (when its motion is adjusted to the sun's mean motion) was found to be at 25\u00bd\u00b0 Cancer (Ptol. book 10 Chapter 7). Fifth, that the eccentricity of Mars was 20,000 where its semidiameter is 100,000 (in the same chapter). Sixth, that the ratio of the epicycle (in Ptolemy) or of the annual orb (in Tycho and Copernicus) to the orb of Mars was as 100,000 to 151,900. Hence, where the semidiameter of the sun's orb (or the earth's) is 100,000, Mars's eccentricity will be 30,380 (Ptolemy book 10 Chapter 8).","410":" We shall proceed as in Chapter 5. Let A be the point about which the earth's orb is described, C Mars's equalizing point, B the center of the sun's orb. And because AB is at 5\u00bd\u00b0 Gemini, while AC is at 25\u00bd\u00b0 Cancer, CAB is 50\u00b0. By supposition, AB is 4153, while AC is 30,380 of the same units. Therefore, since two sides and the included angle are given, the angle CBA is given as 123\u00b0 27'. And because BA is directed towards 5\u00bd\u00b0 Sagittarius, the direction of BC (by subtraction of the angle 123\u00b0 27\u2019) will be about 2\u00b0 3' Leo, for Ptolemy's time. Also at that time, the eccentricity of the equant, after transposition to the sun's true motion, was 18,353. I discovered this above by transposition of the Tychonic value, 18,342, with one change: for the size of Mars's orbit, instead of 151,386 I took 152,500, which is nearer the truth. But this is tangential to the matter at hand, to which we return.","411":" On the motion of the aphelia","412":"Because the precession of the equinoxes was exceedingly high around Ptolemy's time, while before and after there remains not the least suspicion of any such thing, I shall exclude this, and relate the position of the apsis to the fixed stars. At that time, Cor Leonis was at 2\u00b0 30' Leo. The apsis or aphelion of Mars therefore preceded this star by 27', in about A. D. 140. In our times, in 1587, Tycho Brahe found this star at 24\u00b0 5' Leo. Since the aphelion progressed to 28\u00b0 49' Leo, it was 4\u00b0 44' east of Cor Leonis. If you add the above mentioned 27' to this, the sum (5\u00b0 11') is the motion over the 1447 years between 140 and 1587. Therefore, the annual motion is very nearly 13\": 6' 29\" every 30 years. If, in turn, you add to this the Tychonic value for the motion of the fixed stars or of precession, which is quite uniform and is the same for all times (Ptolemy's alone excepted), namely, 25' 30\" in 30 years, you will get the sum of 31' 59\". Therefore, the annual motion of Mars's aphelion with respect to the equinox is 1'4\" in our time.","413":" On the motion of the nodes","414":"Although it is not really necessary, we shall consider this subject here because it is related to the motion of the aphelia. And because Ptolemy says in Book 13 Chapter 1 that Mars's northern limit is \"near the end of Cancer, and almost at its apogee,\"2 it will therefore be at 29 Cancer, that is, 3\u00bd\u00b0 before Cor Leonis. Despite this, in book 3 Chapter 6 Ptolemy put the northern limit exactly at the position of the apogee (25\u00bd\u00b0 Cancer) because it made calculations easier. And today it is at about 16\u00b0 20' Leo, about 7\u00b0 45' before Cor Leonis. By subtraction of 3\u00b0 30', the northern limit, and consequently the nodes, are found to have retrogressed 4\u00b0 15' from Cor Leonis. This accords well with the motions of the moon, whose apogee likewise has a progressive motion with respect to the fixed stars, while its nodes retrogress. So the annual westward motion is 10\" 34'\", or 5' 17\" in 30 years. Subtract this from the motion of precession, 25' 30\". The remainder is 20' 13\". And the nodes of Mars are moved the same number of minutes with respect to the equinoctial point in 30 of our present years, likewise eastward.","415":"Chapter 18","416":"Examination of the twelve acronychal positions using the hypothesis we have found.","417":"I shall use that form1 of calculation which I explained above in Chapter 4, because it is more succinct. Also, it is indubitable that not a minute and a half (actually somewhat less) would be gained or lost by using the Copernican or Tychonic form, as I noted in the same place.","418":"You see, then, O studious reader, that the hypothesis found by the method developed above, is able in its calculations not only to account, in turn, for the four observations upon which it was founded, but also to comprehend all the other observations within two minutes\u2014a magnitude which this star, when in its acronychal position, always fills and even exceeds in the size of its body. This shows that if anyone were to repeat the above method, taking in turn various sets of four observations, the same eccentricity with the same division, an identical aphelion, and very nearly the same mean motion, would always result. I therefore proclaim that the acronychal positions displayed by this calculation are as certain as the observations made with the Tychonic sextants can be. As I have said before, these observations are subject to some degree of uncertainty (at least two minutes), owing to Mars's small but appreciable diameter, and refraction and parallax, which are not yet known with complete certainty. ","419":"Finally, you see how nothing prevented the transposition of acronychal observations from the mean to the apparent motion of the sun, so as to keep me from, not just imitating, but even surpassing, the certitude of the Tychonic calculation, which was raised as an objection against my abandoning the sun's mean motion. 4 5","420":"Chapter 19","421":"A refutation, using acronychal latitudes, of this hypothesis constructed according to the opinion of the authorities and confirmed by all the acronychal positions.","422":"Who would have thought it possible? This hypothesis, so closely in agreement with the acronychal1 observations, is nonetheless false, whether the observations be considered in relation to the sun's mean position or to its apparent position. Ptolemy indicated this to us when he teaches that the eccentricity of the equalizing point is to be bisected by the center of the eccentric bearing the planet. For here neither Tycho Brahe nor I have bisected the eccentricity of the equalizing point. Now for Copernicus* it was a matter of religion not to neglect this anywhere. For he made very little use of observations, perhaps thinking that Ptolemy used no more than are referred to in his Great Work. Tycho Brahe balked at this. For in imitating Copernicus, he set up this ratio of the eccentricities, which the acronychal observations required. But when this was gainsaid not only by the acronychal2 latitudes (for these still underwent some increase arising from the second inequality) but also, and much more forcefully, by observations of other positions with respect to the sun which are affected by the second inequality, he stopped here and turned to the lunar theory, and I meanwhile stepped in.","423":"Now the method by which the whole theory of Mars could easily be acquitted of error, if the premises were correct, and by which it is demonstrated to be incorrect, is this.","424":"First, through the latitudes at acronychal positions. In the Copernican form, let the line DE be set out in the plane of Mars's eccentric, upon which let A be the sun, D the northern limit, E the southern limit, or the point nearest it. And through A let the straight line HL be drawn, lying in the plane of the earth's eccentric orb. Now let AH and AD be conceived as lying in the one plane of a circle of latitude, and likewise AL and AE. And let the earth in 1585 lie at B on line AH, and in 1593 let it be at the point C on line AL. Now AB and AD are directed toward 21\u00b0 Leo, so that the sun A seen from B appears at 21\u00b0 Aquarius. And on the other hand, E and C are at 12\u00b0 Pisces, so that the sun A seen from the earth C appears at 12\u00b0 Virgo. But 12\u00b0 Virgo is nearer to the sun's apogee than is 21\u00b0 Aquarius. Therefore, BA is shorter than AC. I shall take these lines from vol. I p. 98 of Tycho Brahe's Progymnasmata 3 and shall suppose them to be correct, although below (by a method developed for the purpose) I am going to be showing them to be slightly different. In Tycho BA is shown as 97,500, and AC as 101,400, while in the correction which is to follow, BA turns out a little longer and AC a little shorter; they are not equal, however. Now because in Chapter 13 above, through two procedures independent of the present enquiry, the angle BAD was found to be about 1\u00b0 50' at the limit (about 16\u00b0 Leo), therefore, at four or five degrees from the limit it will be 1\u00b0 49\u00bd\u00b0. But HBD, the apparent latitude in 1585, was 4\u00b0 32' 10\". Hence, given the angles HBD and BAD, their difference BDA of 2\u00b0 42' 40\" is also given. Now as the sine of BDA is to the known length BA, so is the sine of DBA to DA. So that if BA is taken to be 97,500, DA comes out to be 163,000. But if BA is 100,000, DA will be 167,200.","425":" Likewise, since in 1593 C and E are in Pisces, and Mars is 26\u00b0 from its limit, 64\u00b0 from the node, consequently, the whole sine is to the sine of the maximum inclination 1\u00b0 50' as the sine of 64\u00b0 is to the sine of CAE, the inclination at this position. CAE is therefore 1\u00b0 39'. But the apparent latitude LCE was 6\u00b0 3'. Therefore the angle AEC is 4\u00b0 24'. And now, as before, as the sine of AEC is to the known length AC, so is the sine of ACE to AE. So that if AC is taken to be 101,400, AE comes out to be about 139,300. But if AC is 100,000, AE comes out to be about 137,380. Now since 21\u00b0 Leo is about 8\u00b0 from the aphelion, the line AD will be about 150 parts longer at aphelion (as will be clear to anyone who computes the distances from the hypothesis we have found and substitutes these numbers); that is, either 163,150 or 167,350. And since 12\u00b0 Pisces is about 13\u00b0 from the perihelion, AE when right at the perihelion will be about 300 parts shorter; that is, either 139,000 or 137,080. So we have the lengths of the lines AD and AE when right at the apsides, when they are parts of the same straight line DE. So let them be added: ","426":"Let these numbers be substituted for the original ones, where the radius of the eccentric was 100,000. Thus, as 151,075 is to 100,000, so is 12,075 to 8000,\nand as 152,215 is to 100,000, so is 15,135 to 9943. ","427":"Therefore, the exact eccentricity of the eccentric, as indicated by the acronychal latitudes, is somewhere between 8000 and 9943, where the radius of the eccentric orb is 100,000. But our hypothesis based upon the acronychal observations of longitude resulted in an eccentricity of the eccentric of 11,332, far different from that which is near the mean between 8000 and 9943. Therefore, something among those things we had assumed must be false. But what was assumed was: that the orbit upon which the planet moves is a perfect circle; and that there exists some unique point on the line of apsides at a fixed and constant distance from the center of the eccentric about which point Mars describes equal angles in equal times. Therefore, of these, one or the other or perhaps both are false, for the observations used are not false.","428":"The same demonstration also holds against that hypothesis established by observations adjusted to opposition to the sun's mean position, because the latitudes remain about the same over the time interval between the two moments. Thus they show an eccentricity of the eccentric of 9943, while in Chapter 5 above, 12,600 was taken from the Brahean revision, and 12,352 in the Ptolemaic equant, where the whole eccentricity of the equalizing point was 20,160 or 19,763.","429":" For the transformation of our diagram to the Ptolemaic form, let DE be the line of apsides, A the earth, D and E the center of the epicycle at the highest and lowest apsis, and from both points D and E let straight lines be drawn toward the earth A parallel to the plane of the ecliptic BC. On these, let DF, EG be taken as radii of the epicycle, equal to BA, AC, and let the planet be at F and G. Therefore, the inclination FDA will be equal to the inclination BAD, and the line of vision AF will be parallel to the original line BD. Therefore, the observed latitude, HAF and HBD, is the same. The same must be said of the congruent triangles ACE and EGA. And thus the demonstration and the magnitudes of the corresponding lines are the same.","430":"It will occur to the reader to question why I make the semidiameter of Mars's epicycle unequal to itself, namely, DF equal to BA, which is longer, and EG equal to CA, which is shorter. I answer from Part I, that this happens because of the transposition of observations from opposition to the sun's mean position to opposition to the sun's apparent position. If we should stay with the sun's mean motion\u2014and the present line of argument prevails even then\u2014DF and EG will remain equal at least up to this point. But for this see Part I Chapter 6.","431":"For the Brahean form, leaving one of the triangles (DBA, say) the same, so that B may be the motionless earth and A the sun in 1585, let AB be extended so that BH is equal to AC, and let H be the sun in 1593, at 12\u00b0 Virgo, and let HI be made equal and parallel to AE in the same direction, so that Mars at perigee is at I, and at apogee at D; HBA the ecliptic, BHI, BAD the inclination; IBA the latitude at perigee, and DBH at apogee. So again the sum of DA and HI will come out the same, whose half is DK, and the eccentricity will be KA. The only difference is this, that for Ptolemy the plane of the epicycle, and for Tycho the plane of the eccentric, is moved from north to south and back, remaining parallel to itself, while in Copernicus both stay in the same place.","432":"Meanwhile, consider this also. In chapter 16 I had found the combined eccentricity to be 18,564, whose half, 9282, is just about the mean between 8000 and 9943. And Ptolemy too, as was remarked above, had taught us that half of the eccentricity found by acronychal observations is to be assigned to the eccentricity of the eccentric. So it was no lack of reason that so moved him, and we should not rashly reject this bisection, since the observed latitudes support it.","433":"Now if, on the other hand, we bisect the eccentricity of 18,564 that we found, we shall indeed represent the acronychal positions near the middle longitudes on the eccentric accurately enough, but not so well the positions around the octants and towards the apsides.","434":"Take, for example, the opposition in 1593. From the preceding chapter, the simple anomaly was 6s 11\u00b0 3' 16\". I multiply the sine of 11\u00b0 3' 16\", namely, 19,174, by 9282 (before, it was to be multiplied by 7232). This gives 1780, the sine of the arc 1\u00b0 1' 12\", which is part of the equation. When this is added to 11\u00b0 3' 16\", the sum is the semiequated anomaly, 6s 12\u00b0 4' 28\". The supplement of this is 167\u00b0 55' 32\", whose half is 83\u00b0 57' 46\". The tangent of this is about 945,500, which, multiplied by the perihelial distance of 90,718 and in turn divided by the aphelial distance of 109,282, gives a tangent of 784,880, whose arc is 82\u00b0 44' 20\". This, subtracted from the 83\u00b0 57' 46\" found earlier, leaves 1\u00b0 13' 26\", which is the other part of the equation. If this be added to the semi-equated anomaly, and the sum be added to the aphelion, it puts the planet at 12\u00b0 13' 37\" Pisces, which differs from the former hypothesis by three minutes, and is more distant from the observed position. For it should have been 12\u00b0 16' Pisces.","435":"This appears more clearly at 17\u00b0 Cancer in 1582. For with the eccentricity bisected, Mars falls at 17\u00b0 4\u00be\u2019 Cancer, and this calculated value differs from ours by 7\u2154\u2019 at about 45\u00b0 from aphelion, but by about 9' from the observation.","436":"And from this difference of eight minutes, so small as it is, the reason is clear why Ptolemy, when he made use of bisection, was satisfied with a fixed equalizing point. For if the eccentricity of the equant, whose magnitude the very large equations in the middle longitudes fix indubitably, be bisected, you see that the very greatest error from the observations reaches 8', and this in Mars, which has the greatest eccentricity; it is therefore less for the rest. Now Ptolemy professes not to go below 10', or the sixth part of a degree, in his observation. The uncertainty or (as they say) the \"latitude\" of the observations therefore exceeds the error in this Ptolemaic computation.","437":"Since the divine benevolence has vouchsafed us Tycho Brahe, a most diligent observer, from whose observations the 8' error of this Ptolemaic computation is shown in Mars, it is fitting that we with thankful mind both acknowledge and honor this favor of God. For it is in this that we shall carry on, to find at length the true form of the celestial motions, supported as we are by these proofs showing our suppositions to be fallacious. In what follows, I shall myself, to the best of my ability, lead the way for others on this road. For if I had thought I could ignore eight minutes of longitude, in bisecting the eccentricity I would already have made enough of a correction in the hypothesis found in Chapter 16. Now, because they could not be ignored, these eight minutes alone will have led the way to the reformation of all of astronomy, and have become the material for a great part of the present work.","438":"Chapter 20","439":"Refutation of the same hypothesis through observations in positions other than acronychal.","440":"I shall now proceed to the other argument whereby the eccentricity of the eccentric as found in Chapter 161 is proven false, despite its providing true longitudinal motions. This argument is based upon observations of Mars at positions with respect to the sun other than opposition, when the planet was observed in the region of the eccentric's apsides.","441":"On 1600 March 5\/15 about midnight, Mars was observed at 29\u00b0 12\u00bd' Cancer with latitude 3\u00b0 23' north. Its mean longitude, corrected by our addition, was 4s 29\u00b0 14' 58\", while the aphelion was at 4s 29\u00b0 2' 45\". Therefore, the anomaly was 0s 0\u00b0 12' 13\", requiring an equation of 2', to be subtracted, according to the hypothesis of eccentric positions established above. Therefore, Mars's eccentric position was 29\u00b0 13' Leo, and the sun's position, 25\u00b0 45' 51\" Pisces.","442":" In the diagram, let A be the sun, B Mars, and C the earth. By subtraction of CB (29\u00b0 12\u00bd\u2019 Cancer) from AB (29\u00b0 13' Leo) the angle CBA will be 30\u00b0 0' 30\", while by subtraction of CA (25\u00b0 45' 51\" Pisces) from CB (29\u00b0 12' 30\" Cancer), BCA will be 123\u00b0 26' 39\". But as [sin] CBA is to CA, so is [sine] BCA to BA. But CA, the distance of the sun from the earth, is 99,302 from Tycho's table. (Although this is incorrect, the true value is nevertheless between this and 100,000, as we shall learn below in Chapter 30). Therefore, AB is between 165,680 and 166,846.","443":"At perihelion, let the observation be taken that was made on the night following 1593 July 30 at 1h 45m. ","444":"Mars was found to be at 17\u00b0 39\u00bd\u2019 Pisces with latitude 6\u00b0 6\u00bc\u2019 south. Mars's mean longitude was 10s 26\u00b0 16' 38\", aphelion 4s 28\u00b0 55' 43\", so Mars was 2\u00b0 39' 5\" from perihelion, to which corresponds an equation of 32', to be subtracted, in accordance with the above hypothesis, making Mars's eccentric","445":"position 10s 25\u00b0 44' 30\", and the sun's apparent position 17\u00b0 3' 0\" Leo.","446":"In the diagram let BA be extended to D, and let AD be at 25\u00b0 44' 30\" Aquarius and ED at 17\u00b0 39' 30\" Pisces. Therefore, EDA is 21\u00b0 55' 0\". And because ED is at 17\u00b0 39' 30\" Pisces and EA at 17\u00b0 3' Leo, AED is therefore 149\u00b0 23' 30\". Now as [sin] EDA is to EA, so is [sine] AED to AD. But EA, the sun's distance from the earth, is 102,689, from Tycho's table, an erroneous figure, to be sure, but it is surely greater than 100,000. Therefore, AD is between 140,080 and 136,409. But since the star Mars is 2\u2154 degrees from perihelion, AD will be shorter by about 15 at the perihelion itself, that is, between 140,065 and 136,394. Distances for both apogee and perigee must be increased, because they were computed using observations related to the ecliptic. Thus AD and AB are lines in the plane of the ecliptic. On which point, take this","447":"Protheorem4 to be used frequently below","448":"By observations of the star Mars related to the ecliptic, and by lines in the plane of the ecliptic found through those observations, to show the length of lines corresponding to them and next to them in the plane of Mars's own orbit.","449":"Let the line BAD be set out in the plane of the ecliptic, and through A, which denotes the sun or center of the world, let the straight line LAM be so drawn in the plane of the orbit that the star be at L and M. Now let the earth be at C, and the triangle CAB be part of the plane of the ecliptic, to which the plane of the triangle LBA is to be understood to be perpendicular. Let the points C, L, and B be joined, and lines be extended to the surface of the sphere of the fixed stars: AB to \u03b2, AL to \u03bb, AC to k; and let \u03ba\u03b2 be an arc of the ecliptic, \u03b2\u03bb an arc of the circle of latitude, and \u03ba\u03bb a transverse arc. Thus the observation of the star's position beneath the fixed stars is referred to the ecliptic, by means of an arc of the circle of latitude drawn at right angles to the ecliptic \u03ba\u03b2 through the observed position of the star, and the triangle CLB is part of the plane of that circle. But \u03bb\u03b2 is also by supposition the circle of latitude, perpendicular to the ecliptic \u03ba\u03b2. Therefore, the planes CLB and LBA of the two circles perpendicular to the same ecliptic intersect each other at the line LB. Therefore, by Euclid XI. 19, the line of intersection LB will be perpendicular to the plane of the ecliptic CBA and to the line BA contained in it; that is, LBA will be a right angle. Therefore, once the length of BA on the ecliptic is found, and the angle LAB is known, it will be impossible for the length of LA which is sought to fail to be known. QEF","450":"Now in the matter at hand, since the inclination, or angle LAB, is 1\u00b0 48' at this position, LA is 82 parts longer than BA (in the present units), and AM 72 parts longer than AD. 5","451":"In new units, taking KL or KM as 100,000, the eccentricity of the eccentric is between 8377 and 10,106. But our hypothesis postulated 11,332, which exceeds both of these. Therefore, it postulated something false.","452":"You should not let it disturb you that the second number, 10,106, which was arrived at through the assumption that AC and AE are equal, comes rather close to 11,332. For since I have related these observations to the sun's apparent positions, constructing the eccentricity from the center of the sun's body, AC and AE will therefore not be equal. Consequently, this eccentricity will be much less than 10,106, and would in fact be 8377 if the sun's distances were correctly given as 99,302 and 102,689, which the requirements of this demonstration leads us to take as 100,000 and 100,000. But since these Tychonic distances will be corrected below, and will be brought closer to the average radius, the eccentricity being sought here certainly lies between these two limits (8377 and 10,106). In fact, it is approximately half the total eccentricity found previously (18,564); that is, 9282.","453":" To go through the same demonstration also in the Ptolemaic hypothesis of the second inequality, proceed as in the previous chapter. Draw AI, BI, AF, DF parallel to CB, CA, ED, EA in the larger diagram, and fix the earth at A, and the center of the epicycle (or, more correctly, the point about which the epicycle is rotated, distant from the center of the epicycle by the whole of the sun's eccentricity) at D and B; the sun at H and G; in such a manner that AH is equal and parallel to EA and AG to CA, so that the angles of equated anomaly of commutation are HAD and GAB; let Mars be at I instead of B or L and at F instead of D or M; and the lines AG, AH parallel to BI and DF (the lines of the planet's position on the epicycle) will be the sun's position. The rest is obvious.","454":"For the Tychonic form and hypothesis of the second inequality, let A remain the earth, H and G the sun; and let HF, GI be drawn parallel and equal to AD, AB, so that Mars is again at F and I. The lines of vision, AF and AI, will therefore also be the same as in Ptolemy, and will be parallel to ED, CB, the lines of vision in the larger diagram. Therefore, they point in the same direction from the sun, and the sum of the lines HF, GI will equal the prior BD. Because the lines are parallel, the proof will be the same as the one at the beginning of the chapter.","455":"Now, as in the previous chapter, I shall accommodate the proof that the eccentric's eccentricity has been falsely determined to the Brahean revision as well, which depends upon the sun's mean motion. This is done so that no one will think the discrepancy is a result of my having wrongheadedly transposed the observations from the sun's mean motion to its apparent motion.","456":"On 1600 March 5, Mars's mean longitude was, in Tycho's reckoning, 4s 29\u00b0 11' 3\", apogee 23\u00b0 41' Leo. Therefore, the simple anomaly was 5\u00b0 30', which, in his reckoning, requires an equation of 1\u00b0 7' 11\", to be subtracted, so as to give Mars an eccentric position of 4s 28\u00b0 3' 52\", with the sun's mean position 23\u00b0 44' 31\" Pisces. In the above diagram let A be the point of the sun's mean motion, distant from the center of the sun by the whole of the sun's eccentricity. Therefore, the angle CBA is 28\u00b0 51' 22\", and BCA is 125\u00b0 28' 0\". Also, this demonstration requires that AE and AC be assumed equal, namely, 100,000, retaining those suppositions made by Tycho and the ancients. These will be given an airing in Part III below, where it will be shown that the distance of the earth from the point of the sun's mean position is somewhat less; that is, that the Ptolemaic epicycle or the Copernican-Tychonic annual orb is not placed evenly about that point about which equal angles are traversed in equal times. But for now let us hold to the fundamentals as given: and let CA be 100,000; therefore, AB will be 168,760.","457":"At perigee, on 1593 July 30, since (in Brahe's reckoning) Mars's [mean] longitude was 10s 26\u00b0 12' 43\" and the apogee was at 23\u00b0 34' Leo, the simple anomaly was 182\u00b0 38' 43\", which requires an equation of 35' 52\", to be added. Therefore, Mars's eccentric position was 10s 26\u00b0 48' 35\", and the sun's mean position 18\u00b0 24' 31\" [Leo]. Therefore, in the diagram, EDA will be 20\u00b0 50' 55\", and AED will be 158\u00b0 45' 0\". Let EA again be 100,000, although below (as has just been remarked) it will turn out to be somewhat greater. AD is therefore 137,300. This you shall diminish by 15 so as to fit right at perigee: let it be 137,285. The other you shall increase by about 100, so as to fit exactly at the apogee, so it will be 168,860. But we shall increase both (as before) because of the inclination of the planes, 82 being added at apogee and 72 at perigee. The final values will be: \nthe eccentricity from the point of the sun's mean motion, or (in the Ptolemaic form) on the line of apsides drawn through the center of the epicycle.","458":"Now where BK is 100,000, KA is 10,312. However, the Tychonic revision based upon acronychal observations and presented in Chapter 8 required BK to be greater, namely, 12,352.","459":"It has therefore been shown that the Tychonic revision is also subject to the same incongruity, that the eccentric has one eccentricity that results from acronychal observations, and a different one that results from the others.","460":"And meanwhile, the observations in this Tychonic rendition lead the way to bisection. For Tycho's figure for the whole eccentricity of the equalizing point is 20,160, half of which is 10,080, or in the form of the Ptolemaic equant, 9882. And here we have found it to be 10,312, which closely approximates half the Tychonic value. Indeed, it will approach much nearer, decreasing to a value less than the Tychonic (that is, to a very exact 9282), when AC in the greater diagram (BI in the smaller, on the left) is diminished, and along with it AB or GI (the distance at apogee); and, in turn, when AE of the right diagram (and its equal and equivalent DF of the left) is increased, and along with it AD or HF (the distance at perigee). For when the lesser part is increased, and the greater diminished, the difference between the two is decreased.","461":"The blame for this discrepancy among the different ways of finding the eccentricity (I am repeating this over and over so that it will be remembered) falls entirely upon the faulty assumptions deliberately entertained by me, in common with Tycho and all who have ever devised hypotheses. For the necessary consequence of this enquiry is that there is no single fixed point on the planet's eccentric about which the planet always sweeps out equal angles in equal times. We would instead have to make such a point reciprocate up and down along the line of apsides\u2014if, indeed, we could keep the other assumption of a circular orbit. And how such a reciprocation could be reconciled with natural principles, I do not see.\nBut in fact the other assumption will be demolished, in Chapter 44 below; that is, the orbit of the star is not a perfect circle, but an oval, and its greatest diameter is the line of apsides, while its least is that passing through the center at the middle elongations. No wonder, then, that the other observations at points not at opposition to the sun do not accord with the hypothesis constructed in Chapter 16, since we have made two false assumptions in it.","462":"Chapter 21","463":"Why, and to what extent, may a false hypothesis yield the truth?","464":"I particularly abhor that axiom of the logicians, that the true follows from the false, because people have used it to go for Copernicus's throat, while I am his disciple in the more general hypotheses concerning the system of the world. I therefore considered it particularly worth while now to show the reader how it does happen here that the true follows from the false.","465":"First, you have already seen that what has followed is not completely true. For the path of the planet through the single plane of the ecliptic was considered in two ways: first, in respect to its longitude beneath certain degrees and minutes of the circle of the zodiac, and second, in respect to its altitude or distance from the center of the world about which it moves, which it shows to be different by means of other zodiacal positions. Therefore, our false supposition, although it does put the planet in the right longitudinal position at the right time, does not give it the right altitude. So what follows from this false hypothesis is not completely true.","466":"Further, even concerning the longitude alone, the fact that the result appears identical to the senses does not prove that the as yet unknown true hypothesis and the false one assumed by us have an identical result. For there can be a very small discrepancy which the senses do not perceive.","467":"There are, however, occasions upon which a false hypothesis can simulate truth, within the limits of observational precision, with respect to the longitude. These I shall now demonstrate.","468":" Through the center of the world A let the straight line MP be set out, falling upon opposite parts of the zodiac (29\u00b0","469":"Leo and Aquarius, say). And let it so be that according to some true hypothesis the planet spends half its time between lines AM and AP on the left and the other half on the right, so that after successive halves of its periodic time it is always alternately on the lines AM and AP. And let it be assumed that this particular effect of the true hypothesis is expressible by some other hypothesis that has been discovered. And so let a circle of any kind or some other curvy line be described about a center taken on the line MP, with the sole provision that it go around the center of the world A and that it be cut into two equal parts by the line MP. What is proposed will happen if the planet traverse the circle with a uniform motion (one which is regular about any one particular point on the line MP, whether fixed or movable); as, for instance, if the circle OP were described about center A and moved uniformly about it. So all these circles and other figures have something in common through which what was proposed occurs, namely, that they move around the center of the world, and move regularly around some point on the line MP. Now the figure, whether this or that circle, whether one or another point of uniform motion, out of all those comprehended under the same genus, can be false. But we have brought about what was proposed, not through this false specific model, but through that which, within this false one, was comprehended within a general truth.","470":" Now let us continue, and let it happen that after successive quarter periods the planet lies on the lines AM, AK, AP, AL, the angles MAK, MAL, being less than right angles. Here, then, the former circle OP will be in error at the sides. For since the motion was supposed regular about A, a straight line drawn through A perpendicular to MP (namely, VX) will make the angles MAV, MAX measures of quarter periods. And accordingly, this hypothesis would put the planet on the lines AV, AX, when it should have been on AK, AL.","471":"Now experience testifies that the planets' motion closely emulates circularity (although it may perchance not exactly attain it), and it is the nature of motions of this kind to undergo gradual intensification and remission, admitting nothing sudden. Therefore, the error of this hypothesis of the circle OP will begin little by little from the line AM, will grow continually greater, becoming a maximum at AK, and will again gradually decrease and vanish at AP. Therefore, the uniform and concentric hypothesis OP will never be more in error than it is at AK, AL, by the angles KAV, LAX, which, for Mars, are 10\u00bd\u00b0.","472":"So let there now be another hypothesis which, in addition, also shows us the lines AK, AL. But again, there can be several hypotheses that do this. For we might connect the points where AK and AL intersect the circle OP, and where this straight line intersects the straight line MP we may place the center of uniform motion of the circle OP, so that the motion of the circle OP becomes nonuniform. We would then also have [the planet on] the lines AK, AL [at the appropriate times]. But since we have a certain inclination towards choosing the simplest and most regular, we shall therefore seek out that circle that moves uniformly about its own center while effecting for us what is proposed. Therefore, beginning from A, mark off equal lengths AK, AL upon the lines AK, AL; let the points K and L be joined by a straight line intersecting MP at C; and about C with radius CK let the eccentric circle MN be described, whose motion shall be regular about its center. This hypothesis will represent the planet in the correct position, on the four lines AM, AN, AK, AL. But it is not this hypothesis alone, but many others as well, that could have this effect. For they have this feature that is general, and is indeed perfectly true, that the point of uniform motion is on the line that connects the positions of the planet falling upon the lines AK, AL, and at that point upon it where the line intersects MP. Now it follows from the premises that this hypothesis has absorbed the entire maximum error of the former hypothesis OP, namely, KAV, LAX, at about the quarters of the period, nor does it commit a new error (since at AM, AP it is equivalent to the former). Therefore, if this hypothesis is still in error, that error will be much smaller than KAV. And since it has done its job at CM, CN, CK, CL, the error (if any) will retreat to the four regions intermediate to those just mentioned, and will occur at the eighths of the period, since the time is measured about C. Therefore, angles MCK, KCN being bisected, let two new lines be drawn through C intersecting the circumference at Q, T, R, S. The maximum error, if any, will be about these points. But the hypothesis will also place the planet on the lines AQ, AR, AS, AT, at the eighths of the period. Now suppose (as is true for Mars) that after the eighths of the periodic time the planet should not appear on the lines AQ, AR, AS, AT, but should instead be, for the former two, on the lines AF, AE, higher up, and for latter two on the lines AG, AD, lower down. Therefore, if the former error KAV was 10\u00bd\u00b0, the present error QAF will hardly amount to a few minutes. For Mars, the magnitude of QAF or RAE is observed to be about 9\u2019, while SAG or TAD is about 28'.","473":"Now as a third step, let this hypothesis too be corrected. As this can happen in a variety of ways (specifically, by a reciprocation of the point C along the line CA), we are not prevented by any scruple from keeping the point of uniform motion C fixed at distance CA, on account of the angle KAV, and also keeping the planet's path circular. These three, taken by choice and not forced by demonstration, will compel us to move the center of the eccentric downwards to B from the point of uniform motion C. Hence, HI is substituted for MN, and the body of the planet departs from the points Q, R, S, T, nevertheless remaining on the lines CQ, CR, CS, CT (because the measure of time stays at C), and arrives at the points marked F, E, G, D. And QF, ER, SG, TD would be such as to make QAF, EAR 9\u2019 and SAG, TAD 28'. With this done, that error at the eighths of the period will also be absorbed, and the hypothesis will exhibit the longitude perfectly accurately at eight places. Thus if again some error remains, it will be at the sixteenths of the period, the points in between. Also, since this third eccentric HI is equivalent to the first at positions AM, AP, as well as to the second at the additional positions AK, AL, it introduces no new error. And because the error of the second was greatest at the eighths of the period, and this is now absorbed, at the sixteenths there will therefore be a much smaller error remaining from the old error. Let us estimate it proportionally: just as the error of the first eccentric was 10\u00bd \u00b0 while that of the second was 9' or 28', that is, one seventieth or one twenty-fifth of the former, let us now make the errors of the second that many times the errors of the third. Plainly, already at the sixteenths of the period, we will have driven the business down to within the limits of observational accuracy.","474":"It is at least now clear to what extent and in what manner the truth may follow from false principles: whatever is false in these hypotheses is specific to them and can be absent, while whatever endows truth with necessity is in general aspect wholly true and nothing else. ","475":"Further, as these false principles are fitted only to fixed positions throughout the whole circle, so the truth does not invariably follow outside those very positions, except to the extent that happens in this procedure, that the difference can no longer be appraised by the acuteness of the senses.","476":"Also, this same dullness of the senses hides the following additional small error which remains at the eighths of the period. That there is such a remainder I prove thus:","477":" Once again, let a perfect eccentric be described about B so that BD, BE, BF, BG are equal, and let us have made BC such that the angle QAF is of the required magnitude. Now it is not likewise left to our discretion how great we want angle SAG to be, since it will be completely determined. From A draw a perpendicular to QT, and let this be AZ. Now, as above, let AC be 18,564 where CQ is 100,000. And because ACZ is 45\u00b0, AZ or ZC will be 13,127 (both in the same units). Therefore, ZQ is 113,127, and AQZ 6\u00b0 37' 5\", and QAZ 83\u00b0 22' 55\", whose tangent is 864092. Now let FAZ be taken as 9' less: its tangent FZ will be 844900. But where AZ is 13,127, ZF will be 110,910. Therefore, QF will be 2217. Now QF is larger than TD, which I prove thus: QT is the diameter of the circle, and is therefore equal to the two semidiameters FB, BD taken together. But BF, BD taken together are greater than FD. Therefore, QT is also greater than FD. Let the common part FT be subtracted. The remainder QF is then greater than TD. And yet, over and above what is required, we shall allow them to be equal. Let CZ, 13,127, be subtracted from CT, so as to leave ZT, 86,873. Now from AZ, AT, ATZ is known, and it is 8\u00b0 35' 33\". So ZAT is 81\u00b0 24' 27\". And because ZT is 86,873, I shall add to it a magnitude equal to QF, as if it were TD, namely, 2217. This will makes ZD 89,090. But where AZ is 100,000, ZD will become the tangent of the angle ZAD, 686,291. Thus this angle is 81\u00b0 42' 35\". But ZAT was 81\u00b0 24' 27\". Therefore TAD, or SAG, is less than the difference, 18' 8\", since TD is less than 2217.","478":"This then is the required angle TAD, which ought to have been 27\u2157\u2019. And so if you make QAF 12' instead of 9', TAD becomes 24'. And in both places the planet is made to be 3 minutes higher than it should be. The equation therefore will be seen to be too large, and thus the eccentricity [of the equant] is too large. It will straightaway be diminished, then, so that the planet is about 1\u00bd\u2019 lower at the lines AK, AL, and the same amount (that is, 1\u00bd\u2019 ) higher at D, E, F, and G. ","479":"This mutual tempering of various influences causes one error to compensate for another, brings the calculation within the limits of observational precision, and makes it impossible to perceive the falsity of this specific hypothesis. And so this sly courtesan cannot gloat over the dragging of truth (a most chaste maiden) into her bordello. Any honest woman following the lead of this prostitute would stay closely in her tracks owing to the narrowness of the streets and the press of the crowd, and the stupid, bleary-eyed professors of the subtleties of logic, who cannot tell a candid appearance from a shameless one, judge her to be the prostitute's maidservant.","480":"This is without doubt the reason for the remaining discrepancies of one or two minutes in Chapter 18, in Cancer, Leo, Scorpio, and several other places. But the error is not easy to see, since the observations used do not fall at the apsides and at the quarters and eighths of the period.","481":"Conclusion of Part II","482":"Up to the present, the hypothesis accounting for the first inequality (in which Brahe and Copernicus are in agreement, both differing somewhat in form from Ptolemy) has been presented using the sun's mean motion, which all three authors had substituted for the sun's apparent motion. Thereafter, it was shown that whether we follow the sun's apparent motion and the hypothesis found in Chapter 16, or the sun's mean motion and the hypothesis proposed in Chapter 8 according to Brahe's revision, in both instances there result false distances of the planet from the center, whether of the sun (for Copernicus and Brahe) or of the world (for Ptolemy). Consequently, what we had previously constructed from the Brahean observations we have later in turn destroyed using other observations of his. This was the necessary consequence of our having followed (in imitation of previous theorists) several things that were plausible but really false.","483":"And this much of the work is dedicated to this imitation of previous theorists, with which I am concluding this second part of the Commentaries.","484":"Chapter 22 ","485":"The epicycle, or annual orb, is not equally situated about the point of equality of motion.","486":"This, then, is the way our predecessors measured the first inequality. With this calculation established, which would represent the planet's eccentric position at any desired moment, they next turned to exploring the second inequality (which depends upon the sun), comparing the observed or apparent position with that which the eccentric and the planet's first inequality alone would assign.","487":"When I was on this same path and was confronted with this equivocal fork in the road (in Chapters 19 and 20 above), and the observations (most faithful guides) were seen to be at war with observations, I had to give thought to altering completely the way the path was set out, using the method which follows.","488":"First, in this third part I shall approach the second inequality. Here I shall use unquestionable observations to demonstrate, with either a confirmation or a refutation, all that I have hitherto supposed as principles but had doubts about. Once this is found it will be like a key: the rest will be opened up. Afterwards, in Part IV, I shall proceed to the first inequality.","489":"In Chapter 22 of the Mysterium cosmographicum, when I was giving the physical cause of the Ptolemaic equant or of the Copernican-Tychonic second epicycle, I raised an objection against myself at the end of the chapter: if the cause I proposed were true, it ought to hold universally for all planets. But since the earth, one of the celestial bodies (for Copernicus), or the sun (for the rest) had not hitherto required this equant, I decided to leave that speculation open, until the matter were clearer to astronomers. I nevertheless entertained a suspicion that this theory might perchance also have its equant. After I gained the recognition of Tycho, this suspicion was confirmed in me. For in a letter to me in Styria in 1598 Brahe said the following:","490":"The annual orb according to Copernicus, or the epicycle according to Ptolemy, does not appear always of the same size, in comparisons made to the eccentric, but introduces a perceptible alteration in all three superior planets, so much so that for Mars the angle of difference reaches one degree 45'. ","491":"He also touched upon this point at the same time in the appendix to his Mechanica, an account of his studies. Also, his words in volume I p. 209 of his letters4 are not much different, where he states the opinion that as an effect of the solar eccentricity a certain amount of nonuniformity is also mixed in with the eccentric equations and the acronychal positions. This is, in fact, refuted in Part I: it is not reflected in the acronychal positions, or at least very little. But it appears that this needs to be understood through a certain correction regarding Mars at 90\u00b0 from the sun.","492":" Now when I heard that the annual orb grows and shrinks, an inspiration said to me that this illusion arises thus: Copernicus's annual orb, or Ptolemy's epicycle, is not everywhere distant from that center about which by supposition it is sweeping out equal angles in equal times. For what physical cause could make the circuit of the center of the planetary system (Tychonic) or of the circuit of the earth (for Copernicus) or of the epicycle bearing the star (for Ptolemy) grow and shrink? What, I ask, is this novelty unprecedented in astronomy, this unlikely absurdity? Wouldn't it seem more worthy of belief that the sun (for Copernicus) or the center of the planetary system (Tychonic) or the body of the planet (for Ptolemy) would in certain places be farther from, and in others nearer to, the selected point of uniform motion (at rest for Copernicus and Tycho, and moving around on the circumference of the eccentric for Ptolemy)\u2014and this especially on the line of apsides? And for this, that suspicion of mine arising from my Mysterium cosmographicum\u2014that an equant might be introduced into the theory of the sun (or, as I call it, the theory of the Ptolemaic epicycle)\u2014seemed to provide a convenient occasion.","493":" Let us suppose that the second inequality starts from the line of the sun's mean motion, as the practitioners have hitherto been pleased to hold (lest anyone hold suspect here my innovation of using the sun's apparent motion), and in the present diagram let the planet's eccentricity for Copernicus originate not from the sun's center A, but from the point C about which the earth's motion is supposed regular. But let that point C not be the center of the earth's orb DE but only of uniformity of motion, and let its distance from the sun A be greater than that of B, the center of the earth's orb ED. I say that, these things being granted, such observations will be produced from which one might suspect the annual orb of growing and shrinking. Let a line CF be drawn from C perpendicular to DE, and let the star Mars be at F twice: once when the earth is at D and again when it is at E; and let F be joined with the points D and E. Now because C is the point of uniformity of the earth's motion on DE, FCD and FCE will be the anomaly of commutation, and (as we suppose) equal on both sides. Now if CD and CE were equal (as has hitherto been thought) then the angles DFC and EFC, parallaxes of the orb, would be equal on both sides, for both anomalies of commutation. But because CE is greater than CD, the angle CFE will also appear greater than the angle CFD. Therefore, anyone not noticing that this growth occurs only at or about E, and that the contrary diminution occurs only at the contrary position D, will think that the entire annual orb sometimes gets larger, with radius CE, and sometimes smaller, with radius CD. This is because such a person presupposes, along with astronomy as it has hitherto been practiced, that the point of equal motion C is at the same time also the center of the circle DE.","494":"In the Ptolemaic form, let the earth be at C, and the lines of the sun's mean motion be CK, CL, in place of DC and EC in the preceding Copernican arrangement. And let the center about which the epicyclic motion is regular be at F, and IH be equal and parallel to ED so that, CI being drawn, it is parallel to DF and CH to EF. For when the earth (or the observer) is transferred to the center of the world C, as Ptolemy has it, Mars at F is likewise transferred to H. Similarly, owing to the translation of D to C, F is transferred to I. Now Ptolemy, thinking that the point F, about which the motion of the epicycle IH is uniform, is also the center of the epicycle IH, supposed FI and FH to be exactly equal. Consequently, for both of the equated anomalies HFC as well as IFC (that is, at both 90\u00b0 and 270\u00b0, in this diagram) he posited one and the same equation of the epicycle, namely, the equal angles HCF and ICF. So if observation affirms that HCF is greater than ICF, then the center of the epicycle will not be at the point of uniform motion F, but at G, in the direction of H. Further, on the supposition that F be nevertheless considered the center of the epicycle, the epicycle will appear distinctly enlarged at anomaly 90\u00b0 at H, and diminished at 270\u00b0 at I, while in both instances Mars, in its eccentric position (that is, on the line CF), is in the same position with respect to the fixed stars.","495":"In the Tychonic form, let C remain the earth, DE the sun's circle with center B, but let the center of uniform motion be A. And let the lines along which the planet is seen (namely, CI and CH) be the same as in Ptolemy. Accordingly, let HL, IK descend from H and I parallel to FC. In order that K and L be the center of the planetary system, let the center of its circuit be M, which is in the direction of the sun's perigee, so that the point M, the center of the circuit KL (in which is found the point from which Mars's eccentricity originates) descends as far below C as the point B, the true center of the sun's circuit (contrary to common opinion), descends below A, the putative center of the same circuit of the sun. And let AC and BM be equal. The line of equated motion on the eccentric (that is, KI, LH) will be parallel to itself after an integral number of returns of the planet. Therefore Tycho, thinking that the earth C is in the middle of the circuit KL bearing the planets' eccentrics, will make angles CIK, CHL equal when the angles of commutation CLH, CKI are equal. But if these are perceived to be unequal, CHL being the greater, CL will be longer than CK: and the orb KL, the deferent of the center of the system, will appear to grow at L and to shrink at K, because M, the center of the orb which is the planetary system's deferent, is not believed to be elsewhere than at the earth C, about whose center the motion of that orb is uniform.","496":"Now what greatly contributes to obscuring the true cause of this difference, namely, to freeing the sun's eccentricity from suspicion, is the fact that thereby* the distance CK of the center of the system from the earth becomes short just where the distance CE of the sun from the earth becomes long; and conversely, that the former, CL, becomes long where the latter, CD, becomes short.","497":"The reason why the apsides have thus been reversed is this. For Copernicus, the earth traverses the regions opposite the Tychonic sun and the Ptolemaic epicycle, and also DC, CE, the distances of the earth from the sun, of the sun from the earth, and of Mars H or I from the epicycle's center of uniform motion F, subtend angles of the same magnitude in all three forms of hypothesis. Therefore, it also happens that the Copernican distances of the sun and the earth will be transferred to the opposite sides by Brahe and Ptolemy; that is, CE to CL or FH, and CD to CK or FI.","498":" Next, in order either to confirm or undermine this speculation by observations, this is the road upon which I set out. Since the sun's apogee is at 5\u00bd\u00b0 Cancer, I enquired whether there might exist an observation in which Mars, reckoned by the first inequality, would be twice at 5\u00bd\u00b0 Libra or Aries, while the sun would be at 5\u00bd\u00b0 Cancer at one time, and then at 5\u00bd\u00b0 Capricorn. As it turns out, this is not possible within such a short space of time (20 or 30 years). For the periodic motions of Mars and the sun are incommensurable, nor do they ever coincide at 90\u00b0 from one another, or at opposition, after a certain number of complete periods, or quarter or half periods, of either. I therefore had to choose the next best thing, which was to find many days throughout those 20 years on which the planet was observed, and in which the equated anomaly of commutation was 90\u00b0 or 270\u00b0 or nearly that much, with Mars at 6 Aries or Libra or thereabouts. Afterwards, it was necessary to look up all those dates in the catalog of observations of Mars, so I could see whether it had been observed at those moments. Had the indefatigable Tycho Brahe not observed Mars very frequently, the selection would have been so exclusive that I would not have been able to accomplish what I wished. Now since Tycho put the apogee of Mars at 23\u00bd\u00b0 Leo, while the required position of Mars, corrected by the eccentric equation, was 5\u00bd\u00b0 Libra, an equated anomaly of 42\u00b0 was required. And from his table, to an equated anomaly of 42\u00b0 there corresponds an equation of 8\u00b0 15\u2157'; therefore, a mean anomaly on the eccentric of 50\u00b0 16' was required, and through this I was shown twelve points in time in the twenty years between 1579 and 1600.","499":"Thus, what had to be skillfully tracked down is whether for any of these times there was an equated anomaly of commutation that was at one time 90\u00b0 and again 270\u00b0, or if the former were greater or less, the latter would be correspondingly less or greater.","500":"One revolution of Mars has 687 days, and two of the sun have 730\u00bd. The difference is 43\u00bd days, to which corresponds 42\u00b0 54' 23\" of the sun's mean motion. This, therefore, is how much the anomaly of commutation changes at the end of any revolution of Mars. Therefore, when in any two year period one seeks two anomalies of commutation that are equal, with Mars at the same eccentric position, each angle of commutation should be 21\u00b0 27'. Over four years 42\u00b0 54' is required; over six years, 64\u00b0 22'; over eight years, 85\u00b0 49'. And we were supposing 90\u00b0, if it were possible. Therefore, we had to look for our two observations eight years apart. However, a team of two such observations is not to be found in the catalog of observations we had.","501":"I next turned to the interval of six years, and found at length that from 1585 May 18 and 1591 January 22 suitable observations exist. For they corresponded to 1585 May 30 at 5h and 1591 January 20 at 0h. For both, the mean longitude of Mars was 6s 22\u00b0 43'. The Tychonic equation was 9\u00b0 14' 52\", to be subtracted. Therefore, Mars's eccentric position was 13\u00b0 28' 16\" Libra. The equated commutation for 1585 was 8s 4\u00b0 23' 30\", by which it was shown, Ptolemaic style, that the planet was 64\u00b0 23' 30\" beyond the perigee of the epicycle. Similarly, the equated commutation for 1591 was 3s 25\u00b0 36' 30\", by which it was shown that the planet was 64\u00b0 23' 30\" before the perigee of the epicycle. Therefore, both the angles of commutation, FCD and FCE (or CFI, CFH) in the diagram, are equal. However, in 1585 the sun was in 18\u00b0 Gemini, 18\u00b0 before apogee, and in 1591 in 9\u00b0 Aquarius, 33\u00b0 beyond perigee, and this inequality could not be avoided.","502":"Now to the observations: on 1585 May 18 at 10\u00bdh at night Mars was observed at 0\u00b0 50' 45\" Virgo with latitude 1\u00b0 19' 30\" north. Magini puts it at 1\u00b0 5' Virgo, 14' or 15' too much. Therefore, when on the 30th at 5h in the evening he puts it at 6\u00b0 48' Virgo, we shall again subtract the discrepancy of eleven days previously. So he will be left with 6\u00b0 34' Virgo. Here we will assume that some very few minutes are in error because the deduction over 12 days is too great, and the diurnal motion is not exactly the same as that obtained here from Magini. For consider that on April 18 preceding at 10h Mars was found at 17\u00b0 37\u00bd\u2019 Leo, while Magini puts it at 18\u00b0 0' Leo. The difference is 22\u00bd\u2019. Over the 33 days elapsed to May 18 this difference was diminished to the measure of 14\u00bc\u2019. So if we extrapolate, since over 33 days eight minutes vanished, in the same ratio, over the 12 following days 3 minutes will vanish. Therefore, on May 30 the difference will be 11\u00bc\u2019. So, more accurately, Mars is at 6\u00b0 37' Virgo.","503":"Similarly, on 1591 January 22 at 7h in the morning, Mars was 34\u00b0 32' 45\" from Spica with declination 17\u00b0 25' south, at an altitude of 16\u00b0. Therefore, after correction for horizontal variations, the declination was 17\u00b0 30'. Hence, the right ascension was 230\u00b0 23' 12\", longitude 22\u00b0 33' Scorpio, latitude 1\u00b0 0' 30\" north. Now this time differs from ours by 1 day 19 hours, and the diurnal motion, from Magini, is 33'. Therefore, 59' are required for the intervening time. Therefore, the remainder is the position of Mars on January 20 at 0h (which, as we said, corresponds to the other time): 21\u00b0 34' Scorpio. ","504":" Witness the great difference in the equations on the annual orb, despite the promise that the two anomalies of commutation would be equal. The Copernican hypothesis shows us the cause. The earth at D and E was considered to be equally distant from the point of uniform motion C, but it was found to be at unequal distances, in such a way that the center of its circuit is at B towards the sun A. And by equivalence, in the Ptolemaic form the epicycle HI is not equally situated about point F, whose eccentric path the acronychal observations have been describing to us, and about which the epicycle's motion is regular. And the center of the epicycle G draws towards E, on the same side as the solar perigee. Similarly, in the Tychonic form the deferent KL of the planetary systems does not go around the earth C at a constant distance, although its motion is regular about this point, but the center M of its circuit draws towards the region of the sun's perigee.","505":"Chapter 23","506":"From the knowledge of two distances of the sun from the earth and of the zodiacal positions and the sun\u2019s apogee, to find the eccentricity of the sun's path (or the earth\u2019s, for Copernicus).","507":"From this it is also not difficult for us to measure the line BC tentatively. Let FC be 100,000. And because DFC is 36\u00b0 51' and FCD is 64\u00b0 23' 30\", therefore the remaining angle FDC is 78\u00b0 45' 27\". And as the sine of this angle is to FC (100,000), so is the sine of DFC to DC, 61,148.","508":"In the same way, because EFC is 38\u00b0 5\u00bd\u2019 minus a little, and FCE is 64\u00b0 23' 30\", FEC will be 77\u00b0 31' 0\" plus a little. Therefore, EC is 63,186 minus a little.","509":" Let the earth's orb NED be set out, and on it let CBN be the line of apsides, and N perihelion, R aphelion, B the center, C the point of equality of motion, E and D the positions of the two observations, and let these be joined to C and to B. Now EC and CD are known in the same units, and angle ECD is known, viz. 128\u00b0 47' 19\". Let EC be extended, and let DO be dropped perpendicular to it from D; and also to DE let two perpendiculars CP, BQ be dropped from C and B.","510":"DCO is therefore 51\u00b0 12' 41\", and CDO is 38\u00b0 47' 19\". Therefore, where DC is 61,148,","511":"DO will be 47,660 and CO 38,305. And this placed on the end of CE makes EO 101,491. But from the two given sides DO, OE about a right angle, the magnitude of DEO is obtained: 25\u00b0 9' 20\". Therefore DE is 112,125. The half of this, 56,062\u00bd, is the magnitude of DQ, since DB, BE are equal. And because DEC was 25\u00b0 9' 20\", EDC or PDC will be 26\u00b0 3' 21\". Therefore, where DC is 61,148, CP will come out to be 26,858 and PD is 54,932. Subtract this from QD, and the remainder, PQ, is 1130\u00bd. And now from the known inclination of the lines ED and NC the length CB is easily obtained. For since CR is the line of the aphelion, at 5\u00b0 30' Capricorn, while CD is at 17\u00b0 52' Sagittarius (because the sun is at 17\u00b0 52' Gemini), DCR will be 17\u00b0 38'. But EDC was 26\u00b0 3' 21\". Therefore, after subtraction, there remains the inclination of the lines in question: 8\u00b0 25' 21\". From P let PS be drawn parallel to CB. This will be equal to CB, and CP will be equal to BS. So in right triangle PQS, as the whole sine is to the tangent and secant of the angle QPS, 8\u00b0 25' 21\", so is the known magnitude of PQ to QS, 167, and SP, 1143, which is CB. And because PC and SB are equal, with magnitude 26,858, add QS, and QB will come out to be 27,025. So in the right triangle DQB, given the sides about the right angle, DB will also be given, 62,237. Therefore, the ratio of DB to BC (the radius to the eccentricity, which is being sought) is the same as 62,237 to 1143. And as 62,237 is to 100,000 so is 1143 to 1837. This, at last, is the eccentricity sought. It would have been less if we had accounted for the","512":"precession of the equinoxes, for then CE would have been less.","513":" And so from these two observations and the accepted true position of the sun's aphelion [sic] there is provided the distance of our equalizing point C or F (which we were considering as center) from the orbit's true center B or C or M, which is 1837 where the radius of its orbit is 100,000. In contrast, Tycho Brahe found the sun's eccentricity, that is, the distance of the equalizing point C from the center A of the solar body (in Copernicus) or the distance of the equalizing point A of the solar motion from the cen","514":"ter of the earth C (in the Tychonic-Ptolemaic supposition) to be 3584, whose half, 1792, is only slightly different from 1837. It is therefore fitting that the halving of the eccentricity hold in the theory of the sun, which halving previously held for the eccentric of Mars (in Chapter 19 and 20). For owing to the large corrections and the use of a controverted value for the diurnal motion, the observations which I have presented are not exact enough to allow anything to be concluded from 45 parts in one hundred thousand, not to mention the ignoring of precession in Mars's and the sun's eccentric motions for the time interval involved.","515":"What has been demonstrated here of the circuit of the earth can be demonstrated in exactly the same way concerning the Ptolemaic epicycle and the Tychonic deferent of the system, provided only that in the diagram the apsides be reversed.","516":"I have supposed here that the sun's apogee established by Tycho was in the right place, and that the orbit of the sun (or earth) which it bodily traverses is arranged in a circle. Although in Chapter 44 below the analogy with other planets will declare something different, the small breadth of the deflection nonetheless does not in the least vitiate our demonstration.","517":"Chapter 24","518":"A clearer proof that the epicycle or annual orb is eccentric with respect to the point of uniformity.","519":"Such, then, was the beginning of this enquiry, timid and encumbered with so many concerns that the anomaly of commutation be equal on both sides.","520":"Now that we have once made a hazard of this, we are buoyed by audacity and will begin to be more free on this battlefield. For I shall seek out three or more observed positions of Mars with the planet always at the same eccentric position, and from these find by trigonometry the distances of that number of points on the epicycle or annual orb from the point of uniform motion. And since a circle is defined by three points, I shall use triplets of such observations to investigate the position of the circle, its apsides (previously taken as a presupposition), and its eccentricity with respect to the point of uniform motion.","521":"If a fourth observation will be at hand, it will serve as a test.","522":" Let the first time be 1590 March 5 at 7h 10m in the evening, since then Mars had hardly any latitude, and thus no one looking at the demonstration could be troubled by any irrelevant suspicions about the intermingling of latitude. To this there correspond these moments, in which Mars returns to the same sidereal position: 1592 Jan. 21 at 6h 41m; 1593 Dec. 8 at 6h 12m; 1595 Oct. 26 at 5h 44m. For the first of these times, Mars's [mean] longitude, according to Tycho's revision, is 1s 4\u00b0 38' 50\", and for subsequent times 1' 36\" greater for each. For this is the motion of precession corresponding to the periodic time of one return of Mars. And since Tycho places the apogee at 23\u00bd\u00b0 Leo, its equation will be 11\u00b0 14' 55\", and consequently the equated anomaly in 1590 will be 1s 15\u00b0 53' 45\".","523":"Now at the same time, the commutation, or difference of the mean, motions of the sun and Mars, is reckoned to be 10s 18\u00b0 19' 56\", so the equated [commutation], or the difference between the sun's mean motion and Mars's equated eccentric motion, is 10s 7\u00b0 5' 1\".","524":"We shall present this first in the Copernican form, since it is simpler to perceive.\nLet \u03b1 be the point of uniform motion of the earth's circuit, and let this be considered to be the circle \u03b4\u03b3 described about \u03b1, and let the sun be on the side \u03b2, such that the line of the sun's apogee \u03b1\u03b2 lies in the direction of 5\u00bd\u00b0 Cancer, even though we are going to investigate this degree freely, as if unknown, in Chapter 25. And let the earth be on \u03b1\u03b8 in 1590, \u03b1\u03b7 in 1592, \u03b1\u03b5 in 1593, and \u03b1\u03b6 in 1595. And the angles \u03b8\u03b1\u03b7, \u03b7\u03b1\u03b5, \u03b5\u03b1\u03b6 are equal, since \u03b1 is the point of uniform motion and the periodic times of Mars are presupposed equal. And let the planet at these four times be at \u03ba, and its line of apsides be \u03b1\u03bb. Thus the angle \u03b8\u03b1\u03ba, according to the measure of the equated anomaly of commutation, is 127\u00b0 5' l\". ","525":" As for the observed position of Mars, at the same time on the preceding day, the fourth, it was at 24\u00b0 22' Aries. Its diurnal motion for the day would be 44'. Therefore, at our time it was seen at 25\u00b0 6' Aries, which is the position of the line \u03b8\u03ba. But \u03b1\u03ba is directed towards 15\u00b0 53' 45\" Taurus. Therefore, \u03b8\u03ba\u03b1 is 20\u00b0 47' 45\". So the remainder \u03b1\u03b8\u03ba to make up two right angles is 32\u00b0 7' 14\".","526":"Now as the sine of \u03b1\u03b8\u03ba is to \u03b1\u03ba, which we shall say is 100,000 units, so is [the sine of] \u03b8\u03ba\u03b1 to \u03b8\u03b1, which is what is sought. Therefore, \u03b8\u03b1 is 66,774.","527":"Now if the remaining lines \u03b7\u03b1, \u03b5\u03b1, \u03b6\u03b1, will turn out to be of the same length, my suspicion will be false, but if they are different, my triumph will be complete.","528":"Second, then, in 1592 at our moment the equated longitude was 1s 15\u00b0 55' 23\"; the equated commutation was 8s 24\u00b0 10' 34\" \u2014 that is, the angle \u03b7\u03b1\u03ba is 84\u00b0 10' 34\". It was observed on January 23 at 7h 15m at 11\u00b0 34\u00bd\u2019 Aries, with the correction for parallax. And its motion over two days was 1\u00b0 25'. Therefore on the 21st at 7h 15m it was seen at 10\u00b0 9\u00bd\u2019 Aries. Let the remaining parts of an hour deduct the half minute. Therefore, the angle \u03b7\u03ba\u03b1 is 35\u00b0 46' 23\", and \u03b1\u03b7\u03ba is 60\u00b0 3' 3\", and \u03b1\u03b7 is 67,467, now longer than \u03b1\u03b8. This is doubtless because the sun has descended towards perigee, and the earth has been moved from \u03b8 to \u03b7; thus, in this region the sun is found beyond \u03b2 at a nearer point.","529":"Third, in 1593 at our moment the equated longitude was 1s 15\u00b0 56' 56\", the equated commutation was 7s 11\u00b0 16' 16\", which makes \u03b5\u03b1\u03ba 41\u00b0 16' 16\".\nIt was observed December 10 at 7h 20m at 4\u00b0 45' Aries, parallax corrected. Its motion over two days was 1\u00b0 8'. Therefore, on December 8 at 7h 20m it was seen at 3\u00b0 37\u2019 Aries, while at our time of 6h 12m it was at 3\u00b0 35\u00bd\u2019 Aries. Hence, \u03b5\u03ba\u03b1 is 42\u00b0 21\u2019 30\" and \u03ba\u03b5\u03b1 is 96\u00b0 22\u2019 14\", and \u03b1\u03b5 is 67,794, again longer, for it is yet closer to the sun's perigee.","530":"Fourth, in 1595 at our moment the equated longitude was 1s 15\u00b0 58' 30\" and the [equated] commutation was 5s 28\u00b0 21' 55\", which makes the angle \u03ba\u03b1\u03b6 1\u00b0 38' 5\".","531":"It was observed on October 27 at 12h 20m at 18\u00b0 52\u2019 15\" Taurus, retrograde. Its diurnal motion was 23'. And so on the 26th at 12h 20m it was at 19\u00b0 15' 15\" Taurus, while at our time it was at 19\u00b0 21\u2019 35\" Taurus. Therefore, \u03b1\u03ba\u03b6 is 3\u00b0 23\u2019 5\", and the supplement of \u03b1\u03b6\u03ba is 5\u00b0 l\u2019 10\", and \u03b1\u03b6 is 67,478. But this last operation is untrustworthy, owing to the small angles of the triangle. For if an error of one or two minutes is made either in observing or in computing Mars's eccentric position using Tycho's hypothesis, the ratio of the angles is easily changed perceptibly. But for now I shall present all four lines for inspection. ","532":"So the longest is \u03b1\u03b5, which is also the closest to the sun's perigee; the shortest is \u03b1\u03b8, which is also the farthest from the sun's perigee; and \u03b1\u03b6 and \u03b1\u03b7 are about equal, because they are also nearly equally removed from perigee.","533":"Moreover, even if \u03b1\u03b6 is a little longer than \u03b1\u03b7 which is nearer the perigee, this should be attributed to the smallness of the angles at \u03b6, through which such a small error as this could easily be introduced. Therefore, the circle \u03b4\u03b3 which Copernicus described about the point x of uniformity of the earth's motion, is not the earth's path. There is instead some other circle \u03b8\u03b7\u03b5\u03b6 on which the earth is found, whose center lies in the same direction as the sun\u2014that is, at \u03b2. ","534":"In the Ptolemaic form, let the earth be at A, the sun's sphere be \u039e\u039f\u0399\u03a4, \u039a the putative center of the epicycle; that is, the center about which [is described] the epicycle \u0394\u0393, itself putative, which is equal to the [circle of the] theory of the sun. The total equivalence between the hypotheses of Copernicus and Brahe requires this to be done, although for the present demonstration it doesn't matter what","535":"ratio the sun's orb and the planet's epicycle have, provided that they have equal periods. And let \u0391\u039b be Mars's line of apsides. Let \u0391\u039a, \u0391\u039b be parallel to \u03b1\u03ba, \u03b1\u03bb in the preceding Copernican form. From the center of the earth \u0391, let the lines \u0391\u0398, \u0391\u0397, \u0391\u0395, \u0391\u0396 be drawn parallel to the previous lines \u03ba\u03b8, \u03ba\u03b7, \u03ba\u03b5, \u03ba\u03b6, and equal to them, so that Mars is at \u0398 in 1590, at \u0397 in 1592, at \u0395 in 1593, and at \u0396 in 1595; and at the same time the sun's mean position at those times is \u0391\u03a4, \u0391\u0399, \u0391\u039f, \u0391\u03a7, respectively, so that \u039a\u0398 and \u0391\u03a4 are parallel (as is known from the Ptolemaic hypothesis), and so","536":" on for the rest. With \u0398, \u0397, \u0395, and \u0396 connected with \u039a, it will be demonstrated as before, with exactly the same numbers, lines, and angles, that these lines are unequal, contrary to common opinion, and consequently that Mars does not traverse the circle \u0393\u0394, whose center is at the center of uniform motion \u039a, but the circle \u0396\u0397\u0395\u0398 instead, whose center lies in the direction of \u0392 from \u039a, very near to the line \u039a\u0392, which should be parallel to the line drawn from the earth \u0391 through the sun's perigee.","537":"Therefore, the epicycle's apogee lies in the direction of the sun's perigee. And because the epicycle, owing to the total equivalence just mentioned, is to be supposed equal to the sun's circuit, with \u0396\u039a parallel to \u03a7\u0391, \u0395\u039a to \u039f\u0391, \u0397\u039a to \u0399\u0391, and \u0398\u039a to \u03a4\u0391, it is also likely that \u03a7\u0391, \u039f\u0391, \u0399\u0391, and \u03a4\u0391 are unequal, and that the point of the sun's mean position (the center of the sun's epicycle, in the Brahean conception) does not stand at the same distance from the point of uniform motion throughout its circuit. This I remark only in passing: it has no effect upon the present demonstration, but serves as an extension to it. ","538":"In the Tychonic form, let A be the earth, and about it let the sun's concentric CD be described, and let this be considered to be the deferent of the system of the planets, since A is the point of uniform motion of the sun's concentric. Therefore the sun itself will be on another eccentric circle. Let its center be in the direction of B from A. Now let AL be the reference for Mars's line of apsides, so that the line of apsides, in the circling and translating of its eccentric, remains ever parallel to AL. And let the lines of the sun's mean motion at our four moments be AH, AT, AE, AS, and from A let the lines along which Mars is observed be extended in the direction of one or another degree of the zodiac in accordance with the description above. And since at all four ","539":"times Mars is supposed to be at the same place on the eccentric, its distances from the points of the sun's mean position will all be equal and parallel. Let them be GH, FT, IE, and KS, all equal, and let the angles LHG, LTF, LEI, LSK, all be equal to the previous angle \u039b\u0391\u039a or \u03bb\u03b1\u03ba, so that at our moments Mars might be at G, F, I, K. I would note in passing that these four points G, F, I, K, in actual fact will make an arc exactly equal in length and placement to the previous arc \u0398\u0397\u0395\u0396 in the Ptolemaic form. This is because there is no difference between the two other than that Ptolemy has an epicycle, equal to the [circle of the] theory of the sun, carried around on an eccentric, while Tycho has the eccentric carried around on the [circle of the] theory of the sun or on a circle equal to the Ptolemaic epicycle. ","540":"Once again, then, with the angles and numbers remaining the same, it will be demonstrated, contrary to common opinion, that the lines AH, AT, AE, AS are unequal. And so that point on the eccentric whence originates the eccentricity of Mars and all the planets (which is here considered to be on the line of the sun's mean motion, following earlier practitioners) does not go around on that circle DC about whose center A it makes equal angles in equal times, but on the circle HTES whose center lies in the direction opposite the center of the sun's eccentric B, as has thus far become roughly clear from the lines themselves. ","541":"Chapter 25","542":"From three distances of the sun from the center of the world, with known zodiacal positions, to find the apogee and eccentricity of the sun or earth.","543":"I shall now once again test the quantity of the eccentricity and the position of the apogee in a single circle adapted to all three forms. For it is easily seen that they are simply opposites: for example, in the Copernican form the longest line is towards Gemini, while in the other forms it is towards Sagittarius. This is because Copernicus's observer is looking towards the center, and the others are looking away from it. Thus Copernicus too looks across the center at the same parts of the zodiac as the others.","544":" Let the circle \u03b8\u03b7\u03b5\u03b6, with center \u03b2, be set out, in which from the given point \u03b1 there are the given lines \u03b1\u03b8, \u03b1\u03b7, \u03b1\u03b5, and \u03b1\u03b6, as before, with the angles about \u03b1 also given, for each of them is 42\u00b0 52' 47\". What is sought is both the magnitude \u03b1\u03b2 and the direction of that line with respect to the fixed stars or to the other lines. Let \u03b8, \u03b7, and \u03b5 be selected and joined with one another, since three points suffice for investigating this. ","545":"First, in the triangle \u03b8\u03b1\u03b7 the sides and the included angle are given, \u03b8\u03b7 is sought, and is shown by trigonometry to be 49, 1691 in the same units in which the sides \u03b1\u03b8 and \u03b1\u03b7 were expressed.","546":"Second, in triangle \u03b1\u03b5\u03b7 the angle \u03b1\u03b5\u03b7 is sought, and found to be 68\u00b0 12' 26\".","547":"Third, in the triangle \u03b8\u03b1\u03b5 the angle \u03b1\u03b5\u03b8 is sought, and is found to be 46\u00b0 39' 10\", which, subtracted from \u03b1\u03b5\u03b7 leaves 21\u00b0 33' 16\", which is the angle at the circumference \u03b8\u03b5\u03b7. Therefore, twice this amount, 43\u00b0 6' 32\", will be \u03b8\u03b2\u03b7, the angle at the center, because \u03b2 is by supposition the center of the circle. So in the isosceles triangle \u03b8\u03b2\u03b7 the angles are given, as well as the side \u03b8\u03b7 found previously. The size of \u03b8\u03b2, the radius of the circle, is sought, and found to be 66,923. And since \u03b2\u03b8\u03b7 is 68\u00b0 26' 44\", while before, when \u03b8\u03b7 was being found, \u03b1\u03b8\u03b7 was 69\u00b0 18' 46\", \u03b2\u03b8\u03b1 is therefore 0\u00b0 52' 2\". Next, in the triangle \u03b2\u03b8\u03b1, from the sides and the included angle, \u03b8\u03b1\u03b2 and \u03b1\u03b2 are sought. And the angle \u03b8\u03b1\u03b2 is found to be 97\u00b0 50' 30\", so that \u03b1\u03b2 is at 15\u00b0 8' 30\" Gemini, because \u03b1\u03b8 is at 22\u00b0 59' Virgo. Tycho, however, places the sun's apogee at 5\u00bd\u00b0 Cancer. So you see that this very free enquiry has brought us within 20\u00b0 of the correct Tychonic position. Also, \u03b1\u03b2 is found to be 1023, and if \u03b8\u03b2 be taken as 100,000, \u03b1\u03b2 will become 1530. But the whole solar eccentricity is 3592, and its half is 1796 or 1800. So here somewhat less than half of the solar eccentricity is claimed for the eccentricity of our circle. But you should bear in mind that observations near minimum values may be somewhat in error, and that use was made of a questionable mean longitude and equation from Tycho. This will become quite clear if you will carry out the same operation with the angle \u03b8\u03b7\u03b6, and then with \u03b7\u03b5\u03b6 and \u03b8\u03b5\u03b6. For each time, \u03b1\u03b2 has a somewhat different magnitude, and its position beneath the fixed stars will fall one side or the other of 5\u00bd\u00b0 Capricorn and Cancer.","548":" Consequently, we shall look into this more carefully below, where in many instances, through a fine demonstration, the eccentricity will be found to be half the solar eccentricity, and the apogee very near the Tychonic one.","549":"It has thus been demonstrated in the Copernican form that the center of the earth's circuit is halfway between the sun's body and the point of uniform motion of that circuit. That is, that the earth proceeds nonuniformly on its orbit, slowing when it recedes far from the sun and speeding up when it approaches. This is in conformity with physical principles and with the analogy with other planets.","550":"In the same way it has been demonstrated in the Ptolemaic form that the epicycle is eccentric with respect to the point about which its motion is uniform, and that its eccentricity is half of the sun's eccentricity as it is commonly determined, and in the opposite direction.","551":"Finally, in the Tychonic form it has been demonstrated that the point whence originate the planets' eccentricities does not move on the sun's concentric, but throughout its course is unequally distant from the earth, about which it moves regularly and uniformly; and that it is farther away near the sun's perigee, and nearer at apogee, again by half of the sun's eccentricity. And so, since this Ptolemaic epicycle and this Brahean deferent have so great an analogy with the theory of the sun, it is likely that they also have a greater analogy: that is, the sun's true eccentricity, too, will only be half of that computed from the maximum equation; or, what is the same thing, the sun will make use of an equant, whose eccentricity is twice the eccentric's eccentricity.","552":"I admit that this line of argument is a little weaker when applied to the Ptolemaic and Tychonic form, insofar as we are using the sun's mean motion, following the authorities. The argument consequently will become clearer when, moved by those reasons adduced in Chapter 6 above, I measure out the planet's motion by the sun's apparent motion.","553":"Chapter 26","554":"demonstration from the same observations that the epicycle is eccentric with respect to the point of attachment or axis, and that the annual orb (and so also the earth's path around the sun, or the sun's around the earth) is eccentric with respect to the body of the sun or earth, with an eccentricity just half that which Tycho Brahe found through equations of the sun's motion.","555":"We shall now go through the observations1 again, carefully: On 1590 March 4 at 7h 10m, Mars was found by careful observation and calculation to be at 24\u00b0 22' 56\" Aries with latitude 0\u00b0 3' 20\" S. At that time, 8\u00b0 Aries was setting, so Mars was rather low. Therefore, it was raised up towards the east by refraction, so it seems right that without refraction it would have been seen at 24\u00b0 20' Aries. Its parallax can only be very small, particularly at this distance, for Mars was near the sun and therefore had receded very far from the center of the earth.","556":"On 1592 January 23 at 7h 20m, using only one stellar distance from Mars, without the confirmation of another, Mars was found at 11\u00b0 32' 44\" Aries with latitude 0\u00b0 1' 36\" S. And so we shall make no changes for horizontal variations, although we suspect an uncertainty of one or two minutes.","557":"On 1593 December 7 at 8h 0m Mars was found at 3\u00b0 6' 50\" Aries with no danger of horizontal variation, with latitude 7\u2019 9\" S. However, the right ascension found using three stars showed a discrepancy of 4', and the value taken as true was the mean between the extremes.","558":"On 1595 October 25 at 8h 10m, the planet's distance from three fixed stars was observed, and by unanimous consensus the planet was found to be at 19\u00b0 39' 25\" [Taurus] with latitude 0\u00b0 12' 41\" S. ","559":"Now we shall reduce the three subsequent times to the first. Accordingly, to its sidereal position ","560":"The motion over three days and 35 minutes of time in 1592 was 2\u00b0 9' 4\", according to Magini. Therefore, at our time Mars was seen at 9\u00b0 23' 40\" [Aries].","561":"In 1593 the motion over 1 hour 45 min., from the diurnal motion of 33', is 2' 25\". And so at our time Mars's position comes out to be 3\u00b0 4' 27\" Aries. Likewise, in 1595 the motion over 2 hours 25 min., from a diurnal motion of 22' 11\", is 2' 14\". Therefore at our time the position of Mars comes out to be 19\u00b0 41' 39\" Taurus.","562":"FROM THIS THERE FOLLOWS THE TABLE OF POSITIONS ","563":"Now because we have proposed to enquire how far the earth is from the center of the sun, we will need first of all to use the hypothesis constructed above in Chapter 16, from oppositions to the sun's apparent position, to find the position of the line drawn from the center of the sun through the body of Mars to the zodiac. And on 1595 October 25 at 5h 45m that line is found at 14\u00b0 19' 52\" Taurus. Therefore, for the three other times it is set back each time by 1' 36\". Thus, in 1593 it was at 14\u00b0 18' 16\" Taurus; in 1592, at 14\u00b0 16' 40\" Taurus; and in 1590, at 14\u00b0 15' 4\" Taurus.","564":"Let the First Diagram be in Copernicus\u2019s Form","565":"And let \u03b1 be the center of the sun; \u03b2 the center of Mars's eccentric, drawn through \u03bf; \u03c7 the center of uniform motion for Mars's eccentric; \u03b3 the center of the earth's eccentric; \u03b4, \u03b5, \u03b6, \u03b7 four positions of the earth, opposite the sun's apparent positions; \u03b8 the position of Mars on its eccentric. Let all points be connected with one another. ","566":" 3","567":"We shall now test the magnitude of the eccentricity that may be deduced from these distances. If the sun's theory lacks an equant, the eccentricity of its circle will turn out to be about 3600. This is because we used the true or apparent positions of the sun, whose point of uniform motion has to be that far (namely, 3600) from the center of the world, as Brahe has proved from solar observations. But if, on the other hand, the eccentricity turns out to be less, and is about half the Brahean value, we have won, and vindicated our contention, that the point of uniform motion found by Brahe is not the center of the sun's eccentric.","568":"You can see at a glance (I would note in passing) that \u03b1\u03b6 is the shortest, it being near the sun's perigee; next, that \u03b1\u03b5 is longer, it being in Aquarius, 34 degrees from perigee; then \u03b1\u03b7, it being 54 degrees from perigee; and lastly, that \u03b1\u03b4 is longest, because it is 80 degrees from perigee. And since \u03b1\u03b6 is almost at perigee, it will be hardly any longer than the shortest. Similarly, since \u03b1\u03b4 is near the middle elongation, it will be only a little less than the mean distance. Therefore, the eccentricity will come out to be a little greater than 1038, which is the difference between \u03b4\u03b1 and \u03b6\u03b1. And if \u03b4\u03b1 is assigned the measure of 100,000, 1038 will have the value 1539; and this is about what the eccentricity will amount to (it is actually a little greater). And this is much closer to 1800, half the Tychonic value, than to the full value of 3600. ","569":"The same is to be said of the sun's apogee. For because \u03b6\u03b1 is shortest, the perigee is about 25\u00b0 53' Sagittarius. And because \u03b5\u03b1 is shorter than \u03b7\u03b1, the perigee is closer to 10\u00b0 17' Aquarius than to 11\u00b0 42' Scorpio. But the mean is 25\u00b0 57' Sagittarius. Therefore, the perigee is beyond 25\u00b0 57' Sagittarius and before 10\u00b0 17' Aquarius; that is, in Capricorn.\nThis I wanted to give as a foretaste to make up for the coming labors. For now I shall follow the geometric path to investigate the position of the apogee and the eccentricity. And since three points determine a circle, I shall at first use the points \u03b4, \u03b6, and \u03b7.\nI proceed as in Chapter 25 above. Since the points \u03b4, \u03b6, \u03b7 are placed on a single circumference with center \u03b3, the angle \u03b4\u03b7\u03b6 will be half of the angle \u03b4\u03b3\u03b6, and its measure will be the arc \u03b4\u03b6. Therefore, the ratio of \u03b4\u03b6 to the radius \u03b4\u03b3, and to the eccentricity \u03b3\u03b1, will be given, together with the angle \u03b4\u03b1\u03b3, because \u03b1\u03b3 lies in the direction of the apsides. But for the knowledge of the angle \u03b4\u03b7\u03b6 and of the line \u03b4\u03b6, we need to solve the three triangles. 4","570":"And yet it was said before that using \u03b4 and \u03b6 the eccentricity comes out to be a little greater than 1539, supposing that \u03b6 is nearest to perigee. But since here (letting \u03b7 be received into the company in place of \u03b6) the eccentricity comes out much greater, this hints, although erroneously, that there is some line at perigee which is still shorter than \u03b1\u03b6. In order that this distance at perigee might be able to be shorter than \u03b1\u03b6, this line of argument has moved the perigee to 16 Capricorn, that is, farther from \u03b1\u03b6. ","571":"But since we already know that the sun's perigee is not in 16 Capricorn but in 6 Capricorn, it is appropriate to assign the cause of the slight error to the point \u03b7 and to the excessive length of the line \u03b1\u03b7. For the result of this is that the circle \u03b4\u03b5\u03b7 would be too large, and its radius \u03b4\u03b3 too long; and consequently \u03b3\u03b1 would be too long, and \u03b3 would move perpendicularly away from the line \u03b4\u03b7, but obliquely from the point \u03b6. Thus, the line \u03b3\u03b1 would now be placed too far to the east. Therefore, letting \u03b4 and \u03b3 remain, let \u03b1\u03b7 be supposed shorter. Then the center \u03b3 moves perpendicularly toward the line \u03b4\u03b7, and \u03b4\u03b3 thus becomes shorter. And because \u03b3 approaches \u03b4\u03b7 perpendicularly, it recedes obliquely from the present \u03b3\u03b1. Hence, if a straight line be drawn from \u03b1 through the new position of \u03b3, it will be inclined back towards \u03b4. ","572":"You thus see how shortening \u03b1\u03b7 helps us in two ways. But to shorten \u03b1\u03b7 is a very easy, small change, because the angles are small: it can be done by saying that the planet was seen in a slightly prior position along a line drawn from \u03b7 below \u03b8. 6 For example, let the observed position of Mars be 19\u00b0 40' Taurus, and the supplement \u03b1\u03b7\u03b8 be 7\u00b0 58' 26\", and \u03b7\u03b8\u03b1 5\u00b0 20' 8\": then \u03b1\u03b7 will be 67,030. The second and third triangles are then changed, \u03b1\u03b7\u03b4 becoming 23\u00b0 53' 0\", and \u03b1\u03b7\u03b6 67\u00b0 15' 32\". Therefore, \u03b4\u03b7\u03b6 is 43\u00b0 22' 26\", and \u03b4\u03b3\u03b6 86\u00b0 44' 52\". The remaining angles are 93\u00b0 15' 8\", whose half, \u03b3\u03b4\u03b6, is 46\u00b0 37' 34\", and \u03b3\u03b4\u03b1 is 1\u00b0 10' 12\". Hence \u03b4\u03b3 is 67,892. And where this is 100000, \u03b1\u03b4 will be 99,416, and \u03b4\u03b3\u03b1 73\u00b0 24' 39\". Consequently the perigee is in 10\u00b0 36' Capricorn, and the eccentricity is still about 2100.","573":"Thus, as the true perigee was approached the eccentricity decreased, and so when we get exactly to the correct perigee we shall also get exactly to the halving of the eccentricity.","574":"But it is nonetheless worthwhile to find out, in addition, how much we gain by changing the line \u03b1\u03b8, adding one minute to the computed eccentric position of Mars while keeping fixed the point of observation (that is, the point \u03b7) for the year 1595. With \u03b1\u03b8 accordingly moved forward, if these lines of sight \u03b7\u03b8, \u03b6\u03b8, and the rest were to stay the same, it would come about that \u03b1\u03b8 would be intersected by \u03b7\u03b8 at a place higher than \u03b8; and on the other hand it would be intersected by \u03b6\u03b8 and its counterparts at a place lower than \u03b8. 7 So \u03b1\u03b8 would not keep the same length. But since we are supposing that Mars is at the same eccentric position all four times, the length of \u03b1\u03b8 will also be the same for all four times. Therefore, in order that the point of intersection \u03b8 be the same, and nonetheless the lines of sight go towards their original zodiacal positions, it will be necessary to draw a line parallel to \u03b7\u03b8 somewhat lower, thus making \u03b1\u03b7 shorter; and also a line outside \u03b6\u03b8 and parallel to it, by which \u03b1\u03b6 would be lengthened; and so on for the rest. Next, the whole labor is to be repeated from the beginning. For \u03b4\u03b5\u03b8\u03b1 will now be 19\u00b0 56' 4\", \u03b5\u03b8\u03b1 34\u00b0 53' 40\", \u03b6\u03b8\u03b1 41\u00b0 14' 46\", and \u03b7\u03b8\u03b1 5\u00b0 21' 8\". Therefore \u03b4\u03b1 will be 67,522, \u03b5\u03b1 66,660, \u03b6\u03b1 66,451, \u03b7\u03b1 66,963. Hence, \u03b1\u03b4\u03b6 will be 45\u00b0 26' 37\", \u03b1\u03b7\u03b4 23\u00b0 54' 30\", \u03b1\u03b7\u03b6 67\u00b0 20' 48\". And \u03b4\u03b7\u03b6 will be 43\u00b0 26' 18\", and \u03b4\u03b3\u03b6 86\u00b0 52' 36\", \u03b3\u03b4\u03b6 46\u00b0 33' 42\", and \u03b3\u03b4\u03b1 1\u00b0 7' 5\"- a different angle from different beginnings. Now, when \u03b1\u03b6 is divided by the sine \u03b1\u03b4\u03b6, and the quotient multiplied by the sine of \u03b4\u03b1\u03b6, the product is \u03b4\u03b6, 93,252. Again, when this is divided into the sine of \u03b4\u03b3\u03b6, and the quotient is multiplied by the sine of \u03b4\u03b6\u03b3, the product will be \u03b4\u03b3, 67,823. Hence the angle \u03b4\u03b3\u03b1 is 76\u00b0 37\u2019 30\", and the perigee is at 7\u00b0 23' Capricorn, with an eccentricity about 1880, which would clearly be 1800 if the perigee were brought back to 5\u00bd Capricorn, as could happen through the combined effect of both causes. ","575":"For if you now subtract only half a minute from the apparent position for 1595, we will be within range. And there could easily be a one minute error in the equations of the eccentric found by the hypothesis of Chapter 16.","576":"So, since the result is easily vitiated by the data from 1595, let us now omit them and use the remaining three points \u03b4, \u03b5, \u03b6, keeping the most recent correction of eccentric position, and forming the new triangles \u03b4\u03b1\u03b5 and \u03b5\u03b1\u03b6. 8","577":"And because \u03b4\u03b6 remains 93,252, as before, dividing the sine of \u03b3\u03b4\u03b6 by the sine of \u03b4\u03b3\u03b6 and multiplying the quotient by \u03b4\u03b6 gives \u03b3\u03b4 as 67,873. But \u03b1\u03b4 is 67,522. From this and \u03b3\u03b4, \u03b4\u03b3\u03b1 is found to be 75\u00b0 8' 40\" and the perigee in 8 51 45 Capricorn, about what it was before. ","578":"The eccentricity, somewhat more than 2000, is to be decreased to 1800 (as before), if the perigee were to be moved back to 5\u00bd\u00b0 Capricorn. This is accomplished by lengthening \u03b1\u03b5. And \u03b1\u03b5 is lengthened if we say that the planet appeared one or two minutes before 9\u00b0 24' Aries. For then, from the point \u03b8 set up by the other lines of observation, some line would be drawn outside \u03b8\u03b5 towards \u03b8\u03b6.","579":"One might, however, hold suspect such license in making small changes in the data, thinking that by taking the same liberty in changing whatever we don't like in the observations, the full Tychonic eccentricity might also at last be obtained. Anyone who thinks this way should make a try at it, and, comparing his changes with ours, he should judge whether the changes remain within the limits of observational precision. He also needs to beware lest, elated by a trust in one such iteration, he render himself all the more guilty afterwards, because of the very divergent apogees found for the sun.","580":"I, to be sure, have laid all my prejudices and preferences out in the open here, so that I am more afraid of appearing to the reader to be importunate than I am of seeming untrustworthy.","581":"And by the way, a remark for future use: if \u03b3\u03b4 is made 100,000, \u03b1\u03b8 will come out to be 147,443, or even greater when the figures that are still missing are obtained correctly.","582":"Finally, to avoid prolixity, if \u03b1\u03b8 be 147,700, and the eccentric position of Mars in 1595 be 14\u00b0 21' 7\" Taurus, and the eccentricity of the earth be 1800, and the earth's path oval, as will be said in Chapters 30 and 44, the appearances will come out to be ","583":"And thus it has been demonstrated that \u03b1\u03b3 is about 1800, although it ought to have been 3600 if Tycho's discoveries were accommodated to the Copernican form and the sun's apparent motions. Consequently the point \u03c0 of the earth's uniformity of motion must be sought on the line \u03b1\u03c0, so that \u03b3\u03c0 and \u03b3\u03b1 are equal. For if the earth is moved uniformly about \u03c0, that is, if \u03b4\u03c0\u03b5, \u03b5\u03c0\u03b6, \u03b6\u03c0\u03b7 are equal, Tycho's observations of the sun will remain unaltered, and \u03c0\u03b9\u03b1 will be 3600. And at the same time, since the earth will be at the same distance from point \u03b3 when at the points \u03b4, \u03b5, \u03b6, \u03b7, the observations of Mars will also remain unchanged. ","584":"Second Diagram: Ptolemaic Form9","585":"In the Ptolemaic form, the delineation can be done in two ways. In the first, let the earth replace the solar body at \u03b1, and then lines of vision be drawn out from \u03b1 parallel to \u03b4\u03b8, \u03b5\u03b8, \u03b6\u03b8, \u03b7\u03b8, so that the Copernican positions of the earth \u03b4, \u03b5, \u03b6, \u03b7 coalesce into one Ptolemaic position of the earth. Meanwhile, let the star Mars, which for Copernicus had stayed at one point \u03b8, now be placed about \u03b8 at four locations: \u03b9, \u03ba, \u03bb, \u03bc. The description of this circle is as follows. Upward through \u03b8 let \u03b8\u03bd be drawn equal and parallel to \u03b3\u03b1. And about center \u03bd, with radius \u03b3\u03b5, let the circle \u03b9\u03ba\u03bb\u03bc be described. Thus the point \u03b8, which we can call the point of attachment, moves around on the eccentric, which before, in Copernicus, the planet bodily traversed. While the epicycle is thus being borne around, the center \u03bd is driven around \u03b8, so that at one time it is inside \u03b8\u03b1 and at another it is outside; however, \u03b8\u03bd is always parallel to itself and to the line \u03b1\u03b3. And the epicycle will be moved uniformly, not about \u03b8 to which it is attached, nor about its center \u03bd, but about a higher point, \u03bf, such that \u03b8\u03bf is twice \u03b8\u03bd. For the earth too will thus be moved10 uniformly about \u03c0, not about the center of its orb \u03b3, nor about the sun at \u03b1. ","586":" That these characteristics belong to the Ptolemaic epicycle, is properly demonstrated. But that they are carried over from the epicycle to the theory of the sun is shown by a probable argument only, pieced together from Ptolemaic opinions. For, keeping everything as it was, let \u03b1\u03c4 be set up equal to \u03b1\u03c0 and in the same line but in the opposite direction, so that \u03c4\u03b1 may be the center of the sun's uniform motion, which the theorists had thought to be the center of the sun's orbit. Therefore, the line \u03b8\u03bd\u03bf will always be parallel to the line of the sun's apogee \u03b1\u03c4. Now, if you decide that the diurnal parallax of Mars should be kept in the same ratio to the sun's parallax as that given out by Tycho, \u03b9\u03ba\u03bb\u03bc will also be equal to the [circle of the] sun's theory, and consequently \u03b8\u03bf will also be equal to the eccentricity of the point \u03c4, about which the sun moves uniformly. But \u03b9\u03ba\u03bb\u03bc also moves in the same direction in which the sun moves on its circle, according to Ptolemy, and at the same times both the sun on its eccentric and the planet on its epicycle are found in the same, or at least corresponding, places, so that lines from \u03c4 through the sun and from \u03bf through the planet are ever parallel, again as taught by Ptolemy. So, since all other things are in agreement, why not this, too: that, just as \u03b9\u03ba\u03bb\u03bc is moved uniformly, not about the center \u03bd but about the point \u03bf above it, as is demonstrated here by the transposition of the eccentric of the earth into the epicycle, where we took \u03b8 in place of \u03b1, \u03bd for \u03b3, and \u03bf for \u03c0: so likewise, these points are distinct in the sun, so that the eccentricity \u03b1\u03c4, which is found from solar observations, is to be bisected at \u03be, with \u03be the center of the sun's eccentric \u03bb\u03c1\u03c3\u03c5? For Ptolemy made use of such a procedure to make it appear that if the sun's apparent positions were used, exactly the same eccentricity would be used on the planet's epicycle as was found in the sun. So since the observations give evidence of the double eccentricity of the Ptolemaic epicycle (because, as was said, the parallel relationships of the lines leave the triangles the same as in the Copernican form), the spirit of Ptolemy urges us to bisect the sun's eccentricity as well, so that the lines \u03bb\u03b9, \u03c1\u03ba, \u03c3\u03bb, \u03c5\u03bc remain parallel.","587":"So by this reasoning even Ptolemy will be persuaded that \u03b1\u03c4, the eccentricity of the sun's motion found by Tycho, should be bisected at \u03be, so that the center of the sun's orbit is at \u03be, and the uniformity of motion at \u03c4.","588":"Now this argument in the Ptolemaic form (as I just now began to say) is no firmer than the Ptolemaic world system itself. For anyone who believes Ptolemy, thinking that for the three superior planets there are three theories of epicycles, exactly equal to the theory of the sun, in quantity and quality, in lines as well as motions, in absolutely all respects\u2014this same person will not admit this one inconsistency, but will also gladly derive this bisection for the solar theory from the epicycle, as if from an image in a mirror to the face itself.","589":"And finally, when a comparison of hypotheses has been made, and it has appeared that four theories of the sun (or rather, six, as will be said elsewhere) can be generated from a single theory of the earth, like many images from one substantial face, the sun itself, the clearest of truth, will melt all this Ptolemaic apparatus like butter, and will disperse the followers of Ptolemy, some to Copernicus's camp, and some to Brahe's. ","590":"Here, one might raise a question. The Ptolemaic epicycle has three notable points: the center \u03bd, the point \u03b8 which we have called the point of attachment, and the point \u03bf about which its motion is uniform. Now since it is said that the line \u03b8\u03bf remains parallel to \u03b1\u03c4 throughout the entire circuit, what are the properties of the circuits described by the other two points \u03bd and \u03bf? In order to show this, let lines be drawn from \u03be and \u03c4\u03b1 parallel to \u03b1\u03b2, and from \u03b2 and \u03c7 parallel to \u03b1\u03c4, and let them be extended so as to intersect one another; and let the intersection of the lines from \u03be and \u03b2 be \u03c6, from \u03be and \u03c7 be \u03c8, from \u03c4 and \u03b2 be \u03c2, and from \u03c4 and \u03c7 be \u03c9. Now, just as the point \u03b8 is moved regularly around \u03c7, traversing an eccentric described about \u03b2, so also \u03bd is moved regularly around \u03c8, traversing an eccentric described about \u03c6. Also, \u03bf is moved regularly about \u03c9, traversing a third eccentric likewise equal to the others, described about \u03c2. For all three of these eccentrics the zodiacal position of the apogee is the same, owing to the lines' \u03b1\u03c7, \u03be\u03c8, \u03c4\u03c9 being parallel. But the word \"apogee\" cannot be applied properly to any of these, apart from the first, that belongs to the point \u03b8, since its line of apsides \u03b1\u03b2\u03c7 is drawn through the earth itself, which was placed at \u03b1, but not at \u03be or \u03c4.","591":"It is indeed true that straight lines can be drawn from the earth \u03b1 through the centers of the remaining two eccentrics \u03c6 and \u03c2, which may properly be called \"lines of the apogee\". These will fall to the west of the apogee \u03b1\u03c7; that is, \u03b1\u03c6 will be at 24\u00b0 Leo, and \u03b1\u03c2 at 19\u00b0 Leo, approximately. But then these lines will not pass through the pertinent point of uniform motion of each eccentric. Thus, if any of Ptolemy\u2019s followers does not wish to attach the epicycle to the eccentric at the point \u03b8, but prefers to relate it to the center \u03bd, he will be driven to use two lines of apsides: one, \u03b1\u03c6, for the eccentric, and the other, \u03b1\u03c8, for the equant; and also two eccentricities, \u03b1\u03c6 and \u03b1\u03c8. How intricate and inconvenient this is (for of its absurdity enough has been said in Chapter 6), let anyone so inclined judge.","592":"The same will happen if anyone wishes to attach the epicycle to the eccentric at the point \u03bf, about which the epicycle revolves uniformly. For then the eccentric bearing the point \u03bf will have two apogees and eccentricities, one for the center on the line \u03b1\u03c2, and the other for the point of uniformity of motion on the line \u03b1\u03c9. The remaining possibilities are either to attach the epicycle at \u03b8, or improperly to take apogees for the eccentrics that bear the points \u03bd and \u03bf, and to compute the eccentricities from the points \u03be and \u03c4, not from the reference point, the earth \u03b1.","593":"What has been given so far is the first delineation in the Ptolemaic form. The other can be constructed as follows. Let the Copernican positions of the earth \u03b4, \u03b5, \u03b6, \u03b7, merge, not in \u03b1, but in \u03b3, so that in this diagram, \u03b3, not \u03b1, denotes the earth, the center of the world. Here, the epicycle too, as well as its three eccentrics belonging to the points \u03b8, \u03bd, \u03bf, will be shifted from their positions by the amount \u03b1\u03b3, and a perfect equivalence will result. I shall forego further explanation, lest the reader become too confused, for this has really only been mentioned for smatterers or curiosity seekers.","594":"Tychonic Form11","595":"In the Tychonic form, there is no need of any new delineation. A very brief sketch is sufficient. The eccentric's point of attachment is set at the four different positions \u03bb, \u03c1, \u03c3, \u03c5, so the planet is at \u03b9, \u03ba, \u03bb, \u03bc, and \u03b9\u03bb, \u03ba\u03b1\u03c1, \u03bb\u03c3, \u03bc\u03c5, and \u03b8\u03b1 parallel. Now Tycho himself made the center of the circle of Mars carry the double epicycle, and said that it goes around a uniformly on a circle concentric with the sun: for this idea he was indebted to Ptolemy. And in this matter, he, along with Ptolemy and Copernicus, was strongly urged by me (in Chapter 6 of Part I) to seek that point of attachment, whether it is the center of the concentric or of an eccentric, in the center of the solar body, this being supported by physical arguments and by the demonstration of its geometrical possibility. Additional support is provided by the valid argument of Chapter 22 and 23, that unless this were done, even if the observations were referred to the sun's mean position, the Ptolemaic epicycle and the Brahean deferent would be made eccentric, in directions exactly opposite to the sun's eccentricity. I have also promised stronger arguments, deduced from Brahe's own observations, for abandoning the sun's concentric, and in Chapter 52 and 57 below I will produce them. But it has already been proved here in Chapter 26 that this center of Mars's concentric (or the point from which Mars's eccentricity originates) is found, not on an equal eccentric described about the center of the sun's point of uniform motion \u03c4, as Brahe along with the other authorities had believed, but on an eccentric described about \u03be, which is in the middle position between \u03b1 and \u03c4.\nTherefore, if the center of Mars's concentric goes around with the sun, it nevertheless goes around on an eccentric described about \u03be, and consequently the sun itself will go around on an eccentric described about \u03be. But its motion is uniform about \u03c4. Therefore, the sun's eccentricity \u03b1\u03c4 must be bisected at \u03be. For it is not likely that, although the centers of the concentrics of Mars and the sun go around in the same way, reach apogee at the same time, transpose their apogees in the same way, go slowly or quickly in the same way, and describe the same circumferences, their circles would nevertheless make different digressions from the earth in the same direction.\nLet it be enough for now, to present this form of demonstration in the three hypotheses. In what follows, whenever there is need of the same demonstration, I shall use Copernicus's form alone, it being simpler, so as not to be too long-winded. Now, in contrast, the industrious reader has seen how any of these diagrams can be transformed into either the Ptolemaic or the Copernican form using parallel lines. ","596":"Chapter 27","597":"From four other observations of the star Mars outside the acronychal situation but still in the same eccentric position, to demonstrate the eccentricity of the earth's orb, with its aphelion and the ratio of the orbs at that place, together with the eccentric position of Mars on the zodiac.","598":"Hitherto we have almost exclusively used the aphelion of Mars, along with the correction of the mean motion and the hypothesis of the equations found above. If these should err by a single minute in defining the planet's zodiacal longitude, as can easily happen, this creates considerable difficulty for us in the present undertaking.","599":"So now, at this point, we shall take for granted nothing whatever except Mars's periodic time, concerning which there can be no doubt, and the sun's zodiacal positions from Tycho's calculation. We shall also, I grant, assume an eccentric position, as is the common procedure in demonstrations leading to an impossibility, but we shall test that very position using repeated suppositions.","600":"These are the observations. ","601":"Since in 1589 there is but a single day that can be related to the others, and nothing else was observed for a long time before and after, let us refer the other times to this one. The catalog of them, along with the apparent positions of the sun and Mars, and with the eccentric position of Mars, is this: 1","602":"Let the diagram be made as before, with \u03b1 the sun, \u03b2 the center of the earth's eccentric, \u03b6, \u03b4, \u03b5, \u03b3 the four positions of the earth, \u03b7 the position of Mars on its eccentric, and let each point be connected with all the others. Now, from the data, ","603":"Now because there stand two angles upon the arc \u03b6\u03b5 at the circumference of the circle, namely \u03b6\u03b4\u03b5, \u03b6\u03b3\u03b5, by Euclid III. 21 these must be equal. And in order that they come out equal, \u03b1\u03b7 must be moved forward and backward above \u03b1 beneath the zodiac as long as [is necessary]. And since in this first trial the zodiacal position for \u03b1\u03b7 is given, let it therefore be tried whether \u03b6\u03b4\u03b5, \u03b6\u03b3\u03b5 might be equal, for then it will be established that the position of \u03b1\u03b7 is correct. ","604":"In the four triangles \u03b6\u03b1\u03b4, \u03b4\u03b1\u03b5, \u03b5\u03b1\u03b3, \u03b6\u03b1\u03b3, the same number of angles is sought, namely \u03b6\u03b4\u03b1, \u03b5\u03b4\u03b1, \u03b5\u03b3\u03b1, \u03b6\u03b3\u03b1 in order to get \u03b5\u03b4\u03b6, \u03b5\u03b3\u03b6.","605":"Now in any of these triangles the angles at \u03b1 are given by the position of the sun from Tycho, and the correction for the precession of the equinoxes. But the sides comprehending those angles have just been found. Therefore, the angles, too, will be given. ","606":" Since these angles didn't quite come out equal, I made a second trial with \u03b1\u03b7's sidereal position moved forward 2'. And I found \u03b5\u03b4\u03b6 to be 21\u00b0 40' 9\", \u03b5\u03b3\u03b6 21\u00b0 22' 14\", differing by 18', which is twice the previous discrepancy. Whence it is understood that \u03b1\u03b7 should have been moved backwards to a lesser longitude, rather than forwards.","607":"For a third trial, then, supposing Mars\u2019s eccentric position in 1585 was 5\u00b0 20' 2\" Libra, \u03b5\u03b4\u03b6 came out to be 21\u00b0 15' 54\", and \u03b5\u03b3\u03b6 21\u00b0 13' 54\". There remains a difference of 2\u2019, which we may safely ignore. Nevertheless, by extrapolating we realize that at this place Mars's eccentric position has to be moved back3 through 2\u00bd\u2019, just as previously in Chapter 22 it was moved forward 1' on the opposite semicircle. Both of these are brought about by an increase of the eccentricity and a slight retraction of the aphelion. ","608":"Let us now proceed to the investigation of the rest. And because each of the angles in question has decreased, they will decrease further when \u03b1\u03b7 is moved back. Therefore, let each be 21\u00b0 13', and \u03b6\u03b2\u03b5, the double angle at the center, be 42\u00b0 26'. Therefore, \u03b6\u03b5\u03b2 is 68\u00b0 47'.","609":"In triangle \u03b6\u03b1\u03b5 angle \u03b6\u03b1\u03b5 is 42\u00b0 6' 57\", and the sides are given by a new correction, so that \u03b1\u03b6 is 62,177 and \u03b1\u03b5 61,525, approximately. Hence, \u03b6\u03b5\u03b1 is given as 69\u00b0 43' 31\", and \u03b6\u03b5 44,518. But this same \u03b6\u03b5, from angle \u03b6\u03b2\u03b5 (which \u03b6\u03b5 subtends), is 72,379 where \u03b5\u03b2 is 100,000. Therefore, where \u03b5\u03b2 is 100,000, \u03b1\u03b7 is 162,818, and thus \u03b1\u03b5 is 100,174. But when \u03b6\u03b5\u03b2 is subtracted from \u03b6\u03b5\u03b1, the remainder \u03b2\u03b5\u03b1 is 0\u00b0 56' 31\", and \u03b2\u03b1\u03b5 is 83\u00b0 30'. Therefore, the aphelion is at 10\u00b0 19' Capricorn, while the eccentricity \u03b1\u03b2 is 1653.","610":"Again we have come rather close to half of 3600, and would doubtless obtain exactly that if we had the apogee perfectly.","611":"Still, it should be noted that if we suppose that the earth's path is not a perfect circle, but is narrower at the sides, \u03b1\u03b7 comes out here to be a little less than 163,100. And then, with 1\u00bd\u2019 subtracted from the eccentric position, and taking the value 1800 for the earth's eccentricity and 5\u00bd\u00b0 Capricorn as the aphelion, the following apparent positions are produced: ","612":"This supposition also agrees with my observations of 1604 February 29\/ March 10, for on the night following that day with my instruments I found Mars culminating at 26\u00b0 18\u2158\u2018 Libra. And calculation based upon these assumptions puts it at 26\u00b0 17\u00bd\u2019 Libra. Moreover, at 8\u2154h a few hours before the observation, it was again in the same eccentric position.","613":"Besides, since Mars has some latitude here, the value for \u03b1\u03b7 just found is the distance in the plane of the ecliptic of the point \u03b7 from the center of the sun, to which point a perpendicular is drawn from the body of Mars, as was noted in Chapter 20 above. So the true distance of the planet's body itself from the center of the sun is made a little longer by 37 units.","614":"Chapter 28","615":"Assuming not only the zodiacal positions of the sun, but also the sun's distances from the earth found using an eccentricity of 1800, through a number of observations of Mars at the same eccentric position, to see whether by unanimous consent the same distance of Mars from the sun, and the same eccentric position, are elicited. By which argument, it will be confirmed that the solar eccentricity of 1800 is correct, and was property assumed.","616":"The reader should not be surprised that in this third turn I am now not presupposing the eccentric position of Mars as it is given by the hypothesis of acronychal observations found above. For I have said that that hypothesis was only vicarious, not natural, and thus possesses only as much trustworthiness as is derived from the observations; and it could deviate somewhat in the intermediate positions between observations. Besides, it helps us to have at hand various methods of demonstration by which to explore carefully the distances of Mars at all places throughout the entire circle. Accordingly, a new form of demonstration will follow here.","617":"The observations are these. ","618":"When the times of the other observations are adjusted so as to return Mars to the same eccentric position which it had at the last time, we are given the following times. Along with these are added the requisite positions of the sun, and the distances of the sun and the earth computed from the hypothesis so far established. And indeed, we have taken up this labor in order to test these very things. A little later, in Chapter 30, a technique for computing these distances will follow. ","619":"Now for the deduction of the observations from the days of the observations to the times we have chosen. For the first time, the diurnal motion was taken from Magini, since over the space of a few hours there is no danger of error. The other times are supported by observations before and after. However, for the penultimate time I also looked up the sequence of diurnal motions in Magini. Around December 15 the diurnal motion was 30', and around December 5 it was 32'. For the last time, although Mars, being at an altitude of 23\u00b0, is affected by refraction, so that 2\u2019 might easily be wanting in the latitude, this refraction nevertheless hardly affects the longitude of Mars. (Tycho claimed that the refraction of the fixed stars, also applicable to the planets, ceases at this altitude, although the solar refraction reaches higher, and at this altitude is about 4\u2019. This distinction was discussed and demolished in my Optical astronomy p. 137, 2 and would be rendered even more dubious if any changes are to be made in the parallax of the sun.)","620":" Let \u03b1 be the sun's body, \u03b1\u03b2 the eccentricity of the earth's orb (1800), and the line of apsides be at 5\u00bd\u00b0 Cancer, positions of the earth \u03b6, \u03b5, \u03b4, \u03b3, \u03b8, and the body of the planet at the same eccentric position \u03b7 all five times, since the intervals span complete periods of Mars. And let all points be connected with one another. It is desired to find \u03b1\u03b7, and its zodiacal position, that is, angle \u03b7\u03b1\u03b8, \u03b7\u03b1\u03b3, or some other angle at \u03b1. We shall do this from two earth positions in the following manner. For a start, let these be \u03b5, \u03b4. And in triangle \u03b5\u03b1\u03b4, given the sides \u03b5\u03b1 99,770, \u03b1\u03b4 98,613, and angle \u03b5\u03b1\u03b4, the rest are sought, namely, the angles \u03b4 and \u03b5 and the side \u03b4\u03b5. ","621":"With these matters investigated, one goes on up to triangle \u03b5\u03b7\u03b4. ","622":"So, angles \u03b5 \u03b7, \u03b4 and one side \u03b5\u03b4 being given, side \u03b5\u03b7 will also be given. ","623":"Finally, let the triangle \u03b7\u03b5\u03b1 also be solved, in which are given: ","624":"If the other three observations at \u03b6, \u03b3, \u03b8, will allow this same position and length for \u03b1\u03b7, we shall have excellent confirmation of them.","625":"So, in the same manner in which we have hitherto worked with \u03b5 and \u03b4, we shall now work with \u03b6, \u03b3, seeking the same \u03b1\u03b7. \nFor the angles, and line, \u03b3, \u03b6 ","626":"And now in \u03b6\u03b3\u03b7. ","627":"Next, given the angles of the triangle \u03b6\u03b7\u03b3, and side \u03b3\u03b6, the side \u03b6\u03b7 is sought. ","628":" Finally, in triangle \u03b7\u03b6\u03b1 the sides and the comprehended angle are given. ","629":"And so it appears that with two other observations, at \u03b6 and \u03b3, we have come upon the same thing, within the limits of observational accuracy. For an error of a minute and a half in observing, or in deducing the observed position to a day when it was not observed, can be committed.","630":"But let us also see the evidence of the fifth position, \u03b8, that is, of the observation at \u03b8. ","631":"If I lengthen \u03b1\u03b7 subtending this angle, I will thereby move \u03b1\u03b7 farther forward in longitude, and vice versa. ","632":"So let \u03b1\u03b7 be 166,208, as initially determined.","633":"Now as \u03b1\u03b7 is to [sin] \u03b1\u03b8\u03b7 so is \u03b1\u03b8 to [sin] \u03b1\u03b7\u03b8. ","634":"And so through the very slightest shortening of \u03b1\u03b7, it will fall exactly in the same place with the first two observations.","635":"And so it appears from this that we rightly assumed and posited the distances \u03b1\u03b6, \u03b1\u03b5, \u03b1\u03b4, \u03b1\u03b3, \u03b1\u03b8, and the eccentricity \u03b1\u03b2 as well. For it is impossible to take other distances than these, which nonetheless also fit as nearly as possible on a circle and have the appropriate zodiacal positions, and still obtain the same magnitude for \u03b1\u03b7, and its zodiacal position, from all five observations.","636":"But concerning the length of \u03b1\u03b7, we shall put our trust mostly in observations \u03b6, \u03b3, \u03b8. For also in the common method of measuring distances of things on earth, the farther the standing points are from one another, the more accurately distance of the mark is obtained.","637":"For the zodiacal position, however, we shall put our trust in the observations at \u03b5, \u03b4, instead. For if there is some slight error in the length of \u03b1\u03b7, it is presented to the observer at \u03b5, \u03b4 quite obliquely, and the angle does not change perceptibly.","638":"Nor is this to be forgotten: that there is no perceptible lengthening of \u03b1\u03b7 over the seven years from 1583 to 1590, because the pregression of the aphelion is very slow.","639":"Summary: On 1590 October 31 at 6\u00bch in the morning, Mars's eccentric position was 8\u00b0 19' 20\" Virgo, while the hypothesis constructed using acronychal observations places it at 8\u00b0 19' 29\" Virgo. Its distance was 166,180, which must be lengthened because of the latitude, so that from this distance, the distance from the actual body of Mars to the center of the sun comes out to be about 166,228.","640":"Chapter 29","641":"A Method of Deducing the Distances of the Sun and Earth from the Known Eccentricity","642":"I think it has been well enough confirmed that the distances of the sun and the earth are to be deduced by halving the eccentricity obtained by Tycho. This is also abundantly confirmed by observation of the sun's summer and winter diameter, as I have shown in the Astronomiae pars optica chapter 11. But it is also wonderfully confirmed in the Mysterium cosmographicum, in the table in ch. 15 p. 53, where the equations of the center for Mars, Venus, and Mercury were deficient when the lunar orb was interposed, but excessive when it was omitted. Now, the orb of the moon being retained while the eccentricity of the sun is bisected, they come out about right.","643":"Furthermore, the same thing will again be confirmed more frequently and much more clearly when we use the distances forthcoming from the bisection (as we have just begun to do in the last chapter) and shall see the phenomena follow from them. Therefore, in order that these distances be ready at hand for our future use, I shall show how they may easily be computed, using a geometrical demonstration.","644":"On the line \u03b1\u03b4 let \u03b1 be the body of the sun (or the earth for Tycho, or the center of attachment of the epicycle for Ptolemy); \u03b2 the center of the eccentric \u03b6\u03b4\u03b7 of the earth (or of the sun and of the annual orb for Tycho, or of the epicycle for Ptolemy); and \u03b1\u03b2 being extended, let it intersect the eccentric at \u03b4, \u03b5, so that \u03b4 is the aphelion or apogee and \u03b5 the perihelion or perigee; and let \u03b2\u03b3 be equal to \u03b1\u03b2. Also, let \u03b3 be the center of motion or of uniformity, at which the earth (for Ptolemy, the center of the epicycle, for Tycho the sun and the point of attachment of all the eccentrics) sets out equal angles in equal times. And let \u03b1\u03b3 be 3600, from the observations of Tycho and the Landgrave, but \u03b1\u03b2, according to my recently introduced adjustment, be 1800. Now let \u03b6\u03b7 be drawn through \u03b1 perpendicular to \u03b4\u03b5, intersecting the circle at \u03b6, \u03b7, and also through \u03b1 let the straight line \u03b8\u03b9 be drawn, at any inclination whatever, intersecting the circumference at \u03b8, \u03b9; and let the four points \u03b8, \u03b9, \u03b6, \u03b7 be connected with the center \u03b2. And let this also be posited at the start: that although the earth (sun, or planet) is moved uniformly around \u03b3 and thus nonuniformly around \u03b2, it nevertheless remains on the circumference of the circle described about \u03b2. Now, by the equivalence shown in the second chapter (which, for the sake of avoiding confusion, I shall not apply to the general Ptolemaic hypothesis), this is exactly as if one were to say that the earth (or sun) is moved nonuniformly on a concentric with an epicycle4 about center \u03b1, the semidiameter of the epicycle being equal to \u03b1\u03b2; and the arcs described on the concentric by the center of the epicycle being similar to the arcs of the epicycle described by the earth (or sun), so that both the earth (or sun) and the center of the epicycle are moved unequally in equal times, so as to become slow, or again speed up, simultaneously. I am going to postpone for a little while the physical explanation of this hypothesis. ","645":"Now, with these things supposed, I shall proceed to the work of finding the distances. And because \u03b2\u03b4 is 100,000 and \u03b2\u03b1 1800, and \u03b1\u03b2\u03b4 is a straight line, by addition of the two the aphelial distance \u03b1\u03b4 is obtained; and because \u03b2\u03b5 too is 100,000, when \u03b1\u03b2 is subtracted the perihelial distance \u03b1\u03b5 remains. ","646":"And because \u03b2\u03b1\u03b6 is right, and \u03b6\u03b2 is 100,000 (that is, the whole sine), therefore \u03b1\u03b2 is the sine of the angle \u03b1\u03b6\u03b2. Consequently \u03b1\u03b6\u03b2 is 1\u00b0 1' 53\", which is the optical part of the equation of the sun or earth. Now the maximum equation at the middle elongations, which is composed of the optical and physical parts, has the whole eccentricity 3600 (or 3592) as its sine. Therefore, when the sun or earth goes from \u03b4 to \u03b6, it will in fact take two days longer than one fourth of the periodic time, but it nevertheless does only one day's journey beyond one fourth of its total circuit. So in this distance, or quarter of the periodic time, it will take one day longer than it should, owing to the physical weakening.","647":"But to proceed to the distance \u03b1\u03b6. In the right triangle \u03b6\u03b1\u03b2, since one of the acute angles is given, the other, \u03b6\u03b2\u03b1, will be the difference between the first and one right angle, that is, 88\u00b0 58' 7\". And therefore \u03b1\u03b6 will be the sine of this angle, 99984. And the opposite line \u03b1\u03b7 is the same size.","648":" For finding the intermediate distances of two opposite degrees of equated anomaly, let \u03b8\u03b9 be inspected, passing through the body \u03b1 whence the eccentricity is computed. Now \u03b4\u03b1\u03b8 and \u03b4\u03b1\u03b9 are equated anomalies, and are opposite, in that \u03b1 is between them and is collinear with them. Now let a line \u03b2\u03ba fall from \u03b2 perpendicular to \u03b8\u03b9, so as to make \u03b8\u03ba, \u03ba\u03b9 equal. In the right triangle \u03b2\u03ba\u03b1 the base \u03b2\u03b1 is given, as well as the angles \u03ba\u03b1\u03b2 (from the integral number of degrees of equated anomaly chosen) and its complement \u03ba\u03b2\u03b1. Therefore, the sides \u03ba\u03b1, \u03ba\u03b2 will not be unknown. And \u03ba\u03b2 is the sine of the angle \u03ba\u03b8\u03b2 or \u03ba\u03b9\u03b2. This being given, its complement, \u03b8\u03b2\u03ba or \u03b9\u03b2\u03ba, will also be known, and its sine, which is the line \u03b8\u03ba or \u03ba\u03b9. And when \u03ba\u03b1 is added to \u03ba\u03b8, \u03b1\u03b8 is obtained; and when the same is subtracted from \u03ba\u03b9, \u03b1\u03b9 is obtained. The former distance corresponds to the equated anomaly \u03b4\u03b1\u03b8, and the latter to the equated anomaly \u03b4\u03b1\u03b9, which line has a line equal to itself in the preceding semicircle, standing as far from aphelion in semicircle \u03b4\u03b8 as \u03b1\u03b9 itself does in semicircle \u03b4\u03b7. ","649":" Now through \u03b1 let the straight line \u03bc\u03bd be drawn intersecting the circle in \u03bc, \u03bd, and making the angle \u03bc\u03b1\u03b4 equal to the angle \u03ba\u03b2\u03b1. And from \u03b2 let \u03b2\u03bb fall perpendicular to \u03bc\u03bd bisecting \u03bc\u03bd at \u03bb. And let \u03bc, \u03bd be connected to \u03b2. Now, since \u03ba\u03b1\u03b2 is an angle of an integral number of degrees, the remainder \u03ba\u03b2\u03b1, and \u03bc\u03b1\u03b4 equal to it, are also an integral number of degrees, and in the similar triangles \u03b2\u03ba\u03b1, \u03b2\u03bb\u03b1 side \u03ba\u03b1 is equal to side \u03bb\u03b2, and \u03ba\u03b2 to \u03bb\u03b1. But \u03bb\u03b2 is the sine of angle \u03bb\u03bc\u03b2, \u03bb\u03bd\u03b2, and the complement of \u03bb\u03bc\u03b2 is \u03bb\u03b2\u03bc, \u03bb\u03b2\u03bd. And the sine of this is the lines \u03bb\u03bc, \u03bb\u03bd, and the difference between these and \u03b1\u03bc, \u03b1\u03bd is \u03bb\u03b1. But the magnitudes \u03bb\u03b1, \u03bb\u03b2 have just now been found in triangle \u03b1\u03b2\u03ba. Therefore, by the use of one triangle, four distances can be found making equal angles about \u03b1 with the line of apsides and its perpendicular \u03b6\u03b7 drawn through \u03b1. For \u03bc\u03b1\u03b6 is equal to \u03b8\u03b1\u03b4, and \u03bd\u03b1\u03b7 to \u03b9\u03b1\u03b5. ","650":"So the greatest distance is at \u03b4, the shortest at \u03b5, but the mean, which is equal to \u03b2\u03b6, is not at \u03b6\u03b7. Neither is it on a line through \u03b2 parallel to \u03b6\u03b1 which shall be called \u03be\u03bf. For \u03b1\u03b6 is less than \u03b2\u03b6, since \u03b6\u03b2\u03b1 is subtended by a smaller line than \u03b6\u03b1\u03b2, which is right, and, \u03b1\u03be being drawn, it is longer than \u03b2\u03be, since it subtends a greater angle \u03be\u03b2\u03b1. (which is right) while \u03be\u03b2 subtends a lesser angle \u03be\u03b1\u03b2.","651":"But in order that the place of the mean distances be defined geometrically, let \u03b1\u03b2 be bisected at \u03c3, and through this let \u03c0\u03c1 be drawn perpendicular to \u03b1\u03b2, intersecting the circle at \u03c0, \u03c1. I say that these are the points that are equally distant from \u03b1 and \u03b2.","652":"For let one of the points \u03c0 be connected with \u03b1 and \u03b2. The lines \u03c0\u03b1 and \u03c0\u03b2 will subtend equal angles \u03c0\u03c3\u03b1 and \u03c0\u03c3\u03b2 (since they are right), and \u03b1\u03c3, \u03c3\u03b2 will be equal, and \u03c0\u03c3 common. Therefore \u03c0\u03b1, \u03c0\u03b2 are equal. And thus the demonstration taken from Reinhold7 concerning the whole \u03b1\u03b3 and its midpoint \u03b2 remains true for the point \u03c3 and its half \u03b1\u03b2. ","653":"One might think that since at \u03c0 the distance \u03b1\u03c0 becomes equal to the semidiameter \u03b2\u03c0, the angle \u03b2\u03c0\u03b1 is also greater than \u03b2\u03b6\u03b1, and thus the greatest equation occurs at \u03c0, on the argument that the straight line \u03b2\u03b1 is presented more directly from \u03c0 than from \u03b6. However, this proposed line of reasoning is not true. For to the same extent that \u03b2\u03b1 is more oblique with respect to \u03b6, \u03c0 in turn is more distant than \u03b6, since \u03c0\u03c3 is longer than \u03b6\u03b1. For \u03c0\u03b2\u03c3 is greater than \u03b6\u03b2\u03b1, which \u03b6\u03b1 subtends. So Ptolemy, and after him Reinhold in the Theoricae, demonstrated correctly that the greatest equation (eccentric equation alone, or optical part) occurs at \u03b6. I shall, however, set up this demonstration in another, simpler form. Let any point be taken above \u03b6, such as \u03b8, and any below \u03b7 or \u03b6, such as \u03b9. Let them be connected with \u03b1, and from \u03b2 let perpendiculars \u03b2\u03ba fall to \u03b8\u03b1 or \u03b9\u03b1 extended. Now since \u03b4\u03b1\u03b6 and \u03b2\u03ba\u03b1 are equal, they being right angles, and \u03ba\u03b2\u03b1, \u03ba\u03b1\u03b2 together are equal to one right angle, when the same angle \u03b4\u03b1\u03b8 or \u03b2\u03b1\u03ba is subtracted from the equals, the equals \u03b8\u03b1\u03b6, \u03ba\u03b2\u03b1 will remain. And first, let a line be drawn through \u03b1 above \u03b6, as \u03b8\u03b1 was just drawn, whether \u03b8 is next to \u03b6 or remote. At the same time let its perpendicular \u03b2\u03ba be inclined to \u03b2\u03b1. Now \u03b2\u03b1 is greater than any of the perpendiculars \u03b2\u03ba, since \u03b2\u03b1 is subtended by the right angle \u03b2\u03ba\u03b1, while \u03b2\u03ba is subtended by the smaller, acute angle \u03b2\u03b1\u03ba. Now since \u03b2\u03b6, \u03b2\u03b8, \u03b2\u03b9 are equal, and \u03b2\u03b1\u03b6, \u03b2\u03ba\u03b8, \u03b2\u03ba\u03b9 are right, they fit onto the same semicircle, whose diameter is equal to \u03b2\u03b6, \u03b2\u03b8, \u03b2\u03b9. And \u03b2\u03b1, being longer, subtends a greater part of the circumference of any such semicircle than does \u03b2\u03ba or any of the perpendiculars; and for that reason, its angle \u03b2\u03b6\u03b1 will be greater than \u03b2\u03b8\u03ba, or the angle of the equation for any other point above \u03b6, such as \u03c0 or \u03be. Which was to","654":"be demonstrated.","655":"Everything said in this chapter about the computing of distances of the sun and earth will also be valid for Mars, as long as one of the suppositions is that the orbits of planets are perfect circles. When this is seen as false, another method of computing them will be given. ","656":"Chapter 30","657":"Table of the distance of the sun from the earth and its use","658":"We have gathered together here into a three-columned table the distances of the sun accumulated in this way [that is, in the way described in ch. 29], as if they were done for integral degrees of equated anomaly for the whole semicircle (for those in the other semicircle at equal distances from apogee are also equal to these). In the first column, which we have called \"mean anomaly\", are the angles \u03b4\u03b2\u03bc, \u03b4\u03b2\u03b8, \u03b4\u03b2\u03be, \u03b4\u03b2\u03b9, \u03b4\u03b2\u03bd, composed of the integral angles \u03b4\u03b1\u03bc, \u03b4\u03b1\u03b8, \u03b4\u03b1\u03be, \u03b4\u03b1\u03b9, \u03b4\u03b1\u03bd, and their optical or eccentric equations, namely, \u03b2\u03bc\u03b1, \u03b2\u03b8\u03b1, \u03b2\u03be\u03b1, \u03b2\u03b9\u03b1, \u03b2\u03bd\u03b1. In the second, the distances themselves, \u03b1\u03bc, \u03b1\u03b8, \u03b1\u03be, \u03b1\u03b9, \u03b1\u03bd, are placed together opposite [the mean anomalies]. In the third, under the heading \"equated anomaly,\" are tabulated angles not depicted here, but the procedure for whose generation will be revealed in part here and in part in Chapters 31 and 40. For they are constituted by subtraction of the optical equations \u03b1\u03bc\u03b2 and so forth from \u03b4\u03b1\u03bc and so forth. Thus we have given no column to the integral angles like \u03b4\u03b1\u03bc, because they are the arithmetic mean between the angles of the columns at the sides, and thus are easily found in themselves, and are not of any use, as we shall hear.","659":"Therefore, beginning with either the mean or the equated anomaly, either one going into its proper respective column as use will dictate; or, where it exceeds a semicircle, beginning with the full-circle complement of either of these; you find the requisite distance of the sun from the earth, in units of which the radius of the orb is 100,000 and the eccentricity is 1800.","660":"It is true that in this way (that is, in associating the distance \u03b1\u03b6 of the angle \u03b4\u03b1\u03b6 with an angle which is as much smaller than \u03b4\u03b1\u03b6 as \u03b4\u03b1\u03b6 is smaller than \u03b4\u03b2\u03b6) a path is attached to the circuit of the earth (or sun) about \u03b1 which is oval rather than exactly circular. For the distance \u03b1\u03b6 (for example) was determined by the angle \u03b4\u03b1\u03b6, an integral 90\u00b0, and it was assumed in the operation that this angle \u03b4\u03b1\u03b6 was the equated anomaly. Now, however, you are told to get the distances using the angles of the anomaly that is called \"equated\" in our table, which have been diminished by the equation \u03b2\u03b6\u03b1. It thus happens that at 90\u00b0 you do not get 99,984, although you would previously have determined it to be 99,984. For here, opposite 99,984, you find an equated anomaly of 88\u00b0 58' 7\", which is not your value. For it was 90\u00b0 that was proposed, which, standing farther down, shows [a distance of] 99,953, while, according to the law of the circle, \u03b1\u03b6 or \u03b1\u03b7 should be 99,984. So all the distances are diminished at the sides, most greatly about \u03b6, \u03b7, none at \u03b4, \u03b5. Clearly, an oval is thus substituted for the circular path. You will obtain the same result if you begin with a mean anomaly obtained from whatever source. For when the diagram was set out above, the mean anomaly denoted angles about \u03b3. But now you will begin with angles about \u03b2, smaller than the optical equation about \u03b3. And 91\u00b0 1' 53\" of mean anomaly shows you a distance of 99,984. But that was the magnitude of \u03b4\u03b2\u03b6 above. Nor was the mean anomaly there, for it was \u03b4\u03b3\u03b6, which is still greater. So the former mean anomaly of 91\u00b0 1\u2019 53\" had generated a longer distance there than a mean anomaly of the same magnitude, 91\u00b0 1\u2019 53\", shows here. All this, I say, is true. But there is no reason why you should be brought up short. For since we are considering differences of one degree, you see that the distances within one degree vary no more than 31 parts in one hundred thousand. Therefore, there would be no perceptible error, even if what has been done were to be out of the proper order. Below, in Chapter 44 and the following, by analogy with the rest of the planets, you will find the reason for adapting this arrangement to the theory of the sun as well. Therefore, this is not out of order, but perfectly correct, because it concerns the qualitative nature of the figure that the planet describes, which has been substituted.","661":"As regards the quantity of the figure, the cure is excessive. The equated anomaly of 88\u00b0 58\u2019 7\", to which corresponds a mean anomaly of 91\u00b0 1' 53\", ought not to show a value of 99,984, but 100,000, which is the mean between the distances of the figure and of the table. The reason for this assertion must be postponed until Chapter 55 and the following.","662":"It was just said, however, that we shall not err perceptibly if we err by 31 units. It will therefore hurt us much less perceptibly if we err by only half that, or 16 units. So for the time being we safely admit this small error, in order to accommodate ourselves to an understanding of what has been advanced in the reading so far, and to avoid appearing to presuppose what was to be demonstrated. ","663":"Chapter 31","664":"That the bisection of the sun's eccentricity does not perceptibly alter the equations of the sun set out by Tycho; and concerning four ways of computing them.","665":"But lest there remain any suspicions preventing our moving onwards, we shall investigate, in the usual Ptolemaic form of the first inequality,* whether there be any difference in the solar equations consequent upon the now bisected eccentricity.","666":" First let there be an unbisected eccentricity of 3600 on the line of apsides AF, with CE and CD accordingly radii of the orb; and let the anomaly FAE be 45\u00b0, and FAD 135\u00b0. Now it is obvious that, however great the discrepancy may be, it will reach its maximum around these positions of anomaly. For in the middle elongations the equations come out exactly the same, since [an eccentricity of] 3600, when investigated in both the sines and tangents","667":"1 produces the same arc. Therefore, as the radius CE is to the sine of the angle CAE or CAD, so is the eccentricity CA to the equation CEA or CDA, which are both 1\u00b0 27' 31\". And in this first way, Ptolemy computed the equations of the sun, and, following Ptolemy, Copernicus; and, following both, Brahe: each of them using only the eccentricity AC, whose magnitude they found through their observations.","668":"There now follows a second way of computing the same equations, of which Ptolemy made use in the other planets, and of which I should make use. For I have demonstrated in this third part that the center of the eccentric is not at the point C, the center of uniform motion, but at B, the midpoint between the center of the world A and the point of uniformity C.","669":"Therefore let CA be bisected at B, and let EB, BD, be the radius of the orb. By the same method, the part of the equation BDA, BEA, will be 0\u00b0 43' 46\", which, added to EAB, DAB, will result in an angle EBC of 45\u00b0 43' 46\", and DBC, of 135\u00b0 43' 46\". As a result, from the sides and the included angle, BEC comes out to be 43' 38\", and BDC 43' 42\". Thus the whole angle CEA is 1\u00b0 27' 24\", and CDA is 1\u00b0 27' 28\", within a hair's breadth of the previous value. And so, in the appendix to Tycho Brahe's Progymnasmata p. 821, where the difference of the two calculations is given as 1\u2159\u2019, you should read \u2159\u20192. And this is in accord with the conclusions of Chapter 4 applied to the form of the vicarious hypothesis.","670":"And since you may observe that in this particular form of Ptolemaic hy","671":"pothesis the parts of the equation are nearly equal (for the optical part was 43' 46\" and the physical part was 43' 38\" at E and 43' 42\" at D), you see why, in the construction of the table in the preceding chapter, all I did was double the equation to establish the total equation. And this is the third way of computing the sun's equations. For at apogee and perigee both parts of the equation vanish, and in the middle elongations the parts are again equal, as was just now said. Therefore, since these three ways of computing the equations clearly coincide in eight places distributed about the entire circle, they will perceptibly coincide everywhere. This is a result of the smallness of the eccentricity: if it ","672":"were greater, this coincidence would not take place everywhere.","673":"Now I shall prepare myself for finding yet a fourth way to the equation,","674":"to be computed not through an arbitrary hypothesis but from the very nature of things, in eight chapters. Accordingly, this fourth way can finally follow in Chapter 40. ","675":"Chapter 32","676":"The power that moves the planet in a circle diminishes with removal from its source.","677":"I have said above that Ptolemy, well informed by the observations, bisected the eccentricities of the three superior planets, that Copernicus imitated this, and that Tycho's observations of Mars also urge the same conclusion, as has been seen in Chapters 19 and 20, and will appear with much greater certainty below in Chapter 42. In addition, Tycho closely imitated this in his lunar theory. And now the same thing has been demonstrated in the theory of the sun (for Tycho) or of the earth (for Copernicus). Further, there is nothing to prevent our believing the same of Venus and Mercury. Indeed, I now take it as proven that this is the origin of the belief that the centers of these planets' eccentrics move around on a small annual circle. Therefore all planets have this [double eccentricity]. Now in my Mysterium cosmographicum, published eight years ago,* I postponed arguing this case of the cause of the Ptolemaic equant for the sole reason that it could not be said on the basis of ordinary astronomy whether the sun or earth uses an equalizing point and has its eccentricity bisected. However, now that we have the confirmation of a sounder astronomy, it should be transparently clear that there is indeed an equant in the theory of the sun or earth. And, I say, now that this is demonstrated, it is proper to accept as true and legitimate the cause to which I assigned the Ptolemaic equant in the Mysterium cosmographicum, since it is universal and common to all the planets. So in this part of the work I shall make a further declaration of that cause.","678":"And since the declaration will be general, I shall use the word, \"planet\". However, in this and the next few chapters, the reader may always understand by this, in particular, the earth for Copernicus or the sun for Tycho.","679":"First, the reader should know that in every hypothesis constructed according to this Ptolemaic form, however great the eccentricity, the speed at perihelion and slowness at aphelion are very closely proportional to the lines drawn from the center of the world to the planet.","680":"In the diagram of Chapter 29, in which \u03b1 was the center of the world, \u03b2 was the center of the eccentric \u03b4\u03b5, and \u03b3 was the point of the equant, let the equant circle \u03c5\u03c6 be described about center \u03b3, with radius equal to \u03b2\u03b4. And through the center of the world \u03b1, from which the eccentricity is reckoned (and in the business at hand, it is the sun for Copernicus and the earth for the others), let the straight line \u03c8\u03c9 be drawn, intersecting the eccentric at \u03c8 and \u03c9, so that the planet is at \u03c8 and \u03c9, having traversed the arcs of the eccentric \u03b4\u03c8 and \u03b5\u03c9, from apogee or aphelion and from perigee or perihelion, respectively. It is supposed that these arcs appear equal from \u03b1, since the straight line \u03c8\u03c9 makes the vertical angles \u03c8\u03b1\u03b4 and \u03c9\u03b1\u03b5, which are equal. But since \u03b4\u03c8 and \u03b5\u03c9 are taken as minimal arcs, as if at the apsides \u03b4 and \u03b5, they do not differ perceptibly from straight lines. And so, just as \u03b4\u03b1\u03c8 and \u03b5\u03b1\u03c9 were rectilinear triangles, with right angles at \u03b4 and \u03b5, and a common vertex \u03b1, \u03b4\u03b1 will be to \u03b5\u03b1 as arc \u03b4\u03c8 is to arc \u03b5\u03c9. But \u03b1\u03b4 is longer than \u03b1\u03b5. Therefore, arc \u03b4\u03c8 is longer than \u03b5\u03c9. These arcs, which are in fact unequal, appear equal from \u03b1. The question now is, how much time will the planet take to traverse each arc, according to Ptolemy's theory and hypothesis, when it has an equant? So let straight lines be drawn from the center \u03b3 through the points \u03c8 and \u03c9, intersecting the equant at \u03c7, \u03c4. Now Ptolemy will say, \"since the whole circle of the equant \u03c5\u03c6 denotes the periodic time of the planet, then \u03c5\u03c7 is the measure of the time which the planet takes to traverse the arc of the eccentric \u03c8\u03b4, and \u03c6\u03c4 is the measure of the time which the planet takes to traverse the arc of the eccentric \u03b5\u03c9.\" ","681":"And now I myself say that \u03c5\u03c7, thus designated as the arc of the time (as Ptolemy wished) is to the arc \u03b4\u03c8 which the planet traverses, very nearly as \u03b1\u03b4, the distance of the arc \u03b4\u03c8 from the center of the world, is to \u03b4\u03b2 the mean distance of the points \u03c0 and \u03c1 from \u03b1. And likewise, the arc of the time \u03c6\u03c4 is to the arc of the planet's motion \u03b5\u03c9, approximately as \u03b1\u03b5, the distance of the arc \u03b5\u03c9 from the center of the world \u03b1 is to \u03b5\u03b2 and \u03b1\u03c0, the mean distance from the center of the world, which may be found at the points \u03c0 and \u03c1. And now, as before, as \u03b3\u03c5 is to \u03b3\u03b4 so is \u03c5\u03c7 to \u03b4\u03c8, and as \u03b3\u03c6 is to \u03b3\u03b5 so is \u03c6\u03c4 to \u03b5\u03c9. But \u03b3\u03c5 is to \u03b3\u03b4 very nearly as \u03b2\u03b4 (or \u03b3\u03c5) is to \u03b1\u03b4, and this is shown by the fact that \u03b2\u03b4 is the arithmetic mean between \u03b3\u03b4 and \u03b1\u03b4, since Ptolemy makes \u03b1\u03b2, \u03b2\u03b3 equal. And further, the arithmetic mean between two terms whose ratio is near equality is only imperceptibly greater than the geometric mean. For example, the arithmetic mean between 10 and 12 is 11, and the geometric mean is about 1019\/20, so that there is less than the twentieth part of a unit between the two means. Nevertheless, these numbers are of the order of the eccentricity of Mars, which according to Ptolemy has the greatest eccentricity of all the planets.","682":"And therefore, since the ratio \u03b3\u03c5 to \u03b3\u03b4 is imperceptibly greater than the ratio \u03b1\u03b4 to \u03b4\u03b2, \u03c7\u03c5 will also have to \u03c8\u03b4 a ratio imperceptibly greater than \u03b1\u03b4 to \u03b4\u03b2. Likewise, as \u03b3\u03b5 is to \u03b3\u03c6, so is \u03b5\u03c9 to \u03c6\u03c4, but \u03b3\u03b5 is to \u03b3\u03c6 approximately as \u03b5\u03b2 is to \u03b1\u03b5; that is, the former ratio is only imperceptibly less than the latter. Therefore, too, the ratio \u03b5\u03c9 to \u03c6\u03c4 is only imperceptibly smaller than the ratio \u03b5\u03b2 to \u03b1\u03b5.","683":"Now let us permute the ratios. For the ratio \u03b1\u03b4 to \u03b4\u03b2 is imperceptibly less than the ratio \u03b4\u03b2 or \u03b2\u03b5 to \u03b5\u03b1, seeing that \u03b2\u03b4 or \u03b2\u03b5 is the arithmetic mean between \u03b1\u03b4 and \u03b1\u03b5, as before. But it was proved that the ratio \u03c5\u03c7 to \u03b4\u03c8 is greater than the ratio \u03b1\u03b4 to \u03b4\u03b2, from the smaller pair, and the ratio \u03b5\u03c9 to \u03c6\u03c4 is less than the ratio \u03b5\u03b2 to \u03b1\u03b5, from the greater pair, so that in the two ratios \u03b1\u03b4 to \u03b4\u03b2 and \u03b5\u03b2 to \u03b1\u03b5, the amount by which the former terms will be greater and the latter smaller is, in the two ratios \u03c5\u03c7 to \u03b4\u03c8 and \u03b5\u03c9 to \u03c6\u03c4, the same as the amount by which the former is greater and the latter smaller. Consequently, there is some compensation even for that imperceptible difference, so that it is much more nearly true that the ratio of \u03c5\u03c7 to \u03b4\u03c8 is equal within a hair's breadth to the ratio of \u03b5\u03c9 to \u03c6\u03c4.","684":"Therefore, the arcs \u03b4\u03c8 and \u03b5\u03c9 being taken as equal (which hitherto were unequal), either \u03b4\u03c8 or \u03b5\u03c9 will be a mean proportional between \u03c5\u03c7, the increment of time at aphelion, and \u03c6\u03c4, the increment of time at perihelion. Consequently, the ratio \u03c5\u03c7 to \u03c6\u03c4 (\u03b4\u03c8 and \u03b5\u03c9 being equal) will be in the duplicate ratio of \u03b1\u03b4 to \u03b4\u03b2 or \u03b2\u03b5 to \u03b5\u03b1, the former smaller and the latter greater by an imperceptible difference. But since the ratio \u03b1\u03b4 to \u03b1\u03b5 is also the duplicate of either of these (for it is compounded of the two, which are nearly equal, taking the arithmetic mean \u03b4\u03b2 or \u03b2\u03b5), therefore, the arcs on the eccentric \u03b4\u03c8 and \u03b5\u03c9 being equal, the ratio of the times \u03c5\u03c7 to \u03c6\u03c4 will be equal to the ratio \u03b1\u03b4 to \u03b1\u03b5. Or, more clearly, the planet takes a proportionally longer time to traverse some particular eccentric arc at \u03b4 than to traverse an equal eccentric arc at \u03b5, according as \u03b1\u03b4 is greater than \u03b1\u03b5. And this follows from the way the Ptolemaic form* is ordered, and from its equalizing point, by means of a proof that is certain and valid so far as it concerns points near apogee and perigee. At other points there appears a very small discrepancy, which is all the smaller in effect the more obvious it is in the demonstration. This is because (for example) the ratio \u03b1\u03bc to \u03b1\u03bd is smaller, and the ratio \u03b1\u03b8 to \u03b1\u03b9 is much smaller, than \u03b1\u03b4 to \u03b1\u03b5, the greatest ratio, where the effect is also greatest.","685":"Chapter 33","686":"The power that moves the planets resides in the body of the sun","687":"It was demonstrated in the previous chapter that the increments of time of a planet on equal parts of the eccentric circle (or on equal distances in the aethereal air) are in the same ratio to each other as the distances of those same spaces from the point whence the eccentricity is reckoned; or, more simply, to the extent that a planet is farther from the point that is taken as the center of the world, it is less strongly urged to move about that point. It is therefore necessary that the cause of this weakening is either in the very body of the planet, and in a motive force placed therein, or right at the supposed center of the world.","688":"Now it is an axiom of the most common application in all of natural philosophy that of those things which occur at the same time and in the same manner, and which are always subject to like measurements, either one is the cause of the other or both are effects of the same cause. Just so, in this instance, the intension and remission of motion is always in the same ratio as the approach and recession from the center of the world. Thus, either that weakening will be the cause of the star's motion away from the center of the world, or the motion away will be the cause of the weakening, or both will have some cause in common. But no one can think up some third concurrent thing that would be the common cause of these two, and in the following chapters it will become clear that we have no need of feigning any such cause, since the two are sufficient in themselves.","689":"Further, it is not in accord with nature that strength or weakness in longitudinal motion should be the cause of distance from the center. For distance from the center is prior both in thought and in nature to longitudinal motion. Indeed, longitudinal motion is never independent of distance from the center, since it requires a space in which to be performed, while distance from the center can be conceived without motion. Therefore, distance will be the cause of intensity of motion, and a greater or lesser distance will result in a greater or lesser amount of time.","690":"And since distance belongs to the class of related things, whose being depends upon end points, while relation itself, without respect to end points, has no efficacy, it therefore follows (as has been said) that the cause of the variation of intensity of motion inheres in one or the other of the end points.","691":"Now the body of a planet is never by itself made heavier in receding, nor lighter in approaching.","692":"Moreover, that an animate force, which is seated in the mobile body of the planet and imparts a motion to the heavenly body, undergoes intension and remission so many times without ever becoming tired or growing old,\u2014this will surely be absurd to say. Also, it is impossible to understand how this animate force could carry its body through the spaces of the world, since there are no solid orbs, as Tycho Brahe has proved. And on the other hand, a round body lacks such aids as wings or feet, by the moving of which the soul might carry its body through the aethereal air as birds do in the atmosphere, by some kind of pressure upon, and counter-pressure from, that air. ","693":"Therefore, the only remaining possibility is that the cause of this weakening and intension resides in the other endpoint, namely, in that point that is taken to be the center of the world, from which the distances are measured.","694":"So now, if the distance of the center of the world from the body of a planet governs its slowness, and approach governs its speeding up, it is a necessary consequence that the source of motive power is at that supposed center of the world. And with this laid down, the manner in which the cause operates will also be clear. For it gives us to understand that the planets are moved rather in the manner of the steelyard or lever. For if the planet is moved with greater difficulty (and hence more slowly) by the power at the center when it is farther from the center, it is just as if I were to say that where the weight is farther from the fulcrum, it is thereby rendered heavier, not of itself, but by the power of the arm supporting it at that distance. And this is true, both of the steelyard or lever, and of the motion of the planets: that the weakening of power is in the ratio of the distances.","695":"But which body is it that is at the center? Is there none, as for Copernicus when he is computing, and for Tycho in part? Is it the earth, as for Ptolemy and for Tycho in part? Or finally, is it the sun itself, as I, and Copernicus when he is speculating, would have it? This question I began to discuss in physical terms in Part 1. I there supposed as one of the principles what has now been expressly and geometrically proved in Chapter 32: that a planet is moved less vigorously when it recedes from the point whence the eccentricity is computed.","696":"From this principle I presented a probable argument that the sun, rather than being at some other point occupied by no body, is at that point and at the center of the world (or the earth for Ptolemy). Allow me, then, to repeat that same probable argument, our foundation [for it] now demonstrated, in the present chapter. Then, you may remember, I demonstrated in the second part, that the phenomena at either end of the night follow beautifully if we reckon the oppositions of Mars according to the sun's apparent position. If this is done, then at the same time we set up the eccentricity and the distances from the very center of the sun's body, with the result that the sun itself again comes to be at the center of the world (for Copernicus), or at least at the center of the planetary system (for Tycho). But of these two arguments, one depends upon physical probability, and the other proceeds from possibility to actuality. And so in the third place I have postponed, because of its conceptual difficulty, demonstrating from the observations that we cannot avoid referring Mars to the apparent position of the sun, and drawing the line of apsides, which bisects the eccentric, directly through the sun's body, unless perhaps we wish to allow an eccentric such as will by no means be in accord with the parallax of the annual orb. Anyone who cannot tolerate the delay may read about this in chapter 52, and then may carry on here afterwards. For there nothing is assumed but the bare observations. You will find a similar proof in Part 5, from considerations of latitudes.","697":" Therefore, with the sun belonging in the center of the system, the source of motive power, from what has now been demonstrated, belongs in the sun, since it too has now been found to be in the center of the world.","698":"But indeed, if this very thing which I have just demonstrated a posteriori (from the observations) by a rather long deduction, if, I say, I had taken this as something to be demonstrated a priori (from the worthiness and eminence of the sun), so that the source of the world's life (which is visible in the motion of the heavens) is the same as the source of the light which forms the adornment of the entire machine, and which is also the source of the heat by which everything grows, I think I would have deserved an equal hearing.","699":" Let Tycho Brahe himself, or anyone who prefers to follow his general hypothesis of the second inequality, consider by how close a likeness to the truth this physically elegant combination has for the most part been accepted (since for him, too, this substitution of the apparent position of the sun brings the sun back to the center of the planetary system) yet to some extent recoils from his hypothesis.","700":"For it is obvious from what has been said that only one of the following can be true: either the power residing in the sun, which moves all the planets, by the same action moves the earth as well; or the sun, together with the planets linked to it through its motive force, is borne about the earth by some power which is seated in the earth.","701":"Now Tycho himself destroyed the notion of real orbs, and I in turn have in this Third Part irrefutably demonstrated that there is an equant in the theory of the sun or earth. From this it follows that the motion of the sun itself (if it is moved) is intensified and remitted according as it is nearer or farther from the earth, and hence it would follow that the sun is moved by the earth. But if, on the other hand, the earth is in motion, it too will be moved by the sun with greater or less velocity according as it is nearer or farther from it, while the power in the body of the sun remains perpetually constant. Between these two possibilities, therefore, there is no intermediate.","702":"I myself agree with Copernicus, and allow that the earth is one of the planets.","703":"Now it is true that the same objection may be raised against Copernicus concerning the moon, that I raised against Tycho concerning the five planets; namely, that it appears absurd for the moon to be moved by the earth, and to be associated with it and bound to it as well, so that it too, as a secondary planet, is swept around the sun by the sun. Nevertheless, I prefer to allow one moon, akin to the earth in its corporeal disposition2 (as I have shown in the Optics3) to be moved by a power seated in the earth but extended towards the sun, as will be described a little later in Chapter 37, than to ascribe to that same earth as well the motion of the sun and of all the planets bound to it.","704":"But let us carry on in our consideration of this motive power residing in the sun, and let us now again observe its very close kinship with light.} ","705":"Since the perimeters of similar regular figures, even of circles, are to one another as their semidiameters, therefore as \u03b1\u03b4 is to \u03b1\u03b5, so is the circumference of the circle described about \u03b1 through \u03b4 to the circumference of the circle described about the same point \u03b1 through \u03b5. But as \u03b1\u03b4 is to \u03b1\u03b5, so is the strength of the power at \u03b5 to the strength of the power at \u03b4, inversely, by what was proved in Chapter 32. Therefore, as the circle at \u03b4 is to the smaller circle at \u03b5, so, inversely, is the power at \u03b5 to the power at \u03b4; that is, the power is weaker to the extent that it is more spread out, and stronger to the extent that it is more concentrated. Hence we understand that there is the same power in the whole circumference of the circle through \u03b4 as there is in the circumference of the smaller circle through \u03b5. This was shown to be true of light in exactly the same way in the Astronomiae pars optica, Chapter 1. ","706":"Therefore, in all respects and in all its attributes, the motive power from the sun coincides with light.\n And although this light of the sun cannot be the moving power itself, I leave it to others to see whether light perhaps is equivalent to a kind of instrument or vehicle, of which the moving power makes use.","707":"This seems gainsaid by the following: first, light is hindered by opaque things, and therefore if the moving power had light as a vehicle, rest of the moving bodies would follow upon darkness; again, light spreads spherically in straight lines, while the moving power, though spreading in straight lines, does so circularly; that is, it is exerted in but one region of the world, from east to west, and not the opposite, not at the poles, and so on. But we shall perhaps be able to reply to these objections in the chapters immediately following.","708":" Finally, since there is just as much power in a larger and more distant circle as there is in a smaller and closer one, nothing of this power is lost in travelling from its source, nothing is scattered between the source and the movable body. The emission, then, in the same manner as light, is immaterial, unlike odours, which are accompanied by a diminution of substance, and unlike heat from a hot furnace, or anything similar which fills the intervening space. The remaining possibility, then, is that, just as light, which lights the whole earth, is an immaterial species5 of that fire which is in the body of the sun, so this power which enfolds and bears the bodies of the planets, is an immaterial species residing in the sun itself, possessing inestimable strength, seeing that it is the primary activity of every motion in the universe.","709":" Since, therefore, this species of the power, exactly as the species of light (for which see the Astronomiae pars optica Chapter 16), cannot be considered as dispersed throughout the intermediate space between the source and the mobile body, but is seen as collected in the body in proportion to the amount of the circumference the mobile body occupies, this power (or species) will therefore not be any geometrical body, but is like a variety of surface, just as light is. To generalize this, the species of things that emanate immaterially are not by that emanation extended through the dimensions of a body, although they arise from a body (as this one does from the body of the sun). Instead, they are extended according to that very law of emission: they are not bounded by the emanation itself, but just as the surfaces of things to be illuminated cause light to be considered as a kind of surface, because they receive and terminate its emission, so the bodies of things that are moved appear to bring it about that this moving power be considered as if a sort of geometrical body, because their whole masses terminate or receive this emission of the motive species, so that the species can exist or subsist nowhere in the world but in the bodies of the mobile things themselves. And, exactly like light, between the source and the movable thing it does not exist, but will have a quasi-existence.","710":" Moreover, at the same time, a reply can be made here to a possible objection. For it was said above that this motive power is extended throughout the space of the world, in some places more spread out and in others more concentrated, and that the intensification and remission of the motions of the planets are consequent upon this variation. Now, however, it has been said that this power is an immaterial species of its source, and never inheres in anything except a mobile subject, such as the body of a planet. But these appear to be contradictory: to lack matter and yet to be subject to geometrical dimensions; to be poured out throughout the whole world, and yet not to exist anywhere but where there is something movable.\nThe reply is this: although the motive power is not anything material, nevertheless, since it is designed for carrying matter (namely, the body of a planet), it is not free from geometrical laws, at least on account of this material action of carrying things about. Nor is there need for more, for we see that those motions are carried out in space and time, and that this power arises and emanates from the source through the space of the world, all of which are geometrical entities. So this power should indeed be subject to other geometrical necessities.","711":"But lest I appear to philosophize with excessive insolence, I shall propose to the reader the clearly authentic example of light, since it also makes its nest in the sun, thence to break forth into the whole world as a companion to this motive power. Who, I ask, will say that light is something material? Nevertheless, it carries out its operations with respect to place, suffers alteration, is reflected and refracted, and assumes quantities so as to be dense or rare, and to be capable of being taken as a surface wherever it falls upon something illuminable. Now just as it is said in optics, light too, and this motive power as well, does not exist in the intermediate space between the source and the illuminable, even though it passed through it, but quasi-existed there. Moreover, although light itself does indeed flow forth in no time, while this power creates motion in time, nonetheless the way in which both do so is the same, if you consider them correctly. Light manifests those things which are proper to it instantaneously, but so far as it is connected with matter, it too requires time. It illuminates a surface in a moment, because here matter need not undergo any alteration, for all illumination takes place according to surfaces, or at least as if a property of surfaces and not as a property of corporeality as such. On the other hand, light bleaches colors in time, since here it acts upon matter qua matter, making it hot and expelling the contrary cold which is embedded in the body's matter and is not on its surface. In precisely the same manner, this moving power perpetually and without any interval of time is present from the sun wherever there is a suitable movable body, for it receives nothing from the movable body to cause it to be there. On the other hand, it causes motion in time, since the movable body is material.","712":"Or if it seems better, frame the comparison in this manner: light is constituted for illumination, and it is just as certain that power is constituted for motion. Light does everything that can be done to achieve the greatest illumination; nonetheless, it does not happen that color is most greatly illuminated. For color intermingles its own peculiar species with the illumination of light, thus forming some third entity. In like manner, there is no slowing in the moving power to prevent the planet's having as much speed as it has itself, but the planet's speed is not therefore that great, since something intervening prevents that, namely, some sort of matter possessed by the surrounding aether, or the disposition of the movable body itself to rest (others might say, \"weight,\" but I do not entirely approve of that, except, indeed, where the earth is concerned). It is the tempering effect of these, together with the weakening of the motive power, that determines a planet's periodic time.","713":"Chapter 34","714":"The body of the sun is magnetic, and rotates in its space.","715":"Concerning that power that is closely attached to the bodies of the planets and pulls them, we have already said how it is formed, how it is akin to light, and what it is in its metaphysical being. Next, we should contemplate the deeper nature of its source, shown by the outflowing species (acting as an archetype). For it may appear that there lies hidden in the body of the sun something divine, which may be compared to our soul, from which flows that species driving the planets around, just as from the soul of someone throwing pebbles a species of motion comes to inhere in the pebbles thrown by him, even when he who threw them removes his hand from them. And to those who proceed soberly, other reflections will soon be provided.","716":"Now because the power that is extended from the sun to the planets moves them in a circular course around the immovable body of the sun, this cannot happen, or be conceived in thought, in any other way than this, that the power traverses the same path along which it carries the other planets. This has been observed to some extent in catapults and all violent motions. Thus, Fracastoro and others, relying on a story told by the most ancient Egyptians, spoke with little probability when they said that perchance some of the planets, their orbits being deflected gradually beyond the poles of the world, would thus afterwards move in a path opposite to the rest and to their modern course. For it is much more likely that the bodies of the planets are always borne in that direction in which the power emanating from the sun tends.","717":"But this species is immaterial, proceeding from its body out to this distance without the passing of any time, and is in all other respects like light. Therefore, it is not only required by the nature of the species, but likely in itself owing to this kinship with light, that along with the particles of its body or source it too is divided up, and when any particle of the solar body moves towards some part of the world, the particle of the immaterial species that from the beginning of creation corresponded to that particle of the body also always moves towards the same part. If this were not so, it would not be a species, and would come down from the body not in straight but in curved lines.","718":"Since the species is moved in a circular course, in order thereby to confer motion upon the planets, the body of the sun, or source, must move with it, not, of course, from space to space in the world\u2014for I have said, with Copernicus, that the body of the sun remains in the center of the world\u2014but upon its center or axis, both immobile, its parts moving from place to place, while the whole body remains in the same place.\n In order that the force of the analogical argument may be that much more evident, I would like you to recall, reader, the demonstration in optics that vision occurs through the emanation of small sparks of light2 toward the eye from the surfaces of the seen object. Now imagine that some orator in a great crowd of people, encircling him in an orb, turns his face, or his whole body along with it, once around. Those of the audience to whom he turns his eyes directly also see his eyes, but those who stand behind him then lack the view of his eyes. But when he turns himself around, he turns his eyes around to everyone in the orb. Therefore, in a very short interval of time, all get a glimpse of his eyes. This they get by the arrival of a small spark of light or species of color descending from the eyes of the orator to the eyes of the spectators. Thus by turning his eyes around in the small space in which his head is located, he carries around along with it the rays of the spark of light in the very large orb in which the eyes of the spectators all around are situated. For unless the spark of light went around, his spectators would not be recipients of his eyes' glance. Here you see clearly that the immaterial species of light either is moved around or stands still along with that of which it is the species either moving or standing still. ","719":" Therefore, since the species of the source, or the power moving the planets, rotates about the center of the world, I conclude with good reason, following this example, that that of which it is the species, the sun, also rotates.","720":"However, the same thing is also shown by the following argument. Motion that is local and subjected to time cannot inhere in a bare immaterial species, since such a species is incapable of receiving an effect of an applied motion unless the received motion is non-temporal, just as the power is immaterial. Also, although it has been proved that this moving power rotates, it cannot be allowed to have infinite speed (for then it would seem that the power will also impart infinite speed to the bodies), and therefore it completes its rotation in some period of time. Therefore, it cannot carry out this motion by itself, and it is as a consequence necessary that it is moved only because the body upon which it depends is moved.","721":"By the same argument, it appears to be a correct conclusion that there does not exist within the boundaries of the solar body anything immaterial by whose rotation the species descending from that immaterial something also rotates. For again, local motion which takes time is not correctly attributed to anything immaterial. It therefore remains that the body of the sun itself rotates in the manner described above, indicating the poles of the zodiac by the poles of its rotation (by a line extended from the center of the body to the fixed stars through the poles), and indicating the ecliptic by the greatest circle of its body, thus furnishing a natural cause for these astronomical entities.","722":"Further, we see that the individual planets are not carried along with equal swiftness at every distance from the sun, nor is the speed of all of them at their various distances equal. For Saturn takes 30 years, Jupiter 12, Mars 23 months, earth 12, Venus 8\u00bd, and Mercury 3. Nevertheless, it follows from what has been said that every orb of power emanating from the sun (in the space embraced by the lowest, Mercury, as well as that embraced by the highest, Saturn) is twisted around with a whirl equal to that which spins the solar body, with an equal period. (There is nothing absurd in this statement, for the emanating power is immaterial, and by its own nature would be capable of infinite speed if it were possible to impress a motion upon it from elsewhere, for then it could be impeded neither by weight, which it lacks, nor by the obstruction of the corporeal medium.) It is consequently clear that the planets are ill-suited to emulate the swiftness of the motive power. For Saturn is less receptive than Jupiter, since its returns are slower, while the orb of power at the path of Saturn returns with the same swiftness as the orb of power at the path of Jupiter, and so on in order, all the way to Mercury, which, by example of the superior planets, doubtless will move more slowly than the power that pulls it. It is therefore necessary that the nature of the planetary globes be material, from an inherent property, arising from the origin of things, to be inclined to rest or to the privation of motion. When the tension between these things leads to a fight, that planet which is placed in a weaker power overcomes it more, and is moved more slowly by it; that which is closer to the sun overcomes it less. ","723":"This analogy shows that there is in all planets, even in the lowest, Mercury, a material force of disengaging itself somewhat from the orb of the sun\u2019s power.","724":"From this it is concluded that the rotation of the solar body anticipates considerably the periodic times of all the planets; therefore, it must rotate in its space at least faster than once in a three-months\u2019 span.","725":"However, in my Mysterium Cosmographicum I pointed out that there is about the same ratio between the semidiameters of the sun\u2019s body and the orb of Mercury as there is between the semidiameters of the body of the earth and the orb of the moon. Hence, you may plausibly conclude that the period of the orb of Mercury would have the same ratio to the period of the body of the sun as the period of the orb of the moon has to the period of the body of the earth. And the semidiameter of the orb of the moon is sixty times the semidiameter of the body of the earth, while the period of the orb of the moon (or the month) is a little less than thirty times the period of the body of the earth (or day), and thus the ratio of the distances is double the ratio of the periodic times. Therefore, if the doubled ratio also holds for the sun and Mercury, since the diameter of the sun's body is about one sixtieth of the diameter of Mercury's orb, the time of rotation of the solar globe will be one thirtieth of 88 days, which is the period of Mercury's orb. Hence it is likely that the sun rotates in about three days.","726":" You may, on the other hand, prefer to prescribe the sun's diurnal period in such a way that the diurnal rotation of the earth is dispensed by the diurnal rotation of the sun, by some sort of magnetic force. I would certainly not object. Such a rapid rotation appears not to be alien to that body in which lies the first activation of every motion.","727":" This opinion (on the rotation of the solar body as the cause of the motion for the other planets) is beautifully confirmed by the example of the earth and the moon. For the chief, monthly motion of the moon, by the force of the demonstrations used in Chapters 32 and 33, is entirely derived from the earth as its source (for what the sun is for the rest of the planets there, the earth is for the moon in this demonstration). Consider, therefore, how our earth occasions the motion of the moon: while this our earth, and its immaterial species along with it, rotates twenty nine and one half times about its axis, at the moon this emitted species can drive it around only once in the same time, in (of course) the same direction in which the earth leads it.","728":"Here, by the way, is a marvel: in any given time the center of the moon traverses twice as long a line about the center of the earth as any place on the surface of the earth beneath the great circle of the equator. For if equal spaces were measured out in equal times, the moon ought to return in sixty days, since the size of its orb is sixty times the size of the earth's globe.","729":"This is surely because there is so much force in the immaterial species of the earth, while the lunar body is doubtless of great rarity and weak resistance. Thus, to remove your bewilderment, consider that on the principles we have supposed it would necessarily follow that if the moon were not to resist, by its material force, the motion impressed from outside by the earth, the moon would be carried at exactly the same speed as the earth's immaterial species, that is, with the earth itself, and would complete its circuit in 24 hours, in which the earth also completes its circuit. For even if the tenuity of the earth's species is great at the distance of 60 semidiameters, the ratio of one to nothing is still the same as the ratio of sixty to nothing. Hence the immaterial species of the earth would prevail completely, if the moon did not resist.","730":" Here, one might inquire of me, what sort of body I consider the sun to be, from which this motive species descends. I would ask him to proceed under the guidance of a further analogy, and urge him to inspect more closely the example of the magnet brought up a little earlier, whose power resides in the entire body of the magnet, grows with its mass, and is itself also divided when the magnet is diminished. So in the sun the moving power appears so much stronger that it seems likely that its body is of all [those in the world] the most dense.","731":"And the power of attracting iron is spread out in an orb from the magnet so that there exists a certain orb within which iron is attracted, but more strongly so as the iron comes nearer into the embrace of that orb. In exactly the same way the power moving the planets is propagated from the sun in an orb, and is weaker in the more remote parts of the orb.","732":"The magnet, however, does not attract with all its parts, but has filaments (so to speak) or straight fibers (seat of the motor power) extended throughout its length, so that is a little strip of iron is placed in a middle position between the heads of the magnet at the side, the magnet does not attract it, but only directs it parallel to its own fibers. Thus it is credible that there is in the sun no force whatever attracting the planets, as there is in the magnet, (for then they would approach the sun until they were quite joined with it), but only a directing force, and consequently that is has circular fibers all set up in the same direction, which are indicated by the zodiac circle ","733":"Therefore, as the sun forever turns itself, the motive force or the outflowing of the species from the sun\u2019s magnetic fibers, diffused through all the distances of the planets, also rotates in an orb, and does so in the same time as the sun, just as when a magnet is moved about, the magnetic power is also moved, and the iron along with it, following the magnetic force.\nThe example of the magnet I have hit upon is a very pretty one, and entirely suited to the subject; indeed, it is little short of being the very truth. So why should I speak of the magnet as if it were an example? For, by the demonstration of the Englishman William Gilbert, the earth itself is a big magnet, and is said by the same author, a defender of Copernicus, to rotate once a day, just as I conjecture about the sun. And because of this very thing, that it has magnetic fibers intersecting the line of its motion at right angles, those fibers therefore lie in various circles about the poles of the earth parallel to its motion. I am therefore absolutely within my rights to state that the moon is carried along by the rotation of the earth and the motion of its magnetic power, only thirty times slower.","734":"I know that the earth\u2019s filaments and its motion indicate the equator, while the circuit of the moon is generally related to the zodiac \u2013 on this point there will be more in Chapter 37 and Part 5. With this one exception, everything fits: the earth is intimately related to the lunar period, just as the sun is to that of the other planets. And just as the planets are eccentric with respect to the sun, so is the moon with respect to the earth. So it is certain that the earth is looked upon by the moon\u2019s mover as a kind of pole star (so to speak), just as the sun is looked upon by the movers belonging to the rest of the planets, for which see Chapter 38. It is therefore plausible, since the earth moves the moon through its species and is a magnetic body, while the sun moves the planets similarly through an emitted species, that the sun is likewise a magnetic body.","735":"Chapter 35","736":"Whether the motion from the sun, like its light, is subject to privation in the planets through occultations. ","737":"This is a good time for me to take up the objections raised in Chapter 33, where to the kinship of light and motive power were opposed, first, the mutual occultations of the celestial bodies, and then, the different [manner of] emanation of the species of the two.","738":"And concerning the first, it is worthy of consideration whether, just as one opaque body intercepts another's sunlight, mobile bodies similarly impede one another in motion when they lie in line with the sun. If so, light would clearly be the vehicle or instrument of the motive power.","739":" It might appear that in order to avoid this as much as possible, God introduced the relative inclinations of all the eccentrics, deviations from the ecliptic, and transpositions of the nodes, as well as the proportions of the bodies and the attenuation of shadows in a cone. And since it would not be possible completely to prevent the stars' occasionally lining up with the sun, it is tempting to suppose that the very slow motions of the apsides and nodes (which are, as it were, a kind of aberration of the epicycles from their periodic times) derive their origin thence.","740":"But it is answered, first, that the analogy between light and motive power is not to be disturbed by rashly confusing their properties. Light is impeded by the opaque, but is not impeded by a body, because light is light, and does not act upon the body but on the surface (or as if on the surface). Power acts upon the body without respect to its opacity. Therefore, since it is not correlated with the opaque, it will likewise not be impeded by the opaque.","741":"On this account I would nearly separate light from moving power, unless I were to come upon examples in nature which leave to the rays of light, even when impeded, a certain efficacy in those locations where their entry is prohibited. But I am not chiefly concerned here with the association of light with the motive power.","742":"Instead, in order to dissolve this suspicion that the motions are impeded, let us take another example from the magnet. Its power is not at all impeded by the interposition of matter (because, of course, it is immaterial), but passes through sheets of silver, copper, gold, glass, bone, wood, and attracts iron lying beyond these sheets exactly as if no sheets were there. Granted, it is impeded by the interposition of a magnetic plate. But the cause is ready at hand: the plate acts as a counterpart to the magnet. It therefore overcomes, by its strength, the more distant magnet lying beyond it. And although it is impeded by the interposition of the iron plate, this too belongs to the magnetic nature, and it immediately drinks up the magnet's power and uses it as if it were its own.","743":"Therefore, in order that we may deny that the motion of the celestial bodies is impeded by central conjunctions of two of them, we must say that the nature of the sun differs from the nature of the rest of the celestial bodies more than the nature of the magnet differs from the nature of iron. Also, we must deny that the planets drink up the power all at once from the sun in the same way as the iron drinks it from the magnet. The question of whether they drink up any of it at all, I defer to Chapter 57.","744":"In respect to the likelihood of the cause of the apogees' motion, it proves nothing concerning this common solar power being impeded by occultation. For the motion of the apogees could be quite different; for instance, it could have an animate origin. You will find a certain obscure opinion on this point below in Chapter 57. ","745":"Further, if the motion of the apogees were to arise because the motion of the planets around the sun is impeded by the occultation of the motive species emanating from the sun, the motion in longitude would be slowed, either by a progressive motion of latitude (by which the apogees would move back) or equally by slowing of the latitudinal motion. Thus, the apogees would stand still, although the observations testify that they move forward.","746":"But the question whether, once the sun is preserved as the source of motion, the motions proper to the celestial bodies are impeded by occultation, will also be discussed in Chapter 57.","747":"Chapter 36","748":"By what measure the motive power from the sun is attenuated as it spreads through the world.","749":"There follows another, rather more difficult objection, arising from the second argument that was raised in Chapter 33 against the kinship of light and motive power, which seems irreconcilably at odds with our study of immaterial species. This objection wearied me for a long time without offering any prospect of solution.","750":" It was demonstrated in Chapter 32 that the intension and remission of the planets' motion is in simple proportion to the distances. It appears, however, that the power emanating from the sun should be intensified and remitted in the duplicate or triplicate ratio of the distances or lines of efflux. Therefore, the intension and remission of the planets' motion will not be a result of the attenuation of the power emanating from the sun. The logical consequence appears to be proved in the following manner, for light as well as for moving power; however, discussions of light are clearer. The reader should read in \"motive power\". Initially, let \u03b1 be any point on the sun's body. It will therefore spread out its rays to every orb, and from a demonstration in optics, as the amplitude of the greater spherical surface \u03b3, considered as an imaginary terminus for those rays, is to the amplitude of the smaller, \u03b2, so will the density of light at the smaller orb \u03b2 be to its density at the greater \u03b3.","751":"Next, let \u03b4\u03b5 be any luminous great circle on the body of the sun. Thus its individual points, of which there are an infinity, spread out rays to the individual hemispheres \u03b2 and \u03b3 in the same ratio. And the apparent magnitude of the diameter of the circle at the shorter distance (i.e., the angle \u03b4\u03b2\u03b5) is to the apparent magnitude at the greater distance (i.e., the angle \u03b4\u03b3\u03b5), conversely, as the longer distance \u03b1\u03b3 from such a circular line (which at a distance appears straight) is to the shorter \u03b4\u03b2. So, since this diameter appears longer from the nearby point \u03b2 than from the distant point \u03b3, in the same ratio, while the radiation belonging to any given point is denser at a nearby point \u03b2 than at a distant point \u03b3, it is therefore apparent that the density of radiation of the circle at the nearby point \u03b2 will have to the density of radiation at the distant point \u03b3 the [inverse of the] duplicate ratio of \u03b1\u03b2 to \u03b1\u03b3.","752":"Thirdly, let \u03b4\u03b1\u03b5 be the apparent disc of the sun's body, and since similar surfaces (as are the apparent circular discs here) are in the duplicate ratio of their diameters, while the apparent diameters of the sun are in the simple inverse ratio of the distances \u03b1\u03b3, \u03b1\u03b2, the circular discs will therefore appear in the duplicate ratio of the distances \u03b1\u03b3, \u03b1\u03b2. But since the radiation of the circle \u03b4\u03b5 at \u03b3 and \u03b2 was just proved to be in the duplicate ratio of the distances \u03b1\u03b2, \u03b1\u03b3, while both are causes of the density [of radiation], it appears that the radiation of the disc, with respect to density or strength, is in the triplicate ratio of the distances \u03b1\u03b3, \u03b1\u03b2.","753":"For example, if the distances \u03b1\u03b3 to \u03b1\u03b2 were as 2 to 1, the radiations of the point at \u03b1\u03b3 and \u03b1\u03b2 would be as 1 to 2, with respect to the density of light, and the apparent diameters of circles would be 1 at \u03b3 and 2 at \u03b2.","754":"Therefore, the radiation of the diameter of the circle \u03b4\u03b5 would be 1 at \u03b3 and 4 at \u03b2. But the discs are in the duplicate ratio of the diameters. Therefore the apparent magnitude of the disc at \u03b3 would be 1, and at \u03b2, 4, as if you were to say that the disc \u03b4\u03b1\u03b5 when seen from \u03b2 appears to contain four times as many points as when seen from \u03b3. Any of these points illuminates twice as densely at \u03b2 than at \u03b3. Therefore, when the ratios are compounded, the density of radiation of the whole disc \u03b4\u03b1\u03b5 at \u03b3 would have to the density of radiation of the whole disc \u03b4\u03b1\u03b5 at \u03b2 the ratio 1 to 8.","755":"It does not trouble us here that we are reckoning from the sun's apparent disc, although it is a spherical surface. For the ratio of equally many things to each other mutually is the same. But the spherical surface was demonstrated by Archimedes to be four times the area of the greatest circle described in the sphere. Therefore, a body at \u03b3, twice as far away as at \u03b2, should have appeared to be fully eight times as obscurely illuminated at \u03b3 as at \u03b2, not just twice. For it seems that the brightness of the rays ought to be intensified because of the fact that bodies appear to be magnified when they approach, as [for example] Venus, when at the perigee of the epicycle, defines a more evident shadow around bodies than at apogee. Therefore, by the force of the analogy which we have instituted between light and motive force, the same should be thought to apply to the motive force. ","756":"To this objection, I answer decisively, that in the initial supposition involving the point I made a false assumption. Even though I did indeed say something like that in the Optics, you should bear in mind that I was speaking of optics, whose points and lines are not quite indivisibles. For, as concerns the point, since it has no magnitude, while the rays are amplified with the magnitudes of bodies, it follows that the radiation of a point in itself is nothing, and hence what has no radiation has no greater or lesser density. Thus the initial assumption of the ratio of distances \u03b1\u03b2 to \u03b1\u03b3 collapses.","757":"Rather, because of this very thing, we say that any point shines more strongly or weakly to the extent that that point designates to us a greater or lesser quantity.","758":"In the second supposition, concerning the circle, and the third, concerning the disc, there are two false assumptions. The first is that a mathematical circle, lacking breadth, is supposed to shine. For it can no more shine itself than can the point from whose motion the circle is supposed to be generated. You are no closer to having a surface when you posit a line of three stadia1 than you are in positing a line of three feet.","759":"Second, it is supposed that the optical magnification of the diameter or of the disc adds to the strength of the rays, although this is but a deception of the visual faculty, and belongs to the genus of theoretical entities, which lack any efficacy. The physical identity of the circle \u03b4\u03b5, the surface \u03b4\u03b1\u03b5 (when light is in question) and the body \u03b4\u03b5 (when power is in question) remains the same whether it is viewed from \u03b3 or from \u03b2, and will always act the same and have the same effect, spreading the same amount of power or light in the more diffuse orb \u03b3 as in the more compact orb \u03b2. Nothing is lost in the journey: the entire species carries through to any distance, however remote. It is attenuated only in the extensions of the spheres, so that in the individual points of the spheres, such as \u03b3 and \u03b2, it is rarer in the former, and denser in the latter, in the inverse ratio of the distances \u03b1\u03b2 to \u03b1\u03b3. This is the sole cause of the weakening, not the diminishing of the source \u03b4\u03b5, which in fact does not happen, being but an optical illusion.","760":" Indeed, if I might argue from Euclid's Optics here, less light arrives at the nearer, \u03b2, than at the more remote, for the reason that at \u03b2 a smaller circle bounds the visible hemisphere of the luminous body \u03b4\u03b5 than at \u03b3. Therefore, not so many particles of the sun \u03b4\u03b5 can be seen from \u03b2 as from \u03b3. But this is quite imperceptible, hard to express [even] in enormous numbers.","761":"And I, after giving this answer to myself, am having a good laugh at my wretched alarm arising from this obscure business.","762":"But the objection can rebound in the opposite direction, thus. If there is the same amount of light spread out in a large sphere as there is gathered together in a small sphere, there will not be the same amount of power in both places, for power is not considered orbicularly in a sphere as light is, but in the circle in which the planet proceeds. Thus, the magnetic filaments of the sun were supposed, above, to be set up only in longitude, not towards the poles or in other directions.","763":"The answer is that the cause of light and motive power is exactly the same, and there is a deception in the reasoning. For in light, the rays do not flow out solely from the individual points and circles of the body to the corresponding points and circles of the sphere. Thus, the rays from \u03b3 do not come only from \u03b1 (by which arrangement it would be impossible to ascribe a density to light in the spheres, since it would have no quantity in its origin, descending as it does from a point). Instead, the rays flow out from the whole hemisphere of the lucid body to the individual points of the imaginary spherical surface: thus, the rays from both \u03b4 and \u03b5 flow out to \u03b3. The same thing takes place likewise with power. Even though the magnetic filaments of the solar body are ordered according to zodiacal longitude, and even though there is but a single great circle of the sun's body beneath the zodiac or ecliptic, and roughly beneath the orbit of the planet, and (finally) even though the other smaller circles (which are compacted to the size of points beneath the poles) are subordinated to their corresponding circles in the sphere of the planet, nevertheless, the rays from all the filaments of the solar body (standing up from one hemisphere of the body) flow down and converge on the individual points of the path of any of the planets as well as on those poles that are above the poles of the sun's body, and the body of the planet is transported according to the measure of the density of this entire species compounded of all the filaments.","764":" It does not, however, follow that just as the sun shines equally in all directions, the planets, too, as you might fear, are moved indiscriminately in all directions. For the magnetic filaments of the sun, considered in themselves, do not move, but only inasmuch as the sun, rotating very rapidly in its space, carries the filaments around, and with them the moving species spreading abroad from them. Therefore the planet will not go backwards, because the sun always rotates forwards. The planet will not go to the poles (even though there might be some of the species from the sun's body at those points) because the filaments of the solar body are not extended in the direction of the poles, nor does the sun rotate in that direction, but rather in the direction in which its filaments urge it.","765":" On these suppositions, it is so far from being the case that the planets are carried towards the poles that there is instead a single region of the zodiac, the mean between the poles, through which all planets of necessity would move in longitude without deflection, if they were to cease their own proper motions (of which see Chapter 38 below). For the species of the solar hemisphere which takes up a post at some point of the zodiac, such as the point \u03b6 in the present diagram, is the total of the filaments of the semicircle all tending in the same direction in concert, as, from [the region from] \u03b8 through \u03ba, from \u03bb through \u03bc, etc. But when you move off towards the poles of the world, as at \u03b7, then because one pole \u03bd of the sun's body, and the complete circles \u03bb\u03bc that surround the pole \u03bd, are appropriated under the purview of \u03b7\u03bc, the species is made up of filaments tending in opposite directions, for the opposite parts \u03bb and \u03bc of the circles move in opposite directions. Therefore that species \u03b8\u03b7\u03bc descending near the poles is less well adapted to the motion carrying the planets along.","766":"Chapter 37","767":"How the power moving the moon may be constituted.","768":"Since in Chapter 34, in passing, I made mention of the motion of the moon, it would be appropriate to treat the matter in greater detail, so that no reservation introduced by the moon might deflect the reader into withholding his ready assent from me on the entire treatise; so that it might instead be marvelously confirmed, by the clearest consideration of the lunar motion; and finally, to ensure that the physical aspects of astronomy are treated fully in this book. For even though there are a few things in the theory of the moon that must be deferred, as they are to be treated differently or explained in more detail, they nevertheless will have their origin here.","769":"After long and painstaking observations of the moon in every position in relation to the sun, Tycho Brahe expressed the opinion that in the moon, besides the anomaly of the epicycle, and besides that monthly anomaly which was also known to Ptolemy, the mean motion itself (so named in relation to these two inequalities) is not yet quite \"mean.\" That is, it intensifies at the conjunctions and oppositions with the sun, and is remitted at the quadratures. Thus, even if it were undisturbed by epicycles, the moon itself, even though moving on a circle concentric with the earth, would move around nonuni","770":"formly. ","771":"Let S be the body of the sun, M the orb of Mercury, V of Venus, T of the earth, P of Mars, and so on. And let all of them always move from right to left when in the upper part of the diagram. Now let CLOF be the orb of the moon, O the moon at opposition, C at conjunction, L, F at quadratures, and let CLOF remain for now a concentric described about the earth at T, and let it move in the direction OFCL. The question is, by what cause is the moon made to move more swiftly about T at C, O than at F, L, since we have just now mentally removed the eccentricity and the epicycles. Here, I know, the reader expects me to say that it is swifter at O because at that place its motion is in the same direction as the motion of all the planets. But this is not the true cause. For in that case, what would happen at C is what in fact happens, that the moon, in its compounded motion, is slowest, since its proper motion FCL resists considerably this common rightward motion. For it should be noted that the moon, from point C on its orb, is borne less in the leftward direction L than the earth is borne to the right on its orb. Therefore, the moon, by a motion compounded of its proper motion and that which it has in common with the earth, also always moves to the left when above, as when the earth is at \u03b4, but here, when the earth is down low at T, it is carried to the right, but slowly at C and swiftly at O. A motion of this sort is expressed approximately by the spiral lines set out here. ","772":"But perhaps you are expecting me to say instead that this phenomenon arises from the sun's motive power being weaker at O and stronger at C? Much less will I say this. For I would thus make it slow at O and C, and fast at F, L, which is contrary to what is desired. For if it is propelled weakly at O, it therefore is slow, and if it is held back more strongly at C, so that its tendency to move in the opposite direction from C to L is less, it will therefore again move slowly from C to L. Thus it was not correct for us to free the moon from the earth and commit it to the sun. This would cause it at length to wander away from the earth, just as the apogees wander from their places. It would be preferable to attribute to the earth a force that retains the moon, like a sort of chain, which would be there even if the moon did not circle the earth at all. On this supposition, the moon is, as it were, in the same boat with the earth, in the same power of the sun, and now, as if freed from this motion from the sun, it is rotated separately by the earth.","773":"Therefore, I consider the cause of the swiftness at O and C to be none other than this, that the earth T derives its power of moving the moon from the sun S, and preserves it by the continuity of the line TS. Thus SCTO can properly be called the \"diameter of power\" since these two (T and S) are the sources of the entire motion.","774":"On this supposition, that monthly inequality known to Ptolemy also will follow. For if the power coming from the same source T is stronger at C, O than at F, L, then if the apogee is near C, O, there is a greater loss of speed than if the apogee were at F, L. Thus when going from the apogee at O or C the equations at F, L are greater than when going from the apogee at F, L to the conjunctions and oppositions O, C.","775":"You see, then, that these physical speculations are so devised that they can even account for the phenomena of the moon. The moon is not driven to","776":"circle the earth primarily by the sun, but by a power lying hidden in the earth itself, and casting forth its immaterial species to the body of the moon, but more strongly along the line that connects the centers of the sun (the primary source) and the earth.","777":"It is, however, hard to explain more clearly how that diameter comes to be more powerful. For neither the sun's power nor the earth's, going out to the ","778":"moon, is any swifter when the moon falls on that diameter. That the rotations of these bodies (and therefore also of the species) are uniform and forever constant, is a highly reasonable presumption. There remains only the possibility described here, that the power is indeed not swifter, but more robust, when it emanates from the earth in the directions nearer to the line ST, for the reason that it was originally drawn off from the sun to the earth along that line.","779":"Nevertheless, it is the sun which, either immediately or through that faculty which effects annual motion for the earth, is the principal director of the motion which the earth confers upon the moon. This is demonstrated chiefly by the moon's making its circuit beneath the zodiac, like the annual circuit of the centre of the earth, although the earth's diurnal motion, which confers the monthly motion upon the moon, takes place beneath the equator.","780":"Chapter 38","781":"Besides the common motive force of the sun, the planets are endowed with an inherent force [vis insita], and the motion of each of them is compounded of the two causes.","782":"I have spoken of the origin of the motion which rotates the planets around the sun or the moon around the earth; that is, I have spoken of the natural causes of the circle which in the theories of the planets is called either eccentric or concentric, according to the various intentions of the authors. Now, some thing must be said of the natural cause of the eccentricity, or in the particular hypothesis of Copernicus, of the epicycle on a concentric. For the moving power from the sun has hitherto been uniform, having different degrees only at various amplitudes of the circles. Its innate quality is such that if a planet were to remain at the same distance from the sun it would be carried around perfectly uniformly, and would not experience any intension or remission of the solar motion. The modicum of nonuniformity perceived in the working of this power arises from the planet's having been transposed from one distance from the sun to another, so that it encounters one or another degree of strength of this power from the sun. The question therefore arises, if, as Brahe demonstrated, there are no solid orbs, how does the planet come to ascend from and descend towards the sun? Can this, too, really come from the sun? My answer is that it is to some extent from the sun, and to some extent not from the sun.","783":"Examples of natural things, and this hitherto disparaged kinship of celestial things for terrestrial ones, cry out that in a simple body the operations which are more general are simpler, while the variables, if any (such as, in the motion of the planets, the varying distance from the sun, or the eccentricity), arise from the concurrence of extrinsic causes.","784":"Thus, in a river, the simple property of water is to descend towards the centre of the earth. But because its path is not direct, it flows in to those places where it finds a lower bed, stagnates where it meets with level ground, and is carried along with a roar where it comes upon steeper slopes; there is a place where it may be spun into whirlpools, if in a swift drop it should dash into projecting rocks. Where water itself, by its inherent force, endeavors to do nothing but descend towards the centre of the earth, a simple action by a simple property, nevertheless, flow and stagnation and waves and whirlpools and every variety [of phenomena] arise from the causes described, which are adventitious and accidental.\nParticularly happy and better accommodated to our inquiry are the phenomena exhibited by the propulsion of boats. Imagine a cable or rope hanging high up across a river, suspended from both banks, and a pulley running along the rope, holding, by another rope, a skiff1 floating in the river. If the ferryman in the skiff, otherwise at rest, fastens his rudder or oar in the right manner, the skiff, carried crosswise by the simple force of the downward-moving river, is transported from one bank to the other, as the pulley runs along the cable above. On broader rivers they make the skiffs go in circles, send them hither and thither, and play a thousand tricks, without touching the bottom or the banks, but by the use of the oar alone, directing the singular and most simple flow of the river to their own ends.","785":"In very much the same manner, the power moving out from the sun into the world through the species is a kind of rapid torrent, which sweeps along all the planets, as well as, perhaps, the entire aethereal air, from west to east. It is not itself suited to attracting bodies to the sun or driving them further from it, which would be an infinitely troublesome task. It is therefore necessary that the planets themselves, rather like the skiffs, have their own motive powers, as if they had riders [vectores] or ferrymen, by whose forethought they accomplish not only the approach to the sun and recession from the sun, but also (and this might be called the second argument) the declinations of latitudes; and as if from one bank to the other (that is, from north to south and back), travel across this river (which itself only follows the course of the ecliptic) from north to south and back.","786":"It is certain from what has been said above that the power which comes from the sun is simple. But now, the eccentrics of the planets do not just decline from the ecliptic, but go in various directions, intersecting one another and the ecliptic. Therefore, other causes are conjoined with the motive power from the sun.","787":"Chapter 39","788":"By what path and by what means the powers seated in the planets ought to move them, in order that the planet's circular orbit, such as is commonly thought to exist, be brought about in the aethereal air.","789":"And so, let these points of great certainty in what has been demonstrated be axioms for us. First, that the body of a planet is inclined by nature to rest in every place where it is put by itself. Second, that it is transported from one longitudinal position to another by that power which originates in the sun. Third, if the distance of the planet from the sun were not altered, a circular path would result from this transport. Fourth, supposing the same planet to be in turn at two distances from the sun, remaining there for one whole circuit, the periodic times will be in the duplicate ratio of the distances or magnitude of the circles. Fifth, the bare and solitary power residing in the body of a planet itself is not sufficient for transporting its body from place to place, since it lacks feet, wings, and feathers by which it might press upon the aethereal air. And nevertheless, sixth, the approach and recession of a planet to and from the sun arises from that power which is proper to the planet. All these axioms are agreeable to nature in themselves, and have been demonstrated previously. ","790":"Now let us work with geometrical figures in order to see what laws will be required to represent any desired planetary orbit. Let the orbit of the planet be a circle, as has been believed until now, and let it be eccentric with respect to the sun, the source of power. Let that eccentric CD be described about center B with radius BC, and on it let BC be the line of apsides, A the sun, and BA the eccentricity. Let the eccentric be divided into any number of equal parts, beginning from the line of apsides at C, and let the ends of these parts be connected with A. Therefore, CA, DA, EA, FA, GA, HA, will be the distances of the end points of the equal parts from the source of power. Now with center \u03b2, radius \u03b2\u03b3, equal to AB, let the epicycle \u03b3\u03b4 be described, divided into as many equal parts as the eccentric, beginning from \u03b3, and let the line \u03b3\u03b2 be extended so as to make \u03b2\u03b1 equal to BC, and let the point \u03b1 be connected with the end points of the equal parts of the epicycle, by the lines \u03b3\u03b1, \u03b4\u03b1, \u03b5\u03b1, \u03b6\u03b1, \u03b7\u03b1, \u03b8\u03b1. These lines will be respectively equal to the distances drawn to the eccentric from A, this having been demonstrated above in Chapter 2. Next, with center \u03b1 and radius \u03b4\u03b1, let the arc \u03b4\u03b9\u03b8 be described, intersecting the diameter \u03b3\u03b6 at \u03b9 and about the same center \u03b1, with radius \u03b1\u03b5, let the arc \u03b5\u03bb\u03b7 be described, intersecting the diameter \u03b3\u03b6 at \u03bb, and let the end points of the parts equidistant from the aphelion of the epicycle \u03b3 be connected by the lines \u03b4\u03b8, \u03b5\u03b7, which will intersect the diameter at \u03ba, \u03bc, so that \u03b1\u03b4 or \u03b1\u03b9 is longer than \u03b1\u03ba, and \u03b1\u03b5 or \u03b1\u03bb longer than \u03b1\u03bc.","791":" If it were possible for the planet to move on a perfect epicycle by its inherent force, and for its orbit at the same time to be a perfect circle, then we would have to consider similar arcs to be swept out in the same times, on the eccentric and on the epicycle. Consequently, it would now immediately become clear by what means, and by what measure, the distance \u03b1\u03b9 would be made equal to AD. For since \u03b1\u03b9 and \u03b1\u03b8 are equal, when the planet moves from \u03b3 to \u03b8 the distance \u03b1\u03b8 necessarily and without any special contrivance comes out right, equal to AD.","792":" But aside from one appearing to be in conflict with the fifth axiom, who claims that the planet moves locally from \u03b3 to \u03b8 by its inherent force, many additional absurdities are also involved. ","793":"For let AN be drawn parallel to BD, and let AN be equal to BD, and about center N let an epicycle be described which shall go through D. Now since CD is a perfect circle, the same angles are swept out by the planet D about the center of the eccentric B as by the center of the epicycle N about the center of the sun A (through the equivalence demonstrated in Chapter 2), as long as the diameter of the epicycle ND with the planet at D always remains parallel to AB with respect to its position in the world. Therefore, the speed of the center of the epicycle N about the sun A would be made the same as the speed of the planet D about the center of the eccentric B, so that those motions would be intensified at the same time and remitted at the same time. And since intension and remission depends upon greater or less distance of the body of the planet from the sun, therefore the center of the epicycle, remaining at the same distance, would be contrived to move slowly or swiftly on account of the planet's being farther from or nearer to the sun.","794":"And although the power driving the planets is faster than any of them, as is shown in Chapter 34, we are here to suppose in our imagination a single ray of power AN coming from the sun, as if there were a line upon which the center of the epicycle N would always remain. And this line, with that same center N, would be now slow, now swift, again contrary to what was said above, that the power always produces the same speed at the same distance. Moreover, we would be required to suppose that the planet has its own rotatory motion away from this imaginary ray AN in the opposite direction, traversing unequal distances in equal times, according as this ray itself were swifter or slower. In adopting this account, we would indeed approach more closely the geometrical suppositions of the ancients, but we would stray very far from physical theory, as is shown in Chapter 2. And my thoughts on the matter have not sufficed to discover a way in which these things can happen naturally.","795":"Now all this would be conceived more simply were we to consider the diameter of the epicycle ND as always remaining parallel to itself. For then the planet would carry out its motion by a mental image, not of the epicycle, but of the center of the eccentric B, and by keeping itself always at the same distance from that center.","796":"But at the beginning of this work, in Chapter 2, it was said to be most absurd that a planet (even if you give it a mind) may imagine for itself a center and from it a distance, when in that center there is no particular body for the planet to be aware of.","797":"Now you might say that the planet observes the sun A, and already knows beforehand, through memory, which ordered distances from the sun a perfect eccentric would have to attain. But first, this is more indirect, and depends upon something intermediate connecting the effect of the perfectly circular path with the indication of the waxing and waning of the sun's diameter, and this too in some mind. But that intermediate can be nothing but the position of the center of the eccentric B at a certain distance from the sun, which, as has just now been said, cannot be known by an unassisted mind.","798":"I do not deny that a center can be conceived, and about it a circle. But I do say this, that if the center is established by thought alone, without respect to time and without any external indication, it is impossible to set up about it, in reality, the perfectly circular path of some movable body.","799":"Besides, if the planet were to derive its correct distances from the sun, ordained by the rule of the circle, from memory, it could also derive from memory the equal arcs of the eccentric which are to be traversed in unequal times, and which are to be traversed by an extrinsic force originating in the sun, as if it were obtaining the values right from the Prutenic or Alphonsine tables. Thus it would know from memory beforehand what this extrinsic and mindless power originating in the sun was going to do. All of these are absurd:\u2014","800":"Chiefly since, as Aristotle affirms, there is no knowledge of the infinite, while the infinite is involved in this intension and remission.\nBut it is all right, because even the observations themselves will not allow CD to be a perfect circle, below in Chapter 44, so these theoretical arguments, weak though they are thought to be, do not stand alone, and are that much the less vulnerable to scornful rejection.","801":" It is consequently more fitting that the planet itself require no assistance, whether of epicycle or eccentric, but rather that the task which it either performs by itself or has a part in performing is a reciprocating path along the diameter \u03b3\u03b6 directed towards the sun \u03b1.","802":"The question now is, what is the measure by which the planet metes out the correct distances for any given time?","803":" Now to us the measure is clear from geometry and the diagram. For whenever the solar power moves the planet forward to the line DA, we then find the angle CBD and make \u03b3\u03b2\u03b4 equal to it; and thus we say that \u03b1\u03b4, or \u03b1\u03b9 which is its equal, is the correct distance from A to the planet at D. But this measure which we have proposed for humans, we have just now denied the planet when we removed it from the circumference of the epicycle and restricted it to the straits of the diameter \u03b3\u03b6.","804":" Indeed, in this inquiry it is easier to say what is not the case than to say what is. This is because at the moments when the sun has placed the planet on the lines drawn from A through C, D, E, F, G, H, the planet itself is presumed to have produced the distances \u03b3\u03b1, \u03b9\u03b1, \u03bb\u03b1, \u03b6\u03b1, \u03bb\u03b1, \u03b9\u03b1, respectively. Consequently, if the path of the planet is a perfect circle, then to equal parts of the eccentric CD, DE, EF, correspond unequal descents of the planet along the diameter, namely, \u03b3\u03b9, \u03b9\u03bb, \u03bb\u03b6. Moreover, the order is perturbed: the highest is not the smallest, nor the lowest the greatest, but the middle parts \u03b9\u03bb are greatest, and the extremes \u03b3\u03b9 and \u03bb\u03b6 are smaller, and the highest \u03b3\u03b9 are a little smaller than the corresponding lowest, \u03bb\u03b6. For \u03b3\u03ba and \u03bc\u03b6 are equal, and \u03b3\u03b9 is smaller than \u03b3\u03ba, while \u03bb\u03b6 is greater than \u03bc\u03b6. ","805":" Furthermore, this same cause prevents \u03b3\u03b9, \u03b9\u03bb, \u03bb\u03b6, being made proportional to either the times of the equal arcs traversed CD, DE, EF, or to the angles at the sun CAD, DAE, EAF. For the time or duration of the planet on equal parts of the eccentric CD, DE, EF, is continuously diminished from the highest to the lowest points, and the angles at the sun are continuously increased, but the reciprocations \u03b3\u03b9 are increased in the middle regions, such as \u03b9\u03bb.","806":"Therefore, if the path of the planet is a perfect circle, the measure of the planet's descent on the diameter \u03b3\u03b6 is neither time, nor distance traversed on the eccentric, nor the angle at the sun.","807":"And physical theories, too, also decisively repudiate these measures.","808":"What, then, if we should say this: although the motion of the planet does not take place on an epicycle, this reciprocation is measured out in such a way that distances from the sun are produced that are similar to those which exist in an epicycle actually traversed?","809":"First, this attributes to the power belonging to the planet a knowledge of the imaginary epicycle and of its effects in setting out distances from the sun; and further, it attributes knowledge of the future speed or slowness which the common motion from the sun is going to cause. For it is necessary to suppose here an imaginary intension and remission of motion on the imaginary epicycle that is the same as that of the motion on the real eccentric. This is more incredible than the previous accounts, where the motion of the body was combined with knowledge of the epicycle or eccentric. Therefore, the objections raised there should be understood as applying here as well, the judgements being nearly identical.","810":"Nevertheless, for want of a better opinion, we must at present put up with this one. And as for its involving many absurdities, in Chapter 57 below, a certain physicist will quite readily allow this: that on the testimony of the observations the path of the planet is not a circle.","811":"So far, the discussion has concerned the measure relating to the form of this reciprocation. It now remains for us to find the measure of this measure; that is, the measure of the quantity or of motion through place. For it is not enough for the planet to know how far it should be from the sun: it also has to know what to do in order to be at the correct distance.","812":"Now anyone who is so attracted to the supposition of a perfectly circular orbit as to locate in the planet a mind which could preside over the reciprocation, can say only this: that this planetary mind observes the increasing and decreasing size of the solar diameter, and understands, using this as an indication, what distances from the sun it should cause to occur at any given time. For example, sailors cannot know from the sea itself how far they have travelled over the waters, since the course, viewed in that way, has no distinct limits. Instead, they find this either from the amount of time they have sailed, if wind and sea remain constant and the ship does not stop, or from the direction of the wind and the changing elevation of the pole, or from all or several of these in conjunction, or, may it please the gods, by a contrivance of a number of wheels, with paddles lowered into the water (for certain conceited mechanics are proposing an instrument of this sort, who ascribe the calm of the continents to the waves of the Ocean). In just the same way, the mind of the planet cannot by itself measure its position, or the distance between itself and the sun, since between them there is pure aethereal air, devoid of any means of indication. So it makes use either of the increment of time, and an equal exertion of forces through that time (which has just been denied above), or of a physical machine, which is ridiculous (for by the example of the sun and moon we suppose the celestial bodies to be round, and it is therefore also probable that the entire field of the aetheral air moves around with the planets), or finally, of some suitable means of indication that are variable with the altered distance of the planet from the sun. And other than the sun's apparent diameter alone, nothing else presents itself. Thus we humans know that the sun is 229 of its own semidiameters distant from us when its diameter subtends 30', and 222 semidiameters when it subtends 31'. If it were indeed certain that this proper motion of the planet along the diameter of the epicycle could not be carried out by any material and corporeal or magnetic power of the planet, nor by an unassisted animate power, but that it is governed by a planetary mind, nothing absurd would be stated. For that the sun is observed by the planets in other respects as well, the latitudes bear witness. For by these latitudes the planets would depart from the middle and royal road of this solar power, as from the mainstream of a river, and move to the sides, as is said in Chapter 38, unless they meanwhile paid attention to the sun, and carried out their approach and recession along a line drawn to its center. They would then describe circles which, seen from the earth or the center of the world, would appear smaller, parallel to some great circle. But all planets describe great circles that intersect the ecliptic at points that are opposite with respect to the sun, as was demonstrated for Mars from observations above in Chapters 12, 13, and 14. Therefore, the diameter of reciprocation \u03b3\u03b6 is also directed towards the sun, and the latitudes heed the sun in every respect. However, in Part 5 below I will also transfer this characteristic of the latitude from the elements of mind to the elements of nature and magnetic faculties.\n Now one cannot say in reply to me that the solar diameter and its variation is far too small to be used as a standard. For it is certain that there is no planet for which it entirely vanishes. Since on earth it is thirty minutes, on Mars it will be twenty, on Jupiter seven, and on Saturn three, while on Venus it will be forty, and on Mercury at least eighty and sometimes as high as one hundred and twenty. Do not grumble about the smallness of the body, but rather the unapt coarseness of human senses, which cannot follow such small things.","813":"One should on the contrary note that this body, however nonetheless capable of moving such distant bodies in a circle, as is demonstrated in the preceding chapters. The illumination of the world by such a tiny corpuscle is known to all. And so it is credible that if the movers are endowed with some faculty of observing its diameter, this faculty is as much more acute than our eyes as its work, and the perpetual motion, is more constant than our own troubled and confused schemes.","814":"So then, Kepler, would you give each of the planets a pair of eyes? By no means, nor is this necessary, no more than that they need feet or wings in order to move. But Brahe has recently eliminated solid orbs. Now our theorizing has not emptied nature's treasure house: we still cannot establish, through our own knowledge, how many senses there ought to be. There are even examples at hand worthy of our admiration. For tell us, in physical terms: with what eyes shall the animate faculties of sublunar bodies look upon the positions of the stars in the zodiac, so that when a harmonic arrangement (which we call \"aspects\") is found among them, the bodies leap up and become inflamed for their task? Was it with her eyes that my noted the positions of the stars in order to know that she was born with Saturn, Jupiter, Mars, Venus, and Mercury, all in sextiles and trines? And could it have been by the same means that she gave birth to her children, and especially to me her first born, chiefly on those days when as many as possible of the same aspects, especially of Saturn and Jupiter, recurred, or when as many pristine positions as possible were occupied by squares, oppositions, and conjunctions? I have observed those things in all cases whatever that have occurred to this very day. But why do I say these things that are just as absurd as the previous ones, but for those who have exercised themselves in natural matters more diligently than is usual nowadays?","815":"So our hypothetical person who says that the planet's path is a perfect circle will say this: that the planet performs its reciprocation so as to make the diameters of the sun, at the end points of equal arcs of the eccentric, appear very nearly* inversely proportional to the lines \u03b4\u03b1, \u03b5\u03b1, \u03b6\u03b1, or to \u03b9\u03b1, \u03bb\u03b1, \u03b6\u03b1, which are equal to them, taken with respect to the longest line \u03b3\u03b1; and that through this consideration of the diameters of the sun at the chosen moments of time, come the proximities of \u03b9, \u03bb, \u03b6 to \u03b3.","816":"It should be known, however, that the increases of the diameter of the sun and the arcs of the epicycle do not square with each other well, and so the motive mind will have to have a very good memory in order to adjust the unequal versed sines of the arcs on the epicycle to the equal increases of the solar diameter. For this, see Chapter 56 and 57 below.","817":"Let that be enough concerning the means of indicating distances. There remains a third point about the animate faculty that carries the planet around that I briefly draw attention to: anyone who says that the body of a planet is moved by an inherent force is just plain wrong. This we rejected at the beginning. But it is also impossible simply to ascribe this force to the sun instead. For the same force that attracts the planet also repels it in turn, and this is inconsistent with the simplicity of the solar body. But anyone who by some unique argument reduces this motion to the bodies of the sun and the planet in concerted action gives a new cast to the material of this entire chapter, and under this rubric Chapter 57 below is specifically assigned for this topic.","818":"You see, my thoughtful and intelligent reader, that the opinion of a perfect eccentric circle for the path of a planet drags many incredible things into physical theories. This is not, indeed, because it makes the solar diameter an indicator for the planetary mind, for this opinion will perhaps turn out to be closest to the truth, but because it ascribes incredible faculties to the mover, both mental and animate.","819":" But we, who are close to the truth, should find out how to cast these theories (which, though not everywhere perfect, are nonetheless suitable for the sun's motions) in numerical form. In the end, it will be helpful for a more exact discovery of the truth, reserved for Chapter 57, that we had previously worked on them here.","820":"Chapter 40","821":"An imperfect method for computing the equations from the physical hypothesis, which nonetheless suffices for the theory of the sun or earth.","822":"Such a long-winded discussion was necessary to prepare a way for a natural form for the equations, on which I am going to be very busy in Part 4. Now we must return to the equations of the sun's eccentric in particular, which is the main subject of this third part, and for the sake of which the general discussion of the last eight chapters has been presented.","823":"My first error was to suppose that the path of the planet is a perfect circle, a supposition that was all the more noxious a thief of time the more it was endowed with the authority of all philosophers, and the more convenient it was for metaphysics in particular. Accordingly, let the path of the planet be a perfect eccentric, for in the theory of the sun the amount by which it differs from the oval path is imperceptible. Those things that are going to be needed for the other planets, on account of this deviation, follow below in ch. 59 and 60. ","824":"Since, therefore, the time increments of a planet over equal parts of the eccentric are to one another as the distances of those parts, and since the individual points of the entire semicircle of the eccentric are all at different distances, it was no easy task I set myself when I sought to find how the sums of the individual distances may be obtained. For unless we can find the sum of all of them (and they are infinite in number) we cannot say how great the time increment is for any one of them. Thus the equation will not be known. For the whole sum of the distances is to the whole periodic time as any partial sum of the distances is to its corresponding time.","825":"I consequently began by dividing the eccentric into 360 parts, as if these were least particles, and supposed that within one such part the distance does not change. I then found the distances at the beginnings of the parts or degrees by the method of chapter 29, and added them all up. Next, I assigned an artificial round number to the periodic time; although it is in fact 365 days and 6 hours, I set it equal to 360 degrees, or a full circle, which for the astronomers is the mean anomaly. As a result, I have so arranged it that as the sum of the distances is to the sum of the time, so is any given distance to its time. Finally, I added the times over the individual degrees and compared these times, or degrees of mean anomaly, with the degrees of the eccentric anomaly, or the number of parts whose distance was sought. This furnished the physical equation, to which the optical equation, found by the method of Chapter 29 with those same distances, was to be added in order to have the whole. ","826":"However, since this procedure is mechanical and tedious, and since it is impossible to compute the equation given the ratio for one individual degree without the others, I looked around for other means. And since I knew that the points of the eccentric are infinite, and their distances are infinite, it struck me that all these distances are contained in the plane of the eccentric. For I had remembered that Archimedes, in seeking the ratio of the circumference to the diameter, once thus divided a circle into an infinity of triangles - this being the hidden force of his reductio ad absurdum. Accordingly, instead of dividing the circumference, as before, I now cut the plane of the eccentric into 360 parts by lines drawn from the point whence the eccentricity is reckoned.","827":"Let AB be the line of apsides, A the sun (or earth, for Ptolemy); B the center of the eccentric CD, whose semicircle CD shall be divided into any number of equal parts CG, GH, HE, EI, IK, KD, and let the points A and B be connected with the points of division. Therefore, AC will be the greatest distance, AD the least, and the others, in order, are AG, AH, AE, AI, AK. And since triangles under equal altitudes are as their bases, and the sectors, or triangles, CBG, GBH, and so on (standing upon least parts of the circumference and therefore not differing from straight lines) all have the same altitude, the equal sides BC, BG, BH, they are therefore all equal. But all the triangles are contained in the area CDE, and all the arcs or bases are contained in the circumference CED. Therefore, by composition, as the area CDE is to the arc CED so is the area CBG to the arc CG, and alternately, as arc CED is to CG, CH, and the rest in order, so is the area CDE to the areas CBG, CBH, and the rest in order. Therefore, no error is introduced if the areas be taken for the arcs in this way, and substituting the areas CGB, CHB for the angles of eccentric anomaly CBG, CBH.","828":" Further, just as the straight lines from B to the infinite parts of the circumference are all contained in the area of the semicircle CDE, and the straight lines from B to the infinite parts of the arc CH are all contained in the area CBH, therefore also the straight lines from A to the same infinite parts of the circumference or arc make up the same thing. And finally, since those drawn from A and B both fill up one and the same semicircle CDE, while those from A","829":"are the very distances whose sum is sought, it therefore seemed to me I could conclude that by computing the area CAH or CAE I would have the sum of the infinite distances in CH or CE, not because the infinite can be traversed, but because I thought that the measure of the faculty by which the collected distances mete out the times is contained in this area, so that we would be able to obtain it by knowing the area without an enumeration of least parts.","830":" Therefore, from the above, as the area CDE is to half the periodic time, which we have proclaimed to be 180\u00b0, so are the areas CAG, CAH to the times on CG and CH. Thus the area CGA becomes a measure of the time or mean anomaly corresponding to the arc of the eccentric CG, since the mean anomaly measures the time. ","831":"Earlier, however, the part CGB of this area CAG was the measure of the eccentric anomaly, whose optical equation is the angle BGA. Therefore, the remaining area, that of the triangle BGA, is the excess (for this place) of the mean anomaly over the eccentric anomaly, and the angle BGA of that triangle is the excess of the eccentric anomaly CBG over the equated anomaly CAG. Thus the knowledge of this one triangle provides both parts of the equation corresponding to the equated anomaly GAC.","832":"And hence it is manifest why in Chapters 30 and 31 above I said that in the theory of the sun the parts of the equation are very nearly equal. That is, any given arc, and the angle at the center which it subtends (as CG and CBG in the figure), are measured by its area, which is called the \"sector\", as area CBG. Therefore, with one arm of the compass at G, with radius GB, let an arc of the circumference be described intersecting GA at O. Hence, as the area GBC is to the angle GBC, so is the area BGO to the angle BGO. But the angle BGO is the optical part of the equation. Thus the area GOB, through the doubling of the part of the equation, measures the optical part of the equation, for in our account presented earlier the whole area GBA was to be consulted for the physical part of the equation.","833":"Now clearly the genuine measure of the physical part of the equation AGB exceeds the proposed measure of the optical part OGB by the small space or area OAB (while near the perigee the latter in turn exceeds the former by a small area). Nevertheless, if the eccentricity is small, as is that of the sun or earth, with which we are concerned in this third part, this is not perceptible. For the nearer it approaches to the line of apsides, the narrower becomes the whole triangle AGB and consequently also its little part AOB, however much its altitude AO increases at the same time. On the other hand, in the middle elongations, the angle BEA with its sector is at a certain point directly measured by the area BEA, and the excess begins to turn into a defect.","834":"Therefore, the greatest difference that can occur is that accumulated at the octants, or locations intermediate between the apsides and the quadrants. How great that difference is, will now be shown.","835":"Since for some time now I have used the same form of computation by means of areas in the theory of Mars, I could not ignore this difference on account of the planet's great eccentricity, nor did the doubling of the optical equation avoid all perceptible error. It was therefore necessary to investigate the area of the triangle of the equation [triangulum aequatorium]. This can be done in various ways, but I shall go on to state the easiest. ","836":"It is well known that triangles with equal altitudes are proportional to\ntheir bases. I say also, that triangles on equal bases are proportional to\ntheir altitudes.","837":" Let AGB, AHB stand upon the same base AB extended to C. From G let the line GN be drawn parallel to the common base AB and intersecting HB at N, and let N be connected with A. From the three vertices G, \u0397, N, of the triangles let GM, HL, MP be drawn perpendicular to the base, determining the altitudes of the triangles. Therefore, since GN and MP are parallel, and GM, NP are perpendicular [to them], GM and NP will be equal. But GM is the altitude of triangle AGB, and NP is the altitude of triangle ANB. Therefore triangles ANB and AGB have equal altitudes, and since they are both on the same base AB, they are equal. And since ANB is part of AHB, and the base line HB is common, together with the vertex A, the triangles NAB and HAB have equal altitudes. Therefore, as the base NB is to BH, so is NAB to HAB. But NAB and GAB were proved equal. Therefore, as NB is to BH, so is GAB to HAB. But as BN is to BH, so is NP to HL, because NBP and HBL are similar triangles. Therefore also, as NP is to HL, so is GAB to HAB. But NP and GM are equal. Therefore, as GM is to HL, altitude to altitude, so is area GAB to area HAB. q.e.d.","838":" Now let BE be perpendicular to CD, and let the triangle BEA have a right angle at B. BE will be the altitude, and BA the base. Therefore, by Euclid I.42, 900, or half the base BA (which for the sun is 1800), multiplied by the altitude BE, 100,000, which is the radius of the circle, gives the area of the triangle BEA, that is, 90,000,000. But the area of a circle of radius 100,000 (from the most recent investigations of Adrian Romanus, a most expert geometer)2 is 31,415,926,536, with no error in even the last digit. And as this the area of the circle is to the 360\u00b0 of mean anomaly or time (that is, 21,600' or 1,296,000\"), so is the area of the triangle, 90,000,000, to 3713\"; that is, 1\u00b0 1' 53\". So the area BEA has a value of 1\u00b0 1' 53\". But in Chapters 29 and 30, the angle BEA was also 1\u00b0 1' 53\". Therefore, both parts of the equation are equal at this place, that is, near 90\u00b0.","839":"At other degrees of eccentric anomaly, we proceed as follows. Since BEA is 3713\", as its altitude EB is to HL or GM, the altitudes of the other triangles \u2014that is, as the whole sine is to the sines of the eccentric anomaly HBC, GBC\u2014so is 3713\" to the areas of the remaining triangles. So 3713\" will be multiplied by the sines of the angles at B, and, with the last five digits struck, the remainder will be the physical part of the equation expressed in seconds, corresponding to the angle at B. For example, let HBC be 45\u00b0 43' 46\", as it was above in Chapter 31. Therefore, its sine, 71605, multiplied by 3713\", with the last five digits struck, gives 2659\", that is, 44' 19\". In the table above, we assumed this to be equal to the optical part of the equation, 43' 46\".","840":"Therefore, this little area ABO at its greatest does not exceed 33\".","841":"And this is that fourth procedure for computing the eccentric equations, of which I began to speak above near the end of Chapter 34, which closely expresses the very nature of things and the foregoing theories of Chapters 32 and 33. ","842":"Nevertheless, my argument contains a paralogism, not, indeed, of great moment. It arises from this: that while Archimedes did indeed divide the circle into an infinity of triangles, they stood upon the circumference at right angles, so that their vertices were at the center of the circle B. But one cannot proceed in the same way with triangles standing upon the circumference with their vertices at A, because the circumference is intersected obliquely by the straight lines from A in all places other than C and D.","843":"You could have found this error empirically, as I myself did, by taking all the distances AC, AG, AH, at the individual whole degrees of the angle CBG, GBH, and adding them all up. (These distances, though they are presented in the table in Chapter 30, correspond in position to individual whole degrees of the angle at A, and consequently to angles at B cut minutewise. Nevertheless, one could easily find, by interpolation, the distance from A corresponding to any angle of an integral number of degrees about B.) Now the sum comes out to be greater than 36,000,000, although 360 of the distances from B add up to exactly 36,000,000. But, on the contrary, if both sums were measured by the same area of the circle, the sums ought to have been equal.","844":"A demonstration of the error, on the other hand, is as follows. Through B let any straight line other than CD be drawn, intersecting the circumference, and let it be EF; and let the points of intersection E and F be connected with A. Now since the point A does not lie on the line EF, EAF is a figure, a triangle. Therefore, EA, AF together are longer than EF, by Euclid I.22. But the area of the circle contains the sum of all lines EF, and therefore it contains a sum which is less than all the lines EA, AF, since any two opposite points on the eccentric, together with A, determine such a triangle, with the exception of C, D, A, where instead of a triangle there is a straight line.","845":"I would add, in passing, that it is also proved, in the same way, that the distances from A corresponding to all of the 360 integral degrees of the angle at A (which are in the table in Chapter 30 above), added up in one sum, are less than 36,000,000. For through the point A let any straight line other than CD be drawn (let it be EV), and let E and V be connected with B. In the triangle EBV, the straight lines EB, BV together will be longer than EA, AV, two opposite distances. But all 360 of EB, BV taken together are 36,000,000. Therefore, all 360 of EA, AV taken together are less than 36,000,000.","846":"To return to what I was saying, this method of finding the equations is not only very easy indeed, and based upon the natural causes of the motion explained above, but also agrees most precisely with the observations in the theory of the sun or earth. Nevertheless, it errs in two respects. First, it supposes that the orbit of the planet is a perfect circle, which, as will be demonstrated below in Chapter 44, is not true. Second, it uses a plane which does not exactly measure the distances of all points from the sun. Nevertheless, as if by a miracle, each of these exactly cancels the effect of the other, as is demonstrated below in Chapter 59. ","847":" And because at the present time there are first-rate geometers who on occasion labor endlessly on matters whose usefulness is not so evident, I call upon one and all to help me here to find some plane figure equal to the sum of the distances. I have indeed myself found it geometrically - in a broad sense - but let them show me how to express numerically what I have delineated geometrically; that is, let them show how to square the figure I have found. Let the circumference CED be unfolded into a straight line and divided into as many parts as before at the points G, \u0397, E, I, K, and let perpendiculars be set up at the points of division equal to the radius CB, and let the parallelogram be completed. This will be double the Archimedean triangle, by which he measured the area of the semicircle. Now if you were to make individual parallelograms in this manner in the individual sectors, then the whole parallelogram divided into parts will be equivalent to the whole area of the semicircle; that is, the ratio 2:1 holds everywhere.\nNow, in the same manner, let the distances CA, GA, and so on, be set up, and the points A be connected by the conchoid* A, A, A, A, drawn through the individual points (which are potentially infinite): the figure AACD will be equivalent to all the distances from A. For similarly the individual lines AG, AH have approximately made up the one parallelogram, except that the conchoid is not parallel to CD, but inclined to the radii GA, HA, EA, exactly as the distances are inclined to the circumference in the circle itself. Hence, there is nothing wrong with the conchoid's being made longer than the semicircle CD.","848":"But EA is longer than EB, so that, if CA, GQ, HR, EB, IS, KL, DA, be taken, equal to lines determined by perpendiculars drawn from A to the lines between the points and B (as, in the circular diagram, the perpendicular AR is dropped to HB extended, defining HR which is shorter than HA), the figure between the conchoid AQRBSLA and CD would be quite equal to the figure CBBD. For the conchoid would intersect BB at the line EA, and because BA at the top and bottom are equal, and BQ is equal to LB, BR to SB, and so on, the figures BBRQA and BBALS would therefore be congruent. One of these is added to, and the other subtracted from, the equal figures CBBE and EBBD, and therefore, the whole figure between AQRBSLA and CD is equal to that between BB and CD. Hence, the small area between the two conchoids AQRBSLA and AAAAAA is the measure of the excesses of the distances from A over the distances from B \u2014and the standard of measure is the same as that by which the parallelogram is set equal to all the distances from B. ","849":"It should also be noted that this area is not of the same breadth at places equally removed from the line EA, but wider below. For in the circular diagram let HBR be extended to V, so that AH, AV correspond respectively to the upper angle HBE and the lower angle FBV, which are equal and are equally removed from the middle points E and F. And about center A with radius AV let the arc XY be drawn through AH and BH. Now if you connect A and Y, 4 AYR will be exactly congruent to the triangle AVR, for AV and AY and AX are equal, by construction, and are the longer sides, while VR, RY are equal and are the shorter. But from the point H outside the circumference XY two lines are drawn: HX through the center A, and HY not through the center. Therefore, HY is longer than HX, and therefore the greater AV or AX is increased by the shorter XH, and the lesser VR or RY is increased by the longer YH\u2014and nevertheless, the whole RH remains shorter than the whole AH. Therefore, the difference between RH and AH is less than the difference between RY and AX, that is, between VR and VA. And consequently in the conchoid SA is greater, and RA less, although IE, EH are equal. Therefore, the area between the two conchoids is not bisected by EA. However, it appears to be bisected by BB, which some geometer should investigate, who should at the same time show how to square the area between the conchoids, so that it may be expressible in numbers. In Chapter 43 below you will find a rough estimate of this area.","850":"Granted, these general considerations of the computation of the physical equation are not yet well enough supported by the geometrical apparatus. Nevertheless, I wished to given them a preliminary treatment here, so that when all the planetary inequalities are determined (as, in particular, we presupposed that the course of the sun or earth is a perfect eccentric, which will be denied concerning Mars below in Chapter 44 and 53), this operation will not be so sharply divorced from its basis in physical theory. For, touching the theory of the sun, with which we have been concerned to this point, we introduce no discrepancy either by misjudging the area of the conchoid, which we have taken to be less than it really is, or the assumption of a perfect eccentric, in which we appear to be erring in excess\u2014to what extent cannot yet be said, since all has not been presented. But the very things which have been rejected in this chapter as paralogisms will be taken up again below, when we shall have come to a perfectly correct way of expressing the equations, when the thing that gave rise to the paralogism will have been eliminated from that hypothesis.","851":"I have described within a hair's breadth, through most certain observations and proofs, the cause and measure of the second inequality, which makes the planets appear stationary, direct, and retrograde. It has been shown that this second inequality itself shares something in common with the first inequality, and that the theory of the sun or earth (for Copernicus) or of the epicycle (for Ptolemy) is like the theory of the other planets. Also, the physical causes of this inequality have been found, and have been adapted to the calculation in the theory of the sun. Therefore, it is fitting that I now bring to a close this third part, as a morning task followed by lunch, with the master of the Metamorphoses adding his voice to mine:","852":"Part remains of what has been begun, part of the work is finished:","853":"The anchor is cast; here let the craft lie. 5","854":"Chapter 41 ","855":"A trial examination of the apsides and eccentricity, and of the ratio of the orbs, using the observations recently employed, made at locations other than opposition with the sun, with, however, a false assumption.","856":"In the second part, above, I tried to find the aphelion and eccentricity, as well as the distances of the star Mars from the sun on the entire circle, using acronychal observations in imitation of the ancients. And indeed, the eccentric equations corresponded closely to other observations made elsewhere than at opposition with the sun. However, the eccentricity and the distances from the sun were first repudiated by the annual parallaxes of longitude and latitude. Therefore, in order that the distances of the star from the center of the sun could be found throughout the circumference of the eccentric, the second inequality (epicyclic, for Ptolemy, or belonging to the annual orb for Tycho and Copernicus) had to be explored first, in Part 3. But now, if the planet's path were a perfect circle, the planet's first inequality, which exists by reason of the eccentric, could be investigated immediately. For in Chapter 25 above we presented a method by which, given the distances of three points of the circumference from some point within the circumference, and the angles at that point, to find the position and size of the circle with respect to that point, the center and eccentricity, along with the apsides.","857":"Now, in Chapter 26, the distance of Mars from the center of the sun was found to be 147,750 in 14\u00b0 21' 7\" Taurus, at the node, on 1595 October 25. Again, in Chapter 27, the distance of Mars was found to be somewhat less than 163,100 at 5\u00b0 25' 20\" Libra, and that was on 1590 December 31. And because Mars is 41 degrees from the node, multiplying the sine of 41\u00b0 by the sine of the greatest inclination, found in Chapter 13, yields an inclination at that point of 1\u00b0 12' 40\". The secant of this exceeds the radius by 22 parts in one hundred thousand, which, in our dimensions, is 34 units. Therefore, the corrected distance at this place would be somewhat less than 163,134. Let it remain 163,100. But the secant of this inclination divided into the secant of 41\u00b0 gives the secant of an arc 50\" longer. Therefore, 50\" must be subtracted from the position of Mars, so as to make it 5\u00b0 24' 30\" Libra.","858":"Thirdly, in Chapter 28 the distance of Mars was found to be 166,180 at 8\u00b0 19' 20\" Virgo, on 1590 October 31, 68 degrees from the node. So the inclination at that place is 1\u00b0 42' 40\". The secant of this is 45 units larger [than the radius], or 75 in our dimensions. Therefore, the corrected distance is 166,255. The subtraction from Mars's position, to reduce it to the ecliptic, is 16\".","859":"These three positions, referred to the same year, 1590, and the month of October, through corrections for the precession of the equinoxes, are: 1","860":" It is clear that the aphelion is nearer to the eighth degree of Virgo than to the others, because its distance is longer. So, following [the pattern of] the demonstration in Chapter 25, let \u03b1 be the center of the solar body. From it let \u03b1\u03b8, \u03b1\u03b7, \u03b1\u03ba be drawn in the same ratio as the distances are produced numerically here, and let all the points be joined. And let the angle \u03ba\u03b1\u03b8 be 114\u00b0 2' 12\", which is the angle from 14\u00b0 Taurus to 8\u00b0 Virgo. Similarly, let \u03ba\u03b1\u03b7 be 27\u00b0 5' 17\", which is the angle from 8\u00b0 Virgo to 5\u00b0 Libra. And \u03b7\u03b1\u03b8 is the sum of the two. For the sun is assumed to be the center of the zodiac.","861":" We need now to investigate the circle which passes through \u03b7\u03ba\u03b8, so that \u03b7, \u03ba, \u03b8 may be three positions of the planet.","862":"In the Ptolemaic form, \u03b1 will be the earth, the center of the zodiac, and \u03b7, \u03ba, \u03b8 three positions of the point of attachment of the epicycle. Everything else stays the same.","863":"So, in triangle \u03b7\u03b1\u03b8, the angle with its sides being given, the angle \u03b1\u03b8\u03b7 is found to be 20\u00b0 26' 13\". Likewise, in \u03ba\u03b1\u03b8, angle \u03b1\u03b8\u03ba is given as 35\u00b0 10' 17\". Subtracting \u03b1\u03b8\u03b7 from this leaves \u03b7\u03b8\u03ba, 14\u00b0 44' 4\". Let \u03b3 be the center of the circle in question. Let \u03b1\u03b3 be drawn, and extended to the aphelion \u03b5 and perihelion \u03b4; and let \u03b7 and \u03ba be joined to \u03b3.","864":"Now, since \u03b7\u03b8\u03ba stands on the circumference, and \u03b7\u03b3\u03ba on the center, subtending the same arc \u03b7\u03ba, \u03b7\u03b3\u03ba will therefore be twice the angle \u03b7\u03b8\u03ba, or 29\u00b0 28' 8\". And where \u03b7\u03b3 is 100,000, \u03ba\u03b7 will be 50,868, which is double the sine of half \u03b7\u03b3\u03ba.","865":"Now, in triangle \u03b7\u03b1\u03ba, the angle with its sides again being given, \u03ba\u03b7\u03b1 is found to be 78\u00b0 44' 1\", and through this, \u03ba\u03b7 [is found to be] 77,187 where \u03b7\u03b1 is 163,100. Therefore, in the units of which \u03ba\u03b7 formerly was 50,868 and \u03b7\u03b3 was 100,000, \u03b7\u03b1 becomes 107,486. And since \u03b7\u03b3\u03ba is 29\u00b0 28' 8\", \u03ba\u03b7\u03b3 will be half the supplement, because \u03b7\u03b3, \u03ba\u03b3 are equal. Therefore, \u03ba\u03b7\u03b3 is 75\u00b0 15' 56\". Subtract \u03ba\u03b7\u03b1 from this. The remainder is \u03b3\u03b7\u03b1.","866":"Thus, in triangle \u03b3\u03b7\u03b1, the angle with its sides is given. Therefore, \u03b7\u03b1\u03b3 is known to be 38\u00b0 15' 45\". And consequently (since \u03b1\u03b7 is in 5\u00b0 24' 21\" Libra) the line of apsides \u03b1\u03b3 will be in 27\u00b0 8' 36\" Leo. But through the angle \u03b7\u03b1\u03b3 the eccentricity \u03b1\u03b3, 9768, is also found, in units of which \u03b7\u03b3 is 100,000. Finally, in the units of which \u03b1\u03b7 is 163,100, \u03b7\u03b3 will be 151,740. But in the same units, the semidiameter of the annual orb was also 100,000. Therefore, the ratio of the orbs is that which 100,000 has to 151,740.\nHow erroneous all of this is, you can gather from this: that however many times you take, instead of one or more of the distances \u03b1\u03b8, \u03b1\u03b7, \u03b1\u03ba that were used, some other distance, corresponding to another place on the eccentric, and found by an equally certain irrefutable line of argument, each time you do this all of those things come out differently.\nAnd in the following chapter, [the ratio] will be found with greatest certainty to be that which 100,000 has to 152,640, approximately; eccentricity 9264, where the radius is 100,000. The aphelion for 1590 October 31 was found in Chapter 16 above to be at 28\u00b0 53' Leo, which will be confirmed, in the next chapter, to be within 11' of the truth.","867":" Chapter 42","868":"Through several observations at places other than the acronychal position, with Mars near aphelion, and again several others with Mars near perihelion, to find the exact location of the aphelion, the correction of the mean motion, the true eccentricity, and the ratio of the orbs.","869":"You have just seen, reader, that we have to start anew. For you perceive that three eccentric positions of Mars and the same number of distances from the sun, when the law of the circle is applied to them, reject the aphelion found above (with little uncertainty). This is the source of our suspicion that the planet's path is not a circle. On this supposition, one could not use three distances to learn the others. Therefore, the distance at any particular place has to be deduced from its own observations, and especially those at aphelion and perihelion, through the comparison of which we learn the true eccentricity. ","870":"Let \u03b1 be the center of the world, \u03b1\u03b2 the line of apsides, and \u03b9\u03b8 the eccentric upon center \u03b2, with \u03b9 the aphelion and \u03b8 the perihelion. From Chapter 41, or better, from Chapter 16, we understand that the observations in which Mars is nearest to \u03b9 are these.","871":"On 1585 February 17 at 10h the planet was seen at 15\u00b0 12\u00bd Leo, with latitude 4\u00b0 16' North. II. On 1586 December 27 at 4h in the morning, at 29\u00b0 42\u2154\u2018 Virgo, latitude 2\u00b0 46\u2157\u2019 N. ","872":"[IIb.]3 And on 1587 January 1 at 7h 8m in the morning, at 1\u00b0 4' 36\" Libra, latitude 2\u00b0 54\" N. And on January 9 in the morning, at 2\u00b0 51\u00bd\u2019 Libra, latitude 3\u00b0 6' N. ","873":"[III.] On 1588 November 10, at 6h 30m in the morning there was 31\u00b0 27' between Mars and Cor Leonis. Mars's declination was 3\u00b0 16\u00bc\u2019 North. Therefore, Mars was at 25\u00b0 31' Virgo, latitude 1\u00b0 36' 45\" N. [IIIb.] On December 5 at 6h in the morning there were 45\u00b0 17' between Mars and Cor Leonis. The declination was 2\u00b0 5' south. Therefore Mars was at 9\u00b0 19\u2156\u2019 Libra, latitude 1\u00b0 53\u00bd\u2019 north. These observations were not, however, confirmed by fixed stars on the other side of Mars.","874":"[IV] On 1590 October 6, at 4h 45m in the morning, Mars was observed at an altitude of 12\u00bd degrees, [and distances taken] from the Tail of Leo and the Heart of Hydra, with its declination. But since neither of the fixed stars was extended straight from Mars in the direction of the longitude, it happened that the two right ascensions, constructed through the given declination, disagreed by 6'. This can easily happen if some very small amount is wanting in the declination. Indeed, they appear not to have had much confidence in this, with the result that they measured Mars from the Tail of Leo, which is at the same longitude, all the distance being latitudinal, with the aim of knowing Mars's latitude with greater certainty from this rather than from the declination. But let the declination of 6\u00b0 14' stand, as well as the distance from the Heart of Hydra of 34\u00b0 33\u00bd\u2019. Its right ascension would thus be 168\u00b0 56\u00bc\u2019. Therefore, its position would be 17\u00b0 16\u00be' Virgo, latitude 1\u00b0 16\u2154\u2019 north. The table of refraction for the fixed stars shows 4 minutes at this altitude, while the table for the sun shows more. Also, Virgo is rising steeply. Therefore Mars has to be put forward (eastward) about 3 minutes or (using the solar refractions) a little more, whence it was subtracted by refraction. The parallax was quite small, so it hardly removes anything from the refraction. Mars would have been at 17\u00b0 20' Virgo. [V] On 1600 March 5\/15 at 8\u00bdh pm Mars was at 29\u00b0 12\u00bd\u2019 Cancer, latitude 3\u00b0 23' N. And on March 6\/16 at 8\u00bdh at 29\u00b0 18' Cancer, latitude 3\u00b0 19\u00be' N. Now the times that return Mars to the same place on the eccentric correspond to one another as follows:14 \nThe procedure for referring the observations to the appropriate times is this. Since in 1587 the diurnal motions of Mars are decreasing, as is apparent both in Magini and in the observations on the three days, I have assumed the following diurnal motions: 17, 16, 16, 16, 15, 15, 14, 14, 13, 13, 13, 12, 12.","875":"On 1588 November 10 the observation is 39 minutes less than the midday position of Magini. On December 5 it is 33 minutes less. And our moment is between these. Therefore, we have also taken the intermediate difference of 36'.","876":"In 1590 the observation is solitary, and, as was seen, was itself not well made. Nevertheless, the diurnal motion in Magini is a constant 37' over many days. ","877":"Now to the point: and while I have so far presented many methods of finding or testing the eccentric positions and distances, I nevertheless here follow yet another one, it being the easiest. Let \u03b4, \u03b5, \u03ba, \u03bb, \u03b3 be positions of the earth, with \u03b4, \u03b3 on the left and \u03b5, \u03ba, \u03bb on the right side of the eccentric. And since the lines \u03b1\u03b4, \u03b1\u03b5, \u03b1\u03ba, \u03b1\u03bb, \u03b1\u03b3 are given, and also the angles \u03b1\u03b4\u03b9, \u03b1\u03b5\u03b9, \u03b1\u03ba\u03b9, \u03b1\u03bb\u03b9, \u03b1\u03b3\u03b9, I shall take a third element common to all the triangles, namely the side \u03b1\u03bb\u03b9, which is one of the magnitudes sought, and using this side I shall find the angles at \u03b9 and see whether they place the line \u03b1\u03b9 at the same zodiacal position (except to the extent that it is moved forward in the later times by the precession of the equinoxes). From this I am going to know whether the value assumed for \u03b1\u03b9 was any good. ","878":"The basis of the method is this: that as \u03b1\u03b9 is to [the sines of] the angles \u03b4, \u03b5, \u03ba, \u03bb, \u03b3, so are \u03b1\u03b4, \u03b1\u03b5, \u03b1\u03ba, \u03b1\u03bb, \u03b1\u03b3 to [the sines of] the angles at \u03b9. ","879":"The sines of these, multiplied by the earth-sun distance, and divided by the magnitude assumed for \u03b1\u03b9, 166,700, yields the sines of the angles which, added to the observed positions of Mars at \u03b3, \u03b4, and subtracted at \u03b5, \u03ba, \u03bb, put the line \u03b1\u03b9 at the following positions: ","880":"That is, the five positions ought to have differed by no more than the amount occasioned by the precession of the equinoxes. ","881":"You see from the diagram that if, other things remaining the same, you will take \u03b1\u03b9 to be shorter, it is going to be moved forward at \u03b3, \u03b4 and back at \u03b5, \u03ba, \u03bb, but not by an equal distance for all of them. And as soon as you do this, you will make matters worse at \u03b4, \u03ba, \u03bb, and better at \u03b3, \u03b5. The opposite will happen if you will lengthen \u03b1\u03b9. But it ","882":"is fitting to have these small errors distributed among all the positions. Therefore, the distance \u03b1\u03b9 is not to be changed at all, and the planet, at the prescribed times, is at the positions last mentioned.","883":"If you wish to seek confirmation using the method of Chapter 28, to test the consensus, the points \u03b4, \u03b5 being joined, you will find \u03b4\u03b5 to be 74,058, \u03b4\u03b5\u03b1 68\u00b0 36' 0\", \u03b5\u03b4\u03b1 67\u00b0 21' 3\". Therefore \u03b5\u03b4\u03b9 is 88\u00b0 28' 50\", and \u03b4\u03b5\u03b9 is 44\u00b0 36' 46\", and \u03b5\u03b9\u03b4 is 46\u00b0 54' 24\". Therefore, \u03b9\u03b5 is 101,380, and \u03b5\u03b1\u03b9 is 33\u00b0 58' 33\". Therefore, in 1587 \u03b1\u03b9 was at 29\u00b0 19' 49\" Leo (we just now chose 29\u00b0 18' 36\", the difference of one minute keeping it in agreement with other positions). Finally, \u03b1\u03b9 is 166,725, and the position of \u03ba is in agreement [with the former one].","884":"Since 166,666\u2154 is the sesquialter of the radius, 100,000, it is credible that this is the ratio of the mean distance of the earth from the sun to the greatest distance of Mars from the sun. But at present I shall base nothing upon conjecture.","885":"And since the plane of the eccentric is inclined to the ecliptic here at an angle of 1\u00b0 48', whose secant is 49 units above [the radius], or 82 of our present units, the most correct distance of Mars and the sun will be 166,780, as far as can be told from these observations, which, you will recall, were deduced from ones that were rather distant instead of being optimally obtained on the very days in question.","886":"Let us now proceed to the perigee, where the catalog of observations, and a middling knowledge of the mean motion, show the following to be the nearest observations:","887":"On 1589 Nov. 1 at 6\u2159h in the evening, Mars was at 20\u00b0 59\u00bc\u2019 Capricorn, with latitude 1\u00b0 36' south. ","888":"On 1591 Sept. 26 at 7h 10m at 18\u00b0 36' Capricorn, latitude 2\u00b0 49\u2155\u2018 south. ","889":"On 1593 July 31 at l\u00beh am at 17\u00b0 39\u00bd' Pisces, latitude 6\u00b0 6\u00bc\u2019 south, and August 11 at 1\u00beh am at 16\u00b0 7\u00bd' Pisces, latitude 6\u00b0 18\u215a\u2018 south. ","890":"The times correspond thus: ","891":"For 1591 we need to take it on faith that the diurnal motions are the same as","892":"those of Magini, since the observation is solitary. And since in Magini it moves","893":"4\u00b0 16' in 7 days, on September 19 at 7\u2159h Mars will be at 14\u00b0 20' Capricorn, and at 6\u2159h it will be at 14\u00b0 18\u00bd\u2019 Capricorn. About the station on July 16 or 17, Mars was about 1\u00b0 16' farther forward in the calculation than in Magini. Now, on September 26, it is still 0\u00b0 53' farther forward. Therefore, over 70 days the difference has been diminished by about 23 minutes. So if we interpolate, this difference will be about 2 minutes greater on September 19. We shall therefore believe that at our given moment Mars is at 14\u00b0 20' Capricorn.","894":"In 1593 Mars left its station. And on midnight of July 30 the position of Mars disagrees with Magini's midday position by 3\u00b0 25\u00bd\u2019, and on August 10 by 3\u00b0 59\u00bd\u2019,so that the difference is increased, but gradually less so. Therefore, I have assumed a difference of 3\u00b0 46' for August 6, so that at 11 hours after midnight it would be at 16\u00b0 52' Pisces. And the diurnal motion was 10'. This is 8 hours 30 minutes past our time, which would account for about 4' of Mars's retrograde motion. Therefore, at our time it was at 16\u00b0 56' Pisces. It is certain that (on this point at least) we are no more than one minute high or low.","895":"It was not observed more frequently at perigee. For in 1595 its arrival at perigee fell in the middle of summer, when twilight lasts all night in Denmark.","896":"In 1597 Tycho Brahe was travelling. And when it is near the sun in its winter semicircle it is long hidden, since its speed is not much less than the sun's.","897":"In the diagram, let Mars's eccentric position be \u03b8, the positions of the earth, \u03b6, \u03bc, \u03b7; and let ","898":"But if it was 55' 20\" at \u03b6, it should have been 56' 56\" at \u03bc, and 58' 32\" at \u03b7, for that is the amount of the precession of the equinoxes. It can thus be seen from the diagram that the line \u03b1\u03b8 determined through \u03b7 goes too far forward, and through \u03bc, \u03b6, too far back, in relation to that through \u03b7. Other things remaining unchanged, this happened because I assumed too small a value for \u03b1\u03b8. Therefore, if I make it a hundred parts longer, namely, 138,500, the following positions come out: ","899":"21","900":"So now the positions of \u03b1\u03b8 have been made to be too close to one another, and the error is now more so in closeness than it was before in remoteness. Therefore, the most correct length of \u03b1\u03b8 will be about 138,430.","901":"At this point the plane is inclined 1\u00b0 48' (as it was before at the opposite position), and the secant is 49 units greater than the radius. But as 100,000 is to 138,430, so is this 49 to 68. Therefore, the correct length of the radius is approximately 138,500, at least from these observations involving long interpolations.","902":"Investigation of the apsides, from the above","903":"With all three observations taken into account, let the position of the line \u03b1\u03b8 on 1589 November 1 at 6\u2159h pm be taken as 29\u00b0 54' 53\" Aquarius, so that in 1591 it would be 29\u00b0 56' 30\", and in 1593, 29\u00b0 58' 6\" Aquarius. The vicarious hypotheses of Chapter 16 shows it to be at 29\u00b0 52' 55\" for the first of the times. ","904":"But previously and in like manner we took at \u03b1\u03b9 1588 November 22 at 9h 2\u00bdm to be 29\u00b0 20' 12\" Leo.","905":"From 1588 November 22 at 9h 2\u00bdm to 1589 November 1 at 6h 10m are 344 days diminished by 2h 52\u00bdm while a whole revolution to the same fixed star has 687 days diminished by 0h 28 min. Therefore, our interval appears to exceed half the periodic time by a few hours.","906":"Consider: ","907":"And from the position at the earlier time, 29\u00b0 20' 12\" Leo, to the position which Mars held at the later time, 29\u00b0 54' 53\" Aquarius, is 180\u00b0 34' 41\", or 180\u00b0 33' 53\" with the precession of 48\" subtracted. Therefore, if the excess of 33' 53\" beyond the semicircle is sufficient for the 10 hours 6\u00bd minutes from Mars's diurnal motion on the eccentric, the aphelion would consequently be understood to be at 29\u00b0 20' 12\" Leo.","908":"But we know the diurnal motions of Mars on the eccentric near apogee and perigee from the distances just found and from the demonstrations of Chapter 32. For the diurnal motions are approximately in the [inverse] duplicate ratio of the distances. At apogee the diurnal motion is about 26' 13\", at perigee 38' 2\", since the mean diurnal motion is 31' 27\".","909":"Consider, then: if Mars, in moving from its apogee point, expends half its periodic time, at the end of this time, having traversed exactly 180 degrees, it is going to be at the perigee point. But now if it begins this space of time one day after it was at apogee, it will begin its course 26' 13\" beyond apogee and will end it at 180\u00b0 38' 2\". Therefore, in half the time it will traverse 11' 49\" more than half the path. The opposite will happen if it were to begin one day before apogee. ","910":"Therefore, since our time, too, shows an arc greater [than a semicircle], our aphelion also should be moved forward. First, we shall credit half of our hours to the time before aphelion, and half after perihelion. The planet will then begin from 5' 16\" before aphelion, which is thus put at 29\u00b0 25' 28\" Leo, and it will come to 8' 1\" after perihelion, the amount of travel being 13' 17\" beyond 180\u00b0. But its path was seen to be 33' 53\" beyond 180\u00b0. Therefore, it is still faster by 20' 36\". Therefore, since to increase the path by 11' 49\", one day, or the promotion of the planet to 26\u2019 13\" beyond aphelion, is needed, how much will the planet be promoted from aphelion to increase the path by 20\u2019 36\"?","911":"The rule of proportions shows it to be 1 day 17h 54m, or a distance from aphelion of 45' 42\". Therefore, the aphelion is to be moved forward 45' 42\" from the position we just gave it, 29\u00b0 25' 28\" Leo. ","912":"To which of the investigations of the aphelion one ought to give more trust, is uncertain. For it can easily happen that in the positioning and assuming of the lines \u03b1\u03b9, \u03b1\u03b8 we have erred by 4 minutes, two for the one and two for the other, owing to difficulties in the observations. And this is all that needs to be accumulated, through the compounding of errors, to change the aphelion by 11 minutes. Here, however, it is reasonable for us to trust the present operation.","913":"Correction of the mean motion","914":"When the aphelion is changed, the mean motion is changed as well. For at the same time at which in the previous investigation of the aphelion Mars is thought to fall at aphelion, with no equation, it has now passed the aphelion by 11 minutes. Therefore, it has an equation of 4 minutes, subtractive. Thus in its mean motion it has passed that original position by 4'. ","915":"Investigation of the eccentricity","916":"First, the distances found previously should be corrected, if necessary, to the extent that they are some small amount distant from the apsides just found, the aphelia by 40 minutes, perihelia by 75 minutes. But there is no perceptible change so close to the apsides. ","917":"And as 152,640 is to 100,000, so is 14,140 to the eccentricity 9264. But half the eccentricity of the equating point was 9282. The difference of 18 is clearly of no importance. You see how precisely the eccentricity of the equating point is to be bisected in Mars in order to establish the distance between the centers of the eccentric and the world. Above, in Chapter 32, I took this to be fundamental, and in the following chapters postponed its demonstration. Now, however, that obligation is discharged. ","918":"Chapter 43","919":"On the defect in the equations accumulated by bisection of the eccentricity and the use of triangular areas, on the supposition that the planet's orbit is perfectly circular.","920":" What was proved in Part III concerning the bisection of the eccentricity in the theory of the sun has now likewise been demonstrated with perfect certainty for Mars. And now that our evidence of this is complete, it would at last be time to proceed to the physical theories of Chapters 32 and the following, seeing that they are going to apply to all planets in common, had I not seen fit to present them earlier. I did so because there, in the theory of the sun or earth, the procedure for computing the equations from physical causes had to be completed with full perfection, and because I knew that where that method of constructing the equations is to be applied to the theory of Mars, much more difficult physical theories were to follow.","921":"Now when the true configuration of the orbits is found, the eccentric equations, upon which alone the vicarious hypothesis found in Chapter 16 has hitherto depended, must necessarily follow by the same means. We shall therefore explore it in turn here.","922":"Therefore, following what was demonstrated in Chapter 40 (all of which, in every detail, is to be understood as holding here), let the orbit of the planet, in accord with the well-worn opinion, be circular, even though Chapter 41 has just urged us to doubt it. Therefore, at the eccentric anomaly of 90\u00b0 the eccentricity 9264 found in Chapter 42 will be the tangent, which will give the optical part of the equation, 5\u00b0 17' 34\". And since at the eccentric anomaly of 90\u00b0 the area of the triangle is right-angled, the radius multiplied by half the eccentricity, 4632, gives the area of the triangle, 463,200,000. Now as the area of the circle, 31,415,926,536, is to 360 degrees or 1,296,000 seconds, so is this area just found, 463,200,000, to 19,108\", or 5\u00b0 18' 28\", the physical part of the equation. Consequently, the whole equation is 10\u00b0 36' 2\", so that to the mean anomaly of 95\u00b0 18' 28\" corresponds the equated anomaly of 84\u00b0 42' 26\".","923":"But according to the method of Chapter 18, the vicarious hypothesis, accurate enough for the longitudes, shows us that to the mean anomaly of 95\u00b0 18' 28\" there ought to correspond the equated anomaly of 84\u00b0 42' 2\". The difference is 24\".","924":"Now let our eccentric anomaly be taken as 45\u00b0 and 135\u00b0. And as the whole sine is to the sine of these angles, so is 19,108\", the area of the greatest triangle of the equation, to the area at this position, 13,512\", or 3\u00b0 45' 12\", so that by addition of this the physical part of the equation to the eccentric anomaly the mean anomalies of 48\u00b0 45' 12\" and 138\u00b0 45' 12\" are constructed. But from the given sides of the given angles, the angles of equated anomaly corresponding to these mean anomalies come out to be 41\u00b0 28' 54\" and 130\u00b0 59' 25\". But by the vicarious hypothesis, as in Chapter 18 of this work, the same simple anomalies of 48\u00b0 45' 12\" and 138\u00b0","925":"45' 12\" being taken, the equated anomaly for the former comes out to be 41\u00b0 20' 33\", less than by the area of the triangle, the excess being 8' 21\"; and for the latter, 131\u00b0 7' 26\", more than by the area of the triangle, the defect being 8'. So, since it is certain that an error of this magnitude cannot be attributed to our vicarious hypothesis, I had to accept that this procedure for finding the equations was still imperfect.","926":"Indeed, in Chapter 19 as well, when I tried out the bisection on Mars and computed the equations using a motionless point of the equant in the Ptolemaic manner, a difference was found at about 45\u00b0 of eccentric anomaly of nearly the same amount, but in the opposite direction. For in the upper quadrant, the planet was closer to the aphelion, and in the lower to the perihelion, than it should have been; while here in the upper quadrant it was farther from the aphelion, and in the lower from the perihelion, than it should be. And so in the upper quadrant it is moving too swiftly away from the aphelion, and the same from the perihelion below. Therefore, it is slower than it should be in the middle elongations.","927":"I believe it has just occurred to the reader that the cause of these errors might perhaps lie in the flaw to which this operation with areas is subject, mentioned in Chapter 40: that the areas are not equivalent to the distances that modify the swiftness and slowness. But the present error cannot arise thence. For first of all, the excess of the sum of the distances over the area of the circle is small: just about as small, that is, as the little space between the conchoids. Then, too, the area makes all the distances a little smaller than they should be, and most of all those that are at the middle elongations. So if any error flows from this, it lies in our not having made the planet take enough time in the middle elongations. But the errors we are now seeing are in the opposite direction, for we have made the planet take too much time in the middle elongations.","928":"The same can be raised in objection to anyone who might conceive a suspicion that the error arose because we rejected the double epicycle of Copernicus and Tycho, which makes the orbit of the planet oval, and took up the Ptolemaic perfect circle in the present account. For it was said at the end of Chapter 4 that the Copernican orbit moves outwards from the center by 246 parts, which would only increase the error, rather than making an incursion towards the center, as would suit our purposes, since we are now following the idea that the time increments are proportional to the distances.","929":"But to make it clear to the eye that the area of the conchoid of Chapter 40 is made very small, consider that the secant of the angle 5\u00b0 19' (the maximum optical equation) is 100,432, which is the line EA. So from this excess of 432, which is the small line BA, part of the line EA, we will be able to get an approximate idea of the sum of all these excesses (such as, QA, RA, BA, SA, LA) in this way.","930":"The secant of 89\u00b0, and its tangent, taken together, are as great as the sines of all degrees of the whole semicircle, as Cardano helps us see in the part of De subtilitate in which he explains the properties of the circle. A proof of this is given by Justus Byrgius. ","931":"Therefore, if all our remaining excesses (other than the greatest, 432) were [to the greatest]3 as [all] the sines in one semicircle are to the semidiameter, then as 100,000 is to the sum of the secant and the tangent of 89\u00b0 (that is, 11,458,869), so, approximately, would 432 be to 49,934, the approximate sum of all the excesses at integral degrees of the semicircle. For the excesses of the distances in the upper quadrant are longer than those excesses of the secants, to about the same extent that they are shorter in the lower quadrant.","932":" But nevertheless, it is not true that the excesses QA, RA, SA, and so on, are to one another as the sines of the corresponding number of degrees. Instead, they are approximately in the duplicate ratio of the sines. As for example, the sine of 90\u00b0 is twice the sine of 30\u00b0. Now the optical equation of 90\u00b0 is 5\u00b0 19', and half of its sine gives an arc which is likewise about its half, that is, 2\u00b0 39\u2019 15\", for the optical equation at 30\u00b0 of eccentric anomaly, whose secant is 100,107. And here 107, the excess of the secant over the radius, is about one fourth of the former, 432, while the sine of 30\u00b0 would be half the sine of 90\u00b0. Some geometer should see whether this argument be demonstrable. For me it suffices at present to answer those very small questions with which I am occupied.","933":"Therefore, to arrive at 432, parts are accumulated that are not proportional to the sines, but are always smaller, and at the 45th degree or thereabouts are but their halves. Before that point they are less than the halves, so that about 30\u00b0 they are only the fourths, and at length become imperceptible.","934":"And so, of the sum of 49,934, we retain only one seventh, or about 7000. This is also shown empirically, by computing all the distances degree by degree and adding them up. ","935":"And because one distance of 100,000 has the value of 60', this little sum has a value of no more than 4\u2155\u20197 which is nonetheless spread all around the circumference, so that about 45\u00b0 and 135\u00b0, where it is greatest, this tiny error turns out to be imperceptible even in Mars.","936":"Consequently, we must seek another occasion for this discrepancy.","937":"Chapter 44","938":"The path of the planet through the ethereal air is not a circle, not even with respect to the first inequality alone, even if you mentally remove the Brahean and Ptolemaic complex of spirals resulting from the second inequality in these two authors.","939":" With the eccentricity and the ratio of the orbs established with the utmost certainty, it might appear strange to an astronomer that there remains yet another impediment in the way of astronomy's triumph. And me, Lord knows, I had triumphed for two full years. Nevertheless, by a comparison of the things which have been established in Chapters 41, 42, and 43, preceding, it is readily apparent what we are still lacking. The positions of the aphelia, eccentricity, and the ratio of the orbs, as constituted in the several places, differed greatly from one another. Nor were the computed physical equations in agreement with the observations (which the vicarious hypothesis represents). Let the diagram of Chapter 41 be brought back. And because, in that diagram, in units of which \u03b3\u03b7 was 151,740, \u03b3\u03b1 would have been 14,822, when \u03b3\u03b1 and \u03b3\u03b7 or \u03b3\u03b5 are added, \u03b1\u03b5 would be 166,562. In Chapter 42 this was found to be 166,780. Likewise, when \u03b3\u03b1 is subtracted from \u03b3\u03b4, the remainder, \u03b1\u03b4, would be 136,918, which in Chapter 42 was found to be fully 138,500.","940":"Again, the true length of the lines \u03b3\u03b5, \u03b3\u03b1, \u03b1\u03b5, and \u03b1\u03b4 was found in Chapter 421 If, therefore, what was supposed and used in Chapter 41 is true, that the path of the planet is a circle, it is not difficult to say how much \u03b1\u03ba, \u03b1\u03b7, \u03b1\u03b8 ought to be. Since in Oct. 1590 \u03b1\u03b5 was at 28\u00b0 41' 40\" Leo, and \u03ba, \u03b7, \u03b8 are as given in Chapter 41, the angles \u03ba\u03b1\u03b3, \u03b7\u03b1\u03b3, \u03b8\u03b1\u03b3 will be given. Therefore, the optical equation will also be given: \u03b1\u03ba\u03b3 0\u00b0 53' 13\", \u03b1\u03b7\u03b3 3\u00b0 10' 24\", \u03b1\u03b8\u03b3 5\u00b0 8' 47\". And as the sine of these angles is to the truest eccentricity \u03b1\u03b3, 14,140, so are the sines of \u03ba\u03b3\u03b5, \u03b7\u03b3\u03b5, \u03b8\u03b3\u03b1 to \u03b1\u03ba, \u03b1\u03b7, \u03b1\u03b8.\n3\nIf anyone wishes to attribute this difference to the slippery luck of observing, he must surely not have felt nor paid attention to the force of the demonstrations used hitherto, and will be shamelessly imputing to me the vilest fraud in corrupting the observations of Brahe. I therefore appeal to the observations of subsequent years, at least those that experienced observers made. For if in any respect I have given free rein to my inclinations in one direction, it will only go so much the farther into error on the other side. But there is no need of this. I am addressing this to you who are experienced in matters astronomical, who know that in astronomy there is no tolerance for the sophistical loopholes that beset other disciplines. To you I appeal. You see at \u03ba a small defect from the circle, at \u03b7, \u03b8 on both sides, a rather large one, enough so that we cannot excuse it by uncertainties in observing (for in Chapter 42 I reckon an uncertainty of perhaps 200, or at most 300 units).\nWhat, then, is to be said? Could this actually be the situation described in Chapter 6 above, in which by transposition of the reference point from the sun's mean motion to its apparent motion I set up another eccentric that makes an excursion towards the side of the sun's apogee? By no means. For there, it would approach from the one side by the same amount as it moves away on the other. Here, however, you see that the planet approaches the center from the circular orbit on both sides. This is confirmed by many other observations, some of which follow below in Chapters 51 and 53.","941":"Clearly, then, [what is to be said] is this: the orbit of the planet is not a circle, but comes in gradually on both sides and returns again to the circle's distance at perigee. They are accustomed to call the shape of this sort of path \"oval.\"","942":"This same thing is also proved from Chapter 43 preceding. There it was supposed that the area of a perfect eccentric is very closely equivalent to all the distances of the equal parts of the circumference of that eccentric from the source of the motive power, however many they are. Thus, the parts of the area measure the amounts of time which the planet spends on the parts of the corresponding eccentric circumference. But if that area about which the planet marks a boundary is not a perfect circle, but is diminished at the sides from the amplitude it has at the apsides, and if nevertheless this area circumscribed by an irregular orbit still measures the times which the planet takes to traverse the whole and its equal parts, then this diminished area measures a time equal to that measured by the previous undiminished area. So the parts of the diminished area nearest aphelion and perihelion will measure a greater time, because in those regions the diminution is narrowest, but the parts at the middle longitudes measure less time than before, because the greatest diminution in the whole area occurs there. So if we now use the diminished area in adjusting the equations, the planet will become slower near aphelion and perihelion than it was in the previous faulty form of equation, and swifter near the middle elongations, because here the distances are lessened. Therefore, the times, when they are abstracted from the area and adjusted upward and downward, will be accumulated at aphelion and perihelion in much the same manner as, if one were to squeeze a fat-bellied sausage at its middle, he would squeeze and squash the ground meat, with which it is stuffed, outwards from the belly towards the two ends, emerging above and below his hand. ","943":"And indeed, if contraries remedy one another, this is plainly the aptest medicine for purging the faults under which, in Chapter 43 above, our physical hypothesis was perceived to be laboring. For the planet is going to be swifter at the middle elongations, where previously it was perceived to be going slower than it should, and it will be slowed down above and below, near the apsides, where previously it did violence to the equations belonging to the eighths of the period through its excessive fleetness.","944":"This, then, is the other argument by which it is proved that the orbit of the planet really is deflected from the established circle, making ingress towards the sides and the centre of the eccentric.","945":"But for all that, this argument still did not have enough effect upon me to let me go beyond it and think about the planet's departure from the orbit. When I had sweated for the longest time trying to reconcile equations of this sort, I was finally discouraged by the absurdity of the measurements, and abandoned the whole enquiry until I was informed by the distances (found in the way shown in Chapter 41) about the departure from the [circular] orbit, and once more took up this problem of the equations.","946":"And from this, what I promised I would prove, in Chapters 20 and 23 above, is now done: that the orbit of the planet is not a circle but of an oval shape.","947":"Chapter 45","948":"On the natural causes of this deflection of the planet from the circle: first opinion examined.","949":"When I was first informed in this manner by Brahe's most certain observations that the orbit of the planet is not exactly circular but is deficient at the sides, I judged that I also knew the natural cause of the deflection from its footprints. For I had worked very hard on that subject in Chapter 39. And I suggest to the reader that he reread that entire chapter carefully before going on. For in that chapter I assigned the cause of the eccentricity to a certain power that is in the body of the planet. It therefore followed that the cause of this deflecting from the eccentric circle should also be ascribed to the same body of the planet. But then what they say in the proverb, \"A hasty dog bears blind pups,\" happened to me. For in Chapter 39, worked very energetically on the question of why I could not state a probable enough cause for a perfect circle's resulting from the orbit of the planet, as some absurdities always had to be attributed to the power that has its seat in the planet's body. Now, having seen from the observations that the planet's orbit is not perfectly circular, I immediately succumbed to this great persuasive impetus, believing that from those things which were called absurd in fabricating the circle in Chapter 39, now transmuted into a more probable form, an orbit of the planet that would be both correct and in agreement with the observations would be effected. If I had embarked upon this path a little more thoughtfully, I might have immediately arrived at the truth of the matter. But since I was blind from desire, and did not pay attention to each and every part of Chapter 39, staying instead with the first thought to offer itself\u2014a wonderfully probable one, owing to the uniformity of the epicyclic motion\u2014I entered into new labyrinths, from which we will have to extract ourselves in this Chapter 45 and the following ones all the way to 50.","950":" Let the diagram from Chapter 39 be repeated. The weaker opinion in that chapter was that, in order to describe a perfect circle, the planet effects an epicycle by its inherent force, thus disengaging its body from the ray of power from the sun. As, for example, if the ray of power from the sun be AC, and move forward at an unequal pace from AC to A\u03b3, while the planet be initially at C, and from that time forth, by its inherent force, it disengage itself from [the ray] AC or A\u03b3. Thus, at the time when AC comes to A\u03b3, the planet from C or \u03b3 would come to D, and would also do this at a nonuniform pace, more swiftly or slowly in the same proportion as [the length of] AC. For in this fashion the line ND through the center of the epicycle and the planet remains ever parallel to the line AB. However, I said in Chapter 39 that it appeared absurd to me that the planet [in moving] from \u03b3 to D at a nonuniform pace disengages itself from the ray of the solar power, and thus accommodates itself by its own force to the extrinsic force from the sun, and has foreknowledge of its speed and the decrease thereof. Therefore, to avoid this absurdity, let AC still go nonuniformly, but let the planet go uniformly from \u03b3 to D. Let us see whether what follows is anything like what we have proved in the preceding chapter from the observations.","951":" While the center of the epicycle N and its aphelion [in moving] from the line AC to A\u03b3 will be slow from C to \u03b3, it being near the eccentric's aphelion C, let the planet [in moving] from \u03b3 to D be supposed not to be slow but to proceed with its mean motion. Consequently, the angle \u03b3ND will be greater than the angle \u03b3AC. So ND will not be parallel to AB but will be inclined towards AC. Thus the planet D will not stay on the circle which it began to describe from C, the one, that is, which goes through CF, but will encroach from the circumference D and the parallel ND towards CA. And this same thing was also affirmed in the previous chapter by the distances AD computed from the observations, namely, that they do not reach all the way to the circumference of the circle CF. This same thing was also affirmed by the physical equations constructed through the summation of the distances AC, AD, namely, that the planet ought to be faster at the sides of the eccentric; that is, that its distances from the sun ought to be supposed smaller. Therefore, since this consensus brought to bear a considerable force of persuasion, I concluded forthwith that the planet's incursion at the sides is the result of this: that the power moving the planet and administering the distances according to the law of the circle supersedes the power of the sun, in that the former made equal progress in equal times thus sending the planet down uniformly towards the sun according to the law of the epicycle, while the latter, in its varying degrees [of speed obtained] through the varying distances, moved the planet in its care forward nonuniformly, and more slowly when it is high. It thus happened that the distances of equal arcs on the epicycle were accumulated near the aphelion C and the perihelion F, and were more sparsely scattered about the middle elongations. In this way, all the shorter ones were drawn back upwards from the correct [circular] distances from perihelion to the place of longer ones. That error therefore began to become rooted in me which I had happily begun to refute in Chapter 39 above, that it is a property of the planetary power to lead the body of the planet around in the path of an epicycle. Had the diameter of the epicycle ND remained equidistant from AB, I could have shed my erroneous opinion, and could have ascribed (as is perfectly correct) all promotion in zodiacal longitude to the sun, leaving to the planet only the reciprocation on the diameter \u03b3\u03b6, as in part of Chapter 39. But because the observations testified that the diameter of the epicycle is inclined in the middle elongations, this error concerning the motion of the planet on the circumference of the epicycle, whose motion would be regular measured with respect to the line \u0391\u039d\u03b3 going from the sun A through the center of the epicycle N, was admirably confirmed in me. Think yourself, reader, and you will feel the force of the argument. For I did not think it possible for the planet's orbit to be rendered oval in any other way.\nWhen, therefore, these things occurred to me, quite certain of the quantity of the incursion at the sides (that is, that the numbers would be in agreement), I celebrated another triumph over Mars. Nor did it appear to me any difficulty, if there were some discord among the numbers, to dissipate it through some slight adjustment in the equations of the center1 all around, so as to make it imperceptible.\nAnd we, good reader, can fairly indulge in so splendid a triumph for a brief day's respite (for the following five chapters, that is), meanwhile repressing the rumors of renewed rebellion, lest its splendor die before we enjoy it. If anything will be left of it afterwards, we shall go through it in the proper time and order. We are merry indeed now, but [will be] active and energetic then.","952":"Chapter 46","953":"How the line of the planet\u2019s motion can be described from the opinion of Chapter 45, and what its properties are.","954":"In the preceding chapter a cause was stated by which it could happen that the planet depart from a circular orbit. However, the geometrical description of the path cannot be carried out using this model. For the epicycle is inclined according to the length of the distances, while the multitude and length of the distances is in turn dependent upon the rotation of the epicycle. And because the sum of the distances is contained in the plane of the eccentric, as was demonstrated in Chapter 40, that sum cannot be found unless the epicycle be transformed into an eccentric. But it was demonstrated in Chapter 2, and repeated in Chapter 39, and used in Chapter 40, that if a concentric be described about center \u03b1 with semidiameter equal to \u03b2\u03b4, and on it an epicycle with semidiameter \u03b1\u03b2; and then about center \u03b2 an eccentric \u03b4\u03bb with eccentricity \u03b1\u03b2; and afterwards the circumferences of both the epicycle and the eccentric \u03b4\u03bb be divided into similar parts; the distances of the points of division, both of the epicycle and the eccentric, from the chosen point \u03b1 are respectively equal to one another in length. On this premise, since in Chapter 40 we posited an eccentric to present a plain and easy demonstration, and a method of computing the distances, here, too, we can examine the distances on the eccentric, even though we are supposing them to be meted out by the uniform motion of the planet\u2019s epicycle. This procedure seems to open a way to us to a geometrical description of the planetary path that follows from the hypothesis of Chapter 45. Let us therefore say, for the sake of understanding, that in the circuit of the epicycle the planet makes digressions from the sun \u03b1 of the same magnitude as if it were on the circumference of a perfect eccentric \u03b4\u03bb (and let this be a semicircle defined by the straight line \u03bb\u03b1\u03b2\u03b4) describing equal arcs in equal times, such as \u03b4\u03b5, \u03b5\u03b6, \u03b6\u03b8, \u03b8\u03b9, \u03b9\u03ba, \u03ba\u03bb. It does this in such a manner that the angles at \u03b2 are equal, and \u03b2 is the point of uniform motion, at least for this position for which the distances are being sought. Let the points of division be connected to \u03b1 and \u03b2. Now this semicircle is purely fictitious: it should be drawn only for computing the sum of a number of distances. If the planet were moved forward with the same degree of power from the sun at both \u03b4 and \u03bb, in the same manner as the epicyclical rotation is supposed to be always uniformly set in motion, then it really would traverse these equal parts of the eccentric, from which we have taken the distances, in equal times; and also, the distances corresponding to the times denoted by the points of division would be these very ones, \u03b1\u03b4, \u03b1\u03b5, \u03b1\u03b6, \u03b1\u03b8, \u03b1\u03b9, \u03b1\u03ba, \u03b1\u03bb, not only in quantity, but also in their identical position. In a word, the path of the planet would be the circle \u03b4\u03b8\u03bb.","955":"The planet does in fact represent quantitatively the reported distances resulting from the uniform rotation of the epicycle, but is itself moved forward unequally in equal times by the sun, less at \u03b4, more at \u03bb. Thus in the time signified and measured by \u03b4\u03b2\u03b5*, it does not traverse the space \u03b4\u03b5, although it does attain the distance \u03b1\u03b5. And in [that same] time (measured by the angle \u03bb\u03b2\u03ba, equal to \u03b5\u03b2\u03b4) it traverses more space than \u03ba\u03bb, although it attains the length of the distance \u03b1\u03ba. Therefore, the planet has a length of distance \u03b1\u03b5 before it is actually moved forward to \u03b5, and a length of distance \u03b1\u03ba before it is moved forward to \u03ba; and inversely, when it is moved forward to \u03b5 or \u03ba, it has already been at the distances \u03b1\u03b5 and \u03b1\u03ba, and for that reason it will now be somewhat nearer. Thus the planet, when at \u03b5, \u03ba, and all the other points of this sort, is nearer to the point \u03b1 than are the points \u03b5, \u03ba on the circumference. So the planet moves inward from the established distance of the circle \u03b4\u03bb towards the point \u03b1 which is near the center \u03b2, never coinciding with this circle at any points other than \u03b4, \u03bb. For the manner of the incursion is the same in the opposite semicircle. ","956":"Also, the plane \u03b4\u03b1\u03b5, \u03b4\u03b1\u03b6, and so on, contains in itself the sum of the distances of all the points on the arc of the epicycle, which, by Chapter 40, is similar to the arc \u03b4\u03b5. And yet the planet, in equal times (which are now being measured by \u03b4\u03b5, \u03b5\u03b6), describes unequal arcs on its real path, short when it is far from the sun \u03b1, long when it moves near to the sun, in such a way that the arcs of the planetary path which are traversed in equal times are in the inverse ratio of the distances, by Chapter 32. It thus happens that the arc \u03b5\u03b4 (which is here the measure of the time) exceeds the arc of the path traversed, which let be \u03bc\u03b4, to about the same extent that the area \u03b5\u03b1\u03b4 exceeds the sector \u03b5\u03b2\u03b4, whose measure is the angle \u03b5\u03b2\u03b4 or the arc \u03b5\u03b4.","957":"If you declare the entire plane area to be 360\u00b0 in number, just as the circumference of the circle, and the periodic time as well, then the number of the time, or \u03b4\u03b5 (at this position) is approximately the mean, either arithmetic or geometric (for they hardly differ) between the number of the sum of the distances or the area \u03b5\u03b1\u03b4, and the number of the planetary path or \u03bc\u03b4. There occurs here a multiple obstacle to calculation. First, that the plane of the circle is not perfectly equivalent to the sum of the distances, as was demonstrated in Chapter 40, even though it was said at the end of Chapter 43 that the defect is quite small. ","958":" Second, that the proportion just described is not exactly geometrical. True, the individual distances are to the individual mean distances in the inverse ratio of the individual arcs of the planetary path to the mean arcs. But the sum of a certain number of distances does not maintain the same ratio to the sum of the same number of mean distances, as the sum of the same number of arcs to the sum of the mean arcs, inversely. As you will see from an example. Let the two distances be 12 and 11, the mean 10, and let the mean arc be the same. And let it be that, as the distance 12 is to the mean distance 10, so is the mean arc 10 to the arc 8\u2153 belonging to the distance 12. Let it also be that as the distance 11 is to 10, so is the arc 10 to 9$\\frac{1}{11}$\u00b7. Compound the distances 12 and 11 into one sum, which will be 23. The sum of the two means is 20, the sum of the two arcs is 17$\\frac{14}{33}$. Here, 10 was indeed the mean proportional between 12 and 8\u2153, and between 11 and 9$\\frac{1}{11}$, but now the sum 20 is not the mean proportional between 23 and 17$\\frac{14}{33}$, but between 23 and 17$\\frac{19}{23}$, which is greater. ","959":"However, this ratio is valid for the arithmetic mean. For example, let 10 be the arithmetic mean between 12 and 8; likewise, between 11 and 9. Compound 12 and 11: they make 23. Compound 8 and 9: they make 17. Therefore, 20 is again the arithmetic mean between 17 and 23. And since it was demonstrated in Chapter 32 that there is hardly any difference for the present undertaking between the arithmetic and geometric means, what is here denied to be true for all cases will therefore be only slightly different from the truth.","960":"Third, even if the area \u03b5\u03b2\u03b4 were the exact geometric mean between \u03b5\u03b1\u03b4 and \u03bc\u03b2\u03b4, nonetheless, it cannot be constructed geometrically. For the triangle \u03b1\u03b5\u03b2 ought to be equal to the sector \u03b5\u03b2\u03bc. But geometers have yet to devise a method by which a given angle can be cut in a given ratio.","961":"Fourth, if none of the above deter us, the sector \u03bc\u03b2\u03b4 of the circle is still not the same as the so-called \"sector\" \u03bc\u03b2\u03b4 of the oval plane. Therefore, even if the arc \u03bc\u03b4 were defined as if it were on the circumference of a circle, nevertheless, nothing would follow concerning \u03bc\u03b4 defined as if it were an arc on the path of the planet, which is not a circle. Therefore, even though this is helpful to those who would like to make use of numbers, to know that \u03b5\u03b2\u03b4 is a mean between \u03b5\u03b1\u03b4 and \u03bc\u03b2\u03b4; nevertheless, for us, who strive after a geometrical path, this passage does not lie open.","962":"We shall therefore try another way. And on our fictitious eccentric \u03b4\u03b8\u03bb the measure of the time is \u03b4\u03b5, \u03b4\u03b6 for finding out the distances \u03b1\u03b5, \u03b1\u03b6, while the ratio of the sectors \u03b4\u03b2\u03b5, \u03b4\u03b2\u03b6 to one another is the same as that of the arcs \u03b4\u03b5, \u03b4\u03b6. However, on the true path of the planet, the plane [area] lying between the arc of the path and the sun \u03b1 is likewise the true measure of the time during which the planet is found on the arc lying above it, by Chapter 40. Therefore, from the point \u03b1 of the diameter let straight lines be projected enclosing spaces equal to \u03b5\u03b2\u03b4, \u03b6\u03b2\u03b4, so that the space \u03b5\u03b7\u03bc, which is subtracted from the space \u03b5\u03b2\u03b4, is equal to the space \u03b7\u03b1\u03b2, which is added to that same space \u03b5\u03b2\u03b4. And let these lines be \u03b1\u03bc, \u03b1\u03bd. And about center \u03b1, with radii \u03b1\u03b5, \u03b1\u03b6, let arcs \u03b5\u03bc, \u03b6\u03bd be drawn intersecting these lines at \u03bc, \u03bd. Will the points \u03bc, \u03bd, o, \u03c0, and so on, constructed in this way, be thus obtained correctly, so that in the times \u03b4\u03b5, \u03b4\u03b6, \u03b4\u03b8, \u03b4\u03b9, \u03b4\u03ba the planet will arrive at them? This is indeed approximately true, but here, too, three things are wanting. First, as above, the plane is not exactly equivalent to the sum of the distances. Second, there is no geometrical way showing how to cut a given semicircle in a given ratio with a straight line drawn from a given point on the diameter. Third, it is not known whether the shortfall for any of the planes \u03bc\u03b1\u03b4, \u03bd\u03b1\u03b4, and so on, produced by the deflection of \u03bc, \u03bd, from the circumference, is in the same ratio as the rest. Nevertheless, these will still be useful to those who wish, contrary to the geometrical usage, to proceed using least parts, with the aid of numbers.","963":"Since geometry has left us destitute, in order that we may have a description of the line which has been born to us out of the theory of Chapter 45, let us go seek the assistance of a contrivance4 by fetching our vicarious hypothesis from Chapter 16, which places the lines \u03b1\u03bc, \u03b1\u03bd, and so on, on which the planet stands, at the correct zodiacal places at the correct times, combining it with the present fictitious eccentric \u03b4\u03b8\u03bb, from which the theory of Chapter 45 has persuaded us that we have derived the correct length of the lines \u03b1\u03b5, \u03b1\u03b6; that is, \u03b1\u03bc, \u03b1\u03bd.","964":"And besides, it is a good idea for the sake of shedding some light to compare the two hypotheses with one another, combined into one diagram. For although both are deceptive on certain points, each is useful for investigating particular truths (to the extent that they can be known at this point). And in this diagram, many things which have been said so far are brought together into a single view.\n Let A be the center of the earth (or of the sun, for Copernicus), AI the line of apsides, AD the eccentricity of the point of the equant. And although it was denied in Chapter 19 that the point D could remain fixed, and AD the same, this is to be understood as true only if DA is bisected. But if we remain at liberty to divide DA as we please, as in Chapter 16, then this point can remain fixed. Therefore, let AD be divided in the ratio found in Chapter 16. Let the point of division be C, and let AC be 11,332, CD 7232. And about center C, with radius CH equal to 100,000, let the eccentric be described, as sketched out by the dotted line passing through H. This, then, will be the hypothesis of Chapter 16. For, taking any known angle of mean anomaly, let a dashed straight line, DH, be drawn from the center of the equant D bounded by a point on the circumference, containing between itself and the line of apsides the required angle,* which is the measure of the proposed time. And let the point H be connected to A. The angle IAH will thus be the equated anomaly and the true zodiacal position of AH, and the planet most certainly will be on the line AH at the given time and anomaly, by Chapter 16 and 18. But the distance AH will be false, and the planet will not be at the point H, because the division of AD at C and the eccentric H described about C are false by Chapter 19, 20, and 42. There it was shown that AD is to be bisected at B, so as to describe a more correct eccentric IL about B, but it will not be a perfect circle. Now let the other hypothesis be delineated. And let AD be bisected at B, so that AB is 9282 (or, according to the numbers of Chapter 42, 9264), and about center B with radius CH, let another eccentric IL be described, which in this chapter I have also called a fiction,* for computing the correct distances. This is the one which in the diagram before the last was described as \u03b4\u03b8\u03bb, about center \u03b2. And let the mean anomaly, which was previously proposed to us in the form of time, be transferred from D to B, and the straight line BF be drawn from B parallel to the former DH. And let the point of intersection F of the new eccentric be connected with A.","965":"Therefore, by what was said in this Chapter 46, AF will be the distance of the planet at F from the center of the sun at A, which the hypothesis of Chapter 45 requires. But the angle BAF is false, and the zodiacal position of AF is false. For at the selected time and mean anomaly the planet is not found on AF. Before, however, the true line of the planet was AH, and the distance AH was false. So about center A with distance AF let the arc FG be described, intersecting AH at G. Thus the line AG, constituted by two manifestly false hypotheses, is nevertheless true in its position beneath the zodiac, and its length is consonant with the hypothesis of Chapter 45.","966":"Thus through the vicarious hypothesis of Chapter 16, which consists of the points A, C, D, and the eccentric H, we have made up for the defect of geometry, which was unable to show us the position of the line AG (onto which the correct distance AF is to be transferred) which we required of the hypothesis of Chapter 45.","967":"One might ask, \"Couldn't we, in the former diagram just as well as in the latter, take as given the point \u03b3 of uniform motion, and from it draw \u03b3\u03bc, \u03b3\u03bd, \u03b3\u03bf, \u03b3\u03c0, \u03b3\u03c1 parallel to \u03b2\u03b5, \u03b2\u03b6, \u03b2\u03b8, \u03b2\u03b9, \u03b2\u03ba, and draw the arcs \u03b5\u03bc, \u03b6\u03bd, \u03b8\u03bf, \u03b9\u03c0, \u03ba\u03c1 intersecting these parallels? And then understand the points of intersection to be the determinate places and positions of the distances?","968":"The answer is no. For in so doing we shall err considerably in transferring the distances too high up, as is easily seen in the latter diagram. For in it the line AH containing the true distances AF is lower than the line DH from the point of uniform motion D parallel to BF.","969":"Whichever of the described ways is used for delineating the line possessing the body of the planet, it now follows that this way indicated by the points \u03b4, \u03bc, \u03bd, \u03bf, \u03c0, \u03c1, \u03bb, is truly oval, not the elliptical one to which the mechanics give that name from the egg (ovum), contrary to correct usage. For an egg is rotated about two vertices, one more blunt, the other sharper, and is visibly inclined at the sides. It is such a figure, I say, that we have created. For the planet is fast at \u03bb, slow at \u03b4, and less fast at the former than it is slow at the latter. For there are more of the long distances exceeding the semidiameter than there are of the short ones (for they are longer up through 92\u2154\u00b0. and then shorter for 87\u2153\u00b0, as can be demonstrated from the theory presented in Chapter 29). But in addition, that greater number of long [distances] is crowded into a narrower arc of the eccentric by being translated upwards, while these fewer [shorter] ones are spread out into a larger arc. So that to a mean anomaly of 92\u2154\u00b0*, which contains 92\u2154\u00b0 of distance, there corresponds an eccentric anomaly of about 87\u2153\u00b0. The remaining 87\u2153\u00b0 of mean anomaly, with the same amount of distances shorter than the radius, is scattered over the remaining angle at the center of the eccentric, 92\u2154\u00b0. Consequently, the short distances near perihelion are farther from one another than are the longer ones at aphelion. So even if the ratio between two neighboring perihelial distances remained constant, the part cut off from the circle would nevertheless be thinner about \u03b5, \u03bc, \u03b4 than about \u03c1, \u03ba, \u03bb. For the short ones are transposed into the position of the longer ones in a shorter space at \u03b4 than at \u03bb. But in addition, the distances themselves of the equal parts of the epicycle near to perihelion are in a greater ratio to one another than the distances of the parts near aphelion. For it was demonstrated in Chapter 40 above that the conchoidal area is wider in its lower part than in its upper. Therefore, the conchoid must be thinned in greater steps over a shorter space at its lower point than at its upper, and in addition those greater intervals are compared to shorter lines. So on both counts the ratio is increased. With so many causes concurring, it appears that the part cut off from our eccentric circle is much wider below than above, at an equal distance from the apsides. Anyone can easily explore this using numbers, or by a mechanical delineation, by assuming some appreciable eccentricity.* ","970":"Chapter 47","971":"An attempt is made to find the quadrature of the oval-shaped plane which Chapter 45 produced, and which we have been busying ourselves to describe in Chapter 46; and through the quadrature, a method of finding the equations","972":"We have accomplished nothing if, from the hypothesis we have taken, and the physical causes of Chapter 45, which we follow here as true, we shall not have constructed the correct equations, no less than the distances. But the equation is compounded from the parallaxes of the points on the eccentric and the elapsed time. The former of these I am accustomed to call the optical part of the equation, and the latter, the physical part. Now the elapsed time, even if it is really something different, is certainly measured most easily (if not most perfectly) by the plane area circumscribed by the planet's path. We therefore turn to the measurement of the plane area of the eccentric ovoid, the rules for delineating which have been laid down already. Now there is going to be something lacking in our account that prevents our stating the true measure of this time. (For at the circumference of the ovoid the lines that join the parts of its circumference with the source of power are even more inclined than on the circle. This is even true as well of the lines that are drawn from the center of the eccentric to those same parts of the ovoid, although otherwise the radii from the center to the circumference of a perfect circle make perfect right angles.)1 But the consequence of this is that the sum of the distances is not exactly measured by the plane surface, nor are the arcs of the ovoid exactly proportional to the distances. All these things will be clear from a rereading of Chapters 40 and 32. A guess as to how small this discrepancy is going to turn out to be, however, can be grasped from Chapter 43.","973":"And how else can we measure this plane surface, compare it to the plane surface of a circle, and divide it into prescribed parts, unless we find a square equal to the trimmed-off part, or the lunule cut off? Here we will have to summon up from tragedy a deus, or rather a sort of ratio, ex machina, to teach us how to manufacture a quadrature of the ovoid, or of its border in the last diagram but one\u2014that is, the lunule \u03b4\u03bf\u03bb\u03b8\u2014whose removal from the surface of the circle generates the ovoid \u03b4\u03bf\u03bb. And just as I called upon the geometers before in Chapter 40 for the area of the conchoid, and begged their assistance, I do so again now for the ovoid (or, if you prefer, the \"metopoid\"3). ","974":" If our figure were a perfect ellipse, the job would have been done by Archimedes, who demonstrates in his book On Spheroids, prop. 6, 7, and 8, that the area of an ellipse is to the area of a circle sharing a common major diameter with the ellipse, as the rectangle contained by the diameters (or the \"figure\"4 of the section) is to the square on the circle's diameter.","975":"But let the figure be a perfect ellipse, for they hardly differ. Let us see what follows.","976":"I say, therefore, that the lunule \u03b4\u03bf\u03bb\u03b8 cut off from the semicircle will turn out to be imperceptibly greater than the small semicircle whose semidiameter is the eccentricity itself, \u03b1\u03b2, or 9264. For let \u03b1\u03b2 be bisected at \u03c3 (as in Chapter 29), and from \u03c3 let \u03c3\u03c4 go out perpendicular to \u03b1\u03b2. Let \u03b1 and \u03b2 be connected to \u03c4. Now let \u03b3\u03c6 extend parallel to \u03b2\u03c4, and let \u03b2\u03c6, \u03b1\u03c6 be joined. And about center \u03b1 with radius \u03b1\u03c4 let the arc \u03c4\u03c8 be drawn, intersecting \u03b1\u03c6 at \u03c8 and \u03b2\u03c6 at \u03be.","977":" Now since the point \u03c4 is equally remote from \u03b1 and \u03b2, we are (following the Arabs in using the term most properly) at the middle elongation, that is, at the average distance of the planet \u03c4 from the sun \u03b1. And because \u03b3\u03c6 is parallel to \u03b2\u03c4, the point \u03c8 on the line \u03b1\u03c6 (in the diagram of the previous chapter) is the genuine and most true position of the translation of \u03b1\u03c4 to \u03b1\u03c8. Therefore \u03c8 is also the point of the planet's average distance. Hence, the little part of the line \u03b2\u03c8 between \u03c8 and the circumference is the measure of the breadth of the lunule about the middle elongation, while the small line \u03be\u03c6 is greater than this breadth by some imperceptible magnitude.\nLet a perpendicular be drawn from \u03b2 to \u03b1\u03c4, and let it be \u03b2\u03c5. I say that \u03be\u03c6, a part of the line \u03b2\u03c6, is twice \u03b1\u03c5.","978":"For let \u03c4 and \u03c6 be joined, and from \u03c4 let \u03c4\u03c7 come out perpendicular to \u03b2\u03c6. Similarly, from \u03be let \u03be\u03c9 come out perpendicular to \u03b1\u03c4. Since the straight line \u03b1\u03b3 intersects the parallels \u03b3\u03c6, \u03b2\u03c4, [angles] \u03b2\u03b3\u03c6 and \u03b1\u03b2\u03c4 will be equal. And \u03b3\u03b2 is equal to \u03b1\u03b2 by construction. But also, \u03b2\u03c6 is equal to \u03b1\u03c4, for both are equal to \u03b2\u03c4 by construction. Therefore, triangle \u03b3\u03c6\u03b2 is congruent with triangle \u03b2\u03c4\u03b1. Thus, \u03b3\u03c6 will be equal to \u03b2\u03c4. But they are parallel by construction. Thus \u03b2\u03b3 and \u03c4\u03c6, which connect the ends of equal parallels on the same side, will also be parallel and equal. But \u03b2\u03b3 is also equal to \u03b1\u03b2. Therefore, \u03b1\u03b2 and \u03c4\u03c6 are also equal and parallel. Consequently, \u03b2\u03c6 and \u03b1\u03c4 will also be parallel. And because the angles at \u03c7 and \u03c5 are right, and the base \u03c4\u03c6 is equal to the base \u03b2\u03b1, and the angle \u03b2\u03b1\u03c4 or \u03b2\u03b1\u03c5 is equal to the angle \u03c4\u03c6\u03b2 or \u03c4\u03c6\u03c7, \u03b1\u03c5 and \u03c7\u03c6 will therefore be equal, as well as \u03b2\u03c5 and \u03c4\u03c7 perpendicular to them.","979":"Again, because the lines \u03c4\u03c7 and \u03be\u03c9 are equal, being parallels between parallels, while \u03b2\u03c4 and \u03b1\u03be are equal, and the angles at \u03c7 and \u03c9 are right, the remaining sides of the triangles \u03b2\u03c7 and \u03b1\u03c9 will also be equal. But \u03b2\u03be and \u03c5\u03c9 are also equal, for they are parallels between the parallels \u03b2\u03c5, \u03be\u03c9. Therefore, when the equals \u03b2\u03be and \u03c5\u03c9 are subtracted [from \u03b2\u03c7 and \u03b1\u03c9], the remainders \u03be\u03c7 and \u03b1\u03c5 will be equal. But before, \u03c7\u03c6 and \u03b1\u03c5 were also equal. Therefore, \u03be\u03c6 is twice \u03b1\u03c5.","980":"With these things demonstrated, we shall draw nearer to our proposition. And to the diameter \u03c6\u03b2 of the circle (which should be understood to be extended to the other circumference), a straight line is drawn perpendicular from a point \u03c4 on the circumference, namely, \u03c4\u03c7. Therefore, as \u03c6\u03c7 is to \u03c7\u03c4, so is \u03c7\u03c4 to the remainder of the diameter. Therefore, the rectangle contained by \u03c7\u03c6 and the remaining part of the diameter is equal to the square on \u03c4\u03c7.","981":"And because the square on \u03c4\u03c6, that is, \u03b1\u03b2, is equal to the [sum of the] squares on \u03c4\u03c7 and \u03c7\u03c6, when equals are added, the rectangle contained by \u03c7\u03c6 and the entire diameter is equal to the square on \u03b1\u03b2.","982":"And because \u03c6\u03be is twice \u03c6\u03c7, the rectangle contained by \u03c6\u03be (which is imperceptibly greater than the breadth of the lunule \u03c8\u03c6) and the semidiameter \u03c6\u03b2 is equal to the square on \u03b1\u03b2.","983":"But the rectangle contained by \u03be\u03c6, \u03c6\u03b2 is the difference of the rectangle contained by \u03be\u03b2, \u03b2\u03c6 and the square on \u03b2\u03c6. And the lunules are also the difference between the areas of the ellipse and the circle. And as the rectangle contained by \u03be\u03b2, \u03b2\u03c6 is to the square on \u03b2\u03c6, so is the area of the ellipse to the area of the circle, approximately.* Therefore also, as the square on \u03b2\u03c6 is to the rectangle \u03be\u03c6, \u03c6\u03b2, that is, the square on \u03b1\u03b2, so is the area of the circle to the area of the two lunules, approximately. And by permutation, as the square on \u03b2\u03c6 is to the area of the circle, so is the square on \u03b1\u03b2 to the area of the lunules, approximately.","984":"But as the square on \u03b2\u03c6 is to the area of the circle of which \u03b2\u03c6 is the radius, so is the square on \u03b1\u03b2 to the circle of which \u03b1\u03b2 is the radius. Therefore, the area of the circle of which \u03b1\u03b2 is the radius imperceptibly exceeds the two lunules \u03c8\u03c6 cut off. It exactly equals the lunules \u03b3\u03be\u03c6 which are a little wider than they should be, because \u03be\u03c6 is imperceptibly longer than \u03c8\u03c6, as was said at the beginning.","985":" So, granted what we have supposed, namely, that the area of the ellipse differs imperceptibly from the area of our ovoid, as a result of the compensation between the excess of the ovoid over the ellipse in the upper regions, and the defect in the lower regions,\u2014these, as I said, being granted, we have squared our \"new moon\"6 figures, and thus also the ovoidal one. Or, properly speaking, we have \"circled\" it. For Archimedes teaches us the ratio of the circle and the square. ","986":" We shall now put this to use, thus. Since the area of the ovoid is less than the area of the circle by the area of the small circle described by the eccentricity, let the area of this small circle be computed next. Now the ratio of the areas is the duplicate of the ratio of the diameters. And as \u03b2\u03c6, 100,000, is to \u03b2\u03b1, 9264, so is \u03b2\u03b1 to \u03be\u03c6, 858. Therefore the ratio between \u03b2\u03c6 and \u03be\u03c6 is also the duplicate of the ratio between \u03b2\u03c6 and \u03b2\u03b1. Therefore, as \u03b2\u03c6, 100,000, is to \u03be\u03c6, 858, so is the area of the circle, 31,415,900,000, to the area of the small circle, 269,500,000.","987":"Therefore, when the area of the small circle is subtracted, the remainder is the area of the ovoid, 31,146,400,000, equivalent to 360 equal parts of the periodic time.","988":"Those things which have been said so far are entirely consonant with the opinion of Chapter 45. Nevertheless, to use them, it is not enough to know the magnitude of the area of the ovoid. Indeed, we need also understand how to divide it, from the center \u03b2 or the point \u03b1, in a given ratio. For example, in the previous diagram, let the point \u03b8 be taken, and let the planet be observed on the line \u03b1\u03b8, but let it recede from the circumference \u03b8 towards the sun \u03b1. Therefore, given the eccentricity \u03b1\u03b2 and the angle \u03b8\u03b1\u03b2, and supposing that the planet is at the point \u03b8 of the circumference, the angle \u03b8\u03b2\u03b4 will be given, and hence also the sector of the perfect circle, namely, \u03b8\u03b4\u03b2, and the area of the triangle \u03b8\u03b2\u03b1, that is, the whole area \u03b8\u03b4\u03b1 which (with the exceptions in Chapter 40 above) ought to have been the measure of the time elapsed while the planet moves from \u03b4 to \u03b8, if the planet had gone in a perfect circle. But because it describes an oval inside the circle, not embracing the full area of the perfect circle, so, exactly as just a moment ago we needed knowledge of the area of the whole ovoid, we now need to know what portion of the ovoid is contained between the lines \u03b4\u03b1 and \u03b1\u03b8, that is, what portion the area of the part \u03b4\u03b8 of the lunule is of the area that measures the two lunules, namely, of the area of the small circle on the eccentricity. For when this is subtracted from the portion of the circle cut off by the lines \u03b1\u03b8, \u03b1\u03b4, the remainder will be the portion of the ovoid cut off by the same lines \u03b1\u03b8, \u03b1\u03b4. Thus, finally, the whole oviform will be correctly compared to its part \u03b4\u03b1\u03b8 in order to find the time, or the slowing of the planet, which occurs between the lines \u03b1\u03b4 and \u03b1\u03b8.","989":" Once again, now, where is the geometer who will show us how to do this? Let the last diagram of Chapter 40 be presented again, in which CD is the semicircle stretched out into a straight line, divided into equal parts, and DE is a quadrant. And on the line EA from E let some line [E\u03bf] be extended towards A which bears the ratio to the longest line BA (the one on the line CA) which that BA has to BC. And let the rest, G\u03bc, H\u03bd, I\u03c0, K\u03c1, be set up similarly in the appropriate magnitudes, having the breadth of the lunule at any given position, so that G\u03bc is a little shorter than K\u03c1, and H\u03bd shorter than I\u03c0 (although they are the same distance from C and D), in accordance with what was demonstrated in Chapter 46. Thus the lunule, insofar as it shortens the distances, will be delineated and unfolded partwise onto a straight line.","990":"And because the whole space between CD and AA is twice the area of the stretched-out semicircle CD, the geometer should consider whether the small space between the curve C\u03bc\u03bd\u03bf\u03c0\u03c1D and the straight line CED is also going to be twice the lunule cut off from the area of the circle.","991":"Nothing appears to contradict the possibility of this being so. For when the lunule really is a lunule, CD is then curved inwards while retaining its same length. But C\u03bc\u03bd\u03bf\u03c0\u03c1D, which was just constructed longer than CED, is then much shorter. Therefore, the lunule then contains a much smaller area than now. But in fact, O geometers, this is not a demonstration. Therefore, you will assist me. And if this turns out to be true, you will teach me a method by which may be known the magnitude, not only of the whole small area between the straight line CED and the curve C\u03bfD, which I have so far said is equal to the small circle on the eccentricity (for two lunules are equal to the small circle, and this small area is now supposed to be twice one of the lunules), but also of any part of it, at any given length of the parts CG, CH; and by which this may be compared to the area between CD and BB.","992":"So once again, as before in Chapter 46, since there is no way out through geometry, we shall be content with a contrivance. And no wonder, for the opinion born in Chapter 45, which threw us into these difficulties, is false. ","993":"Therefore, let the previous diagram from Chapter 46 be considered again. If the area \u03b4\u03bf\u03bb, which is an ovoid, were a perfect ellipse, when the ellipse \u03b4\u03bf\u03bb and the circular area \u03b4\u03b8\u03bb are described on the common longer diameter \u03b4\u03bb, and the planes of the two figures are divided by lines BC applied ordinatewise (that is, perpendicularly to the longer diameter \u03b4\u03bb) from one side of the longer diameter, the portions of the ellipse \u03bd\u03b4C would always remain in the same ratio to the portions of the circle B\u03b4C. This is demonstrated by the authors who wrote on conics, and Archimedes takes it over in On Spheroids Prop. 5. If this were so, there would indeed be no need to know the oviform area. For then we would substitute the area of the circle for the area of the ellipse, and the parts of the circle for the similar parts of the ellipse.","994":"Let it be that \u03b4\u03bf\u03bb is a perfect ellipse, for it is but slightly different from one. And from any point on the ellipse, \u03bd say, let a perpendicular be dropped to \u03b4\u03bb, which let be \u03bdC, and let it be extended until it intersects the circle at B. And let B and \u03bd be joined to \u03b1. Now as \u03b2\u03c6 is to \u03b2\u03be so is CB to C\u03bd, from the assumption of a perfect ellipse and Prop. 5 of On Spheroids, and also as BC is to C\u03bd so is the area B\u03b4C to the area \u03bd\u03b4C. But also as BC is to C\u03bd, so is the area B\u03b1C to the area \u03bd\u03b1C. Therefore, as \u03b2\u03c6 is to \u03b2\u03be, so is the area \u03b1\u0392\u03b4 to the area \u03b1\u03bd\u03b4.","995":" First, let it be, at the planet's proposed time of departure from \u03b4, that as the periodic time is to four right angles, so is the proposed time to the angle about \u03b2 (\u03b4\u03b2\u03b6, say), and let the distance \u03b1\u03b6, to which \u03b1\u03bd is equal, be computed.","996":" Again, let it be that as half the periodic time is to the known area of the semicircle \u03b4\u03b8\u03bb, so is the proposed time (whose measure we have just now said is something else, \u03b4\u03b6, when the distance \u03b1\u03b6 was computed) to the area \u03b1\u0392\u03b4. Thus the area is given. Now a value for angle \u0392\u03b2\u03b4 must be found, such that its sine BC multiplied by half \u03b1\u03b2 (that is, the area of the triangle \u03b1\u0392\u03b2), together with the sector \u0392\u03b2\u03b4, adds up to equal the area just found from the time. Here one has to proceed by trial and error. When you have obtained the angle \u0392\u03b2\u03b4, then in triangle B\u03b2\u03b1, from the angle \u03b2 and the known sides \u03b1\u03b2, \u03b2B, the angle \u0392\u03b1\u03b4 will become known. And because the ratio B\u03bd to BC is known, B\u03b1\u03bd will also be known, and when it is subtracted, there will remain \u03bd\u03b1\u03b4, the correct equated angle for the time selected.","997":"For example, as in Chapter 43, let the mean anomaly, that is, the artificial or astronomical numbering of time, be 95\u00b0 18' 28\". And because 360\u00b0 is equivalent to the area of the perfect circle, 31,415,926,536, 95\u00b0 18' 28\" will therefore be equivalent to the area 8,317,172,671. Let this be \u03b8\u03b1\u03b4. Now, if the eccentric anomaly \u03b4\u03b8 were 90\u00b0, as I suppose conjecturally, its sector \u03b8\u03b2\u03b4 would be 7,853,981,670. And the sine \u03b8\u03b2 of 90\u00b0 is 100,000. This multiplied by half the eccentricity \u03b1\u03b2, 4632, gives 463,200,000 as the area \u03b8\u03b2\u03b1. The sum of the areas is 8,317,181,670, which is \u03b8\u03b1\u03b4, and which exceeds what it should be by some small amount. We therefore guessed well that the angle or anomaly of the eccentric \u03b4\u03b2\u03b8 is 90\u00b0. And because its sine is 100,000, the lunule \u03b8D cut off at \u03b8 will be 858. Therefore, the shorter semidiameter D\u03b2 will be 99,142, which is to 100,000 as 9264 is to 9344. This is the tangent of the angle \u03b1D\u03b2, 5\u00b0 20' 18\", making the equated anomaly D\u03b1\u03b4 84\u00b0 39' 42\". The vicarious hypothesis shows this to be 84\u00b0 42' 2\", the difference being 2'20\".","998":"The investigation of the eccentricity in Chapter 42 depends upon aphelial and perihelial distances, and in these there can be some slight error which is increased tenfold in establishing the eccentricity. Therefore, it should be noted in passing that if a perfectly reliable way of equating through the physical causes were finally found, a perfectly true eccentricity could afterwards be established, and through it the distances of aphelion and perihelion could be entirely corrected. For example, provided we can trust the vicarious hypothesis for the planet's zodiacal longitude, and suppose that everything we have assumed here and in Chapter 45 is true, the equation has been made 2' 20\" too large here, while the optical and physical effects upon the equation at the middle elongations are equal, as here. So the error being bisected, the half (1' 10\") should be subtracted from the angle last found, 5\u00b0 20' 18\", making it 5\u00b0 19' 8\": this shows a tangent of 9310. Before, it was 9344. The difference of 34 subtracted from the eccentricity 9264 would leave a corrected eccentricity of 9230. But we shall not pursue this now, since our assumptions are wrong on the very small quantities. Let it be enough to have noted it for future use in the chapters following next.","999":"Let us also see what promise this form of computing equations holds at the eighths of the periodic time. Let the mean anomaly be 48\u00b0 45' 12\", as in Chapter 43. And since the numerical measure by which the areas are expressed is a matter of indifference, we shall retain the number 360\u00b0 for the area of the circle, and 19,108\" for the area of the greatest triangle (for just now, in the other system of numbering, it was 463,200,000). Let us guess that the anomaly of the eccentric, or \u0392\u03b2\u03b4 in the diagram, is 45\u00b0. Therefore, the sine, BC, is 70,711. This, multiplied by the greatest triangle 19,108\" and with the zeroes dropped, gives the triangle B\u03b1\u03b2 as 13,512\", or 3\u00b0 45' 12\", for this location. This, added to the sector \u0392\u03b2\u03b4, 45\u00b0, gives 48\u00b0 45' 12\" for the area B\u03b1\u03b4, which is also the mean anomaly we assumed. We therefore guessed the angle \u03b2 well. Now, as the radius \u03b2\u03c6 is to \u03b2\u03be, 99,142, so is BC, 70,711, to C\u03bd, 70,104. And because BC is 70,711, C\u03b2, the sine of its complement, will also be 70,711 at this location. Therefore, C\u03b1 is 79,975. But as this is to 100,000, so is C\u03bd to the tangent of the required angle \u03bd\u03b1C, 41\u00b0 14' 9\". The vicarious hypothesis shows 41\u00b0 20' 33\".","1000":"The same things are also easily investigated in the lower octant. Let the mean anomaly be 138\u00b0 45' 12\", and let the area, whose angle at \u03b1 is sought, be expressed in the same units. We will find that the sine of an angle of 135\u00b0 at \u03b2, which is 70,711, makes the sum of the sector and the area of the triangle this much. And because, as before, the sine 70,711 is shortened in order to constitute a line of the ellipse that is applied ordinatewise11 becoming 70,104, this is now to be combined with the sine of the complement of 135\u00b0, which is 70,711, now not increased by the eccentricity \u03b1\u03b2, as before, but decreased by it, making it 61,447. Thus, as this is to 100,000, so is 70,104 to the tangent of the required angle, 48\u00b0 45' 55\", or its supplement 131\u00b0 14' 5\". The vicarious hypothesis shows 131\u00b0 7' 26\". Compare this with Chapter 43, and with other methods, using this table. ","1001":" So, of the two physical hypotheses for computing the eccentric equations, that one shows equations nearer the truth, which previously, in Chapter 45, also gave truer distances, namely, the last one. And, what may seem strange, by a slight increase in the eccentricity it becomes equivalent to the Ptolemaic method, using a stable equant point and a bisected eccentricity.","1002":"And since we convicted this Ptolemaic method of error above, the physical method, which is in effect the same as the Ptolemaic, must also be somewhat askew of the truth. And it does indeed make the planet slow near the apsides, and too swift about the middle longitudes. This is the first argument by which it is proved that either the opinion of Chapter 45 is erroneous, or it has been transposed into numbers by an erroneous method.","1003":"But because the plane area of the circle is not equivalent to the aggregate of all the distances, nor is the oval figure that, according to the opinion of Chapter 45, Mars describes, a perfect ellipse, as we have assumed, the causes for the departure from the truth are still hidden. For in addition to these two causes of error in the calculation, the error can be consistent with a third cause, of the very foundation, the opinion expressed in Chapter 45. So we have not yet set up equations according to the law of the opinion of Chapter 45, and have not yet done justice to the hypothesis we took up there, because we have been abandoned by geometry. Therefore, we cannot yet charge it with being erroneous. For a calculation that is going to do this points to the rule of innocence for itself. ","1004":"Chapter 48","1005":"A method of computing the eccentric equations by a numerical measure and division of the circumference of the ovoid described in Chapter 46.","1006":"So, since the calculation taken up in the preceding chapter was abandoned by geometry on so many counts, so as to be suspected of culpability for the excesses and defects which we noted in the eccentric equations in that chapter, I finally sought refuge in the numberings of arithmetic, by which I attempted to avoid the obstacles which stood in the way of our describing the path of the planet in Chapter 46. For first, because the area was not the perfect measure of the sum of the distances, I dismissed the area and computed the distances of individual parts of the equally divided circumference. Second, since the ratio did not remain the same when any given number of terms of the geometrical proportion were added, I therefore investigated each individual proportion separately for each distance in relation to its minimal arc. Third, since the sum of any particular number of distances in Chapter 46 could not be established geometrically, I established it arithmetically. For there was no difficulty in that. Fourth, in carrying this out I had no dealings with sectors, whether circular or oval, and therefore it could obviously no longer be a hindrance to me that these sectors differ from one another.","1007":" And so, with renewed effort, I brooded upon it, in order to know at least at the end whether the equations shown to us by the vicarious hypothesis also follow from the hypothesis under consideration, which gives the correct distances (that is, the one following the opinion of Chapter 45.)","1008":"I approached the matter thus. About center B, with radius BD, let the circle DGR be described, in which the line of apsides is DR, and A is the source of power or the center of the sun. On the circle DG let the point G be chosen, and joined to B and A. And at first, let the angle GBD be the measure of the time, for computing the distance. Consequently, GA will be the true distance of the planet from A, although the planet has not moved all the way from D to G. For this method of computing or demonstrating the distance, from Chapter 45, has so far had the status of an assumption. But let DG be a small part of the circle, such as 1\u00b0 out of the 360\u00b0. And since all the distances AG of this sort at the ends D and G of all the degrees DG can be computed in this way, by what was demonstrated in Chapter 29, I gathered all the 360 distances AG into one sum, in a very long addition. This was found to be 36,075,5621 (with an eccentricity of 9165), corresponding to the entire oval path of Mars. Now about center A, with radius AG, let an arc, GC, be described towards D. And because the greater the distance the shorter the planet's path, when the distance of an arc of the circle DG is given (which arc is now, when we are computing the distance GA, nothing but the measure of the time), the length of the oval path DC will also be given, which the planet describes in the time DG under consideration (which is a simple anomaly of 1\u00b0). For as the length of the whole of the oval circumference is to the sum of all the distances, so is the distance of the arc DC (found by the arc DG) to the length of its oval arc DC. For it was proved in Chapter 33 above, and used in Chapter 46 (where the foundations for this operation were laid) that the arcs traversed are inversely as the distances. However, I applied this precaution: that AD and AG, the distances of the ends C and D of the arc from A, be added, and the mean of the sum be taken as the genuine distance of the whole arc DC. For let","1009":"some eccentric circle DK described about center B, be divided into any number of parts, say, at D, G, L, K, \u039c, N, and from the beginnings of the parts let arcs be drawn about the center of the world A until they intersect the lines drawn from A to the ends of the arcs, as DO, GP, LQ, KR,","1010":"MS, NT. The areas on the left semicircle, ADO,","1011":"AGP, ALQ, will be greater than they should be, while the arcs on the right, ANT, AMS, AKR, will be less than they should be. So, when least arcs are in question, the one is compensated by the other, so that TNA and ODA are very nearly equal to the area GDNA.","1012":"Next, the length DC in the previous diagram being thus given, to correspond to the given time DG and to the distance GA or CA, the angle of equated anomaly CAD now also should be found. Let C be joined to B, and let AC be extended to E, where it intersects the circle, and BC to the intersection F. It was therefore not enough to know the length DC. The angle CBD ought to have been investigated as well. For because CD is shorter than FD, CD does not measure the angle FBD, or CBD. And on the other hand, even though CD is shorter than FD, if you place an observer at B it appears from B to be just as much a measure of the angle CBD as is FD. But according to the demonstrations of Chapter 32, it is true within all limits of sense perception that to the extent that FD is farther from B than is CD, FD is also longer than CD. Also, it is true within the same limits of sense perception (for the present purposes it does not matter how acute) that CE and CF are equal (though in the truth of the matter CE is longer than CF, which is drawn through the center, by Euclid Book III prop. 7). I have therefore supposed, first, that CD and FD are equal, and both are a measure of the angle CBD or FBD, or even EBD, as if the arc EF were imperceptible. Thus, from knowing CD, the angle EBD was given. Therefore, in triangle EBA, from the angle EBA and the sides EB, BA, I sought the length of AE, whence I subtracted AC or AG computed earlier, and the remainder was CE or CF, the amount that the other end of CD approached the center B. So, when CE is bisected (for this can be done within the limits of perception) the approach of CD towards B was known, if it were to approach equally at all its points. But from the amount of approach, the optical parallax, or the apparent size of CD, was also given: that is, the angle CBD now corrected, which was previously assumed to be a little smaller, though in our numbers there is no error. Therefore, given the angle CBD, now corrected, which is the supplement of angle CBA, and the side CA, and the eccentricity BA, the equated anomaly CAD, which was sought, was given.","1013":" It was impossible to use this method to establish independently any equation other than the first, at a mean anomaly of 1\u00b0. All the rest, all the way to the 180th, always presupposed that the equation immediately preceding was known. I can't imagine anyone reading this not being overcome by the tedium of it even in the reading. So the reader may well judge how much vexation we (my calculator and I) derived hence, as we thrice followed this method through the 180\u00b0 of anomaly, changing the eccentricity each time.","1014":"But the foundation of this calculation has not yet been laid out. For I said that I presupposed knowledge of the length of the whole oval. Whence, then, is this known? As for me, once I had descended to this clumsy numbering procedure, I did not manage to avoid clumsily presupposing the length, and then, when the whole thing was complete, seeing whether in the 180th operation I came out with an apparent position of more than 180\u00b0, or less. For if it had come out at exactly 180\u00b0, I knew that the length I had assumed for the oval was good, but if it was less, I had assumed it to be too small, and if more, greater.","1015":"Nevertheless, we are not left without a kind of geometrical helping hand for making a good guess as to the oval's length. For as BD is to BA let BA be to DH, extended from D towards B. By what was demonstrated in Chapter 46, the rectangle contained by the breadth of the lunule and the semidiameter of the circle is nearly equal to the square on the eccentricity. Therefore, by Euclid VI. 17, the eccentricity is the mean proportional between the breadth of the lunule and the semidiameter. But this is how the diagram is set up. Therefore, DH is the breadth of the lunule.","1016":"Let the half of DH also be taken, and extended from B towards D, and let it be BI. And about center I, with radius ID, let the circle DK be described, touching the eccentric at D. And also, about center B, with radius BH, let the circle HK be described, touching the previous circle at K. It is obvious that the circle HK is smaller than DK, and the circle DGR is greater than DK. And because circular circumferences are to one another as their semidiameters, as BD is to DI and BH, so is the greater circle DG to the smaller circles DK and KH. But DI is the arithmetic mean between DB and HB, because BI is half of HD. Therefore, the circle DK too, touching the smaller and larger circles described about the same center B, is the arithmetic mean between those circles that it touches.","1017":"If the oval path is continued, by hypothesis it will be tangent to the greater circle at aphelion D and perihelion R, and to the smaller circle HK at the middle elongations. Thus, it is greater than the smaller circle HK, and smaller than the greater circle DR. It is therefore likely that the oval circumference is not much different from the length of the circular circumference DK.","1018":"The following demonstration, however, makes one believe it to be a little larger.","1019":"Let the mean proportional between BH and BD be taken, and let it be BO, and about center B with radius BO let the circle OP be described. Thus, by Archimedes, On Spheroids 5, the area of this circle OP will be equal to the area of an ellipse whose longer semidiameter is BD, and whose shorter is BH. And because the greatest of all figures of equal perimeter is the circle, conversely (in the common significance of the term), of figures with equal area, the one with the shortest perimeter will be the circle. Therefore, since the proposed figures, namely, the ellipse which has semidiameters DB, BH, and the circle OP, are equal in area, from the evidence just presented, the circumference of the ellipse will be longer than the circumference of the circle OP. But BO is imperceptibly less than ID, since BO is taken as the geometrical mean, and ID the arithmetic mean, between the same terms. For by the theory presented in Euclid's Book V, since BO is the mean proportional between HB, BD, as the lesser, HB, is to the greater, BD, so is HO, the excess of the mean, to the defect, OD. 3 Therefore, since HB is less than BD, HO will also be less than OD. But BI is equal to the half of HD. Therefore, BI is greater than HO, and less than OD. Therefore, to the common semidiameter HB of the least circle HK unequals are added, namely, less than the half of DH to make BO, and half of DH to make DI. Therefore, DI is greater than BO. Thus the circle DK is greater than OP. But it is only imperceptibly greater, since DH is less than a hundredth of DB. Therefore, on the supposition that these circles are superabundantly equal, and on the supposition that the oval is a perfect ellipse, the circumference of the oval will be a little longer than the circle DK, and certainly longer than the circle OP. And because in Chapter 47 above DH was 858 where DB was 100,000, let half of DH, 429, be subtracted from DB, 100,000. The remainder will be 99,571. Next, as 100,000 is to 99,571, so almost exactly will the circumference of the circle be to the circumference of the oval, which is sought. And because the circumference of the circle has 360 degrees or 21,600', or 1,296,000\", a small part will be removed which has 5560\" or 92' 40\". And for the semi-circumference of the oval, 46' 20\" are to be subtracted, or even less, if the oval exceeds the circle DK at the place considered in the measuring. As for me, I made no use of any demonstration, but by a most laborious and dogged calculation found the defect of the oval semicircle to be 45' 45\", so that where the perfect semicircle is 180\u00b0, the oval would be 179\u00b0 14' 15\".","1020":"Now this shortening of the oval circumference is necessarily equal to the opposite optical lengthening (for although this oval is shorter, it nonetheless appears contained within two right angles, or exactly 180 degrees, and is judged to be that long). Hence the reader may with good reason doubt whether in this process it is necessary first to shorten the entire oval, and afterwards to lengthen it again by parts optically. For from the diagram it seems apparent that the shortening is at its greatest where the approach towards the center B is greatest, and vice versa.","1021":"If in fact these variations did happen to be equal, the following method for computing the equations would arise.","1022":" The first mean anomaly would be GBD, from which the distance GA would be computed, which, added to AD, the distance of the other end of the preceding from GD (which is always 1\u00b0), and the sum being halved, would give the uniform distance of the arc CD (the same for all its points). And we would then say that as the length of the semicircle is to the sum of all the distances on the semicircle, so is this distance of the arc GD to the length of FD, which is the apparent size of CD seen from B. Now from FD, as if it were a measure of the angle CBD, and from AC, AB, we would find the equated anomaly CAD by a shorter path than before.","1023":"But the reader should know that these two variations do not keep in step. For the optical amplification which arises from the approach of the path DC to the center B, happens chiefly about the middle elongations, and hardly at all at aphelion and perihelion. Contrariwise, the shortening of the oval path, which arises from the incursion of the planet towards the center, is about the same everywhere. For two opposite distances at the middle elongations of the eccentric add up to the sum of two near the line of apsides, one near aphelion and the other near perihelion. But the arcs of the oval circumference are in the inverse ratio of the distances. Therefore, two of these arcs, at the middle elongations, will be equal to two other arcs, one near aphelion and the other near perihelion. If these arcs of the oval path are equal, the diminution of the arcs at all four places will be approximately equal also. This is confirmed by experiment. For if the defect of the oval semicircle is 45' 15\", the defect of the 180th part of the oval would be about 14\" about aphelion. And the amplification from the approach of the oval does not equal one second at aphelion.","1024":"So, as concerns the proposed ocular estimation of the diagram, it is not quite as simple as was said in the objection, that the shortening of the oval and its optical amplification compensate one another. This would indeed be so if all the arcs of the oval path were presented directly when seen from center B. But this happens only at the middle elongations. Near the apsides, on the other hand, these arcs are not at the same distance at both ends. Therefore, they are not made to be as much greater in their appearance, by approaching, as they are made smaller by being shortened.\nSo, following this method, I constructed equations for Mars at all degrees of the eccentric, and I did it three times. For the first time I took an eccentricity of 9165 that was not great enough, thinking that I had thus made this very certain through my treatment of the areas. And then I used more than 180\u00b0 in the figuring, when I should have used less.","1025":"So, when this last operation showed more than 180\u00b0, which was absurd, for the second trial I assumed half the oval to be 179\u00b0 14' 15\". At a mean anomaly of 45\u00b0, the result was: ","1026":"And from this, chiefly from the anomaly of 90\u00b0, I realized that the eccentricity of 9165 was too small. This I corrected, using the method presented in passing in the preceding chapter. For seeing that at the middle elongations we have 3' 50\" too much in the greatest equations, half of it, 1' 55\", is given to the optical part, and the rest to the physical. And since 9165 subtends 5\u00b0 15' 30\", you take 5\u00b0 17' 25\", which yields 9227. And so, with a new eccentricity of 9230 (which is hardly different from the 9264 which I found in Chapter 42, nor is it much farther from 9282, which is half of the eccentricity of the equant in Chapter 16) I went through the whole job again. First, the distances GA or CA were constructed at the individual whole numbers of degrees of equated distance anomaly GAD. Next, these were brought over to GD or GBD, the whole number of degrees of mean distance anomaly. Third, adjacent pairs, such as GA, AD, were added. Fourth, division by those divisors was carried out one hundred eighty times. The sum is 358\u00b0 28' 30\", which is the length of the oval path. Fifth, the individual arcs of the oval path were added to one another in order. Sixth, the optical amplifications were borrowed from the previous unsuccessful operation, since I saw that over two computations they differed hardly at all. So these, too were added to the above sum, in order. Seventh, the sums of the arcs were increased by the sums of the optical amplifications. Eighth, from the angle CBD thus found about the center of the eccentric B, and the distance CA as opposite side, and the eccentricity AB as the third side, I sought out the 180 angles of optical equation ACB, whence the total equations and the equated anomalies were derived. The resulting anomalies were: ","1027":"So the eccentricity still can be increased, and up above, [moving] from aphelion, the planet is made to be slightly slower than it should be, and the same near perihelion, and therefore it is swifter than it should be at the middle elongations, as was also found before in Chapter 47. So too many of the distances seem to be collected near the apsides, and not as many as required, or not as long as required, about the middle elongations. But a consideration of this follows in its proper place.","1028":"So, when I saw that the more skillfully and the more conveniently the physical causes, introduced in Chapter 45, are called upon for directing the principles of calculation, the closer I always have approached to the true equations provided by the vicarious hypothesis of Chapter 16, I greatly congratulated myself, and was confirmed in the opinion of Chapter 45.","1029":"On the other hand, since I was revulsed at the multiple contrivances4 with which I contended in this chapter, I did not rest until I had established a more certain and direct way, and at the same time I began to suspect that what the opinion of Chapter 45 had required had in no way been achieved by the calculation.","1030":"Chapter 49","1031":"A critical examination of the previous method for the equations, and a more concise method, based upon the principles constituting the oval in the opinion of Chapter 45.","1032":"So, in order to see the cause of the contrivance1 in this method, now fully presented, consider upon what foundations it rests. The planet is supposed to move uniformly on the epicycle, and to be swept around by the sun nonuniformly, according to the distance. From these two principles of motion, the oval path arises. But this method does not allow one to know what portion of the oval path corresponds to a given time, even if the distance of that portion be known, unless the length of the whole oval be known from the beginning.","1033":"And the length of the oval cannot be known, except through the measure of the incursion of the planet from the circumference of the circle at the sides. But further, the measure of this incursion is not known before it is known what portion of the oval path is traversed in any given time. This, as you see, begs the question, and in our operation we presupposed what was being sought, namely, the length of the oval. This is not just a fault in our understanding, but is utterly alien to the primeval ordainer of the planetary courses: we have hitherto not found such an ungeometrical2 preconception3 in the rest of his works. Therefore a different approach must be taken for calling the opinion of Chapter 45 to the calculations, or if this cannot be done, the opinion itself will totter owing to its being suspected of circularity of argument.","1034":"From this consideration the implication has occurred to us that in using the uniform measure of time, we divided the composite oval path into unequal parts, and thus measured out the parts of this composite oval, unequal but equated again by the compensation of the distances, by the equal increments of time of the planet on all of them. And indeed, we had among our presuppositions that only the other power (that which is from the sun), intensifies according to distance, the power proper to the planet doing so not at all. But now, in this undertaking, we in a way encumber both forces with this relation to the distances, because we give the common work of the two, the oval, to the planet, according to the measure of the distances to be traversed.","1035":"Therefore, although we have approached rather close to the truth in the effect of this method, we have nothing in which to glory that the opinion of Chapter 45 has been expressed by it if we are abandoned by reason. We would therefore have appeared to be going about our business more correctly if, dismissing the composite oval and the quadrature of its area, the subject matter of Chapters 46,, and 48, we were to convert the calculation to the principles themselves of the oval path, assumed in Chapter 45. Let Chapter 45 be taken up again, and about A, the center of the sun's body, with radius AD, let the circle DG of the center of the epicycle be described; and another, with center A, and radius AB, the circle of the aphelion; in which AGB is the line of apsides, and let the planet, when it is at aphelion, be at B. Now let some time have passed from when the planet was at B, and let its measure be CDE, the angle on the epicycle, in order that as the aphelion of the epicycle B is translated to C, and the center of the epicycle G to D, the planet will have moved on the epicycle from C to E. Therefore, in order to know the angle DAB at the time CDE, consider that the planet has passed across from B to E by two powers. One made it move nearer the sun, and at the same time also drew it away from the line AC or AD, on which it had been previously, when AC was at AB. ","1036":" The other moved the planet forward along","1037":"with the center of the epicycle, so that the center D of the epicycle was on the line AC, although it was formerly on AB. Now the power that drives the center of the epicycle around in a time designated by 360\u00b0, moves through 360\u00b0, or four right angles, about A, owing to the sum of 360 distances. Therefore, the sum of any number of distances being given from the time CDE, as before, the angle DAB will also be given. For the impression which the sun makes upon the body of the planet through the mediation of the distances","1038":"AB, AE, is also presumed to make the same impression upon the center of the epicycle GD.","1039":"This is because if the planet had not disengaged itself from the ray of power AB or AC and move towards B, but only descended towards the sun, it would then still be at the point F* on AC, on which line the center of the epicycle D also lies. And it has disengaged itself by the law of the epicycle, at the distance DE and the angle CDE (for this is prescribed by the opinion of Chapter 45, under which we are operating here). Thus by a kind of fiction for itself it places the center of its epicycle at D. For we have said in Chapter 39 how it is to be imagined that the power or fictitious radii of power AB,","1040":"AC, and so on, serve as a position for the planet. Now, though, the ratio of the arcs BE of the oval path to the whole oval is not quite the same as the ratio of the corresponding arcs GD of the perfect circle to the whole circle. But neither is it true that as BC is to the whole perimeter of the circle BC so is the arc BF of the oval to the whole oval. But this should not stand in our way, for BE, and BF too, are composed of two powers, and if anything is disturbed in the proportion this comes from the planet's making its own descent on the circumference of the epicycle (following the opinion of Chapter 45). For if the planet had remained at the highest point of its epicycle, and had been subject to the same motive force from the sun, adumbrated by AB, AE, which is nonuniform (which indeed cannot happen simultaneously, for when the distance of the planet from the sun remains the same, the motive strength from the sun remains the same), then a perfect arc of a greater circle BC would have been described, whose ratio to the whole BC is the same which the arc GD has to the whole GD.","1041":"I am indeed aware that if the planet is supposed to be on a smaller perimeter, that of the center of the epicycle DG, it will go much faster. But that is not a reason for assigning a greater speed to the center of the epicycle. For, by supposition, the center of the epicycle moves, not in its own right, since it is not a body, but because of the planet. It is thus presupposed that the planet moves its own body away from the solar rays according to the law of the epicycle, and makes use of certain rays of power from the sun for its position (ideas which were indeed rejected in Chapter 39, but taken up again in Chapter 45 in considerably altered form, and which are retained here in order to explain my attempts). On this basis, the foundation of the subsequent calculation is sound, whatever its result might be. For the oval is present here no less than before, since DE and AB do not remain parallel. For to the extent that the long distances AB, AE exceed the medium distances AG, AD, the arc DG, or the angle DAG, is made less than the angle CDE, the measure of the time. Thus DE inclines towards B, and E consequently makes an incursion from the circumference of the circle towards BA. For by Chapter 2, if DE had remained parallel to AB, then E would have been on the circumference.","1042":"This gives rise to the following method. Let the distances be found for each degree of mean anomaly. The method you have in Chapter 39 above, and I also used it above in Chapter 47 and 48. First, distances are found for nonintegral degrees of mean anomaly, or CE. Then, by interpolation, they are carried over to integral degrees of CE. If you find this meandering route annoying, and if the greater labor of the direct way pleases you, and if, further, you want to have the whole thing presented in one overview, proceed thus.","1043":"Measure the time, or the artificial units expressing time, which is the astronomers' mean anomaly, on the epicycle CE, from its aphelion C, in a direction opposite the series of signs. Thus the angle ADE, or its supplement CDE, is given as a whole number of degrees of mean anomaly. The radius AD is also given as 100,000, and the radius of the epicycle DE is 9264. Therefore, part of the equation, DAE, will be given, and the distance AE. Put both of these into a catalog, with the mean anomaly CE adjoined, for future use. In this manner, let all the distances AE be gathered and added, and the sum will be found to be about 36,075,562. For this sum was found using an eccentricity only slightly different from our present one, which is 9264. The 360th part of this has the value 100,210, and the same fraction of four right angles is one degree. Therefore, as each of the distances, in order, is to the distance 100,210, so is the arc of this distance 100,210 (60\u2019) to the arcs belonging to the other distances. For, as was frequently announced in Chapter 39, 47, and 48, the ratio is inverse. Next, multiplying 60 minutes, or 3600 seconds, by 100,210, and dividing the product by each of the distances in the semicircle, 180 times, or better, by half the sum of adjacent pairs of distances (following the advice given in Chapter 48), yields the angles of the center of the epicycle, DAG. Next, beginning from the two least values of DAG, add them, and to the sum add the third, and again add the fourth to the sum of the three preceding, and so on, until you have accumulated all the 180 degrees. And if your final sum comes out to be exactly 180\u00b0 it will prove to you that you did everything right, never departing from the instructions. And let these sums of yours, which are the angles DAG, again be inscribed in a catalog with the corresponding mean anomalies in the margin, for ready reference.","1044":"So, since an integral equation is to be computed, that is, the equated anomaly for a given mean anomaly, first, with the mean anomaly CDE measured on the epicycle, you extract the angle DAG or CAB from the latter catalog, the one with the sum of the angles. And with the same mean anomaly, you also extract the part of the equation CAE from the previous catalog. And when this is subtracted from the angle DAB, the remainder is the equated anomaly EAB. The variations in the other semicircle are known. ","1045":"In this manner, at 6","1046":"Near the apsides the planet is made to be slower than it should be, and near the middle elongations swifter than it should be.\nYou will say that we have come out worse, since in Chapter 48 we came nearer the truth in our results. But, my good man, if I were concerned with results, I could have avoided all this work, being content with the vicarious hypothesis. Be it known, therefore, that these errors are going to be our path to the truth. Meanwhile, let us be assured that at last we have brought the physical causes, which we supposed in Chapter 45, at least once to a calculation entirely free of error. At the same time, moreover, the calculation of Chapter 47, above, is confirmed, since this one is equivalent; and it is certain that what we held in suspicion there as ungeometrical, have not obstructed us in any perceptible way. Thus if there remains any discrepancy between these equalities and the truth, it is to be attributed, not to the method of applying numbers, but to the opinion of Chapter 45, whence these numbers flow. This is to say, not that the opinion itself has immediately become totally false, but just that we have been hasty. For instead of waiting for the plenary judgement of the observations, when we understood the planet's path to be oval we immediately seized upon a certain quality for that oval, solely on account of the elegance of the physical causes and the graceful uniformity of the epicyclic motion, which was falsely given credence.","1047":"Now the manner in which the ultimate and truest opinion is to be brought to a calculation, and made to conform most closely to these chapters, will be told in its place.* I am now going to complete the unfolding of my remaining trials. ","1048":"Chapter 50","1049":"On six other ways by which an attempt was made to construct the eccentric equations.","1050":"How small a heap of grain we have gathered from this threshing! But you also see what a huge cloud of husks there is now. They ought to have been hauled back to the beginning of Chapter 48, since before I investigated the arcs of the oval path I would have dealt with them. But for the sake of bringing light, it was appropriate to winnow them. Besides, we might end up finding a few useful grains.","1051":"In the first and second method, the procedure was this","1052":"First, with eccentricity 9165, which is a little less than the correct value, I sought out all the distances, according to the procedure shown in Chapter 29. These corresponded to integral degrees of an anomaly occupying a middle position between the mean anomaly and the true equated anomaly. Although I am calling it \"equated\" for the time being, I nevertheless add a condition, that it be used only for the distances. I therefore name it \"anomaly of distance\".* In the second diagram of Chapter 46 it is the angle FAB, and in the following one, CAD.","1053":" Second, I sought out the third proportional lines, each of which was to its distance as this distance was to the radius, 100,000.","1054":"Third and fourth, I added the lines so found one by one, and the sum of the distances was 35,924,252, less than 36,000,000. The cause of this you have in Chapter 40. But the sum of the proportional lines was found to be 36,000,000, which holds me in wonder. And because it is delightful, I wish some geometer would prove it to be necessary. About centers A, B let two equal circles IH and DC be described, and let the centers A, B be joined, and AB extended so as to intersect the circle about A at I and K and the circle about B at D and L. Then let the circle about A be di","1055":"vided into any number of equal parts, such as 360, beginning from I. And from A let lines AI, AH, AK, and so on, be drawn through the points of division I, \u0397, K, and so on, intersecting the circle about B at the points D, C, L. Then let it be that as AI is to AD, so is AD to AG; and as AH is to AC, so is AC to AF; and finally, as AK is to AL, so is AL to AM; and so on for all the rest. Let, I say, a geometer demonstrate that the sum of the last 360, AG, AF, AM, joined together, is equal to the sum of the first 360, AI, AH, AK, joined together.","1056":" So, in this first method using the sums of the distances, I intended one thing (although erroneously and irrelevantly, namely, to add up the arcs CD or angles CBD, even though they were given at the beginning), but accomplished another, again erroneously. For I obtained, not the arcs, nor the angles, nor the path lengths, but the times on the unequal arcs of the planet's path, as if they were equal. And, following the rule of proportion, I said that as the sum of the means, AD, AE, AL, which is 35,924,252, is to the elapsed time of 360\u00b0, so is the sum of any number of distances to its elapsed time over the length of the path that includes these distances. Let A be the sun, B the center of the eccentric CD, BC the semidiameter. Let B, A be joined to C. Here the distances CA were found corresponding to integral degrees of the angle CAD, and thus to unequal arcs on the circle CD, something that escaped my notice. So let CAD be 45\u00b0. From CB, BA, the angle CBD is given as 48\u00b0 42' 59\". Therefore, if there were no physical cause of the equation, and CBD were the measure of the time or mean anomaly, then there would correspond to it this value for CAD, truly equated. But the planet is slower on CD, owing to its greater distance from A, and the distances are the measure of the elapsed time. Therefore, at the anomaly CAD of 45\u00b0 I collected 45 distances at the beginning of the arcs, which are longer. Their sum was 4,869,307. I also collected 45 shorter ones, the ones at the ends of the arcs, by subtracting the longest, AD, 109,165, from the sum of the 46 distances, 4,975,577. The remainder was 4,866,412. The mean between the two sums, 4,867,852, I reduced to degrees, where 35,924,252 have the value 360\u00b0, or 99,790 have the value 1\u00b0. The result of this procedure was 48\u00b0 46' 51\". And this ought to have been the time corresponding to the angle CAD. But the arc CD or the angle CBD was also found to be about that much, 48\u00b0 42' 51\". This is absurd, and contrary to the hypothesis, which requires the planet to be slower at CD. The cause of this absurdity was immediately clear, namely, that in order to know the elapsed time for CD it would have been proper to take the distances corresponding to equal arcs of CD, while these distances just taken correspond to unequal arcs of CD, and are greater to the extent that the distances themselves are longer, by Chapter 32. Therefore, these distances had too small a numerical value. Nevertheless, in order that I not lose all this labor, I subtracted the excess of this number of the elapsed time over the number of the angle CAD, from CAD, so as to leave as a remainder EAD, 41\u00b0 13' 9\", and so that AC, AE might be equal. Here it was supposed that in the time CBD the planet traverses an angle EBD about the center of the eccentric B equal to CAD, and therefore, that as many distances from A were collected for the equal arcs ED of its eccentric as we found here on equal degrees of CAD. Thus, the same number of arcs which, in this calculation of ours, were spread out over CD, which were unequal and, in this locality, too great, are now understood to be compacted within the confines of arc ED, now divided into equal parts. Therefore, the angle CBD would here be the mean anomaly of distance, giving the angle CAD, for finding the distances CA, from which the angle CAE, the physical slowing and translation of CA to EA, is deduced.","1057":"Although it cannot show much of a discrepancy from the prior method of Chapter 49, this procedure assumes without demonstration that CAD and EBD are equal, and consequently, that CA and EB are parallel, which was refuted above in the second diagram of Chapter 46. But now see how close this operation comes in its results. For ","1058":"The eccentricity was charged with being too small, and indeed, it really is greater, 9264 instead of 9165. Also, the planet was made to go too slowly near the apsides, and too fast near the middle elongations. But, dismissing this first method, which we seized upon by chance in thinking over the error committed at the beginning, let us turn to the implementation of the second method, born from a consideration of the same error.","1059":" The distances scattered over CAD had approximately the same numerical value as the sector CBD, and led the argument to an absurdity (for just as the area CAD, an approximate measure of the distances, is greater than the area of the sector CBD, the numerical value of the distances CD also had to have been greater than the sector CBD). The question therefore followed: did the lines AF, AG proportional to AC, AD correctly express the elapsed time over CD, thus allowing CAD to remain the true equated anomaly? The answer is, they did not. For if so, AC will remain in its place, which is the same at which its length was computed. Therefore, the orbit will be a perfect circle, which was refuted in Chapter 44. Therefore, the distances at the middle elongations, coming out longer than they should be, will make the planet slower than it should be there, and hence it will be faster at the apsides. But look at the result of the operation, which testifies to this itself.","1060":"For ","1061":" First, the eccentricity is charged with being too small, since the maximum equation comes out to be 10\u00b0 29\u2155\u2019, which is 10\u00b0 34\u00bd\u2019 in the vicarious hypothesis. Second, at the time 52\u00b0 39\u2154\u2019 the planet is found to have traversed as much of its path from the apsis as was traversed in the vicarious in the longer time of 52\u00b0 53'. If the eccentricity were adjusted, all the values of the equated anomaly would be increased, so that, in the lower quadrant also, in the time of 37\u00b0 44' (the supplement of 142\u00b0 16', which has been adjusted by increasing the eccentricity), the planet will traverse as much of the path as it traverses in the vicarious in the longer time of 37\u00b0 51', which is the supplement of 142\u00b0 9': that is, both will traverse 45\u00b0, which is the supplement of 135\u00b0.","1062":"By the way, this false hypothesis has come very close to giving us true results: the difference, after correction, is not more than 8' or 7' at either place. You thus see that results must not be trusted. And you will note again what was observed in Chapter 47, that the truth lies at the midpoint between these two methods (the latter of which describes a perfect circle, and the former an oval, following the opinion of Chapter 45). Therefore, you can at least conclude now, as well as before in Chapter 47, that the lunules to be cut off from the perfect circle have only half the breadth of the one that follows from the opinion of Chapter 45.","1063":"Third and fourth method","1064":"So, since this second method did not accord with reason either, and in the first I learned that the distances are to be sought out corresponding to integral degrees of the angle CBD or to equal arcs of the eccentric CD, I also proceeded to the distances.","1065":"So, fifth (I am enumerating for you only those operations each of which is performed 180 times), I made use of interpolation to relate the distances found previously by dividing the mean anomalies CBD minutewise, or unequally, to the mean anomalies which are equal or are of an integral number of degrees. But now, CBD no longer remained the [mean] anomaly, as it formerly was in the first method, but was made the eccentric anomaly by this relating of the distances, as it also is in the second method.","1066":"Sixth, using the same distances as before, I sought out their proportionals, that is, the lines that are to the distances as the distances are to the radius, 100,000. But this was unnecessary. Still, I wanted to be aware of all the possibilities.","1067":"Seventh and eighth, I again added the individual magnitudes, both of the distances AD, AC, and of their proportionals AG, AF. The sum of the distances came out to be 36,075,562. The reason for its coming out greater than 36,000,000 you have in Chapter 40. The sum of the proportionals came out to 36,384,621.","1068":" Now, in the previous diagram, we shall proceed demonstratively, using the equated anomaly CAD to obtain the eccentric anomaly CBD, and further, through this eccentric anomaly CBD learning the sum of the distances found on the arc CD. And by this sum of the distances, we shall obtain the elapsed time over the arc CD, or the mean anomaly. Or, in reverse order, for convenience's sake, if an angle CBD of an integral number of degrees (such as 45\u00b0) is used to find CAB, and 45 correct distances are obtained, these things, I say, follow demonstratively. But again, as before in the second method, CAD becomes the true equated anomaly, and thus CA remains in place, and the orbit DC will be a perfect circle. Since this is false, as was shown in Chapter 44, it necessarily follows that the distances at the middle elongations are taken to be too long here, and consequently that the times are made longer than they should be, and shorter at the apsides.","1069":"This method will in all respects be almost exactly equivalent to the previous method, using proportionals. For the number of distances we have gathered is now greater than before, to about the same extent that the proportionals, the same in number as the distances, were longer than the distances then. But, for safety's sake, witness the result of this calculation. For ","1070":"The eccentricity is again charged with being less than it should be. In other respects, the errors are the same as in the preceding one. The reason why the signs for excess are turned into signs for defect, is that here the difference shows errors in the equated anomaly, while there they are in the mean anomaly. And this is the third method.","1071":" In substituting the proportionals AG, AF for the distances AD, AC, which is the fourth method, we are going to make the two parts of the equation into three. For the area CAD measures the sum of the distances CA, DA. It is therefore much less than the sum of the lines FA, GA. And even if we attempt a remedy like the one used on the first method, we shall still have doubled our errors. For since the distances themselves cannot be admitted, owing to their excessive length at the middle elongations, the proportionals will be even less tolerable, since they are longer. And if you want to test them with the results of a calculation, you will find that to a mean anomaly of 53\u00b0 23' 56\" there corresponds an equated anomaly of 46\u00b0 0', which in the vicarious hypothesis comes out to be about 45\u00b0 27', a difference of 33', clearly absurd.","1072":"Fifth and sixth method","1073":"So, since I accomplished nothing with these four methods, I then took the mean anomaly and the distances assigned to it (in the fifth operation) and went over to the table of the vicarious hypothesis of Chapter 16, and the true equated anomaly. Let the second diagram in Chapter 46 be taken up again. Then, the distances AF belonging to integral degrees of mean anomaly IBF or IDH, also belong to the degrees and minutes of the equated anomaly IAH, which in the table mentioned corresponded to the mean anomaly IDH itself. Therefore,","1074":"Ninth, I related these distances obtained from the minutewise equated anomalies of the vicarious hypothesis of Chapter 16, that is, from the unequal angles HAI, to the individual integral degrees of the equated anomaly HAI, that is, to equal parts.","1075":"Tenth, with the same distances, thus set up, I sought the [third] proportionals, as in the second and sixth operations.","1076":" Eleventh and twelfth, I added each, according to its kind, and the sum of the distances was 35,770,014, and the sum of the proportionals, 35,692,048. In this translation of distances we moved all the long ones upwards, and made them fewer, establishing large arcs IG of the oval path above, at aphelion, and thus attributing the individual distances to the individual degrees, not of the anomaly FAB, as in the first method, but of HAB, which is the true equated anomaly. There are no more of these degrees in the upper semicircle than in the lower. Therefore, there are now more of the short distances than there are of the long ones, whence it not only happens that the sum of the 360 distances comes out smaller than the sum of the 360 diameters, but also the sum of the proportionals comes out smaller than was the sum of the distances themselves.","1077":"So, in the matter of the fifth method and the sums of the distances, reason again cries out against the method of basing equations upon it. Let the diagram belonging to this chapter be repeated, and let it be recalled to mind what was said about the first method. For in it the equated anomaly of distance CAD was divided into equal degrees. Hence it happened that CD was divided into unequal and [excessively] large parts, and had too few distances. From this, seizing upon a sort of accidental remedy for the error, we concluded from the sum of the distances on CD that to those distances belonged a shorter arc ED, so as to transfer AC over to AE, and thus ED could be obtained divided into equal parts with a distance established at any of its degrees. However, it was not from the sum of the distances found on CD, but from a mingling of the vicarious hypothesis with the hypothesis of the distances framed in Chapter 46, that the translation of AC to AE was now made and perfected, and the arc ED was attributed to the mean anomaly (which we have numbered on the arc CD for finding the distance CA or EA). And this was nonetheless done in such a way that BE and AC are not exactly parallel, as in the first method. And now that this has been done by a mingling of hypotheses, as I was saying, there is no need to go through the operation again, as in the first method. Instead, only one thing needs to be found: do the few distances AC, AE gathered into one sum in this fifth method, produce the same physical equation that resulted artificially from the two mingled hypotheses?","1078":"Consider here how the distances are arranged in this last procedure. The angle EAD, whose terminus E is at distance AC from the sun, was divided into equal degrees in this last procedure, to each of which was given a distance. In this way the arc ED of the oval path standing upon the angle EAD ends up divided into unequal parts, and it receives too few distances. So the mean anomaly already given from the vicarious hypothesis cannot be had from the sum of the distances on EAD.","1079":"  Now in the first method above, when CD received too few distances, with the angle CAD divided into equal degrees we substituted ED for CD as the arc suited to those distances. So similarly, in this fifth method, since ED has received too few distances, with the angle EAD divided into equal degrees, if it is permissible again to make use of a clumsy remedy, we would substitute ND for ED, as the arc to which those distances belong. In order to find the distance CA let the mean anomaly CBD be 48\u00b0 44'. Given angle B, and CB, BA, CA is given as 105,784, and CAB as 45\u00b0. The vicarious hypothesis requires AC to be transferred to AE. And we now divide ED, which the vicarious hypothesis declares to be 41\u00b0 22', into equal degrees, and through them we collected no more than 41 distances and part of a 42nd. And these, gathered into a single sum, will constitute a mean anomaly which is by no means equal to the one first taken, DC, but is equal to another, DO, which shows the distance AO, to be transferred to AN. \"Work was begun on an amphora; why has a pitcher come forth from the whirling wheel?\"4 For the question was whether all the distances on the equal degrees of ED, gathered into a sum, would show the mean anomaly DC. But the operation gave me an answer concerning ND, and the anomaly DO.","1080":" Finally, let us turn to the sixth method, and the proportionals that are adapted to the demonstration of Chapter 32. For the true quantities of orbital arcs that appear equal from the center of the sun are in the ratio of the distances: thus to the extent that AE is longer, so is ED.","1081":"But the times of truly equal arcs, as measured on the orbit, are also in the ratio of the distances. For to the extent that ED is farther away from A, the planet also takes longer to traverse the arc ED.","1082":"Therefore, the times that the planet accumulates on those arcs that appear equal from the center of the sun, are in the duplicate ratio of the distances.","1083":"But AF is to AH likewise in the duplicate ratio of the distance AC or AE to the mean AH. And so the measures of the times that the planet accumulates over equal degrees of the angle EAD are the proportional lines AG, AF, belonging to the integral degrees or equal parts of the angle of true equated anomaly EAD.","1084":"Therefore, let the proportionals to the distances at equal degrees of equated anomaly be tested, just as other distances were also tested above in this chapter. As, since 35,692,048, the sum of all 360 distances, at all 360 equal parts of the angle at the sun, is equivalent to an elapsed time of 360\u00b0, what is the value of a just and correct sum at any given degree of equated anomaly?","1085":"In this manner is found ","1086":"The eccentricity is again charged with being less than it should be. With this corrected, the difference above, at 41\u00b0, will be about 8'+, and below, about 7\u00bd\u2019\u2014. Thus here, too, the planet is not made to go fast enough at the apsides, and so there are too many distances near the apsides, and consequently less than there should be at the middle elongations. But it comes quite close to the truth, and clearly coincides with the method of Chapter 49. For if you consider well, the same thing is done here as was done in Chapter 49. There we computed the optical part of the equation by itself, and the physical part also by itself, while here we are computing them both together. There we had introduced fictitious rays of power in order to be able also to ascribe to the epicycle its own task of disengaging itself from those fictitious rays (for in the truth of the matter no rays go around as slowly as the center of the planetary epicycle goes, as was said in Chapter 39). And nevertheless, we left all the physical force carrying the planet around (as concerns the effect) to the sun, so that the epicycle would only serve to adjust the distances. Here, we made use of the same power of the sun for the physical translation, while we again computed the distances from the epicycle, and gave its equal parts to equal times, that is, to equal degrees of mean anomaly, as the opinion of Chapter 45 maintains. And if we ended up taking as many distances in any given part of the time as there are degrees of equated anomaly, they are still derived from the distances of the mean anomaly, and in length they are the same. And this form is so much easier that we can here put aside the other persuasion concerning the planet's epicyclic motion, and take one step closer to the truth of the physical cause, leaving to the epicyclic mode nothing but a reciprocation on the diameter\u2014although this is still flawed, as has was clear from these equations, at least. For, as was noted just above in considering the second method, this preoccupation with epicyclic motion is excessive, showing distances at the middle elongations that are too small, from which it happens that at that place the planet exceeds its measure of velocity, and at the apsides falls short of its measure. But it suffices us to express the opinion of Chapter 45 in our calculation. Therefore, although one might here raise the objection from Chapter 32 that this ratio of diurnal motions cannot be constant, since the parts of the eccentric near the apsides are presented directly to the sun, and the intermediate are presented obliquely, so that they thus appear differently from the way they would if they were presented directly\u2014 if anyone, I say, were to raise this objection, I shall answer as I did in Chapter 49, that this obliquity at the intermediate parts is added by the planet in its own right, and is produced through its descent. Thus, it is not to be imputed to the motive cause arising from the sun, nor is it affected by that cause.","1087":"Therefore, studious reader, from such a great number of chapters and methods, you have only two methods of equating that conform to the opinion of Chapter 45. One is by the physical hypothesis intermingled with an epicycle set up on the longitude, described in Chapter 49. The other is by this chapter, and its sixth method, in accordance with a more purely physical hypothesis, where the epicycle governs nothing but the descent towards the sun, or if anyone should wish to set it up to affect the latitude, [the epicycle should be] perpendicular to the plane of the ecliptic. And both of these two use different means to produce the same effect. You will thus be able to place confidence in them more safely when examining the opinion of Chapter 45.","1088":"And through a hitherto empty trust in the true physical causes that have been discovered, let a triumph over Mars once again be celebrated. Now, some rumor, I know not what, calls me to new tumults and new labors. ","1089":"Chapter 51","1090":"Distances of Mars from the sun are explored and compared, at an equal distance from aphelion on either semicircle; and at the same time the trustworthiness of the vicarious hypothesis is explored.","1091":"While I am thus celebrating a triumph over the motions of Mars, and fetter him in the prison of tables and the leg-irons of eccentric equations, considering him utterly defeated, it is announced in various places that the victory is futile, and war is breaking out again with full force. For while the enemy was in the house as a captive, and hence lightly esteemed, he burst all the chains of the equations and broke out of the prison of the tables. That is, no method administered geometrically under the rule of the opinion of Chapter 45 was able to emulate in numerical accuracy the vicarious hypothesis of Chapter 16 (which has true equations derived from false causes). Outdoors, meanwhile, spies positioned throughout the whole circuit of the eccentric\u2014I mean the true distances\u2014have overthrown my troops of physical causes called forth from Chapter 45, and have shaken off their yoke, retaking liberty. And now there was not much to prevent the fugitive enemy's joining forces with his fellow rebels and reducing me to desperation, unless I had sent new reinforcements of physical reasoning in a hurry to the scattered, straggling veterans, and, informed with all diligence, had stuck to the trail without delay in the direction whither the captive had fled. In the following few chapters, I shall be telling of both these campaigns in the order in which they were waged.","1092":"And, to speak initially of the first of these, I shall begin by seeking out the distances of several places on the eccentric where the evidence is most trustworthy. Therefore, let it be our intention to explore the distances near the mean anomaly of 90\u00b0 and 270\u00b0.","1093":"On 1589 May 6 at 11\u2153h Mars was observed at 27\u00b0 7\u2153\u2019 Libra with latitude 0\u00b0 6\u2154\u2019 north. The true position of the sun at this time is calculated as 25\u00b0 48\u2154\u2018 Taurus, and its distance from earth 101,361. The mean longitude of Mars was 7s 26\u00b0 0' 36\", and therefore its eccentric position was 15\u00b0 32' 13\" Scorpio. But our vicarious hypothesis of Chapter 16 did not come nearer than 2\u2153\u2018 the true or observed position of Mars in an acronychal situation, and thus in so sensitive a procedure the computation of the equated anomaly cannot be trusted. Therefore, to the method of Chapters 27, 28, and 42 I shall add another observation, which nevertheless uses a freer method. Indeed, as I remarked in Chapter 12 above, Mars was not very often observed twice in this region. Therefore, we should be content with two observations. For with this one just now presented, there is associated another, from 1594 December 28. At 7\u00bch on the morning of that day, Mars's mean longitude is calculated to be 7s 26\u00b0 13' 39\", a few minutes beyond the other. And at that time Mars, at an altitude of eight or nine degrees, was observed to be 50\u00b0 34' from Spica Virginis. So, since it stood very close to the ecliptic, in the right triangle between Spica, its ecliptic position, and Mars, the base is given as 50\u00b0 34' and the side between Spica and the ecliptic is 1\u00b0 59', which is Spica's latitude. Therefore, the remaining side is 50\u00b0 32' 18\". Thus, since Spica was at 18\u00b0 11' Libra, Mars fell at 8\u00b0 43' 18\" Sagittarius. The declination of this position from the equator was 21\u00b0 50' 20\" [S.].","1094":"However, Mars was found to have a declination of 21\u00b0 41' [S.]. Therefore, it displayed a small amount of north latitude, 9' 20\". And on the following 1595 January 4 it still had 3' of north latitude. Our observation is hereby confirmed. But if you assume this to be the true latitude of Mars, its ecliptic position will not be changed perceptibly. So you may safely pronounce its position to be 8\u00b0 43' Sagittarius. And because Mars was near the sun, it was very far from earth, and thus had a much smaller parallax than the sun, which we shall ignore. But we cannot similarly ignore the refraction, which I shall now remove. For the sun\u2019s position was 16\u00b0 47' 10\" Capricorn, distance from earth 98,232, and its right ascension was 288\u00b0 12'. Therefore, 306\u00b0 57' on the equator was rising, and along with it 29\u00b0 Sagittarius, at which the angle between the ecliptic and the horizon is 26\u00b0, its complement 64\u00b0. And because the altitudinal refraction as shown by the table of refractions of the fixed stars is 6' 30\", and from the table of the sun, 11', when the star is at an altitude of 8\u00bd\u00b0, 5' 51\" or 9' 53\" are to be subtracted from the latitude. The latitude from the former would be 3' 29\" N., and from the latter 0' 33\" S. And the longitudinal refraction is 2' 39\" or 4' 34\". ","1095":" Of these two methods of determining the refraction, I shall follow the one which is confirmed by the latitudes, as follows. In the earlier observation, the observed latitude was 6\u2154 North. And because Mars was near the earth, and the angle at the sun was 10\u00b0 17', while at the earth it was 28\u00b0 41', this latitude requires an inclination of 2' 30\". Therefore, in our later observation the inclination will also be 2' 30\", or a little less, since we are 8' closer to the node. But with the inclination assumed to be 2' 30\", since here the angle at the sun is 61\u00b0, and at earth 38\u00b0, a latitude of about 1' 50\" N. must follow, as indicated by our parallactic table. But by using the refraction of the fixed stars, we were left with a latitude of 3' 29\" N., while by using the solar refraction we were moved down to 0\u00b0 33' S. Thus in the second our refraction was greater than would be correct, and in the first, less. So the correct refraction is between the two, namely, 3' 36\". That is, for us Mars will be put at 8\u00b0 46\u2153\u2018 Sagittarius. Let O be the sun, B and A points on the earth\u2019s orbit, A the position of the earth in the earlier observation, B in the later, M Mars. Let the lines be connected. And although Mars does not return to exactly the same place, let it nonetheless be represented in both instances by the line OM. Thus MAO is 28\u00b0 41' 14\", and AO is 101,365. Let MO, the distance of Mars from the sun (which is being sought here), be taken as if known, and let it be 154,200. Thus OM will fall at 15\u00b0 31' 3\" Scorpio. And if OM is assumed to be 154,200 in the earlier observation, it ought to be taken as shorter in the later one. Now one degree at this place on the eccentric changes the distance by 240 units, whatever form you use for constructing the distances. Therefore, since the mean longitudes here differ by 13 minutes, and when a subtraction is made to account for precession, only eight, the proportional part of 240 is 32. Consequently, in the second observation, we have assumed OM to be 154,168. But OBM is also known, it being 38\u00b0 0' 40\", and OB is 98,232. Therefore, OMB is given as 23\u00b0 6' 11\". Consequently, on the second occasion OM was at 15\u00b0 40' 9\" Scorpio, differing from the earlier eccentric position by 9 minutes. It should have differed by somewhat more. For the mean anomalies differed by 8' 3\", to which corresponds 7' 49\" in the equated anomaly of the eccentric at this place. Add to these the intervening precession of the equinoxes of 4' 48\". Thus 12' 37\" are accumulated. Mars therefore ought to fall at 15\u00b0 43' 40\" Scorpio. Therefore, we should take a somewhat different value for the distance OM, and should change it so that the lines represented by OM move about another 2\u2154\u2018 apart from each other. For when the earth is at A, OM should move back in longitude, and forward when the earth is at B. But this occurs if you increase OM. So let it be 154,400 in the first instance, and in the second, 154,368. For then OM falls first at 15\u00b0 29' 34\" Scorpio, and second at 15\u00b0 42' 18\" Scorpio.","1096":"And for the first time the mean anomaly is 87\u00b0 9' 24\", and 87\u00b0 16' 30\"8 for the second. This will do for the mean longitude of the earlier one.","1097":"For the other mean longitude, an observation in the month of December 1595 will serve, being well supported by the consensus of a number of consecutive days, and at a place where the vicarious hypothesis also exactly represented the acronychal position of Mars in the preceding October. For the sake of consensus, we shall also add an observation of October 1597. In the other years it was not observed at this eccentric position. For the eccentric position falls at 10\u00b0 Gemini. Thus Mars was observed last at this place in November 1580. In 1582 its arrival at the place fell in October, when [Tycho's] great interest in observing was not yet aroused. In 1584 it came in September, in 1586 in July, in 1588 in June, in 1590 in April, and in 1592 in March, at which times, being near the sun owing to the short, bright nights in Denmark, it was neglected, while, whenever there was an opportunity, they were intent upon the fixed stars, the moon, and the other planets. But at the end of 1593 and the beginning of 1594, when it was at quadrature with the sun, the observation was not continued beyond this aspect because astronomers are usually chiefly interested in the quadrature. So, in 1595 Dec. 17 at 7h 6m in the evening the planet was observed at 11\u00b0 31' 27\" Taurus, with latitude 1\u00b0 40' 44\" N. The sun's position was 5\u00b0 39' 3\" Capricorn. Its distance from earth was 98,200.","1098":"The mean longitude of Mars is concluded to be 2s 2\u00b0 4' 22\".","1099":" And since the aphelion is 4s 28\u00b0 58' 10\", the distance of the position from aphelion is 86\u00b0 53' 48\" backwards. Previously, it was nearly the same as that, namely, 87\u00b0 9' 24\". Therefore, these two positions are nearly the same distance from aphelion. Now from our vicarious hypothesis, there corresponds to this simple anomaly an equated anomaly of 76\u00b0 25' 48\". Which, subtracted from the position of the aphelion, leaves Mars's eccentric position, 12\u00b0 32' 22\" Gemini. Let A be the earth, O the sun, M Mars. AO is given as 98,200. And because OM is at 12\u00b0 32' 22\" Gemini, while AM is at 11\u00b0 31' 27\" Taurus, therefore AMO is 31\u00b0 0' 55\". And because AO is at 5\u00b0 39' 3\" Capricorn, while AM is at 11\u00b0 31' 27\" Taurus, therefore, the supplement of OAM is 54\u00b0 7' 36\". Hence, since the sine of OAM is to OM as the sine of AMO is to AO, OM comes out to be 154,432. And since this position is 15 minutes closer to the apogee than the one from 1589, and at this position on the eccentric 1\u00b0 causes a change of 240 units, therefore 60 units are to be subtracted for 15 minutes, since at positions farther removed from aphelion the distances are shorter. The distance thus comes out to be 154,372. On the other hand, since the node is at about 16\u00b0 20' Taurus, and the eccentric position is 12\u00b0 32' Gemini, the planet is 26\u00b0 12' from the node. And the maximum inclination of the planes is 1\u00b0 50'. Therefore, the inclination at this place is 48' 32\". The secant of this exceeds the radius by 10 units, or in our dimensions, 15\u00bd. So the distance of the actual point on Mars's orbit from the sun is 154,387. And previously, at this distance from aphelion it was found to be approximately 154,400 from the sun. Therefore, the distances of these two points on the eccentric from the sun is equal to within a hair's breadth. For the 13 units that are wanting in the latter distance are of no importance. I shall rejoice if I am able to come within an uncertainty of 100 units everywhere.","1100":"And now I shall add [the observation from] 1597, not so much to confirm the previous ones, which are certain in themselves, as to give the reader an opportunity to compare the observations of Tycho with the observations of others, by which means he will at length understand how much that man has benefited us. There do indeed exist observations of that author from the last days of October of 1597, but they were taken with a radius while travelling, and not brought to a calculation by the author himself, who knew how to correct distances taken with a radius, by applying a kind of parallactic table for the eye, as he informed us in the Progymnasmata. And so, since very different distances are ascribed to the same moment (possibly because they were supposed to be corrected immediately after the observations), they are to be dismissed. But at the same moment, I, though absent in Styria, made an observation, and (strange to say) did so with the eyes of Tycho Brahe, standing by the shore of the Baltic Sea. Here is the series of observations\u2014can you hold your laughter, friends?12","1101":"On 1597 November 8, Saturday, or October 29, in the morning, Mars was not yet on the line from the twelfth of Gemini to the fourth. On the following day, it had already left it: it was nearer the ninth than the twelfth, and precisely on the line from 11 to 9, and also on the line from 1 to 5, or a little farther east. And the fifth was halfway between the first and Mars.","1102":"From these the position of Mars can be elicited, when certain stellar positions are assumed from Tycho Brahe's catalogue, which I had just now been professing to be my eyes. But because the Ninth is not described in Brahe's catalogue (since in its stead in the ninth place is another, distant by more than three degrees from the Ptolemaic, and less than all of them), we shall call upon the latitude of Mars as our counsel. For an approximate knowledge of it will suffice. Now the mean longitude of Mars on the morning of October 29 at 5h (which is an estimate, since I did not write down the time) is found to be 1s 29\u00b0 10' 43\". Therefore, its eccentric position was 9\u00b0 43' Gemini, 23\u00b0 20\u2019 from the node. Therefore, the inclination was 43' 52\". But the sun was at 15\u00b0 40' Scorpio, and the apparent position of Mars, by anticipation, was about 12\u00bd\u00b0 Cancer. Therefore, the latitude was 1\u00b0 36\u2019 24\". Let a computation be made of the longitude of a point on the line from the twelfth to the fourth, having a latitude of 1\u00b0 30\u00bd' North. Since the fourth is at 9\u00b0 54' Cancer, latitude 7\u00b0 43' N., and the twelfth is at 12\u00b0 56' Cancer, latitude 0\u00b0 13\u00bd' S., the longitude of our point, by interpolation, will be 12\u00b0 16' 17\" Cancer. But on 29 October, Mars was not yet here, and on the 30th it had already passed it. The diurnal motion was no more than 5 minutes, half of which is 2\u00bd', so that on the morning of the 30th it was at 12\u00b0 18\u00bd' Cancer. And so indeed it was at the end of 1600, but in 1597 it was at 12\u00b0 16' Cancer. Three minutes of error in latitude barely produce one in longitude. So the position is certain enough. If you also explore it using the first and the fifth, and use the point on that line whose latitude is 1\u00b0 30\u00bd', it falls at 12\u00b0 9' Cancer. And Mars was farther east, that is, forward in longitude, at about 12\u00b0 16' or a little before, also intermediate. Therefore, the latitude computed by us is confirmed. For it should be approximately intermediate, and indeed it is. That is, between the Martian latitude of 1\u00b0 30\u00bd' and the 5\u00b0 42\u00bd' of the fifth there is 4\u00b0 12', and between this and the 10\u00b0 2' of the first there is an intermediate 4\u00b0 20'.","1103":"Therefore, let Mars be at 12\u00b0 16' Cancer. On 1597 October 30 at 5h am the position of the sun is found to be 16\u00b0 38' 8\" Scorpio, distance 98,820. [Mars's] mean longitude was 1s 29\u00b0 42' 10\", aphelion 4s 28\u00b0 57' 10\", supplement of the mean anomaly 89\u00b0 15', of the equated anomaly 78\u00b0 43' 23\", eccentric position 10\u00b0 13' 47\" Gemini. Therefore, a distance of 153,753 is called forth from these. And since we are 2\u00b0 6' lower from aphelion than before, we will add twice 240, the sum of the units corresponding to one degree: ","1104":" If you subtract three minutes from the position of Mars, so that it would be at 12\u00b0 13' Cancer, which could be done in our observation, especially if the time were different, this difference would be reconciled.","1105":"Second, I shall prove the same thing at parts closer to aphelion. On 1589 April 5 at 11h 33m Mars was observed at 7\u00b0 31' 10\" Scorpio with latitude 1\u00b0 28' 13\" N. It was very near the meridian, and consequently there were no horizontal variations. The mean longitude is concluded to be 7s 9\u00b0 46' 8\". And the ","1106":"aphelion is at 4s 28\u00b0 51' 8\". Therefore, the mean anomaly is 70\u00b0 55' 0\", to which corresponds an equated anomaly of 61\u00b0 17' 35\", by the vicarious hypothesis. And so the eccentric position is 0\u00b0 8' 43\" Scorpio; the sun's position, 25\u00b0 52' 43\" Aries; its distance from earth 100,560; the angle at the earth 11\u00b0 38' 27\"; at the planet 7\u00b0 22' 27\". Therefore, the distance of Mars from the sun is 158,090. 16 But again, so as not to trust the eccentric position, on account of the error of about two or three minutes which the vicarious hypothesis commits at this position on the eccentric, we shall appropriate a counterpart from 1591 Feb. 19, when, at 5\u00bdh am Mars was observed to be 28\u00b0 11' from the southern pan of Libra (which in that year was at 9\u00b0 23\u00bd' Scorpio), with a latitude of 0\u00b0 26' North. So Mars fell at about 7\u00b0 34\u00bd' Sagittarius, approximately. But since that eccentric position has a declination from the equator of 21\u00b0 39' 10\", [while] the observed declination of Mars was 20\u00b0 50' 30\", its latitude was therefore 48' 40\". From this, the longitude is corrected, which becomes 7\u00b0 34\u2153\u2018 Sagittarius. But the mean longitude is 7s 8\u00b0 21' 47\", to which corresponds an equated 59\u00b0 57' 38\", and an eccentric position of 28\u00b0 51' Libra. Therefore, the angle at the planet is 38\u00b0 43' 20\"; the sun's position 10\u00b0 14' 25\" Pisces; therefore, the angle at the earth, 87\u00b0 20' 0\"; and the distance of the sun from the earth, 99,210. Thus the distance of Mars from the sun comes out here to be 158,428, longer than before, because here we are also nearer to aphelion by 1\u00b0 26' 30\". But at this place on the eccentric for one degree about 220 units are owed from the distance, or for the entire angular difference, 317, so that this place, if we carry it back to an anomaly similar to the preceding, has a distance of 158,111 rather precisely. Whence it is proved that these two eccentric positions, treated by the method given above, will show exactly the same eccentric position as our vicarious hypothesis, except that on account of our nearness to 17\u00b0 Scorpio we run the risk of being in error by one or two minutes. Moreover, in the latter of these, the distance from Aquila comes out to 54\u00b0 12', which is not within 12' of agreeing with the other observed data, and consequently this observation is not perfectly certain. Also, some small quantity should be added, owing to the latitude.","1107":"A suitable observation at a similar longitude on the other semicircle occurred on 1582 November 12 at 6\u00beh am, when the sun's position was 29\u00b0 35' 17\" Scorpio. Its distance was 98,503, mean longitude of Mars 2s 15\u00b0 10' 20\", aphelion 4s 28\u00b0 44' 20\". Hence, the full-circle complement of the mean anomaly was 73\u00b0 34', and of the equated anomaly, 63\u00b0 45' 18\". Hence, the eccentric position was 24\u00b0 59' 2\" Gemini. Then, I say, the planet was observed at 26\u00b0 35' 30\" Cancer, making the angle of vision, the one at the earth, 57\u00b0 0' 13\", and at the planet, 31\u00b0 36' 28\". By these data it is determined that the planet's distance from the sun was 157,631. And because the anomaly was previously 70\u00b0 55', and is now 73\u00b0 34', we are therefore lower by 2\u00b0 39'. For this, in the previously mentioned ratio, 586 units are owed. So from the analogy of this observation with the previous one, at a similar anomaly, a distance of 158,217 is appropriate, where again about the same amount as before, or a little more, is to be added for the latitude. The difference is about 127, which is excused owing to the uncertainty of the prior observations. For it is very small, and may be neglected in our present undertaking, where we are considering magnitudes of 1800 or 3600 or still more.","1108":" But let us move yet higher towards aphelion, and explore those places","1109":"what was shown in Chapter 6, the dislocation of the eccentric occasioned by exchanging the sun's mean motion for its true motion can occur most clearly, namely, at the sun's apogee and the sign of Cancer.","1110":"On 1596 March 9 at 7h 40m pm, when the sun's position was 29\u00b0 31' 24\"","1111":"Pisces, the distance from earth 99,764, Mars's mean longitude 3s 15\u00b0 35' 0\", aphelion 4s 28\u00b0 58' 31\", the full-circle complement of the mean anomaly 43\u00b0 23' 31\", equated anomaly 36\u00b0 40' 2\", eccentric position from the vicarious hypothesis 22\u00b0 18' 29\" Cancer\u2014the planet was observed at 15\u00b0 49' 12\" Gemini, latitude 1\u00b0 47' 40\" N. Therefore, the angle at the earth was 76\u00b0 17' 48\", at the planet 36\u00b0 29' 17\". Therefore, the distance of Mars from the sun was 162,994, or more correctly, this was the distance of the point in the plane of the ecliptic perpendicularly beneath the body of Mars.","1112":"But, for safety's sake, let another observation be added. And Mars was at precisely the same sidereal position on 1584 Nov. 25 at 10h 20m, when the sun was at 14\u00b0 0' 3\" Sagittarius, distance from earth 98,318. The mean anomaly was not perceptibly different from the previous one, because the motion of the aphelion is only very slightly faster than the motion of the fixed stars. Therefore, the eccentric position is the same, 22\u00b0 8' 44\" Cancer, if you subtract the precession of 9' 45\". But the planet was observed on Nov. 12 at 13h 26m at 23\u00b0 14' 15\" Leo, with latitude 2\u00b0 12' 24\" N. On the 20th of Nov. following, at 18h 30m astronomically, it appeared at 26\u00b0 0' 30\" Leo. Thus in 8 days 5 hours it was moved forward 2\u00b0 46' 25\". In Magini, this is 2\u00b0 48'. Therefore, since our time follows by 4 days 15h 49m, to which corresponds 1\u00b0 28' of motion from Magini, we shall add 1\u00b0 27' according to the above ratio. So Mars could have been observed at 27\u00b0 27' 30\" Leo, approximately. Therefore, the angle at the earth was 73\u00b0 27' 27\", at the planet 35\u00b0 18' 46\". Hence, the distance of Mars from the sun here was 163,051, exceeding the previous one by 57 units. This can be absorbed by a very slight change in the eccentric position, as, indeed, the vicarious hypothesis is not trustworthy to within one minute. Furthermore, I could easily have made some slight error in the application of the observation.","1113":" For a similar longitude in the other semicircle, we shall again take up the observations of Chapter 27. There I derived a distance somewhat less than 163,100 using the equation of the observations, but from the bare observations themselves I obtained 162,818, in the plane of the ecliptic as before. Now for one of the times introduced in that chapter, 1589 Feb. 1227 at 5h 13m am, the mean longitude was 6s 12\u00b0 38' 44\", aphelion 4s 28\u00b0 50' 57\"; and consequently, the mean anomaly was 43\u00b0 47' 48\", lower than our previous one by 24 minutes. To this there corresponds [an adjustment of] about 64 units at this position on the eccentric. So the distance which was less than 163,13729 at an anomaly of 43\u00b0 48' will again be increased at anomaly 43\u00b0 24' according to this ratio, so as to make it almost exactly 163,100 in this semicircle. In the previous one it was 163,051 and 162,996. Again, not an outstanding fit.","1114":"It should be noted, however, that in Chapter 27, to which I am referring here, the observations led us to subtract 1' 30\" from the eccentric position computed from our vicarious hypothesis at 5\u00bd\u00b0 Libra, and this was through the observations of 1585, 1587, 1589, and 1590. Second, in Chapter 18 above, the acronychal observation of 1589, at 5\u00b0 Scorpio, gave the same testimony, namely, that our vicarious hypothesis needs to be diminished by 2\u2155\u2019. And in 1591, at 26\u00b0 Sagittarius, there still was one minute to be subtracted. Third, in this very chapter, about 16\u00b0 Scorpio, the observations of 1589 and 1594 required 3\u00bd minutes to be subtracted from the eccentric position computed from our vicarious hypothesis. So therefore this is constant about the middle elongation of this semicircle.","1115":"And likewise, near aphelion, we shall again take up the observations of Chapter 28, where at a mean anomaly of 11\u00b0 37' a distance of 166,180 or 166,208 was found (without correction for latitude). This was in the descending semicircle. But it was at a similar anomaly on the ascending semicircle at the following times.","1116":"On 1585 January 24 at 9h, when the position of the sun was 15\u00b0 9' 5\" Aquarius, its distance from earth 98,590, the mean longitude of Mars 4s 16\u00b0 50' 10\", the aphelion 4s 28\u00b0 46' 41\", the remainder of the mean anomaly to complete the full circle 11\u00b0 56' 31\", and the consequent eccentric position from the vicarious hypothesis 18\u00b0 49' 0\" Leo: the planet was observed at 24\u00b0 9' 30\" Leo, latitude 4\u00b0 31' 0\" N. The angle at earth was therefore 9\u00b0 0' 25\", and at the planet 5\u00b0 20' 30\". Therefore, the distance of Mars from the sun was 165,792. But if you subtract 1' 30\" from the vicarious hypothesis here, as appeared necessary above in Chapter 18 in the computation of the acronychal opposition, the angle at the planet will be 5\u00b0 19', and the distance of Mars from the sun 166,580. And the distance here is easily changed to this extent, because earth and Mars are close to one another. Therefore, for insurance, we shall bring in other positions.","1117":"On 1586 December 16 at 6\u00bdh am, when the sun was at 4\u00b0 16' 51\" Capricorn, 98,200 distant from earth, mean longitude of Mars 4s 18' 39' 9\", remainder of mean anomaly 10\u00b0 9' 41\", eccentric position from the vicarious hypothesis 20\u00b0 20' 30\" Leo: the declination of Mars was found to be 3\u00b0 54', right ascension from Arcturus and Spica 177\u00b0 27'. Thus its longitude was 26\u00b0 6' 24\" Virgo, latitude 2\u00b0 35'. Hence, the angle at earth was 81\u00b0 49' 33\", at the planet 35\u00b0 45' 54\". And the distance was 166,311, but by subtraction of 1' 30\" from the eccentric position, 166,208. And at the previous distance from aphelion, 11\u00b0 37', it would be about 70 units less. So it would be either 166,241 or 166,138.","1118":"On 1588 Nov. 6 at 6h 50m am, when the sun's position was 24\u00b0 3' 43\" Scorpio, 98,630 distant from earth, mean longitude of Mars 4s 20\u00b0 47' 35\", remainder of the anomaly 8\u00b0 2' 51\", eccentric position from the vicarious hypothesis 22\u00b0 7' 48\" Leo: Mars was observed at 23\u00b0 16' Virgo, latitude 1\u00b0 37', {marginal:","1119":"254} Hence, the angle at earth was 60\u00b0 47' 43\", at the planet 31\u00b0 8' 12\". And thus the planet's distance from the sun was 166,511. But by subtraction of 1' 30\" from the position of the vicarious hypothesis, the distance becomes 166,396. And by this analogy, at the greater distance from aphelion of 11\u00b0 37', where it is less by about 110, it is either 166,401 or 166,286. There is a discrepancy of 150 between this and the previous one. And if, keeping the correction of the eccentric position, we take a mean between the two, 166,230, as if saying that in the two observations of 1586 and 1588 there were some small observational errors in opposite senses [in the determination of the distance], we will hardly differ at all from the distance in the descending semicircle. Even this small difference will be able to be abolished by a slight retraction of the aphelion, of which more later. Thus, near aphelion also, as far as the senses can judge, we find the same distances from the sun at the same relationship to the aphelion in the two semicircles.","1120":"All three observations were made when Mars was in the east, and none with Mars in the west. For the rest of the observations [sc. of Mars at these mean longitudes] are lacking. Therefore, we will probably be safer to stay with the distance in the descending semicircle.","1121":"Third, let the same things we have explored above the middle elongations now be explored below, near perihelion.","1122":" In 1591 on the night following May 13, at 1 hour 40m past midnight, when the sun was at 2\u00b0 8' 43\" Gemini, distant from earth 101,487, while the mean longitude of Mars was 8s 22\u00b0 18' 4\", anomaly 113\u00b0 24' 4\", equated 103\u00b0 15' 48\", consequent eccentric position from the vicarious hypothesis 12\u00b0 9' 48\" Sagittarius (or, by analogy with 26 Sagittarius nearby, just now mentioned, 12\u00b0 8\u00be\u2019","1123":"Sagittarius): Mars was observed at 2\u00b0 24\u00bd' Capricorn, latitude 2\u00b0 15' S. Therefore, the angle at the earth was 30\u00b0 15' 44\", and at the planet either 20\u00b0 14' 39\" or 20\u00b0 15' 42\". Hence, the distance of Mars (or of the point on the ecliptic) from the sun was 147,802, or more correctly, 147,683. Here, you see that an error of one minute in eccentric position causes the loss of 120 of our units at this distance of Mars from earth, and at this distance of the sun in the opposite position. So these slight discrepancies need no further attention. Besides, this observation is well supported by others on many of the days nearby, right up to the day of opposition with the sun. But since 12\u00b0 10' Sagittarius is about 26\u00bd\u2019 from the node, the secant of the inclination at this place exceeds the radius by about 11 units, or 15 or 16 in our dimensions, so as to make the distance of Mars from the sun almost exactly 147,820 or 147,700.","1124":"For a similar distance from aphelion in the other semicircle, we shall take up again the observations of Chapter 26, where I derived a distance of Mars from the sun of about 147,443 or 147,700 or 147,750. And at one of the times noted therein, namely, 1590 March 4 at 7\u2155h, the [mean] longitude of Mars was 1s 4\u00b0 11' 20\". Hence, the full-circle complement of the mean anomaly was 114\u00b0 41'. We are thus lower down from aphelion than we were before by one degree and 17 minutes. And to one degree correspond 230 units at this position on the eccentric. Therefore, the distance of 113\u00b0 24' on the ascending semicircle would be 147,743 or 148,000 or 148,050 (extrapolating from the observations of Chapter 26). But on the descending semicircle, 147,820 or 147,700 was found here. The difference is about 350 or 180 units, or none; it is rather uncertain. For the observations with Mars at perigee are rather poorly obtained, on account of the low elevation of the zodiac and many other causes. And you see in Chapter 26 that the true distance, hesitantly accepted, fluctuates between 147,443 and 147,750, a difference of 300 units which, in our present undertaking, are of no great importance, since Mars is so low and close to the sun or the center of the world.","1125":"But let us descend here even farther towards perihelion, and explore the same thing about 22 degrees before and after perihelion.","1126":"On 1589 Dec. 3 at 5h 39m, when the sun's position was 21\u00b0 44' 56\" Sagittarius, and its distance from earth was 98,248, and the mean longitude of Mars was 11s 16\u00b0 27' 53\", full-circle complement of the anomaly 162\u00b0 24' 11\", and the equated eccentric position 20\u00b0 4' 32\" Pisces: Mars was observed at 15\u00b0 25' 33\" Aquarius, latitude 1\u00b0 11' 47\" S. But because it was found above in Chapter 42 that our vicarious hypothesis errs somewhat near perihelion, we shall admit other positions, as many as we can obtain, and inquire of them, using the method of Chapter 42, the distance of Mars from the sun, and at the same time a more correct eccentric position as well.","1127":"So, on 1591 Oct. 16 at 6h 28m, when the sun was at 2\u00b0 39' 15\" Scorpio, 99,142 distant from earth, Mars's mean longitude 11s 13\u00b0 53' 57\", full-circle complement of the anomaly 165\u00b0 0' 9\", eccentric position from the vicarious hypothesis 16\u00b0 59' 14\" Pisces: Mars was observed at 1\u00b0 27' 18\" Aquarius, latitude 2\u00b0 10' 52\" S. 35","1128":"Also, on 1593 Sept. 8 at 10h 38m, when the sun was at 25\u00b0 41' 0\" Virgo, 100,266 distant from earth, Mars's mean longitude 11s 17\u00b0 10' 17\", full-circle complement of the anomaly 161\u00b0 45' 28\", and eccentric position from the vicarious hypothesis 20\u00b0 53' 54\" Pisces: the planet was found at 8\u00b0 53' 51\" Pisces with latitude 5\u00b0 14' 30\" south. ","1129":"Finally, on 1595 July 22 at 2h 40m am, when the sun was at 7\u00b0 59' 52\" Leo, 101,487 distant from earth, Mars's mean longitude 11s 14\u00b0 9' 5\", and anomaly 164\u00b0 48' 55\", and consequent eccentric position from the vicarious hypothesis 17\u00b0 16' 36\" Pisces: the apparent position of Mars, from the most select observations, was 4\u00b0 11' 10\" Taurus, latitude 2\u00b0 30' S. Thus we twice have Mars in the most opportune position, in quadrature with the sun, while the positions of earth and Mars are also distant by a quadrant.","1130":"And so, following the method of Chapter 42, I shall make a test of the eccentric positions of the star, and to","1131":"begin with I shall suppose that the distance of Mars at the first time was 139,212. Hence, the following ones were 139,033, 139,258, 139,045. For in such close proximity, the connection of the anomalies is easily known, as before. Let A be the sun, D, G, F, E, positions of the earth in 1589, 1591, 1593, 1595. Let K be the position of Mars, the same all four times (even","1132":"though it is not quite the same in the observations). Let the points be connected. AD, AG, AF, AE are given in position and length. And the length of AK is introduced four times. Moreover, the lines of observation DK, GK, FK, EK have known positions. Therefore, ADK, AGK, AFK, AEK are given. So, through the opposition of the sides to the angles, DKA, GKA, FKA, EKA are also given. Hence, so is the position of KA, four times. ","1133":"So, since the first and third position here agree rather closely, some less thoughtful person will think that it should be established using these, the others being somehow reconciled. And I myself tried to do this for rather a long time. But since the second and fourth could not be reconciled, while the force of these observations was great, because in each the planet was observed in quadrature with the sun, and in the quadrilateral AEKG all the sides and angles are about equal, I therefore settled it as follows. From the vicarious hypothesis, you see that AK in the second observation ought to be distant from AK in the fourth by 17' 22\". But by the assumption of this length, the two positions of AK are 30' 55\" apart. So this is too much by 13' 33\". And since all angles of the quadrilateral are about equal, I divided the excess in two, and added 6' 46\" to the angles EKA, GKA. For in the observation at E, the line AK had moved forward too much, and not enough at G. So, with the two AKs moved back towards E and G, EK and GK staying fixed (for we are supposing the observations to be most certain), the angles at K will in all cases be increased. So now, given the angles GKA, 45\u00b0 35' 13\", and EKA, 46\u00b0 51' 16\", the other angles G, E, and the lines EA, GA remaining the same, AK comes out to be 138,765, and 138,787, differing from our assumed value by 258 units. So if we also subtract that much from the other two AKs, so as to make them 138,954 and 139,000, the resulting angles are DKA 34\u00b0 43' 47\" and AK 20\u00b0 9' 40\" [Pisces]; while FKA is 12\u00b0 1' 24\" and AK 20\u00b0 55' 15\" [Pisces]. But since I previously added 6' 46\" at G and subtracted the same amount at E, I have therefore repositioned the eccentric positions at 17\u00b0 2' 31\" Pisces at G, and 17\u00b0 19' 54\" Pisces at E, increasing the position given by the vicarious hypothesis by 3' 17\". Therefore, the same amount also ought to result at D, ","1134":"And so I have also brought the other two positions near enough together. For their errors lie both past and short of the truth, which lends security. And to attribute an error of two minutes to observations at these positions, owing to the low elevation of the zodiac and horizontal variations, is not excessive.","1135":"At a similar anomaly on the descending semicircle, the observations at hand are no more than one, but it is certain enough. For on the night following June 29 in 1593, at 1h 30m after midnight, when the sun was at 17\u00b0 25' 42\" Cancer, 101,760 distant from earth, the longitude of Mars 10s 10\u00b0 1' 29\", anomaly 161\u00b0 5' 29\", and the consequent position of Mars 6\u00b0 10' 5\" Aquarius, it was observed at 13\u00b0 37' 22\" Pisces, with latitude 4\u00b0 37' S. Hence, the supplement of the angle at earth was 56\u00b0 11' 46\", at the planet, or the parallax of the annual orb, 37\u00b0 27' 23\". From which the distance of Mars from the sun comes out to be 139,036. But above, at an anomaly of 161\u00b0 45' 28\", where Mars was 40 minutes farther from aphelion than here, the distance was found and established as 139,000. And at this position on the eccentric these 40 minutes effect a change of 52 units. So here too, by extrapolation from our anomaly, there results a distance of 138,984 at an anomaly of 161\u00b0 45\u00bd', an admirable consensus much to be suspected. For they can hardly all be so certain and neat. Furthermore, both distances must be increased somewhat owing to the inclination of this position on the eccentric, which is at a maximum.","1136":"So, from this long induction, using a great many positions on the eccentric, it is clear that those distances of Mars from the sun are mutually equal whose points on the orbit are equally remote from aphelion, a question which we have investigated in Chapter 16 and 42. This is an evident way of showing that the aphelion we have obtained is correct, by Euclid III. 7.","1137":"At the same time, the distances of the sun from earth are confirmed, which were derived in Chapter 29 above and employed here in various ways. Nor is there any great discrepancy in the numbers that could testify to any flaw in them.","1138":"The implications of the observations presented in this chapter, and of the distances found through them, for the shaping of the planetary path, for which purpose we have produced them in this chapter, we shall postpone until Chapter 55. First, there is something that must be proved in Chapter 52 following, and in Chapter 53 many more observations are going to be called upon to testify.","1139":"Chapter 52","1140":"Demonstration from the observations of Chapter 51 that the planet\u2019s eccentric is set up, not about the center of the sun\u2019s epicycle, or the point of the sun\u2019s mean position, but about the actual body of the sun; and that the line of apsides goes through the latter rather than the former.","1141":" It is a happy accident that the distances found in Chapter 51 also inform us about this, which, though promised in Chapters 6, 26, and 33,","1142":"I deliberately postponed until this point. For if I was correct in constructing the eccentric of Mars about the body of the sun, it is necessary that the planet really be at its greatest distance from the sun in the parts around 29\u00b0","1143":"Leo, and that those parts which are at","1144":"equal intervals from 29\u00b0 Leo in either semicircle be at equal distances from the sun, and at unequal distances from the point that stands for the sun, which for Brahe is the center of the sun's epicycle. More specifically, the distances should be less in the descending semicircle. When this is proved, it will follow in addition that the parts around 24\u00b0 Leo are neither the most distant from the sun's body, nor from the Copernican center of the world, which for Brahe is the center of the sun's epicycle, and also the center to which the planetary circle is attached; and the parts at an equal arc's removal from 24\u00b0 Leo in either semicircle are at unequal distances from the sun and from the point that stands for it. For let there be set out the sun's center A, Mars's line of apsides AC, eccentricity AC, and the eccentric ED with center C, and let the point F above AC be the point of uniform motion, G the aphelion, GFE and GFD equal angles, and let EA, DA be connected, which will be equal, as has now been proven. And through A let the line AB be drawn towards Capricorn, and let AB be extended from A towards Capricorn until its length be 1800 of the units of which AC was 14,140 in Chapter 42, and AE, AD, 154,400; and let B be the center of the earth's orb. Now because BA is directed towards 5\u00bd\u00b0 Cancer, and AE towards 15\u00bd\u00b0 Scorpio, the angle EAB is about 50\u00b0 and acute, and EBA obtuse. Hence, EA is longer than EB. Likewise, since BA is directed towards 5\u00bd\u00b0 Cancer, while AD is directed towards 12\u00bd\u00b0 Gemini, BAD is therefore 157\u00b0 and ABD is quite acute. Hence, AD, or AE which is equal to it, is shorter than BD. Therefore BE is much shorter than BD, and the difference is quite perceptible. For who are we to neglect AB, which is 1800 or even more, who could not tolerate observations with a mere 200 units' error? Hence, regions on opposite semicircles of the eccentric that are equidistant from G, such as E and D, are not equidistant from any points not at the center other than those on the line CA that goes through the body of the sun. ","1145":" You may reply, however, that if BC is connected and extended, a new apsis is created where that line intersects the circle, and the point D is closer to that apsis than is the point E. So is it any wonder that BD is also longer? I answer that whatever lines are drawn, AE and AD always stay the same, since they are proven from the observations, in all three forms of hypotheses, and thus absolutely nothing in this derivation was assumed that could be subject to controversy. And so, with AE and AD remaining the same, let BC be drawn exactly as is objected against me. Nevertheless, that line BC by no means gives rise to a hypothesis that will fit the acronychal2 observations, as I proved in Chapter 6. Instead, to save the acronychal positions, one needs to substitute for BC a line FH through F parallel to CB, passing through F and H, the centers of uniform motion for Mars and the sun. But when this is done, the center of the eccentric is at the same time transferred from C to I, and there is more than a semicircle on the side of E, and less on the side of D. Nor are AE and AD left unchanged, but AE is lengthened, and AD is shortened. And since these lines are altered, the observations at positions other than acronychal will never be saved, since they give evidence that AE and AD are equal. I don't think there is any need for computation. Nevertheless, if there is anyone who enjoys this labor (even though no astronomer should try anything with numbers whose foundations he has not previously seen in geometry, and geometry has just overturned the foundations of such an undertaking), he has an example of it in Chapter 24 above. There, I computed the distances of earth from H, the point of uniformity of the earth's motion, and the distance of Mars from the same point H, simultaneously in a single operation, using the same observations by which I afterwards computed the distances of earth and Mars from the center of the sun A, in Chapter 26.","1146":"For the peculiar cleverness of the method I have used is this: that it shows that whatever point in the plane of the earth's circle is chosen that has, with respect to the sun's body, a position described and determined through a number of observations both in zodiacal longitude and in distance from the sun, also shows the distance of earth and Mars from that chosen point; and it does these things without any knowledge of the equated anomaly on the eccentric corresponding to that point. In fact, the only reason why I used that knowledge in Chapter 26 was that it is a short cut. ","1147":"But in addition, there is another way to argue the point. It was proved in Chapter 44 above that the planet's orbit is not a circle but an oval, such that the diameter on it which is called the [line] of apsides is the longest. Just now, in Chapter 51, it was proved that regions that are equally removed from the point of the aphelion G also make an equal incursion at the sides. There is thus a real oval situated about the line AC, and therefore, it is not situated about the line FH. And one who would compute the various distances of Mars from the point H by the method just recommended will find a great irregularity in the distances, incapable of being included by any means in a circle or in any other possible figure set up about FH.","1148":"So again the faith that was pledged in Chapter 6 and in many other places in this work, I have redeemed from all tincture of circular reasoning, and have shown that the eccentric of Mars cannot be referred to anything but the sun itself; and that, in consequence, it is not only reason that stands with me, but the observations themselves, in my releasing the observations of Mars from the sun's mean motion and measuring them out by the apparent motion of the sun.","1149":"Chapter 53","1150":"Another method of exploring the distances of Mars from the sun, using severed contiguous observations before and after acronychal position: wherein the eccentric positions are also explored at the same time.","1151":"Since we are establishing new hypotheses here, in that we are enquiring into the natural cause of the eccentric equations, it is appropriate that we should explore everything as carefully as possible, lest in neglecting the foundations we build upon them a building doomed to ruin. And so it furthers us to explore this same thing (perfectly accurate distances of Mars from the sun) by many methods. Let \u03b1 be the sun, \u03b2 the position of earth before opposition of Mars to the sun, and \u03b1\u03b2\u03b4 the angle of vision, or the arc of the elongation of \u03b4 from the sun. Likewise, let \u03b3 be the position of the earth after opposition, and \u03b1\u03b3\u03b4 the angle of vision. Thus, at the first time, let the planet be on the line \u03b2\u03b4, and at the second, on the line \u03b3\u03b4, and let it actually traverse \u03b8\u03b7. So, when the time of the two observations is given, the angle \u03b8\u03b1\u03b7 will be given precisely enough by the vicarious hypothesis, for whatever eccentric position. If the pair of times are not far from each other, or if the planet is near the apsides or the middle elongations, the difference in length of the lines \u03b1\u03b8, \u03b1\u03b7 will also be known tolerably. And we have included this much among our presuppositions only in order that there be no remaining difficulty here.","1152":" And so, in dealing with the angles \u03b8\u03b2\u03b1, \u03b7\u03b3\u03b1, given by observation, and \u03b2\u03b1, \u03b3\u03b1, which are known from Part III, if we were to assume [a value for] \u03b8\u03b1, and consequently also \u03b7\u03b1, it is obvious that if this assumed value were longer than it should be, such as \u03ba\u03b1, \u03b9\u03b1, then the angle \u03b9\u03b1\u03ba would come out less than it should be; and if, on the other hand, it were shorter than it should be, such as \u03b6\u03b1, \u03b5\u03b1, the angle \u03b5\u03b1\u03b6 would come out greater than it should be. So we must assume such distances as will make the angle of motion on the eccentric come out right.","1153":"Any possible remaining error in the eccentric position will also occur in the same way. For let it be that \u03b8\u03b1, \u03b7\u03b1 are in their correct positions, and from that point let \u03b8\u03b1 be carried forward, in error, through the angle \u03b8\u03b1\u03b4. And let \u03b7\u03b1 likewise be carried forward, through the equal angle \u03b7\u03b1\u03b5. You see that \u03b1\u03b4, substituted for \u03b1\u03b8, is going to be very much too long, and \u03b1\u03b5, following \u03b1\u03b7, will be quite short, contrary to what, by hypothesis, is known at the start. Furthermore, the angle \u03b3\u03b1\u03b2 ought not to be as small as possible, so that no error of observation, or at least no minimal one, in opposite directions in the sky (as can happen) could have any great effect. Now, with the help of this method, we must go through the years 1582 in Cancer, 1585 in Leo, 1587 in Virgo, 1589 in Scorpio, 1591 in Sagittarius, 1593 in Pisces, and 1595 in Taurus. For sufficient observations are at hand in all places.","1154":"If it seems a good idea to investigate demonstratively the elongation of the earth from the line through the sun and the planet at which any error in the distance of Mars from the sun would be most evidently perceived, let Chapter 6 be consulted. For, following that chapter, we shall define it as that angle at the sun whose sine has to the radius about the same ratio as the excess of Mars's distance from the sun over the sine of the complement of the angle has to the distance itself.","1155":" For let \u03b1 be the sun, \u03b8 the planet, \u03bd\u03be the earth's orb. From \u03b8 let the straight line \u03b8\u03bc be drawn perpendicular to \u03b8\u03b1. And on \u03b8\u03bc let a number of centers be chosen, about which let circles through \u03b8 be described, until one of them be tangent to the earth's orb at \u03bd. The point \u03bd will be where the defect of \u03b1\u03b8 at \u03b8 appears most evidently, that is, where it subtends the greatest angle. 3 From \u03bd let \u03bd\u03bf be drawn parallel to \u03bc\u03b8, intersecting \u03b1\u03b8 at \u03bf. I say that \u03bf\u03bd is to \u03bd\u03b1 as \u03bf\u03b8 is to \u03b8\u03b1. For as \u03bd\u03bc, or (which is the same thing) \u03b8\u03bc, is to \u03bc\u03b1, so is \u03bf\u03bd to \u03bd\u03b1. But \u03bd\u03bc is to \u03bc\u03b1 as \u03bf\u03b8, and (very nearly) \u03be\u03b8, is to \u03b8\u03b1. Therefore, etc.","1156":"Let \u03b1\u03b8 be 161,000. Thus \u03be\u03b8 will be nearly 61,000. And as 161 is to 61, so is 100,000 to 37,888. This, taken as a sine, shows the angle \u03bd\u03b1\u03b8 to be 22\u00b0 15\u2019, and greater, if instead of \u03be\u03b8 you take \u03bf\u03b8.","1157":"So, many days, nearly 45, pass before the anomaly of relative motion is altered by 22\u00bc degrees. And before or after this time, \u03b1\u03b8 is much different. So at aphelion this angle of relative motion is about 28\u00b0, and at perihelion about 18\u2153\u00b0.","1158":"And now, having found the termini at which any error that may arise will be most evident, owing to an incorrect distance of Mars from the sun, it is easy for us to choose suitable observations, since many are available.","1159":"We shall begin from the opposition of 1582, from which year we shall choose the following observations. 5","1160":"6","1161":"7","1162":"8","1163":"The two intermediate ones differ by 4240. And indeed, the later one \u03b1\u03b7, is shorter, although it should have been longer by 336. So the sum of the two is 322,054. From this I subtract 336, and again add it. The halves of these are 160,859, which is \u03b1\u03b8, and 161,363, which is \u03b1\u03b7. And \u03b1\u03b8 will be at 16\u00b0 5' Cancer, and \u03b1\u03b7 at 17\u00b0 55' Cancer. So here, the vicarious hypothesis would lose 1\u00bd minutes.","1164":"But the distances themselves are not to be trusted, owing to the angle's being too small. For if the angle at \u03b4 be varied by one minute, through an error in observing, as easily happens, we shall be in error by a thousand units in either distance.","1165":"Therefore, let the two more remote ones be taken, which are found to differ by 5236. But we already know that they should differ by about 5570. So by an operation conducted as before, the more nearly correct values resulting are: \u03b1\u03b8 158,792, and \u03b1\u03b7 164,364, placing \u03b1\u03b8 at 0\u00b0 41' 0\" Cancer, and \u03b1\u03b7 at 0\u00b0 8' 30\" Leo. And it becomes certain, through observations on the four days at this position, that about 1\u00bd minutes must be subtracted from the eccentric positions derived from our vicarious hypothesis.","1166":"The distances found before are approximately confirmed as well, both before and beyond opposition, which turn out to have a magnitude between these. Unless, as the comparison indicates, they ought to be somewhat longer. ","1167":"But at the same time it is clear that if the angle \u03b8\u03b4\u03b7 had been off by one minute, both distances would have been off by about 50 units, no more. So in these distances there can barely be an error of the hundredth part of the uncertainty that there was in the previous ones.","1168":"Now, if a longitude that was taken up expresses satisfactorily the observed values for the distances for these four days, it will also express the observed values for the intervening days, namely, November 25, 26, and 27, and December 3, 17, 27, 28, and 29 of 1582, and January 16, 17, 18, 19, 21, and 22 of 1583.","1169":"Let us proceed to the opposition of 1585. For while the sun and Mars were at opposition on January 31 of that year, the planet was observed at many closely-spaced positions over the two months preceding and the same number following. From among them we shall take these four observations. ","1170":"The two intermediate ones differ by 118. They should have differed by 187 in the opposite sense, so that \u03b1\u03b8 would be 166,226 and \u03b1\u03b7 166,412. Therefore, \u03b1\u03b8 falls at 18\u00b0 48' 47\" Leo, and \u03b1\u03b7 at 23\u00b0 34' 48\" Leo. And the contemptibly small alteration of the eccentric position confirms the vicarious hypothesis for this place. But we learn from this that an error of one minute in observation at this place would vitiate the two distances by about 100 units.","1171":"When the more remote ones are consulted, their difference is found to be 1022. From what is known already approximately from the hypothesis, the difference should have been greater, namely, 1275. And, in fact, the fourth [degree] of Leo is close to the eighteenth of Cancer, where previously something had to be subtracted from the eccentric position of the vicarious hypothesis. So, if you will subtract one minute at the fourth of Leo, you will now make \u03b1\u03b8 a hundred units shorter, and if 2\u00bd\u2019, you will make it about 164,934, which is short enough that \u03b1\u03b7 can also keep the length 166,206; and the last observation in the previous year 1583, which showed a length of 164,364, can be reconciled with it. For they should have differed by 488, a certain enough value provided in advance by the hypothesis of the distances, while they do differ by 570.","1172":"Furthermore, it is possible to transfer half of this 2\u00bd\u2019 change in the eccentric position to the observations. For if either of them has erred by one minute, that will be able to effect an error of 50 units in either distance.","1173":"It would be tedious to repeat the same method, using the same words, for all the years of the oppositions. And so, in the following table, I have placed the observations themselves which I have consulted, and added what resulted from the computations. The hypotheses underlying the calculations are these. The sun's position is taken from Brahe. The sun-earth distance is from Chapter 30. The aphelion of Mars for the end of 160014 is 29\u00b0 0\u2154\u2019 Leo. The mean motion at the same time is 10s 7\u00b0 14\u2019 34\". The eccentricity and ratio of the orbs is as in Chapter 54. To this I have added the distances of Mars from the sun as if previously known. So if, using these distances, we match the proposed observations, these distances will be the correct ones, which is what I proposed to show in this chapter.","1174":" These, then, are the distances that will result from an investigation using the method of this chapter from the observations set out here The apparent positions, on the other hand, when Mars's eccentric position is in Cancer, will come out about 4 minutes back from these, and in Sagittarius and Capricorn the same number of minutes forward. These small errors do not come from incorrect distances, for they would then be in opposite senses on opposite sides [of opposition], and not in the same sense. I believe they can be reconciled by changing the sun's apogee by one degree, which is easily permitted by Brahe's observations. Nevertheless, I am not going to say anything definite at present. For the correction of both this apogee and the entire hypothesis is reserved for the book of Tables. ","1175":"Chapter 54","1176":"A more accurate examination of the ratio of the orbs.","1177":" In Chapter 42, we did actually establish the ratio of the orbs from observations at positions other than acronychal, but they were not ones that were in agreement with one another entirely and to our full satisfaction. Moreover, considered in itself, regardless of whether the most exact observations be available, the procedure is incapable of being brought to a certitude of 100 units. So it has to be done by polling and counting the votes. And in Chapter 28, at a mean anomaly of 11\u00b0 37', which, after the correction of Chapter 53 preceding, becomes 11\u00b0 52', the distance of the point on the ecliptic to which a perpendicular dropped from the body of Mars descends, was found to be 166,180, or 166,208. And therefore, since this position is 23\u00b0 from the northern limit, the inclination will be about 1\u00b0 43', and excess of the secant will be 45 units, which will be about 70 in our dimensions. Therefore, the distance of Mars from the sun will be 166,250 or 166,278.","1178":"We shall now also compare the observations of Chapter 51, so as to be supported by an middling consensus. In 1586, with 10\u00b0 9' 41\" of mean anomaly remaining, or 9\u00b0 54' 41\" after correction, we found 166,311. But by subtracting 1\u00bd\u2019 from the position given by the vicarious hypothesis, we found 166,208. So for a subtraction of 3 minutes less than two degrees, about 95 should be subtracted, making it 166,113. For the latitude, 80 must again be added, making it 166,193. Thus in 1588, when the remaining [mean] anomaly was 8\u00b0 2' 51\", or 7\u00b0 47' 51\" corrected, by a subtraction of 1\u00bd\u2019 from the position given by the vicarious hypothesis we found the distance to be 166,396. Thus, a position 4\u00b0 4' lower will be shorter by about 102, making it 166,294. And, corrected for latitude, 166,284. This was previously found to be 166,193, from 1586. The mean is 166,238. In the descending semicircle, however, from 5 observations, we had found 166,250 or 166,278. So, although the difference is imperceptible, let us nevertheless take the mean, 166,260, giving more trust to the descending semicircle, as it is better confirmed by the observations.","1179":"Let it thus be [taken as] certain that at a mean anomaly of 11\u00b0 52' the distance is 166,260. Hence, however great a hypothetical value you may conceive by some rough method, which is to be confirmed shortly thereafter, it follows that where the radius is 100,000, the distances at aphelion cannot increase more than 164 units, and even less if you use the hypothesis of a perfect circle. But those units converted through a preconceived ratio of the orbs, as it is set up in Chapter 42, add about 250, and these added to 166,260 make 166,510. But above, in Chapter 42, we found 166,780, using weaker observations. The difference is 270 units.","1180":"We shall also treat likewise the perihelial distance which in Chapter 42 was found to be 138,500, from observations that were not solid enough.","1181":" Just now, in Chapter 51, at a remaining [mean] anomaly of 161\u00b0 45\u00bd\u2019, or 161\u00b0 30\u00bd' after correction, we found the distance, before correction for latitude, to be 139,000 or 138,984. So let 139,000 be at 21 Pisces. Since this position is 35 degrees from the limit, the inclination is therefore 1\u00b0 31\u00bd\u2019 The excess of the secant will be 35\u00bd, which is equivalent to 49 of our units. And so the true distance of Mars from the sun is 139,049. But if the radius is 100,000, the perihelial distance is 5754 units shorter than that at an anomaly of 161\u00bd\u00b0, which becomes 876 of our units, or less, if you use a perfect circle. And when these are subtracted from 139,049, there remains the perihelial distance of 138,173. The difference from the value 138,500, found in Chapter 42, is 327.","1182":"So, according to this method, these distances are found: ","1183":"And where 152,342 becomes 100,000, 14,169 becomes 9301.","1184":"Nevertheless, because our observations, especially at perigee, do not bear out that great a difference, and since it can happen that the vicarious hypothesis, since it is false, also might introduce some falsity into the eccentricity, let all the votes be tallied before the result is announced.","1185":"And so we shall adapt the aphelial distance found here, 166,510, to the eccentricity of Chapter 42, which was 9265. And as 109,265 is to 90,735, so is 166,510 to 138,274, where the radius is almost exactly 152,400.","1186":"Also, manifold experience has shown that the eccentricity that is most true and best fitted to the physical equations is between 9230 and 9300; that is, the eccentricity of Chapter 42, which is 9265.","1187":"Therefore, that we might not unduly abandon the perihelial distance found in this chapter, which is 138,173, nor unduly trust the aphelial distance of 166,510, let us conclude that the truest aphelial is 166,465, and the perihelial, 138,234, where the radius is 152,350.","1188":"Chapter 55","1189":"From the observations of Chapters 51 and 53, and the ratio of the orbs of Chapter 54, it is demonstrated that the hypothesis seized upon in Chapter 45 is in error, and makes the distances at the middle elongations shorter than they should be.","1190":"Indeed, I began to say this in Chapter 51. But since more observations, and more suitable ones, had to be provided to give evidence in Chapter 53, from which at the same time something else was also inferred in Chapter 52, the full demonstration was therefore postponed to this point.","1191":"There is no need for verbosity. At the mean anomalies of all the examples appearing in Chapter 51 and 52 let the distances be computed according to the hypothesis of Chapter 45 and the ratio of orbs of Chapter 54, by the method I used from Chapter 46 through Chapter 50, and let them be compared to the distances of Chapter 51 and 53, found using infallible observations. It will be apparent that the more we descend from the apsides, the more the computed distances fall short of the observed distances, a result quite the opposite of what we saw in Chapter 44 above. For there, the distances computed according to the law of the circle were longer than the observed distances at the middle elongations, while here, the distances resulting from the hypothesis that makes the planet's orbit oval are shorter. It is therefore obvious that the planet's path is neither a circle nor such as to make as great an incursion from the circle at the sides as does the oval that arose from the opinion of Chapter 45 and was described in Chapter 46; but takes a middle course. And if, in turn, using the distances of Chapter 45, you compute the observed positions of Mars, especially those which, in Chapter 53, stood at some distance on either side of opposition, the planet before opposition will fall too far forward, and after opposition, too far back.","1192":"This is most evident in the descending semicircle in 1589 and 1591, and in the ascending semicircle in 1582 and 1595. For in those places, the oval of Chapter 45 is 660 units1 too small, while the perfect circle is too large by the same amount, and this can have an effect upon the appearances of 20 minutes and more. Thus, David Fabricius2 was able to use his observations to charge my hypothesis of Chapter 45, which I had communicated to him as true, with this error of having distances that are too short at the middle elongations, writing at the very time when I was laboring to seek out the true hypothesis with renewed care. He was, in fact, quite close to arriving at the truth before me. And since the perfect circle errs the same amount in the opposite direction, we argue rightly from this that the truth is in the middle, between the two.","1193":"Moreover, the equations computed from physical causes in Chapters 49 and 50 gave the same testimony, namely, that the lunule cut off from the perfect semicircle ought to have only half the breadth of the one which the opinion of Chapter 45 cuts off. Therefore, nothing prevents our saying that the matter is most certainly demonstrated: that the opinion of Chapter 45, in remedying the excess of the perfect circle, falls into the opposite defect.","1194":"So the physical causes of Chapter 45 go up in smoke.","1195":"Chapter 56","1196":"Demonstration from the observations already introduced, that the distances of Mars from the sun are to be chosen as if from the diameter of the epicycle.","1197":"The breadth of the lunule of Chapter 46 above, which the opinion of Chapter 45, which instructed us to cut it off from the semicircle, has produced for us\u2014this breadth, I say, was found to be 858 units, of which the semidiameter of the circle is 100,000. But then, by two arguments, which I have already presented in Chapters 49, 50, and 55, I concluded plainly that the breadth of the lunule is to be taken as only half that, namely, 429, or more correctly, 432, and in units of which the semidiameter of Mars is 152,350, nearly 660. I therefore began to think of the causes and the manner by which a lunule of such a breadth might be cut off. ","1198":"While anxiously turning this thought over in my mind, reflecting that absolutely nothing was articulated by Chapter 45, and consequently my triumph over Mars was futile, quite by chance I hit upon the secant of the angle 5\u00b0 18', which is the measure of the greatest optical equation. And when I saw that this was 100,429, it was as if I were awakened from sleep to see a new light, and I began to reason thus. At the middle elongations the optical part of the equation becomes a maximum. At the middle elongations the lunule or shortening of the distances is greatest, and has the same magnitude as the excess of the secant of the greatest optical equation 100,429 over the radius 100,000. Therefore, if the radius is substituted for the secant at the middle elongation, this accomplishes what the observations suggest. And, in the diagram in Chapter 40, have concluded generally that if you use HR instead of HA, VR instead of VA, and substitute EB for EA, and so on for all of them, the effect on the rest of the eccentric positions will be the same as what was done here at the middle elongations. And by equivalence, in the small diagram of Chapter 39, \u03b1\u03ba will be taken instead of the lines \u03b1\u03b4 or \u03b1\u03b9, and \u03b1\u03bc for \u03b1\u03b5 or \u03b1\u03bb.","1199":"And so the reader should peruse Chapter 39 again. He will find that what the observations additionally testify here was already urged there, from natural causes, namely, that it appears reasonable that the planet perform some sort of reciprocation, moving on the diameter, as if of an epicycle, that is always directed towards the sun. He will also find that there is nothing more at odds with this notion than this: that when we proposed to represent a perfect circle, we were forced to make the highest parts \u03b3\u03b9 of the reciprocation unequal to the lowest \u03bb\u03b6, which parts correspond to equal arcs on the eccentric, the highest being short, and the lowest long. So, now that the planet's circular path is denied, and \u03ba\u03b1, \u03bc\u03b1 are taken instead of \u03b4\u03b1, \u03b5\u03b1, that is, instead of \u03b9\u03b1, \u03bb\u03b1, as was said, it follows further that those parts of the reciprocation, such as \u03b3\u03ba, \u03bc\u03b6, are equal. And that which had tormented us for a long time in Chapter 39 now surrenders to us in the face of the proof of the truth we have perceived. ","1200":" As for the middle parts \u03ba\u03bc still being larger than the extremes \u03b3\u03ba, \u03bc\u03b6, it will","1201":"be said in Chapter 57 following that this is in accord with nature, contrary to what we had been able to understand in Chapter 39.","1202":"But in addition, the difficulty that arose in Chapter 39 through supposing that the increase of the sun's [apparent] diameter serve the planet as an index for its approaching and receding, now vanishes entirely, as will appear in Chapter 57.","1203":"Thus, concerning the eccentric anomaly of 90\u00b0, I easily was able to see in the manner just mentioned, that instead of the distance EA of the perfect circle, EB is to be taken, corresponding to an equated anomaly EAB.","1204":"And although I have drawn a general conclusion concerning all the anomalies using a single one as an example, this was not a consequence just of that one anomaly: there was need to strengthen it using closely spaced observations.","1205":"So now you understand the special capacity in which the observations of Chapters 51 and 53 are appointed to serve us, namely, to give this evidence.","1206":"Come, then: let the eccentric anomalies CBG, CBH be computed at the equated anomalies set out in those chapters, that is, at the angles CAG, CAH, and so on. Nor is there any need to strive after minute parts, nor be concerned about the imperfection of the eccentric equations that still remain in Chapter 19, 29, 43, 47, 48, 49, and 50. Use any of these methods, particularly the one in Chapter 43. You will not err in the equations by more than eight minutes.","1207":"When the angles are set up, seek out the lines, HR corresponding to the equated angle HAC, RV corresponding to the equated VAC, and so on for the others, and transpose them to the dimension of the orbs found in Chapter 54. You will find them as in the following table. ","1208":"From the observations of Chapter 51","1209":"On the descending","1210":"semicircle","1211":"Computed from the reciproca","1212":"tion","1213":"162,994","1214":"163,051","1215":"158,091","1216":"158,111","1217":"148,000","1218":"148,050","1219":"In the observations of Chapter 53, there is no need to do the same thing. For I previously used this same method of reciprocation to find out the distances of Mars from the sun which I called upon in order to compute Mars's apparent positions. And since the observations were represented by these, they will therefore be correct. ","1220":"As you see, therefore, the distances measured on the diameter, found a priori in Chapter 39, are confirmed by closely spaced and very reliable observations throughout the entire perimeter of the eccentric.","1221":"Chapter 57","1222":"By what natural principles the planet may be made to reciprocate as if on the diameter of an epicycle.","1223":" It is clear, then, from the most reliable observations, that the course of the planet through the aethereal air is not a circle, but an oval figure, and that it reciprocates on the diameter of a small circle in the following manner. Suppose that, after describing equal arcs on the eccentric, the planet comes to be at the diametral distances \u03b3\u03b1, \u03ba\u03b1, \u03bc\u03b1, \u03b6\u03b1, instead of the circumferential distances \u03b3\u03b1, \u03b4\u03b1, \u03b5\u03b1, \u03b6\u03b1 (that is, \u03b3\u03b1, \u03b9\u03b1, \u03bb\u03b1, \u03b6\u03b1), upon which the perfect circle lies. It is clear from inspection that a lunule is cut off from the perfect semicircle of the eccentric, whose breadth at any point is equal to the differences between the two diverse distances, such as \u03b9\u03ba, \u03bb\u03bc. This is proposed not on the basis of arguments a priori, but of observations, as I have just said; so now the physical theories will proceed more correctly than they had hitherto.*** For it is not by any ratiocinative or mental process that a planetary mind assigns the equal parts of the reciprocation \u03b3\u03ba, \u03ba\u03bc, \u03bc\u03b6, to equal arcs CD, DE, EF, of the as yet untraversed eccentric, for the former are not equal. Instead, the reciprocation is coordinated with the space traversed on the eccentric in a natural way, which depends not upon the equality of the angles DBC, EBD, FBE, but upon the strength**1 of the ever increasing angle DBC, EBC, FBC, which strength approximates the sine (so called by the geometers). The ascent\u2019s being changed into descent thereby gradually, by a continuous diminution, is more probably than if the planet were said suddenly to turn its prow in the other direction \u2013 which we indeed said in Chapter 39 clearly conflicts with observational results. And since the measure of this reciprocation points the finger at a natural mode, its cause will also be natural; that is, it will be some natural \u2013 or better, corporeal \u2013 faculty, and not a planetary mind.","1224":"Also in Chapter 39, for the best reasons, one of our suppositions was that a planet cannot make a transition from place to place by the bare effort of its inherent forces unless these be assisted or directed by an extrinsic force. This being the case, we must consider whether we should also carry over this reciprocation in part to the solar power itself. In our exertions to this end, we shall be obliged once again to take up our oars which were introduced in Chapter 39. Let there be a circular river CDE, FGH, and in it a sailor who revolves his oar once in twice the periodic time of the planet, by an inherent and perfectly uniform force. Thus at C let the line of the oar be at right angles to the line from the sun, and let it direct now the bow and now the stern forward at alternate returns. At F, however, let the line of the oar be part of the line from the sun, and at other positions let it have an intermediate inclination. Now the stream, flowing down upon the oar at DE, will push the ship3 down towards A, while from C it will push very little, since the ship is also but slightly inclined. The same is true at F, because at this moment the stream strikes the oar directly. At D and E, however, it pushes down more strongly, because here the oar is greatly disposed to such an approach by its inclination. The opposite happens in the ascending semicircle. For the river, coming beneath the oar at G and H, drives it away from the sun. ","1225":"At the same time it will also happen, other things being equal, that the impulse is less at C than at F, since our river is weak at C and strong at F.","1226":"And this is is also in accordance with our wishes, since our reciprocation was following equal spaces on the eccentric, and the planet spends longer in the upper ones than in the lower.","1227":"This example shows only the possibility of this arrangement. In itself it is rather inadequate, since the rotations of the oar and the river are accomplished, not in the same time, but a double time. Furthermore, to those looking at them from earth, the faces of the planets should appear to change, while the face of the moon, although it participates with the planets in that motion which we are discussing, does not change over the course of a month. Instead, it always is turned towards the earth, whence its eccentricity is reckoned. In addition, while the force of a river is material (for its water acts by its weight and material impetus), the force of the sun is immaterial. Therefore, the comparison with the planets ought to be different: they will need no oar, no physical instrument, for catching hold of the force of some weighty thing (for that motive species of the sun has no weight). Nor do we deem it fitting that the stars have a corporeal oar, seeing that we hold them to be round.\n But from this very refutation, there comes another example, which will perhaps be more suitable. The river and the oar are of the same quality. The river is an immaterial species of magnetic power in the sun. So why not have the oar too borrow something from the magnet? What if all the bodies of the planets are enormous round magnets? Of the earth (one of the planets, for Copernicus) there is no doubt. William Gilbert has proved it.\n But to describe this power more plainly, the planet\u2019s globe has two poles, of which one will seek out the sun, and the other will flee the sun. So let us imagine an axis of this sort as a magnetic strip, and let its point seek the sun. But despite its sunseeking magnetic nature, let it remain ever parallel to itself in the translational motion of the globe, except to the extent that over the ages it transfers the polar inclination from one of the fixed stars to another, thus causing the progressive motion of the aphelion. I nevertheless admit the possibility that a mind may be needed for both of these, of such a nature as to be adequately instructed by the animate faculty for performing this motion. For this is a motion, not of the entire body from place to place (which motion, in Chapter 39 above, was rightly denied to a motive cause inherent in the planets), but of the parts about the center of the whole, as if at rest. \nHere again, in the globe of the earth there is an example of this directional property of the axis, from Copernicus. For as long as the axis of the earth, in the annual circulation of its center, remains almost perfectly equidistant from itself in all its positions, summer and winter are brought about.* On the other hand, insofar as very long ages cause it to incline, the fixed stars are thought to move forward, and the equinoxes to retrogress.\nWhy, then, should we have doubts about attributing to all the planets, in order to save the phenomena of eccentricity, something that is observed to be in one of them (that is, the earth) because of the phenomena of the precession of the equinoxes and the sun's annual cycle of rising and falling?\nCopernicus, for example, was deceived here when he thought that he needed a special principle to cause the earth to reciprocate annually from north to south and back so as to produce summer and winter, and to bring about the equality of the tropical and sidereal years (to the extent that they are equal) by constructing it in a manner commensurate with its revolution. For all those effects are obtained by having the earth\u2019s axis, about which the diurnal motion is made, retain a single, constant direction: there is no need for extrinsic causes, except to account only for the extremely slow precession of the equinoxes. And so here, too, there is on no account any need for movers for the planet, which would carry its body about the sun in a parallel position, and at the same time perform the reciprocation. For the one will naturally depend upon the other. The only thing remaining to be considered is the extremely slow progression of the aphelia.","1228":"To continue: when the strip is at C and F, there is no reason why the planet should approach or recede, since it holds its ends at equal distances from the sun, and would undoubtedly turn its point towards the sun if it were allowed to do so by the force that holds its axis straight and parallel. When the planet moves away from C, the point approaches the sun perceptibly, and the tail end recedes. Therefore, the globe begins perceptibly to navigate towards the sun. After F, the tail end perceptibly approaches, and the head end recedes from the sun. Therefore, by a natural aversion, the whole globe perceptibly flees the sun. And when it is opposite A, where the length of the axis is pointed directly at the sun, its approach in the former situation, or its flight in the latter, is strongest. Furthermore, our earlier presuppositions derived from the observations postulated this. For there, of the parts of the reciprocation \u03b3\u03ba, \u03ba\u03bc, \u03bc\u03b6 which correspond to equal arcs on the eccentric, the parts at the middle, such as \u03ba\u03bc, were longest, and those near \u03b3 and \u03b6 were short.","1229":" But it is also consistent that the observations would have \u03b3\u03ba, \u03bc\u03b6 equal, although their arcs \u03b3\u03b4, \u03b5\u03b6, or better, CD, EF on the eccentric, though equal, are traversed in unequal times, longer for CD, so that the part of the reciprocation \u03b3\u03ba is traversed in a longer time than \u03bc\u03b6 which is equal to it. For similarly, magnets approach one another more slowly when they are at a greater interval, and more swiftly and more quickly at a shorter interval.","1230":"In fact, we can transfer that force which keeps the magnetic axis in a parallel position, and does not allow it to remain pointed towards the sun, from the attention of a mind, to which we had entrusted it a little earlier, to the functions of nature. It appears to be an objection to this, that nature always acts in one and the same way, while this retentive force appears to make its exertions differently at different times. This is seen, for example, in the tendency of the axis to incline towards the sun, for the impeding of which the retentive force is ordained, which tendency is evanescent at the middle elongations but most strongly evident at aphelion and perihelion. Nevertheless, what is there to prevent this force of retention's being in many places stronger than the tendency to incline towards the sun, so that the force is either not at all or but little wearied by such a weak adversary? Let us again take an example from the magnet. In it are manifestly mingled two powers, one of directing it towards the pole, and the other of seeking iron. Thus if a strip or nautical needle be directed towards the pole, while some iron approach from the side, the needle quickly declines from the pole and inclines towards the iron, thus indulging somewhat in its intimacy with the iron, but in such a way that it gives most of it to the pole. Indeed, Gilbert thinks this to be the reason why a strip declines from the pole towards the continents of greatest magnitude, the cause of this declination thus lying in the tracts of land, being greater and having a more vigorous power in the vicinity to the extent that they are higher on the right or left.","1231":"Therefore, we can ascribe the same tasks and a uniform action to both natural faculties, and by the interplay of the two we can show a cause for the translation of the aphelia which will be neither obscure, nor, by Hercules, imaginary. For suppose that this force of directing the axis towards the sun does detract somewhat from the retentive power, commensurate with the ratio of the two. Accordingly, in the aphelial semicircle, as at C, the point will gradually incline towards H (that is, backward), and the tail end will turn away from the sun, gradually overcoming the retentive force. Thus the aphelion will become retrograde. But in the perihelial semicircle, as at F, the same point will incline towards G (that is, forward), again overcoming the contrary retentive force. Thus the aphelion will then be made to move forward, and to be fast. But because AF is shorter than AC, and the sun is closer to F than to C, the force tending to turn the magnetic axis towards the sun is therefore stronger at F than at C. Thus more will be detracted from the retentive [force] at F than at C. So the perihelial forward inclination not only compensates for the aphelial backward inclination, but even overcomes it. And so the reason is clear why the apsides progress, and do not retrogress. Thus the aphelion we have found will have that value only at an equated anomaly of 90\u00b0 and 270\u00b0 when the axis of power is directed straight at the sun, which is its correct place. And the motion of the aphelion will be spiral, as will become clear below in Chapter 68 for the motion of precession of the equinoxes also, which exists through another cause. So the direction of the magnetic axis in its parallel position, or the force which is its custodian, will not respect one or another of the fixed stars, but only the position of its body, as it is at any particular time. And, to think the matter through simply: because this direction is more like rest than motion, it is more appropriately sought in the material, and in the disposition of the body, than in some mind. ","1232":" But come: let us track more closely this similarity of the planetary reciprocation to the motion of a magnet, and that by a most beautiful geometrical demonstration, so that it may be clear that magnets have such a motion as that which we perceive in the planet. Let DFA be either a round magnet or the body of Mars, DA the line along which the magnetic power is oriented, D the pole that seeks the sun, A the pole that flees the sun. You will note, first, that in this theory it is all the same whether we consider the entire globe of the magnetic body, or one single physical line of its power, parallel to DA.","1233":"For this magnetic power is corporeal, and divisible with the body, as the Englishman Gilbert, B. Porta, and others, have proved, surely because its globe consists of an infinite number of physical lines, as it were, parallel to DA, whose power is extended in a straight line and in one direction in the world. Therefore, the judgements made about individual parts with respect to the quality of their motion will be the same as those concerning them all joined together, and vice versa. So let the central axis DA be proposed for theorizing, in place of the whole body and all its filaments. Let DA be bisected at B, and let FBI be drawn perpendicular to DA. Thus, when the planet is so positioned that BI points toward the center of the sun, there will be no [tendency to] approach. For the angles DBI, ABI are equal, and thus have equal strength, the former for approaching, and the latter for receding. So this is like an equipoise in mechanics. Under these conditions, the center of Mars B is on the apsides, at aphelion, say, at its greatest distance from the sun. Now let some arc IC be taken, measuring the angle of equated anomaly, and let BC be drawn and extended to K. And let the planet be so situated that BC points towards the sun, which is understood to be indicated by K. The first thing to be sought is the measure of the strength of the planet's approach. Now the approach occurs because the seeking pole D is inclined towards the sun K at the angle DBK, while the fleeing pole A is turned away at the angle ABK. And since the strength of this angle is natural, it will follow the same ratio as the balance. But when a line CP is drawn from C perpendicular to DA, between DP and PA there will be the ratio of the balance. For if a pair of scales is suspended from the balance support KB, and the arms come to rest at the angle DBK, the weight of the arm BD will be to the weight of the arm BA as DP is to PA, just as, if the balance arms were suspended from CP at P, and the weight of the arm BA were applied to PD while the weight of BD were applied to PA, then DA would be at right angles to the hanging balance support CP. See my Optics, and do not be easily swayed by insufficiently careful experimentation. Therefore, as DP is to PA, so is the strength of angle ABC to the strength of angle DBC. Thus DP here measures the force of fleeing, and PA the seeking force. From PA subtract a magnitude equal to DP, and let this be AS. Therefore, SP is the measure of the seeking power alone, with the impediment of fleeing [power] subtracted, and it will be in the same proportion in which AD measures the single greatest force. But where the half DB measures the greatest force, the half of PS, which is PB, or the sine CN of the equated anomaly CBI, measures the net force of the planet's approach towards the sun at this location. So the sine of the equated anomaly is the measure of the strength of the planet's approach towards the sun in this place. And this is the measure of the increments of power. ","1234":" The measure of the distance of the reciprocation traversed by these continuous increments of power is quite another thing. For the observations show that if the eccentric anomaly GI corresponds to the equated anomaly IC, the versed sine IH of the arc GI is the measure of the reciprocation accomplished. If this can also be deduced from the previously indicated measure of the speed CN, then we shall have reconciled experience with the demonstration involving the balance. Since the sine of any arc is the measure of the strength of that angle, the sum of the sines will be an approximate measure of the sum of the strengths or impressions over all the equal parts of the circle. And the completion of the entire reciprocation is the effect of all of these in common. Furthermore, letting IC and IG, though they are unequal elsewhere, be equal here to avoid confusion, the sum of the sines of the arc IG is to the sum of the sines over the quadrant, approximately as the versed sine IH of that arc IG is to the versed sine IB of the quadrant. Approximately, I say. For at the beginning, when both the versed sine and its increments are small, it is less by half than the sum of the sines. For: Let the quadrant be taken as 90 units. The sum of the 90 sines is 5,789,431. In this instance I have added them all in order. The sum of the sines at an arc of 1\u00b0, that is, the first sine, is 1745. And the former sum is to the latter as 100,000 is to 30. On the other hand, the versed sine of the quadrant is 100,000, and the versed sine of 1 degree is 15, which is half of 30.","1235":"The reader should not be at all deterred by this geometrical faux pas10 and fallacious principle. For before this becomes a perceptible portion of the reciprocation, the effects of the two procedures differ imperceptibly. For the sum of 15 sines, which is 208,166, gives 359411 [as a fourth proportional]. And the versed sine of 15\u00b0 gives 3407\/100,000, which is only a little less than the other. Likewise, the sum of 30 sines, which is 792,598, shows, by the rule of proportions, a part of the reciprocation which is 13,691 out of 100,000. And the versed sine of 30\u00b0 shows 13,397. Also, the sum of 60 sines, which is 2,908,017, shows a little more than 50,000, while the versed sine of 60\u00b0 is 50,000. ","1236":"It has been demonstrated that if any magnet be set out as we have supposed the bodies of the planets to be set out in the heavens with respect to the sun, a reciprocation of the magnetic body will result that is such as is measured by the versed sine, as regards the space traversed. And indeed, the observations testify that the planet's body reciprocates according to the measure of the versed sine of the eccentric anomaly. It is therefore perfectly consistent that the bodies of the planets be magnetic, and so disposed to the sun as we have described.","1237":"I must now show that it was not a great mistake to have taken the arcs IC and IG as equal. When I say that the arc IC on the body of the planet is the measure of the equated anomaly, I am speaking properly, and CN is then the genuine measure of the strength possessed by the planet when it has the sun on the line BK. However, when I say that IG is the measure of the eccentric anomaly which corresponds to the [equated] anomaly IC, I am speaking improperly, incorrectly using the circle of the planet's body to represent the eccentric. But on the eccentric's descending semicircle, since a greater arc of eccentric anomaly corresponds to a smaller arc of equated anomaly (namely, IG to IC), we will be adding up considerably more sines on IG than on IC, and rightly so. For since the sine measures the strength, and the strength acts in proportion to the time and to the closeness to the sun (magnets being stronger when closer)\u2014that is, to put it briefly, in proportion to the arc IG\u2014just as many sines should be set up on IC as are found on IG.","1238":"Our only error is this, that we take those many sines to be longer than they should be, as GH is longer than CN.","1239":"But first of all this excess is in itself very small and imperceptible. For at the beginning of the quadrant the arcs IC and IG hardly differ, and the sines are small, and at the end of the quadrant, where the eccentric equation CG is greatest, the sines hardly differ.","1240":"And then this error is in accord with our wishes. For the sums of the sines always come out a little greater than the versed sines; and here we are keen to accommodate and reconcile the libratory and magnetic ratios to those commended by experience. Therefore, this present error of ours, of accumulating long sines instead of short ones, is avoided if we use the versed sines instead of the sums of the sines themselves, for the sums of the sines are not exactly equal to the versed sines, but exceed them because of the effect of the reciprocation.","1241":" Therefore, by the best reasoning at our disposal, we have brought the calculation within the limits of observable error. Let us conclude that the body of the planet, like a magnet, approaches and recedes according to the law of the lever along an imaginary diameter of the epicycle tending towards the sun, and that the body's diameter of power, its true diameter DA, tends towards the middle elongations, so that for this time BD tends toward 29\u00b0 Taurus BA towards 29\u00b0 Scorpio.","1242":" Thus this reciprocational approach is performed without the action of mind, by a magnetic force which, though it inheres in the planet and is independent, nevertheless depends for its definition upon the extrinsic body of the sun. For the force is defined as seeking the sun or fleeing it. And while the force between magnets tending to bring them together ought to be mutual, I have denied, in Chapter 39 above, that the sun has the planets' attracting force: it was instead understood to be purely attractive only, as is clear from the argument presented there. The planets' force, on the other hand, is supposed to be simultaneously attractive on one side and repulsive on the other. Alternatively, one might suppose that the sun, like unmagnetized iron, is only sought after, and does not in turn seek other things. For in the above passage, its filaments were circular, while those of the planets are here supposed to be straight.","1243":" I am satisfied if this magnetic example demonstrates the general possibility of the proposed mechanism. Concerning its details, however, I have doubts. For when the earth is in question, it is certain that its axis, whose constant and parallel direction brings about the year's seasons at the cardinal points, is not well suited to bringing about this reciprocation or this aphelion. The sun's apogee, or earth's aphelion, today closely coincides with the solstitial points, and not with the equinoctial, which would fit our theory; nor will it have remained at a constant distance from the cardinal points. And if this axis is unsuitable, it seems that there is none suitable in the earth's entire body, since there is no part of it which rests in the same position while the whole body of the globe revolves in a ceaseless daily whirl about that axis.","1244":" So indeed, there may be absolutely no material, magnetic faculty that can accomplish the tasks entrusted to the planets individually, since there may be a lack of means, that is, no suitable diameter of the body which remains equidistant to itself as the body is moved around. For this lack has just been made apparent in one of the planets, namely, the globe of the earth. Therefore, let a mind be summoned, which, as was said in Chapter 39, arrives at a knowledge of the distances it traverses by contemplating the growth of the sun's diameter. Let this mind govern a faculty, either animate or natural, that keeps its globe in a parallel position in a manner allowing it to be suitably impelled by the solar power and to reciprocate with respect to the sun. (For a mere mind, unassisted by a faculty of a lower order, could not by itself do anything in a body.16) At the same time care should be taken that the periodic time of the reciprocation not be made exactly equal to the periodic return of the planet, so that the apsides will move. The plausibility of these things is argued in Chapter 39 above. ","1245":" Now that we have obtained from the observations the laws and quantitative characteristics of this reciprocation by which the sun's apparent diameter is varied, matters of which we were still ignorant in Chapter 39, it remains for us to see whether those laws may be such that the planets may plausibly know them. The laws of the reciprocation were that the versed sine of the eccentric anomaly is the measure of the part of the reciprocation completed.","1246":"To begin, therefore, I say that admitting as given the observational evidence, namely, that after equal arcs of the eccentric are traversed, the planet is found at \u03b3, \u03ba, \u03bc, \u03b6 rather than at \u03b3, \u03b9, \u03bb, \u03b6, the increment of the sun's diameter presents a legitimate measure of the versed sine of the equated anomaly, no less so than we know the versed sines of the eccentric anomaly to be a measure of the reciprocation.","1247":"Now, as was said in Chapter 39, the planet's mind (if it has such an adjunct) perceives the spaces it traverses in the reciprocation in no other way than by the evidence provided by the increase of the sun's diameter. It will therefore be fitting that it know the versed sine of the equated anomaly, in order that by approaching it might increase the sun's diameter to its prescribed size.","1248":"The proof is as follows. Let the planet be at \u03b3, \u03ba, \u03bc, \u03b6 after traversing equal arc of the imperfect eccentric CD, DE, EF, and let the points D and H be joined, the line intersecting the diameter CF at I. Therefore, since the straight lines \u03b4\u03ba\u03b8, \u03b5\u03bc\u03b7 cut the {marginal:","1249":"277} epicycle into arcs similar to those on the eccentric, by construction, as CF is to CI, so will \u03b3\u03b6 be to \u03b3\u03ba, one section being a measure of the other. ","1250":"These things being so, I say it will also follow that the diameters of the sun at \u03b1, as observed from \u03b3, \u03ba, \u03bc, \u03b6, will be augmented by the same measure, namely, the measure by which the versed sine of the equated anomaly increases. It would be inconvenient to prove this solidly here. It will, however, easily be understood as holding solidly if we prove it for both ends and the middle. At C the equated anomaly is nothing, and the versed sine is nothing, and the sun, observed from \u03b3, appears at its minimum, so that the amount of its increase is again nothing. At F the equated anomaly is 180\u00b0. The versed sine is equal to the whole diameter, 200,000. And the sun, observed from \u03b6, appears at its maximum, so that it shall have acquired all of its increase.","1251":" Now for an equated anomaly of 90\u00b0, from A let AM be set up perpendicular to CF, and let MB be joined. Also, from \u03b1 let a line be drawn tangent to the epicycle at \u03bd, and the tangent point \u03bd be joined with the center \u03b2. Now since \u03b1\u03bd\u03b2 is right, by Euclid III. 18, and MAB is right by construction, and \u03b2\u03bd, BA are equal by construction, as well as \u03b2\u03b1, BM, therefore, the triangles are equal and congruent. So \u03bd\u03b2\u03b1, ABM are equal. From \u03bd let \u03bd\u03bf be drawn perpendicular to \u03b3\u03b6. Therefore, since \u03bd\u03bf\u03b2 is right, it is equal to MAB, and \u03bd\u03b2\u03bf will be equal to MBA. So the triangles are similar, and as \u03bd\u03b2 is to \u03b2\u03bf, so is MB to BA, and vice versa. And since \u03bd\u03b2, \u03b2\u03b3, and \u03b2\u03b6 are equal, and also MB, BC, and BF, as \u03bd\u03b2, \u03b2\u03bf together, or \u03b3\u03bf, is to \u03bf\u03b6, so are MB, BA together, or CA, to AF. Therefore, since CA is the versed sine of the eccentric anomaly CBM, and is supposed to be the measure of the corresponding part of the reciprocation, \u03b3\u03bf will be that part. Therefore, at this eccentric anomaly CBM, or equated anomaly CAM of 90\u00b0, the planet will be at \u03bf.","1252":"But the versed sine of the equated anomaly of 90\u00b0, the angle CAM, is half the total diameter, or 100,000. I say also that the apparent magnitude of the diameter of the sun at A, \u03b1, as seen from \u03bf, will be a mean between the magnitudes as seen from \u03b3 and \u03b6, so that it shall have acquired half of its increase when the planet is at \u03bf below","1253":"\u03b2.","1254":"For let the diameter of the sun's body be \u03b1\u03be, and the apparent angles formed by joining \u03be with \u03b6, \u03bf, \u03b3, be \u03be\u03b6\u03b1, \u03be\u03bf\u03b1, \u03be\u03b3\u03b1. And because AF, \u03b6\u03b1 are equal, as well as AC, \u03b1\u03b3, and as CA is to AF, so is \u03b3\u03bf to \u03bf\u03b6, therefore, as \u03b3\u03b1 is to \u03b1\u03b6 so is \u03b3\u03bf to \u03bf\u03b6. But \u03b3\u03be differs imperceptibly from \u03b3\u03b1, and \u03b6\u03be from \u03b6\u03b1. Therefore, as \u03b3\u03be is to \u03b6\u03be, so is \u03b3\u03bf to \u03bf\u03b6, within the limits of perception. So in the triangle \u03b3\u03be\u03b6, the angle \u03be is divided by the line \u03be\u03bf so that the base \u03b3\u03b6 is divided in the same ratio as the sides \u03b3\u03be, \u03b6\u03be. Therefore, by the converse of Euclid VI. 3, the angle \u03b3\u03be\u03b6 is divided into two equal parts by the line \u03be\u03bf, and \u03b3\u03be\u03bf is half of \u03b3\u03be\u03b6, the total increase of the sun's diameter. Q. E. D. It is therefore certain at both ends and the middle that in this way, if the diameter of the reciprocation is divided by the planet in proportion to the versed sines of the eccentic anomaly, the sun's diameter would increase in proportion to the versed sines of the equated anomaly.","1255":"To make it more evident, this is clear in part from the following. Let the straight line BL be drawn from B perpendicular to CF, and about center A, with radius equal to BC, let an arc be drawn intersecting BL at L, and let AL be joined. Since the eccentric anomaly CBL is 90\u00b0, the versed sine will be CB, 100,000, half of the whole diameter, and consequently the reciprocation will be \u03b3\u03b2, half of the whole \u03b3\u03b6. Also, the distance will be \u03b2\u03b1. But AL is equal to it by construction. Thus the planet will be at L. And because AL is equal to BC or BM, and BA is a common side, and LBA is right, as well as MAB, therefore, the triangles BMA, ALB are congruent. So BL is equal to AM. But AM is equal to \u03b1\u03bd, as above, and therefore BL is equal to it also. But \u03b1\u03bd, which lies opposite the right angle \u03b1\u03bf\u03bd, is longer than \u03b1\u03bf, which subtends the acute angle \u03b1\u03bd\u03bf. Therefore, BL is also longer than \u03b1\u03bf, and AL is longer than BL. Thus AL is much longer than \u03b1\u03bf. Therefore, the sun appears smaller at the distance AL than at the distance \u03b1\u03bf. But the distance \u03b1\u03bf was just now seen to be the mean between the maximum and the minimum. Thus at distance AL the sun appears less than the mean. So at L, even though half the semicircle of the eccentric has been traversed, less than half of the increase has been added to the sun's diameter. This is, of course, because the equated anomaly LAC is less than the half, 90\u00b0. And this was the problem that had tied us in knots in Chapter 39, as was said in the preceding chapter (Chapter 56). For if the planet's orbit had been a perfect circle, the increase of the sun's diameter would have been a measure of the increases of the versed sines of the eccentric anomaly, whose observation is more foreign to the planet's mind than is the observation of the equated anomaly, as we shall shortly hear. You can see from this contrast just how conveniently this measure is attributed to the planet, and how plausibly.","1256":"We might suppose that the measure of the reciprocation (the versed sine of the eccentric anomaly, as the observations show) is to be grasped directly by the mind. But then the planet's mind would be deprived of the assistance of the variable solar diameter, because it does not adjust itself to the versed sines of this anomaly of the eccentric. For the planet's path is not a circle. And the planet's mind would have to intuit the parts of the reciprocation, or the distances to be traversed in them, without any indicators. This we long ago rejected as absurd. It would also have to intuit the eccentric anomaly, which is the angle between two straight lines projected from the center of the eccentric, one through the aphelial point and the other through the center of the planetary globe. In the diagram, it is DBC (or if the line DK be projected from D parallel to BC, KDB is then the supplement of the same eccentric anomaly). Therefore, if the mind perceives the angle KDB, it must perceive the triad of the points K, D, B. Concerning the point D there is no problem, since it is the center of its globe. I am not much concerned about K, because, owing to the infinite distance of the fixed stars, BC and DK ultimately coincide at the same location among the fixed stars, and the fixed stars are real bodies. Therefore there is no absurdity in holding that the planet\u2019s mind uses some hidden sense to keep in view that fixed star which provides lodging for the aphelion at any planet\u2019s mind, because B is not clothed in any body.","1257":" Furthermore, when one removes the cause for keeping watch on B, the effect is also removed. But B ought to be watched if the circle CD is to be traversed. However, the planets' orbits are not perfectly circular, as was proven from the observations in Chapter 42. Therefore, the planets do not take aim at B. And thus this putative center B is actually secondary to the path CD. But if it were watched by the planet, it would be prior to the path.","1258":"For these reasons, therefore, I deny that the versed sine of the eccentric anomaly provides the planet with a measure of its reciprocation, not because this is not such a measure, but because even if it is, it is not discerned by the planet's mind.","1259":"But if we suppose it to be the increasing and decreasing of the sun's diameter that serves as the means or aid by which the planet arrives at the correct distances (imperceptible in themselves) in its reciprocations, and then for the variation of this diameter of the sun, from the demonstration just completed, we posit a rule or measure, to be perceived by the planet's mind, [namely,] the equated anomaly of the eccentric, DAC, or rather, KDA, in the diagram, we now stand closer to the truth. For both measures are perceptible: for the reciprocation, the increasing and decreasing magnitude of the sun's diameter, and for the measure, or angle, three points clothed in bodies. For at A there is the sun itself, at D the planet, and at K the fixed star that indicates the aphelia.","1260":"Perhaps it ought therefore to be said (as indeed we also just embraced above in Chapter 39, when we supposed that the forces of Nature were insufficient to administer the celestial motions) that there is attributed to the planet an ability to sense the light of the fixed stars and the sun, the confluence of whose radiations at the center of the planetary body gives an estimate of this angle of equated anomaly. ","1261":"There is but one difficulty to clear up. For what reason is it not the angle itself that is made to be the measure of the planetary operation (that is, to make the sun's diameter increase by approaching the sun), but the versed sine in place of the angle?*21 And by what means might the planet perceive the sine of the equated anomaly? Does it proceed in the way humans do, by geometrical reasoning? Nevertheless, hitherto no faculty of administering the celestial motions belongs to the planet's mind that could not have been acquired by a divine inspiration imparted at the very beginning of the world and lasting even to this day, without any reasoning whatever.","1262":"Therefore, what was said just above should be repeated, namely that the sine of the equated anomaly is the index of the strength of the angles KDA: on this point, see Aristotle's Mechanics, and what was said above in this chapter. For when the two balance arms are disposed at an obtuse angle, they are more easily directed than when they are at a right angle, the ease of direction being proportional to the sines. And, on the other hand, when the two arms are connected at an acute angle, they are more easily made to move together into a single line, head-to-head, than if they were connected at a right angle. Refer again to the demonstration contained in what was just presented.","1263":"Thus, in one way, if it makes sense for the planet to have a sense of the strength of the angles, there will be no absurdity in our saying (using our human conception) that the sines of the angles are known to it. But why would it take note of the natural strength of the angles? (We are evidently returning to natural principles). As before, let there be certain regions of the planetary body in which there is a magnetic force of direction along a line tending towards the sun. However, contrary to the previous case, let it be an attribute, not of the nature of the body, but of an animate faculty of the sort that governs the body of the planet from within, that as it is swept along by the sun, it keeps that magnetic axis always directed at the same fixed stars, except to the extent that it turns the axis a little over the ages. The result will be a battle between the animate faculty and the magnetic faculty, and the animate will win. It is no different from what we had said in Chapter 34, that the bodies of the planets naturally seek rest, but are moved by the extrinsic force of the sun.","1264":"Alternatively, here is a more apt example. The weight of the human arm naturally tends towards the center of the earth. However, in a flag bearer the animate faculty takes over and makes the weight extend over his head and wave in a circulatory motion. Here the animate faculty overcomes the natural weight, and would do so forever if the body of the flag-bearer together with all its faculties had not been created mortal.","1265":"On the basis of these presuppositions, the planet's mind will be able to intuit and perceive the strength of the angle from the wrestling match between the animate faculty, which is designed to keep the magnetic axis in line, and the magnetic power of directing it towards the sun.","1266":"This arrangement seems also to be confirmed by the example of the moon, which is incontestably more strongly propelled when it is on the diametral line of the sun and the earth, perhaps because of this strength of the angles.","1267":" The final conclusion, then, will be this. A planet situated at aphelion makes no endeavor in the direction of the sun, but is carried along in proportion to the distance AC. The angle KDA results from its forward motion. In accord with the proportion of strength of this angle, the planet causes the sun's diameter to increase by approaching the sun. In its approach, it diminishes the distance, making it AD. Since the distance is diminished, the forward motion is increased. Therefore, the angle KDA is changed more rapidly. Therefore, the planet causes the sun's diameter to increase more rapidly (other things being equal). Thus is established a perpetual circulation which does not occur by leaps such as we have supposed in our thinking and calculation, ignoring imperceptible errors, but is quite continuous.","1268":" What I have said so far holds conditionally, if the reciprocation supported by the observations cannot be performed by some magnetic power, implanted in the bodies of the planets, and if it has become absolutely necessary for us to have recourse to a mind. Otherwise, if a comparison between the natural motion and the mental one is in order, the former stands on its own, requiring nothing external, while the latter, the mental motion, appears to give evidence of the magnetic one, and to require its assistance, no matter how you equip it with an animate faculty of moving the body. For in the first place, mind by itself can do nothing to a body. It is therefore necessary to provide for the mind an adjunct faculty that performs its functions in making the planet's body reciprocate. This faculty will be either animate or natural and magnetic. It cannot be animate, for an animate faculty cannot transport its body from place to place (as this reciprocation requires) without the power of another assisting body. Therefore, it will be a magnetic, that is, natural, faculty of sympathy between the bodies of the planet and the sun. Thus the mind calls upon nature and the magnets for assistance.","1269":" Second, at the halfway point in its pattern, which is the equated anomaly, it has traversed a greater part \u03b3\u03bf of its reciprocation above, and a smaller part \u03bf\u03b6 below, while completing half of its task, which consists of increasing or decreasing the sun's diameter. Nor do \u03b3\u03bf, \u03bf\u03b6 correspond to parts of the time. For more time is consumed on \u03b3\u03bf than its excess over \u03bf\u03b6 required. Nor do the parts increase continuously from \u03b6 to \u03b3, the ones about \u03b3, \u03ba being smaller, as well as those about \u03bc, \u03b6. The operations of mind, however, are accustomed to being constant.","1270":" There was consequently a need for us to equip the mind with an animate faculty, as well as a magnetic one, and to contrive a battle between the two which would remind the mind of its duties, of which it could not have been reminded by the equality of either the times or the spaces traversed. So again we have asked nature to assist the mind.","1271":"On the other hand, all these modifications really appertain to the workings of the sun's extrinsic magnetic power, and of the magnetic [power] joined to it, which inheres in the planet, as was explained above. If, therefore, the magnetic powers can do the job on their own, what need have they of the directing function of mind?","1272":"Although we have remained uncertain about the magnetic force inherent in the planetary bodies, through our consideration of the earth's axis, which is different from the sun's line of apsides, this difficulty is common to both explanations. For even when we supposed a mind, we were compelled to admit the kind of axis that we wanted in the earth, through whose mediation the mind could apprehend the strength of the angle, or its versed sine. On the contrary, probability strongly urges us to ascribe this reciprocation of the planets, which without doubt is in accord with the laws of nature, entirely to nature, whatever may be the means by which it occupies the planet's bodies.","1273":"Moreover, I do not know whether I have given sufficient proof to the philosophical reader of this perceptual cognition of the sun and the fixed stars, which I myself so easily accept, and bestow upon the planet's mind.","1274":"Furthermore, in those very modes of operation which we have prescribed to the mind, the soundest of all those which were deemed possible appear to involve some geometrical uncertainty. I am not sure whether this might not be repudiated by God Himself, as to this point He has always been seen to proceed by the path of demonstration. For if a planet, insofar as it may have approached the sun partly by its inherent force, comes into one and another degree of extrinsic power from the sun (as it does indeed come); and if the different degrees also reciprocally intensify the planet's own force of approaching while they increase the angle, which is posited as the standard of measure23 of the approach, or of the increase of the sun's diameter; then the planet's own striving finally becomes in part its own measure, and simultaneously prior and posterior in the intensification of the planet['s force]. For in its parts it is unequal, and for this reason it requires a measure. Thus, the search for the forces tempering both powers will be concluded by a kind of iterative method24 rather than deductively, so that they may complete their cycles in the same time, and in the same revolution of the body.","1275":" Someone might, however, want to think he had found the cause of the progression of the aphelia in the ungeometrical25 nature of this measure. But in Chapter 35 we left it undecided whether this category of motion might not exist through another cause, namely, occultation. That is, just as a plate of iron intercepts the force of a magnet on a strip of iron, the bodies of the planets might also mutually intercept the magnetic powers proper to them, by which they incline towards the sun. And so that this might not happen to the solar power\u2014so that, I mean, the solar power, common to all, could not be intercepted for one planet by the interposition of another\u2014we have drawn a distinction between the essence of the solar body and that of the planets' bodies. So, since we have not drawn a distinction between the bodies of the planets themselves, this seems still to be a possible cause. It could, of course, not have been arranged thus unless the exact magnetic disposition of the planet's body, by which the reciprocation is administered, were known.","1276":" But to give an example of reasoning: let the planet have a magnetic disposition of the sort which, though we had introduced it somewhat earlier, we later denied the earth to possess. In this disposition, impediment through occultation does not have any place. For because it was the effect of the magnetic power to tend towards the sun and to recede from the sun, meanwhile keeping the fibers of its magnetic seat in line, if another planet, coming between the sun and the planet, impedes this travel towards the sun, or recession from it, while not impeding the common motion from the sun, the planet will approach or recede less than it should, and thus the size of the circuit will be altered along with the periodic time, over the ages, and will again be corrected by contrary eclipses. However, the aphelion will not change position through this occultation. So the cause for the motion of the aphelia previously proposed by us still reigns alone, without peer or rival.","1277":" Further, if a mind should preside over the reciprocation in the manner described, occultation will still do no harm. For as was said, the mind would use the angle of equated anomaly as its measure for increasing the sun's diameter; and, while deprived of the perception of it for a slight amount of time (that is, while the sun is hidden), it would be possible (the gods willing) to compensate for what the mind would have missed, upon the sun's re-emerging and restoring the equated anomaly into view. For mind (where there is one) is master of the animate faculty, and uses it differently and unequally according to circumstances. So why should it not use the animate faculty in an unusual manner here too, in removing the discrepancy between the measure (the equated anomaly) and the measured quantity (the sun's diameter) which had insinuated itself through the means of the sun's eclipse? What about other slow motions of this kind, such as the precession of the equinoxes arising from the earth's axis being directed at one or another of the fixed stars, and not at the sun? For here, the removal of the sun's light can have no effect, since its presence has no effect either.","1278":"We would like to avoid the inconvenient effects of magnetic occultations upon the reciprocations proper to the planets, just as we did in Chapter 35 for the common revolving effect of the sun. It should therefore be said that the bodies of the planets can indeed be similar in their magnetic dispositions, but either (1) they are so far from one another that the planets' orbs of power would not overlap, or (2) the power coming from the sun is so strong (that which activates the planets' proper powers no less than that which makes them revolve in their orbs) that it could not be impeded at all by the interposition of a small, weak body, so that it would pass on through, just as light passes through a globe of water, or (3) the bodies of the planets are so meager that they would have no effect, nor is the sun ever substantially blocked by any planet for any of the other planets moved by the sun, as the sun is never substantially blocked from the earth by the moon. For although the whole sun can be covered for the moon for several hours, the moon performs its reciprocation with respect to the earth, not the sun, and it can never be deprived of its perception of the earth since there is no body between the earth and the moon.","1279":"Nevertheless, it might appear plausible to someone that the transposition of the apogees is instantaneous, and occurs through the cause of the sun's being eclipsed. Let him say, if he please, that to prevent the reciprocation's undergoing a sudden leap of speed when it is interrupted by an eclipse, during which the planet is moved by the sun to another angle and another degree of its strength, this angular leap is compensated by the planet itself, by having its axis incline towards the sun at the same angle after the eclipse as it was at the beginning of the eclipse. For thus a transposition of the aphelia will be obtained, but one occurring by leaps, and remaining in the same sidereal position for many years, until there happens to be another occultation of the planet.","1280":"On the other hand, the prior cause of the transposition of the aphelia, arising from the reciprocation's aberration from its sidereal circuit, produced by the ungeometrical27 interconnection of its components, favors the uniform transposition of the apogees.","1281":"Finally, if neither of these causes obtains, let the mind, furnished with an animate faculty which presides over the constant direction of the magnetic axis, have the additional task of inclining the axis over the ages. But if none of these causes stands, nor even the general idea of a mind, let us be satisfied with nature, which, as she has allowed everything else to be disentangled, has also shown a splendid occasion for the motion of the aphelia.","1282":"Chapter 58","1283":"In what manner, while the reciprocation discovered and demonstrated in Chapter 56 holds good, an error may nevertheless be introduced in a wrongheaded application of the reciprocation, whereby the path of the planet is made puff-cheeked.","1284":"With an apple Galatea seeks me, the lusty wench:","1285":"She flees to the willows, but hopes I'll see her first. ","1286":"It is perfectly right that I borrow Virgil's voice to sing this about Nature. For the closer the approach to her, the more petulant her games become, and the more she again and again sneaks out of the seeker's grasp just when he is about to seize her through more twists. Nevertheless, she does not cease to invite me to seize her, as though delighting in my mistakes.","1287":"Throughout this entire work, my aim has been to find a physical hypothesis that not only will produce distances in agreement with those observed, but also, and at the same time, sound equations, which hitherto we have been driven to borrow from the vicarious hypothesis of Chapter 16. So, while trying to use a false method to do the same thing through this hypothesis, which is itself perfectly correct, I began once again to fear for the whole undertaking. On the line of apsides, about centers A and B, let the equal circles GD, HK be described. And let AB be the eccentricity of the circle GD. Also, let the eccentric anomaly, or its number of degrees, be the arc GD or HK, by the equivalence established in Chapter 3. Next, about center K, with radius KD which shall be equal to AB, let the epicycle LDF be described, which will intersect the circle GD at D, through the equivalence established in Chapter 3. Let AK be drawn, and extended to intersect the epicycle at L, so that the arc LD is similar to the eccentric anomaly GD or HK. And let BD be joined. Now, from D let perpendiculars be drawn to GA, LA, and let these be DC, DE. Therefore, by what has previously been demonstrated in Chapter 56, AE will indubitably be the correct distance at this eccentric anomaly. The question remains how much time was taken to arrive at it. Now the versed sine of its arc, GC, which, after multiplication, becomes LE, when subtracted from GA, yielded the correct distance AE. These indications persuaded me that the other end of AE should be sought, not on the line DC (which was actually perfectly correct), but at the point I of the line DB, such that if I were to draw the arc EIF about center A with radius AE, it would intersect DB at I. Thus, according to this persuasion, AI would be the correct distance, both in position and length, and IAG would be the true equated anomaly. But it is manifest that the arc EIF would intersect the line DC at a higher place, namely, at F. Thus the angles IAG and FAG differ by the quantity IAF.","1288":" I therefore erred in taking the line AI instead of AF. I first discovered the error empirically. For when I explored the quantity of the area DAG, either using all the distances or using the small area DAB, and then fitted the angle IAG, rather than FAG, to this area DAG, now converted into time, in the upper part of the semicircle I found 5\u00bd\u2019 more, and in the lower half 4' less, than the vicarious hypothesis gave with sufficient certainty. And so, since the equations disagreed with the truth, I began once more to accuse these perfectly correct distances AE, and the planet's reciprocation LE, of the crime for which my false method, which took I in place of F, was to be blamed. What need is there for many words? The very truth, and the nature of things, though repudiated and ordered into exile, sneaked in again through the back door, and was received by me under an unwonted guise. That is, I rejected the reciprocation on the diameter LE, and began by recalling the ellipses, quite convinced that I was thus following an hypothesis far, far different from the reciprocation, although they coincide exactly, as will be demonstrated in the following chapter. The only difference was that the errors I committed in method earlier were corrected by this procedure, and that F was used instead of I, as it should have been.","1289":"My line of reasoning was like that presented in Chapter 49, 50, and 56. The circle of Chapter 43 errs in excess, while the ellipse of Chapter 45 errs in defect. And the excess of the former and the defect of the latter are equal. But the only figure occupying the middle between a circle and an ellipse is another ellipse. Therefore, the ellipse is the path of the planet, and the lunule cut off from the semicircle has half the breadth of the previous one, namely, 429.","1290":"Moreover, if the planet's path were an ellipse, it was clear enough that I could not be taken in place of F, for if this is done, the planet's path is made to be puff-cheeked. For let the angles QBP, SAR in the lower part be equal to GBD, HAK, and about center R let the epicycle PT again be described, equal to the previous one, and from P, the intersection of the epicycle with the eccentric, let perpendiculars PV, PM, be dropped to BQ, AR [respectively], and let PB be joined. And about center A, with radius AM, let the arc MN be described, intersecting PV at O, and PB at N. So, by analogy with the above, just as we are taking I in place of F, let us now take N for O, and let us consider that just as AN is the correct distance in length, it is also correct in position. Now the points I, N, and the like do indeed make the planet's path puff-cheeked. For the arcs GD and QP are equal. And BD, BP, projected from a common center, intersect the lunule cut off. But DI and PN, the breadths of the lunule extended towards the center, are unequal. And DI is smaller, and PN greater. For since ED and MP are equal, and EDI, MPN are right, while EI is a greater circle, since its radius AE is greater, and MN is a smaller circle, since its radius AM is smaller, therefore, PN will definitely be greater, and DI smaller. Therefore, the lunule cut off is narrower above, at D, and broader below, at P. In the ellipse, in contrast, this lunule is of equal breadth at points equally removed from the apsides G and Q. So it is clear that the path is puff-cheeked, so it is not an ellipse. And since the ellipse gives the correct equations, this puff-cheeked path should by rights give incorrect ones.","1291":" Nor was there any need to compute the equations anew from the ellipse. I knew they were going to perform their function without further prompting. I was only concerned about the distances, that if they were taken from the ellipse they might cause me trouble. But although this did happen, I had already prepared a refuge, namely, the uncertainty of 200 units in the distances. Consequently, I did not hesitate much here, either. The greatest scruple by far, however, was that despite my considering and searching about almost to the point of insanity, I could not discover why the planet, to which a reciprocation LE on the diameter LK was attributed with such probability, and by so perfect an agreement with the observed distances, would rather follow an elliptical path, as shown by the equations. O ridiculous me! To think that the reciprocation on the diameter could not be the way to the ellipse! So it came to me as no small revelation that the ellipse is consistent with the libration. This will be made clear in the following chapter, where it will be demonstrated at the same time, through the agreement of arguments from physical principles with the body of experience, mentioned in this chapter, that is contained in the observations and in the vicarious hypothesis, that no orbital figure is left for the planet other than a perfectly elliptical one.","1292":"Chapter 59","1293":"Demonstration that when Mars reciprocates on the diameter of an epicycle, its orbit becomes a perfect ellipse; and that the area of the circle measures the sum of the distances of points on the circumference of the ellipse.","1294":"Protheorems1","1295":"I.\n If an ellipse be inscribed within a circle, touching it at its vertices at opposite points, and a diameter be described through the center and the points of contact, and further, if perpendiculars be drawn to the diameter from other points on the circumference of the circle, all these lines will be cut in the same ratio by the circumference of the ellipse. ","1296":"Using Book I page3 21 of the Conics of Apollonius, Commandino4 proves this in his commentary on Proposition 5 of Archimedes' On Spheroids.","1297":"Let there be the circle AEC with ellipse ABC inscribed in it, touching the circle at A and C, and let the diameter be drawn through the points of contact A and C, passing through the center H. Then from the points K and E on the circumference let the perpendiculars KL, EH be dropped, cut by the circumference of the ellipse at \u039c, B. BH will be to HE as ML is to LK, and so on for all other perpendiculars.","1298":"II","1299":"The area of an ellipse thus inscribed in a circle is to the area of the circle in the same ratio as the lines just mentioned.","1300":"For as BH is to HE, so is the area of the ellipse ABC to the area of the circle AEC. This is Proposition 5 of Archimedes' On Spheroids.","1301":"III. ","1302":"If from a given point on the diameter lines be drawn to the points on the perpendiculars where the circumferences of the circle and the ellipse intersect them, the areas cut off by these lines will again be as the segments of the perpendiculars.\nLet N be the point on the diameter and KML the perpendicular, and let K and M be connected with N. I say that as ML is to LK, or (by Protheorem I) as the shorter semidiameter BH is to the longer HE, so is the area AMN to AKN. For the area AML is to the area AKL as ML is to LK, by Archimedes' assumptions in On Spheroids, Prop. 5, which Commandino demonstrates under letters C and D in his commentary on this proposition. But the altitude NL of the right triangles NLM, NLK is the same, and the bases are LM, LK; and consequently MLN is to KLN as ML is to LK. Therefore, by composition, the whole area AMN is to the whole area AKN as ML is to LK. Q. E. D.","1303":"IV","1304":"If the circle be divided into any number of equal arcs by perpendiculars such as these, the ellipse is divided into unequal arcs; and the arcs that are near the vertices adopt the greatest ratio [to the arcs of the circle], while those in the middle positions adopt the least ratio. ","1305":"For about the vertices the ratio of the arcs is close to the ratio of the perpendiculars cut off [by them], to which they closely approximate in length, although they are less. About the middle positions they are nearly equal, but the elliptical arcs are smaller because they are less curved than the circular ones. This is self-evident.","1306":"V","1307":"The entire elliptical circumference is approximately the arithmetic mean between the circle on the greater diameter and the circle on the smaller diameter.","1308":"For it was proved in Chapter 48 above that that circumference6 is longer [than the ellipse] whose diameter is the mean proportional between the diameters of the ellipse, the area of which circle, by Archimedes, On Spheroids Prop. 7, being equal to the area of the ellipse. But, too, the arithmetic mean is longer than the mean proportional. Therefore, they are approximately equal.","1309":"VI","1310":"The gnomons7 of squares divided proportionally are to one another as the squares.","1311":"Let there be two squares, PL and SH. Let their sides KL, EH be divided proportionally at the points \u039c, B. Let the gnomons KOQ and CRE be described. Therefore, because ML is to LK as BH is to HE, OL will also be to LP as RH is to HS. But the gnomons are the differences of the squares. Therefore also, as LP is to its gnomon, so is HS to its, and permuted, as PL is to HS, so is the gnomon KOQ to the gnomon CRE.","1312":"VII","1313":"If from the end of the shorter semidiameter on the circumference of an ellipse, a line equal to the longer semidiameter be extended, ending at the longer semidiameter, the distance between that point of intersection and the center is equal in square to the gnomon that the square of the longer semidiameter places about the square of the shorter semidiameter. ","1314":"From the end B of the shorter semidiameter HB, let the straight line BN be extended, equal to the longer semidiameter AH. I say that HN is equal in square to the gnomon ERC, that is, that it is the mean proportional between EB and the remainder of the circle's diameter. This was proved in Chapter 46 above. But it is demonstrated more easily and briefly in the pure case here. For the gnomon is the difference of the squares BH and HE or HA, by the sixth of these protheorems. But the square on HN is also the difference of the squares BH and BN, that is, HE or AH, by Euclid I. 46. Therefore, the square on HN is equal to the gnomon ERC. Q. E. D.","1315":"VIII","1316":"If a circle be divided into any number (or an infinity) of parts, and the points of division be connected with some point within the circumference of the circle other than the center, and also be connected with the center, the sum of the lines drawn from the center will be less than the sum of those from the other point.","1317":"Also, a pair of lines close to the line of apsides drawn to opposite points10 from a point other than the center will be approximately equal to two drawn to opposite points from the center, while a pair so drawn at intermediate locations will be much greater than those drawn from the same center.","1318":"This was proved in Chapter 40. So12 that excess does not increase uniformly with the number of lines, much less with the sines. For their differences vanish at the end, while the differences of the said excesses are greatest at the end. And since the area of the circle KNA increases uniformly, its part KHA increasing with the number of lines, by construction, and its part KNH with the sines of the arcs to which the lines are drawn, multiplied by HN, by Chapter 40, the area of the circle is therefore not adapted to the measure of the sum of the distances to its circumference. ","1319":"IX","1320":"If, on the other hand, instead of the lines from the point other than the center, those lines be taken which are bounded by perpendiculars drawn from that point to the lines which are drawn through the center\u2014that is, if, in the terms of Chapter 39 and 57, the diametral distances are taken in place of the circumferential ones\u2014then their sum equals the sum of those drawn from the center. ","1321":" For let any point whatever on the circumference of the circle be chosen, K in the present instance, and from K let a straight line be drawn through H to the opposite part of the circumference I. Now from N let a perpendicular be dropped to KI, and let this be NT. Then KH, HI together, are equal to KT, TI together. And any sum of the pairs KH, HI, is equal to an equal sum of the pairs KT, TI. And since, when AK is divided into any number of equal parts, the sum of the lines AN, KT to those parts increases partly with the number of lines HA, HK and partly with the sines multiplied by HN, the sum therefore increases uniformly with the area KNA, by the foregoing. Thus the area of the circle, and the parts KNA, are a measure of the sums of the diametral distances.","1322":"X","1323":"The ratio of distances from a point not at the center of an ellipse to equal arcs of the ellipse, no less than those on the circle in Protheorem 8, is contrary to the ratio of arcs of the circle and the ellipse to one another, explained in Protheorem 4. For the pair drawn from the point not at the center exceeds the pair drawn from the center in opposite directions, in the least ratio, and nothing at all, at the apsides; but at the middle elongations they exceed the latter in the greatest ratio.","1324":"This is clear from Chapter 40. So, again, as in Protheorem 8, the area of the ellipse is not suited to measuring the sums of the distances of equal arcs of its elliptical circumference. ","1325":"XI","1326":"With these preliminaries completed, I shall now proceed to the demonstration.","1327":"If in an ellipse divided by perpendiculars dropped from equal arcs of the circle, as in Protheorem 4 above, the points of division of the circle and the ellipse be connected to the point that was found in Protheorem 7, I say that those that are drawn to the circumference of the circle are the circumferential [distances], while those that are drawn to the circumference of the ellipse are the diametral, which are established at an equal number of degrees from the apsides of the epicycle.","1328":" From the point I, opposite K from the center H, let IV be dropped perpendicular to AC, intersecting the elliptical circumference at Y. And from the point N found in Protheorem 7 let the lines NK, NM, and also NI, NY be drawn to the points of intersection K, M, and also I, Y made by the two perpendiculars, respectively. Further, let the diagram of Chapter 39 and 57 be brought back, and let the semidiameter of the epicycle \u03b2\u03b3 be equal to the eccentricity HN, and the arc \u03b3\u03b4 beginning from the apsis \u03b1\u03b3 be similar to AK beginning from the apsis, and let \u03b1\u03b2 equal the semidiameter HA. I say that NK is the circumferential distance \u03b1\u03b4 (this was proven in Chapter 2) and NM is the diametral distance \u03b1\u03ba. ","1329":"First, KN is equal in square to the sum of the squares on KL and LN. Likewise, MN in square is equal to the sum of the squares on ML and LN. Let LP be the square on LK, and LO the square on LM. Thus when the square on LN and the square on LM (that is, the square LO), common to both, are subtracted, there remains the gnomon KOQ, by which the square on KN exceeds the square of, or on, MN. Now as KL is to EH, so is KM to EB, by the first protheorem. Therefore also, as KQ, the square on KL, is to EC, the square on EH, so is the gnomon KOQ to the gnomon ERC, by Protheorem 6. And further, as KL, the sine of the arc AK, is to EH or AH, the whole sine, here in the eccentric circle, so is the perpendicular \u03b4\u03ba in the epicycle (from the point \u03b4 of the arc \u03b3\u03b4, which is similar to AK, to the diameter of the apsides \u03b2\u03b3) to the semidimeter of the epicycle \u03b2\u03b3. Therefore also, as the gnomon KOQ is to the gnomon ERC, so is the square on \u03b4\u03ba to the square on \u03b2\u03b3. But HN is equal to \u03b2\u03b3. And the square on HN is equal to the gnomon ERC, by Protheorem 7. Therefore the square on \u03b2\u03b3 is also equal to the gnomon ERC, and consequently, the square on \u03b4\u03ba, the perpendicular from the point on the epicycle just mentioned, will equal the gnomon KOQ. But the square on that perpendicular \u03b4\u03ba is the excess of the square on the circumferential distance \u03b4\u03b1 over the square on the diametral distance \u03ba\u03b1. Therefore also, the gnomon KOQ, equal to it, is the excess of the square on \u03b4\u03b1 over the square on \u03ba\u03b1. But KN is equal to \u03b4\u03b1. Therefore [the square on] KN exceeds [the square on] \u03ba\u03b1 by the gnomon KOQ. But it also exceeds the square on MN by the same gnomon. Therefore, the diametral distances MN and \u03ba\u03b1 are equal. Q. E. D. It also will be demonstrated likewise concerning NY, that it is equal to \u03b1\u03bc, where \u03b6\u03b7 is similar to CI. And so for all.","1330":"XII\nAgain, it is also clear from the same that","1331":"The area of the circle, both as a whole and in its individual parts, is the genuine measure of the sum of the lines by which the arcs of the elliptical planetary path are distant from the sun's center.","1332":"For by Protheorem 9, if the area of the whole circle is set equal to all the diametral distances of all the arcs of the division chosen, the parts of the area, as KNA, bounded at the point N from which the eccentricity originates, are made equal to those diametral distances that belong to the arc KA enclosing that area.","1333":"But by Protheorem 11, preceding, the diametral distances KT, TI, that is, \u03ba\u03b1, \u03bc\u03b1, by Chapter 40, are the same as the distances MN, NY of the points \u039c, Y of the ellipse.","1334":"Therefore, as the area of the circle is to the sum of the distances of the ellipse, so is the part of the area of the circle KNA bounded at the sun's center N, whence the eccentricity is measured, to the sum of the distances on the ellipse belonging to the elliptical arc AM having the same number of degrees as the arc of the circle enclosing the area AK. ","1335":"XIII","1336":" However, the following doubt arises: if the area AKN is equivalent to all the distances of as many points on the elliptical arc A\u039c from N as we have taken to be present on AK, what, then, would that elliptical arc be; that is, where would it end? For it seems that it should not end at the perpendicular line KL. The reason for this is that in this way, by Protheorem 4, unequal elliptical arcs correspond to equal arcs on the circle, and thus the arcs are less about the vertices A, C, and greater about B. However, it appears necessary to take equal arcs of the elliptical orbit, should we wish to estimate and compare the times of the planet to traverse them. To be specific: because it is certain that the end of this arc should be at the distance MN from N, therefore, as in Chapter 58, an arc MZ drawn about center N with radius NM somewhere indicates a point bounding this arc of the ellipse, and it appears that that point is going to be not M but Z, at which the arc intersects the line KH, making that arc of the orbit AZ.","1337":"The reply is made that the arc of the ellipse whose time increments the area AKN measures should by all means be divided into unequal parts, with those near the apsides being smaller.","1338":"For let it be that the actual path ABC of the planet be divided into equal arcs. Since the planet takes a longer time on arc A than on arc C proportionally as NA is longer than NC, while NA and NC taken together equal the longer diameter of the ellipse, and HB is the shorter semidiameter of the ellipse, the planet's amount of time on the arc at B and the opposite arc together will therefore be less than on the equal arcs at A and C together. Therefore, to make the amount of time at A and C shorter, and at B and its opposite longer, thus making the amount of time on any two opposite arcs taken together the same, the arcs at A and C should be made smaller, and at B and its opposite, longer. But this is accomplished by the perpendiculars KML, as is clear from the objection itself.\nBut by this solution we only discover with certainty that the arcs about A, C should be somewhat shorter. Whether the particular arcs determined by the perpendiculars KML are exactly the required arcs, is not yet established. But it will now be made clear, in the following manner.","1339":"XIV","1340":"If someone were to divide an ellipse AMC into any number of equal arcs, assigning to each individually its distance from N, while taking the areas AMN, ABN, ABCNA in place of the sum of the distances on AM, AB, ABC, by Protheorem 10 he would bring about the same error that occured in Chapter 40 above when we tried to do on a perfect circle what we are here supposing to be tried on an ellipse. That is, two lines MN, NY from two points \u039c, Y opposite one another through center H, are taken as equivalent to the shorter line MHY.","1341":"Suppose, however, that that same person were to divide an ellipse AMC into the same number of unequal arcs, contrary to Protheorem 10, according to the following law: the circle AKC being first divided into equal arcs, perpendiculars KL would then be drawn to AC from the ends of the individual arcs, cutting the ellipse AM into arcs also; and the elliptical area would be taken for the distances of these arcs from N. In that case, a remedy would be provided for the error which has been committed: a most perfect compensation.","1342":"I shall prove this for the beginnings of the quadrants, A and C; for the ends, B; and the motion in between.","1343":" At the beginnings of the quadrants A, C, if the two lines NA, NC be taken for the line AHC, there is no error. At the end, however, if for BN (that is, for EH) I take BH, the consequent error or defect is a maximum, by Protheorem 10, the amount being BE. And by Protheorem 7 of this chapter, as HE is to EB, so is the required length to the error committed at this position. Now, suppose that the sum total of all the distances receives a measure erring in defect, namely, the area of the ellipse. Then when the defect is distributed among the individual distances by the force of our operation or computation, it will happen that the distances NA, NC are taken too small with respect to this measure of all of them. We are thus deceived in thinking that all the lines err equally in defect, since NA and NC are not in fact in error. They do indeed add up to the correct sum, but when the sum is in turn distributed, NA and NC will not have received their correct value, because certain lines about B will have defrauded the whole.","1344":"Let us now see how we can remedy this error in the same proportion.","1345":"By Protheorem 4 of this chapter, the least arcs AK, AM about the apsides A or C are in the same ratio as KL to LM, that is, EH to HB. This was the ratio by which the lines about B formerly erred in defect. And at B, in turn, the least arcs of the circle and the ellipse, KE and MB, say, are equal; just as, before, the straight lines AN, NC together were equal to the line AHC. Therefore, as in the previous consideration of the straight lines, so will it be here in the consideration of the arcs: when the mean and uniform measure of the arcs is conceived, the arc at the apsides A or C will be short with respect to it, and the arc at the middle intervals B will be long. Thus where the distances are too short with respect to their erroneous sum, in the faulty area of the proposed ellipse, the arcs will be small with respect to their mean value, as at A, C; and where the distances are too long, the arcs are too long, as at B. And so to the extent that we accumulate too small an interval of time in our calculation, owing to the rather short distances at the apsides, there are that many more distances on that arc, it being cut into small parts each of which has its own assigned distance. And inversely, to the extent that about the middle intervals B more time is accumulated by the individual distances than is fitting in our calculation, when we carried over to the innocent apsides A, C, the part of the defect at this location, the calculation has collected correspondingly fewer distances, they having been obtained from the large parts of the arc by begging. At that place (A and C), what the individual distances cannot do owing to their brevity in the calculation, they accomplish by their being closely spaced, with the result that they accumulate the correct time intervals. And here, the error arising in the calculation occasioned by their excessive length is again removed by their being more widely and loosely spaced.","1346":"Of the beginning and end, I have said that both the arcs of the circle at A and C and the correct distances differ at the beginning from those that the area of the ellipse accumulates, in the same ratio that EH has to HB; and that the arcs at BE and the distances at A and C end up differing by the same ratio, namely, the ratio of equality.","1347":"We must now state the same for the progress in between.","1348":"The lines NA, NC, from small beginnings, by swift increments exceed the lines AHC by a perceptible amount. On the other hand, where the excess is greatest, as that of BN over HB, the increments are slowed down perceptibly. They are greatest in the middle, near an eccentric anomaly of 45\u00b0.","1349":"This is to some extent shown by the angle and secants of the equation. For BN differs from BH by about the same amount as the secant of the angle of the optical equation differs from the whole sine, while the opposite angles of the equations assist each other in the same ratio. Now the increments of the secants of the optical equation at about 45\u00b0 are near a maximum, and are slow at the beginning and end of the quadrant. Concerning these, see the end of Chapter 43.","1350":"Furthermore, the increments of the elliptical arcs marked off by the perpendiculars KL progress in the same ratio. For at the beginnings, A and C, the arc AK, always beginning from A, is to its increment as LK is to KM. But the whole arc itself is small, and so the increment is small as well. At the end, near B, the ratio of AE to AB is reduced nearly to equality, even though the arc AB is large, since it is near the quadrant. Therefore, the increment is again small. So it is at the middle, about 45\u00b0, that the increment of the arcs is most evident.","1351":"It is thus clear that in the intermediate progress, too, the ratios are equal, so far as minute consideration can be carried.","1352":"Although the demonstration is most certain, it is likewise gauche and ungeometrical, at least in that part pertaining to the progress of the intermediate increases. As elsewhere, I would like to have this small part carried out geometrically and with finesse, so that even an Apollonius would be satisfied. Meanwhile, until someone else discovers and provides us with this small part, we should be content with what we have.","1353":"XV","1354":" But let us complete the proof.","1355":"The arc of the ellipse whose time is measured by the area AKN, should end on LK, so it would be AM.","1356":"For hitherto we have been proceeding on the fiction that if anyone had so much leisure as to want to compute the area of the ellipse, it would turn out that in using the area of the ellipse AMN in place of the same number of distances of AM as there are equal arcs on AK, he would not miss the mark. Let this serve us as the previously demonstrated major premise of the proposition.","1357":"I shall now add the minor premise, derived from Protheorem 3 of this chapter. Here it was shown that the area AKN is to the area AMN as the area AKC is to the area AMC. The conclusion therefore is, since the ratio of equimultiples is the same, that the area of the circle AKN also measures the sum of the diametral or elliptical distances (such as KT, TI) on AM, there being as many as there are parts in AK. Whence it is clear that I correctly assigned more closely spaced distances to the parts of the ellipse about A, C: that is, the same number of distances as there were intersections made by the perpendiculars KL coming from equal arcs of AK.","1358":"So that no one may doubt the truth of this, confused by the subtlety and perplexity of the argument, this truth previously came to be known through experience, in the following manner. At the individual degrees of eccentric anomaly, I set up the diametral lines KT, TI in place of the distances from N. I also added each in order to the sum of the previous ones. When all were collected, the sum was 36,000,000, as is fitting. Next, when the individual sums were compared with the whole, following the rule of proportions, the sum 36,000,000 would be to 360 degrees (the nominal value of the whole periodic time) as the individual sums would be to the increments of time they signify. This produced exactly the same results, down to the last second, as would have come out had I multiplied half the eccentricity by the sine of the eccentric anomaly, [added the area of the sector contained by the eccentric anomaly], and compared [the sum] to the area of the circle, which would be given the same value of 360 degrees (the nominal value of the periodic time). Then, when I was of the opinion that the correct distance NM should be applied to the line KH, becoming ZN, and had thus investigated the equated anomaly ZNA, attributing it to the mean anomaly AKN, the equations disagreed obviously with my vicarious hypothesis of Chapter 16. At about 45\u00b0, the difference between the equated anomaly and the true value found through experience in the observations was a defect of about 5\u00bd minutes, and near 135\u00b0 about 4 minutes. But when NM24 was so applied as to end on KL, then when the equated anomaly MNA was applied to the mean anomaly AKN, it agreed exactly with the vicarious hypothesis, that is, with the observations. And when the fact was established, I was afterwards driven, once I had settled on the principles, to seek the cause of the matter which I have revealed to the reader in this Chapter as skillfully and lucidly as possible. And unless the physical causes that I had originally taken in the place of principles had been good ones, they would never have been able to withstand an investigation of such exactitude.\nIf anyone thinks that the obscurity of this presentation arises from the perplexity of my mind, I shall myself only thus far acknowledge to him my guilt, that I was unwilling to leave anything untested, no matter how utterly obscure, and not strictly necessary to the practice of astrology, which many deem the sole end of this celestial philosophy. But as for the subject matter, I urge any such person to read the Conics of Apollonius. He will see that there are some matters which no mind, however gifted, can present in such a way as to be understood in a cursory reading. There is need of meditation, and a close thinking through of what is said.","1359":" Kepler's woodcut of the planet's magnetic body (p. 414), showing the boatman and his oar pulling the planet along.","1360":" Kepler's woodcuts for the diagrams in Protheorem 11 (opposite). Note Urania in her triumphal chariot coming to crown Kepler with a laurel wreath for his conquest of the war god; also, the two angels, one with carpenter's tools and the other bearing what may be the blueprint for the universe.","1361":"Chapter 60","1362":"A method, using this physical\u2014that is, authentic and perfectly true\u2014hypothesis, of constructing the two parts of the equation and the authentic distances, the simultaneous construction of both of which was hitherto impossible using the vicarious hypothesis. An argument using a false hypothesis.","1363":"In Chapters 56, 58, and 59, the planet was assumed to approach the sun and recede from it along a diameter directed towards the sun, thus making an elliptical orbit, and further, it was assumed to spend time at each individual point in proportion to the distance of the point from the sun. Thus we happen upon a most convenient short cut through the preceding Chapter 59, for evaluating the sum of any number of time intervals all at once. For it was shown that when a line is drawn from a circle perpendicular to the longer diameter of an ellipse inscribed in that circle (in the previous diagram, let KL be dropped to AC), so as to intersect the ellipse at M, and supposing that the sun is at N, the sum of all the distances of points on the arc AM from the sun N is contained in the area AKN.","1364":" For an arc AM of the ellipse being supposed, which is defined by the arc of the circle AK, the area AHK is given, which is the sector of the arc AK, by which arc that sector is also measured, in units of which the whole area of the circle is 360\u00b0.","1365":" And because the arc AK is given, its sine KL is also given. But as KL is to the whole sine EH, so is the area HKN to the area HEN, as was proved in Chapter 40. Also, since the eccentricity HN is given, half of it multiplied by HE will describe the area HEN. This value is found at once at the beginning, so that it may be known what this small area amounts to, when the whole area of the circle has the value of 360\u00b0 of time.","1366":"And so, once the area HEN is known, it is very easy to find the area HKN by the rule of proportions. For as EH is to KL, so is NEH to the area NKH, or its value in degrees, minutes, and seconds; and this, added to the value of KHA, establishes a value for KNA, which is the measure of the time which the planet takes on AM. This, then, is one of the parts of the equation, the one I call \"physical\"a, namely, the area HKN. Yet I so arrange the tables that there is no need to mention the equation, nor is there a separate column showing the opticalb part of the equation, that is, the angle NKH. The [meanings of the] terms, \"mean anomaly,\" \"eccentric anomaly,\" and \"equated anomaly\" will be more personal to me. The mean anomalyc is the time, arbitrarily designated, and its measure, the area AKN. The eccentric anomalyd is the planet's path from apogee, that is, the arc of the ellipse AM, and the arc AK which defines it. The equated anomalye is the apparent magnitude of the arc AK as if viewed from N, that is, the angle ANK.","1367":" Now the angle of equated anomaly is found as follows. The arc AK being given, the sine of its complement LH is given. And as the whole is to LH, so is the whole eccentricity to the part to be added4 to 100,000 (subtracted, below 90\u00b0) to give the correct distance of Mars from the sun, namely, NM. So in triangle MLN, the angle at L is right, and MN is given, and LN is also given. For it is made up of LH, the sine of the complement of AK, the distance from apogee, or the eccentric anomaly; and the eccentricity HN. Below 90\u00b0, in place of the sum LH, HN their difference should be taken, and in place of the complement of the eccentric anomaly, its excess. Therefore, the angle of equated anomaly LNM will not be hidden. Here anyone who wishes can easily figure out what has to be changed in the other semicircle.","1368":"On the other hand, given the eccentricity and the equated anomaly, the eccentric anomaly is given, a little more laboriously, whether we proceed demonstratively or by analysis. ","1369":"It can be investigated demonstratively by this method: namely, by measuring the angle under which is viewed the planet's incursion KM made from any point K on the circle as if seen from the center of the sun. This method depends upon several protheorems.","1370":"I.","1371":" The small lines of the planet's incursion towards the diameter of the apsides increase in proportion to the sines of the eccentric anomaly.","1372":"For as EH is to KL, so is EB to KM. This was established in Chapter 59, and demonstrated in the Conics.","1373":"II","1374":"The ends of one of the small lines being connected to the center, and it being supposed that the small line remains the same in quantity at all points of the eccentric, the tangent of the angle at the center decreases approximately in proportion to the sines of the complement of the eccentric anomaly.","1375":" Let DF be the small line, part of DV, the sine of the eccentric anomaly AD. Let the ends D, F be connected to H, and let HF be extended. And let the straight line ED be tangent to the circle at D, intersecting HF at E. Therefore, since DVH is right, VDH will be the complement of VHD, the eccentric anomaly. And since EDH is also right, HED will be less than a right angle by the quantity EHD. This is of hardly any significance, since where it is greatest it does not exceed 8 minutes. And for the same reason, VFH, that is, EFD, is greater than the complement FDH of the eccentric anomaly, but by the quantity FHD, which is clearly of no significance. And since FED is somewhat more acute than a right angle, the arc circumscribed about FED will be somewhat longer than a semicircle. Therefore, ED is to DF as the sine of an angle which somewhat exceeds the complement of the eccentric anomaly, is to the sine which is slightly\u2014really, hardly at all\u2014smaller than the whole sine. Now if FD retains this length throughout the whole quadrant, ED is made approximately proportional to the sines of the complement of the eccentric anomaly. For if FD remains the same in length, and the end D is at A, the angle FDH is right, and thus FHD is a maximum, and then DFH is at its most acute, and consequently the arc above FD is at its longest. From that point, as FD moves down from A, the arc FED decreases and the angle FED increases, until at degree 90 FD becomes part of the line DH. Thus HF belongs to HD, and ED vanishes, and there, by analogy, the arc above FD is equal to the semicircle, and is at its least.","1376":"III","1377":"The ends of the small line of the planet's incursion towards the diameter of the apsides being connected, however long the line happens to be at any eccentric anomaly, the tangents of the angles at the center (and thus the angles themselves as well, when they are at their smallest), increase approximately in the ratio compounded of the ratio of the sines and the ratio of the sines of the complements of the eccentric anomaly; that is, in proportion to the rectangles on the quadrant formed by multiplying the sines of the angles by the sines of their complements. Thus, the greatest rectangle at 45 degrees is to the greatest angle at the same eccentric anomaly of 45\u00b0 as the remaining rectangles are to the angles of the remaining eccentric anomalies.","1378":"For at these angles, such as EHD, two factors are compounded: the length of the incursion, varying from nothing to a maximum, and its apparent magnitude, from nothing to a maximum. But, by I, the incursions increase in proportion to the sines, and by II, the tangents of the angles of apparent magnitude of these incursions, as if viewed from the center of the eccentric, decrease in proportion to the sines of the complement. By the former cause it happens that the angle is nothing at A when the sine is zero, and by the latter cause the angle is zero at an eccentric anomaly of 90, when the sine of the complement is zero; and further, at both places the rectangle has vanished entirely. But at an anomaly of about 45\u00b0, FD has now turned out greater than half [its maximum], because the sine, 70,711, is greater than 50,000, half the whole sine. And its angle EHD is greater than half by still more, because the sine of the complement is also greater than half, namely, 70,711 also. Consequently, the rectangle of the quadrant is the greatest of all, and at the same time is a square, equal to half the square on the radius, namely, 5,000,000,000.","1379":"IV","1380":"The angle of the planet's incursion from the circumference of the circle towards the diameter of the apsides is the same at the eccentric anomaly, about the center of the eccentric, and at the circular equated anomaly, of the same number of degrees, about the center of the sun.","1381":"Let the equated anomaly ANG be constructed equal to the eccentric anomaly AHD at the circumference of the circle G; that is, let NG be drawn parallel to HD. And from G let GX be drawn perpendicular to AC, and on it let GI be the correct incursion of the planet. And let I and N be joined. Now XG is to GI as VD is to DF, by I, while XG is to GN as VD is to DH, because of the similarity of the triangles. Therefore, IG is to GN as FD is to DH. Also, FDH and IGN are equal. So FHD and ING are also equal. And H is the center of the eccentric, while N is the center of the sun. Therefore, the angle, et cetera. Q.E.D.","1382":"V","1383":"The authentic and truest measure of the angle by which the fictitious equated anomaly, which depends upon the circle, differs from the true equated anomaly, which ends on the ellipse, is the rectangle contained by the sine of the fictitious equated anomaly and the sine of the complement of the true equated anomaly.","1384":" In the same diagram, when the sine of the angle AHD was multiplied by the sine of the angle VFH, the authentic measure of the angle FHD was going to result, by III. But by IV, the sine of the equal angles VHD and XNG is the same, and also, the sine of VFH, XIN is the same. Therefore, when the sine of the angle XNG, the fictitious equated anomaly, is multiplied by the sine of the angle XIN, the complement of XNI, the true equated anomaly, there results the authentic measure of the angle FHD, that is, by IV, of the angle ING, which is the difference between XNG and XNI.","1385":"Corollary","1386":"Because the difference ING is small, and is never greater than 8 minutes, the difference between the rectangles of the sines of XIN and XGN is going to be still smaller in effect.","1387":" From this, the following procedure will arise. The angle of the true equated anomaly being given, let its sine be multiplied by the sine of its complement. Let double the product, with the last five digits dropped, be multiplied by the maximum angle of incursion, at an anomaly of 45\u00b0. The product will be the angle of incursion at the given anomaly. This, added to the true equated anomaly XNI, gives the fictitious, XNG. By this angle, and the known sides NH, HG, the eccentric anomaly AHG is found, and the value of the triangle HGN*, as before.","1388":" Moreover, it is not difficult to find the maximum angle at anomaly 45\u00b0. Let VHD be 45\u00b0. Therefore, as the whole sine is to 70,711, so is the maximum incursion, or maximum breadth of the lunule, of 429 (or, more correctly, 432) to FD, 305. And since at 45\u00b0 HV, VD are now equal, subtract FD, 305, from VD, 70,711. The remainder, VF, is 70,406. This, with HV, gives the angle VHF, 44\u00b0 52' 34\", which differs from 45\u00b0 0' 0\" by only 7\u2019 26\". And this is the maximum of the angle ING","1389":".","1390":"The following is another method using analysis, whose fundamentals are these. In the diagram of Chapter 59, given the angle MNL, the ratio of the lines MN, NL is given, and I know that MN and LN are composed of parts in a known permuted proportion. For MN contains the (known) whole sine, and LN contains the known eccentricity HN. The remainder of MN has the same ratio to the remainder of LN, which is LH, as the eccentricity HN has to the whole sine. If you prefer, you may also refer to the diagram in Chapter 58. Therefore, let MN be 100,000 + 1x, LN from the angle MNL, 30\u00b0, be (8,660,300,000 + 86,603x)\/100,000, and NH be 9265 or 926,500,000\/100,000, so that HL would be (7,733,800,000 + 86,603x) \/100,000. But as HN, 9265, is to 1x, so is 100,000 to LH. Therefore, in the second instance, HL is (100,000\/9265)x, that is, (1,079,320\/100,000)x. Previously, it was (7,733,800,000 + 86,603x)\/100,000. With the denominators removed, as well as whatever can be subtracted equally from both sides, what remains is that 992,717x is equal to the number 7,733,800,000. Therefore, the single root is 7790. And MN is 107,790. And because as HN is to this root, so is the whole to LH, LH will therefore be 84,084, which is the sine of KE, 57\u00b0 14', the complement of the eccentric anomaly AK, 32\u00b0 46'. Now that this is found, the area AKN, the measure of the time or the mean anomaly, is found as shortly above. These things are clearest in the diagram of Chapter 58. Let GQ be the eccentric, AB the eccentricity, GD or LD the eccentric anomaly, FAC the equated anomaly, FA or EA the distance. So as AK is to AB, so is BC to KE. And at the equated anomaly CAO, as AR is to AB, so is BV to RM. So EK or RM is supposed to be one root. The rest is as above. ","1391":" But given the mean anomaly, there is no geometrical method of proceeding to the equated, that is, to the eccentric anomaly. For the mean anomaly is composed of two parts of the area, a sector and a triangle. And while the former is numbered by the arc of the eccentric, the latter is numbered by the sine of that arc multiplied by the value of the maximum triangle, omitting the last digits. And the ratios between the arcs and their sines are infinite in number. So, the sum of the two being set out, we cannot say how great the arc is, and how great its sine, corresponding to this sum, unless we were previously to investigate the area resulting from a given arc; that is, unless you were to have constructed tables and to have worked from them subsequently.","1392":"This is my opinion. And insofar as it is seen to lack geometrical beauty, I exhort the geometers all the more to solve me this problem: Given the area of a part of a semicircle and a point on the diameter, to find the arc and the angle at that point, the sides of which angle, and which arc, enclose the given area. Or, to cut the area of a semicircle in a given ratio from any given point on the diameter. It is enough for me to believe that I could not solve this a priori, owing to the heterogeneity11 of the arc and the sine. Anyone who shows me my error and points the way will be for me the great Apollonius.","1393":"Chapter 61 ","1394":"An examination of the position of the nodes.","1395":"The ratio of the orbs of Mars and the earth, the eccentricity of each, and the shape of their paths, have all been found with great certainty in the preceding chapters. Therefore, we can now easily accomplish here what we sought out in an approximate way in Chapters 11, 12, 13, and 14.","1396":"Let us begin with the nodes. On 1593 December 10, at 7h 0m in the evening, Mars was observed at 4\u00b0 44' Aries, with latitude 0\u00b0 1' 15\" south, parallax not accounted for. Its altitude being 35\u00bd\u00b0, it was not subject to refraction. After the 687 days of one complete revolution of Mars, on 1595 October 28 at 11h 30m pm Mars was found at an altitude of 51\u00b0 in 18\u00b0 35' Taurus, with latitude 4\u00bd\u2019 south, parallax not accounted for. And again, 687 days previously, on 1592 January 23, at 10h pm, it again had a southern latitude of 2', with an altitude of 25\u00b0. And finally, subtracting another 687 days, so that we come to 1590 March 7, Mars was observed on March 4 at 7h, at an altitude of 14\u00b0, to have a latitude of 3' 20\" south. This would have appeared larger, except that Mars was low enough to be refracted, and appeared too high. For the refraction at this altitude is 3\u00bd\u2019, of which about 2' is accounted to the latitude; thus, the apparent latitude would be 5' south. But since we are anticipating by three days the date corresponding to the others, three minutes are removed from the inclination by the approach to the node of 1\u00bd\u00b0 made in this space of time. When this is converted into latitude, however, the effect is somewhat less, so that the latitude remaining to Mars on the 7th would be 2\u00bd\u2019, and perhaps a little less, if the refraction were less. For its quantity is not perfectly constant.","1397":"Let the latitude in 1590 be 1 minute; 1592, 1\u00bd\u2019; in 1593, 2\u00bd\u2019; in 1595 at 11h, 4\u00bd\u2019, as we might allow an error from one source or another of one minute either way. These latitudes will indicate to us an inclination of 1\u00bd\u2019, which requires a distance from the nodes of about 40'. This is only for the sake of consensus.","1398":"We will nevertheless accomplish our aim more accurately using the year 1595. For while on October 28 at 12h the latitude was 4\u00bd\u2019 south, six days later, on the following November 3, at the same time, the latitude was 19' 45\" north. Therefore, over 6 days the latitude was changed by 24\u2019. So it changed 4\u2019 per day. And since on October 28 at 12h its eccentric position was 16\u00b0 8\u2153\u2019 Taurus, and the remaining latitude was 4\u00bd\u2019: let this be traversed in one day and one eighth, after which time 37' are added to Mars's position. Therefore, the node will be at 16\u00b0 45\u2156\u2018 Taurus, at the beginning of November of 1595.","1399":" About the other node, there was not such a crowd of observations. Therefore, the year 1589 alone will uphold the trustworthiness of this operation. For since on 1589 May 6 Mars had 6\u2154\u2018 of northern latitude, it traversed this in 2\u2153 days, according to the proportion of the latitudinal motion of the preceding days, [arriving at the node on] May 8 at 20h at which time its eccentric position is found to be 16\u00b0 42' Scorpio. In 1595, this would be 16\u00b0 47' Scorpio, the position of the ascending node, while previously we found the ascending node to be at 16\u00b0 45\u2156\u2019 Taurus. Therefore, at the end of 1595, the nodes are at 16\u00b0 46\u2153\u2018 Taurus and Scorpio.","1400":"Chapter 62","1401":"An examination of the inclination of the planes.","1402":"On 1593 August 25 at 17h 27m, Mars was observed at opposition to the sun at 12\u00b0 16' Pisces. On the 23rd its latitude was 6\u00b0 7' 30\". On the 24th it was 6\u00b0 5' 30\". On the 29th it was 5\u00b0 52' 15\". Therefore, in 5 days the latitude decreased by 13' 15\", while during one day before opposition, by 2'. Therefore, according to this proportion, if the latitude on the day and hour of opposition is taken to be 6\u00b0 2' 30\", there will not be half a minute's error.\n These latitudes were observed when Mars was at an altitude of 22\u00b0, which is now thought to be enough to free the fixed stars from refraction. Now since the equated anomaly was 166\u00b0 36', the distance between Mars and the sun was 138,556, and between the earth and the sun, 100,666. Hence, in the diagram of Chapter 13, if A is the sun, B the earth, C Mars, and AB is 100,666, AC 138,556, and EBC 6\u00b0 2' 30\", the declination BAC of the orbit from the ecliptic at this point is shown to be 1\u00b0 39' 22\". And since the node is at 16\u00b0 43' Taurus, I subtract from this 12\u00b0 16' Pisces. There remains an arc of 64\u00b0 27'. And as the sine of this is to the inclination here of 1\u00b0 39' 22\", so is the whole sine to 1\u00b0 50' 10\", the inclination of the southern limit. ","1403":"But since the position is rather far from the limit, in order to cut off any opportunity for suspicion, let us consult observations at positions other than acronychal, where Mars is near the limit. In undertaking this, I shall also present a universally applicable demonstration of the ratio between the inclination and the observed latitude. On 1593 July 21 at 14h (in astronomical terms), the planet was observed at 17\u00b0 45\u00be' Pisces, with latitude 5\u00b0 46\u00bc\u2019 south. At this hour the eccentric position of Mars is found to be 20\u00b0 1\u00bd\u2019 Aquarius, while the sun's position was 8\u00b0 26' Leo.","1404":" In the present diagram, let EA be at 8\u00b0 26' Leo, and KA at 20\u00b0 1\u00bd\u2019 Aquarius. EAK, the true angle of relative motion, will be 11\u00b0 35\u00bd'. Also, let EK be at 17\u00b0 45\u00be' Pisces. I say that the sine of AEK is to the sine of EAK as the sine of the inclination of K is to the sine of its observed latitude. For let the inclination of K be understood as a straight line dropped perpendicularly from the body of the planet to the ecliptic. So, as the distance EK is to the distance AK, so will the sine of the apparent angle of the line K as seen from A be to the sine of its apparent angle as seen from E. But as the sine of EAK is to the sine of AEK, so is the distance EK to the distance AK. Therefore, as the sine of EAK is to the sine of AEK, so is the sine of the apparent angle of the line K as seen from A to the sine of its apparent angle as seen from E.","1405":" The minor premise is known from trigonometry, and specifically, from Book 3 Number 14 of Lansberg's trigonometry. The major premise requires proof. Therefore, let there be the straight line VO, from two points of which, P and M, let two perpendicular and equal lines PQ and ML be set up. And let the ends Q and L be joined with a point on the line VO, and let this be O. Now, about center O, with radius OL, let an arc be described intersecting QO at N, and from N let a perpendicular NR be dropped to VO. Therefore, as PQ is to QO, so is RN to NO. But ML is equal to PQ. Therefore, as ML is to QO, so is RN to LO. Now ML is the sine of the angle LOM, under which the magnitude PQ or LM is observed from nearby, when LO, which is the shorter distance of the end L, is the whole sine. But QO is the longer distance of the magnitude ML, or of the end of PQ, namely, Q. And RN is the sine of the angle NOR, under which LM is observed, or the more remote line PQ, where NO, or LO, is again the whole sine. Therefore, as the sine of the apparent angle from nearby is to the longer distance, so is the sine of the apparent angle from afar to the shorter distance. And, permuted and converted, as the shorter distance is to the longer, so is the sine of the apparent magnitude from afar to the sine of the apparent magnitude from nearby. And in the present investigation, and universally as well, as the distance of Mars from the earth is to its distance from the sun, so is the sine of the latitude to the sine of the inclination of the planes. And, in turn, as the distance from the sun is to the distance from the earth, so is the inclination to the latitude. Q.E.D.","1406":"Since these things are certain, and since the line designated by K appeared to be 5\u00b0 46\u00bc\u2019 from E, multiplying the sine of this by the sine of EAK, and then dividing by the sine of AEK, results in the sine 3188, whose arc is 1\u00b0 49' 37\". And this is the amount of the inclination of the point K as it would appear from A. And since Mars is at 20\u00b0 1\u00bd' Aquarius, and the node is at 16\u00b0 43' Taurus, and thus the elongation of Mars from the node is 86\u00b0 42', therefore, as the sine of this elongation is to the whole sine, so is the sine of 1\u00b0 49' 37\" to the sine of the maximum inclination, 3200. Therefore, as before, this again gives 1\u00b0 50' 2\" south.","1407":"For the northern inclination, at midnight following 1585 January 31, at an altitude of 53\u00b0, the latitude of Mars, now decreasing, was 4\u00b0 31' north. But the true opposition was 16 hours 46 minutes previously, at 21\u00b0 36\u2159\u2018 Leo. Accordingly, the latitude then was 4\u00b0 31' 10\". And since the [full-circle] complement of Mars's equated anomaly was 7\u00b0 6' 23\", its distance from the sun was 166,334, and the sun's distance from the earth, 98,724. So, again in the diagram of Chapter 13, if AC is 166,334, AB 98,724, and EBC 4\u00b0 31' 10\", BCA comes out to be 2\u00b0 40' 50\". This, subtracted from EBC, leaves BAC, 1\u00b0 50' 20\". But because we are 5\u00b0 from the limit, the inclination of the limit will be about 25\" greater, namely, 1\u00b0 50' 45\". Before, the southern inclination was 1\u00b0 50' 8\". The difference of 37 seconds is clearly of no significance. The average of the two is 1\u00b0 50' 25\", the perfectly correct inclination, the same amount found in Chapter 13 above with various methods and operations, to which I again draw your attention here.","1408":"Now, if I compute the latitudes of Mars at opposition to the sun using this inclination of the limits, I find the following. 13 14 15 16","1409":"In the first, an observation on the day was lacking, as you have seen in Chapter 15. In the second there was an uncertainty of three minutes in the observation, since they occasionally used 34\u00b0 7' as the altitude of the pole, which was 34\u00b0 5\u00bd\u2019. The third has served as our foundation. The fourth agrees to a hair, if you neglect parallax, through which the observed latitude is wrongly corrected so as to be 3\u00b0 41', as you have seen in Chapter 15. In the fifth, we are wanting 2 minutes. It is surely the observation that is too high, on account of refraction, since Mars was no higher than 22\u00bd\u00b0, as you know from Chapter 15. In the sixth, you may note a slight defect of about two minutes. But the quantity of the refraction is not so reliable: what if it was two minutes higher? The seventh, again, served us as a foundation. The eighth without doubt had an erroneous declination, for at that time (8h) Mars was not at the meridian. And the armillary spheres, by which the declination is measured elsewhere than at meridian, err more easily than the quadrants. Furthermore, a comparison with nearby dates, as in Chapter 15, shows that the latitude was 0\u00b0 5' N., the same as we have computed. The ninth observation is not worthy of trust. However, the accurately examined calculation for December 10 closely agrees with the Fabrician latitude19 of 3\u00b0 23', for it gives 3\u00b0 21\u2154\u2019 N. The tenth comes close to the calculation. The eleventh corresponds to a hair when refraction is excluded. The twelfth is barely two minutes greater than the calculation. I believe this is because there is that much uncertainty in my instruments. For in my quadrant of six cubits, two minutes are not easily discerned. We therefore have the acronychal latitudes determined accurately enough throughout the entire circumference of the circle, using this inclination of 1\u00b0 50' 30\". An examination of the remaining latitudes at observations at positions other than acronychal, of which there are many closely spaced examples in this book, I leave to more diligent scholars.","1410":"Chapter 63","1411":"Physical hypothesis of the latitude.","1412":"It was said in Chapter 57 that if the diameter of the body or globe of Mars is supposed to possess a magnetic force, and to be directed towards the middle elongations, and also to remain parallel to itself in that disposition throughout its entire circuit, the physical hypothesis of the eccentricity is complete.","1413":"This supposition is all the more probable in that now the reason for the latitude too is explained using a closely related theory: that there be supposed some diameter of latitude in the body or globe of Mars that is directed towards the sidereal position of the limits, and remains parallel to itself in this disposition throughout its entire circuit. The ratio of this power to the former is that which [the power] of direction towards the pole in our magnets has to the force of attracting iron.","1414":"That is, the former seeks the sun or flees it, while the latter, rather than seeking by sailing towards, or fleeing, those sidereal positions beneath which the limits of the latitude are reached, is only directed towards those positions, as a magnet towards the pole (for likewise, a magnet does not sail towards the polar region even if it floats freely). ","1415":" In fact, the excursion of the planet from the plane of the ecliptic to either side follows the direction towards which this axis of inclination, on the side that leads in the motion of its body, is pointed. Let CBAD be the ecliptic, A, C the nodes;","1416":"B, D the limits. And let the axis of latitudes in the body of the planet be GNH, EAF, LOM, ICK.","1417":"Now since we are supposing this axis to remain equidistant from itself throughout its circuit, it will happen that as the body moves from the ascending node C to the northern limit B, the axis IK of the body, which initially, at the node C, was tangent as it were to the imaginary circle of its circulation through CNAO, later intersects it at right","1418":"angles at the limits N, O, being directed towards the center of the world S, that is, towards the sun. But also, owing to a certain amount of declination from the royal road2 CBA, this axis had thus far been enticing the body of the planet to leave that path in the direction of N, towards which it had turned the preceding part K. Now, at the limits, although it has indeed remained inclined to the plane of the ecliptic CBS (for we have said that it remains equidistant from itself in all positions, and so once it is inclined to the plane of the ecliptic it will always be inclined), it nevertheless does not continue to decline from the royal road itself, that is, from the circumference of the plane of the ecliptic CBAD, once it is placed at GH. For it does not incline forward towards A, nor back towards C, but only to the side or towards the pole, places to which its path is not directed. So when the planet is moved forward beyond B, the other part of the axis, G, which inclines towards the south, is now in front, and it thus leads the planet from its greatest northern inclination N through the descending node A to the greatest southern inclination O.","1419":" This axis of inclination is somewhat like an oar, in that just as boatmen use oars to move from one bank to the other, the planet brings this about through the inclination of the axis, moving it from north to south and back, while the river, that is, the immaterial species of the sun, proceeds along the direct path CBAD.","1420":"As for geometrical dimensions, there is no need for verbosity. A straight line moved on a rectilinear course while remaining parallel to itself creates a plane through its motion. This axis is itself a straight line, and it is moved in the direction it points (this pointing, moreover, presupposes a straight course). It therefore describes a plane. And if this plane be extended, it intersects the sphere of the fixed stars in the shape of a great circle, FEGH in the diagram of Chapter 13, because it intersects the plane of the ecliptic DC at the center of the world or the sun, A. To further convince yourself of this, you should consider that the points of intersection or nodes are in opposite positions about the center of the sun A, as you see in the diagram. This is shown by experience: see Chapter 62. And so since there is a plane in which the orbit of Mars is moved about, its inclination to the plane of the ecliptic will follow a pattern. That is, when two equal circles are described, one, DC, in the plane of the ecliptic, and the other, FE, in the plane of Mars's orbit, about a common center A of the sun (i. e., on one and the same sphere of the fixed stars, concentric with the sun), the sine of the arc BD between the intersection of the circles and some point on Mars's circle, D say, is to the whole sine as the sine of the inclination DF of the point F is to the sine of CE, the greatest inclination, at the limit E. Furthermore, it was proved in Chapter 13 above, using an ingenious treatment of the observations, that the declinations of all the points of the circuit from the plane of the ecliptic are arranged by the same measure. So no instance can be urged against our hypothesis. ","1421":"But there are still two difficult questions to be answered. One concerns how this declination of the axis originates, and the other concerns the axis itself. First, is the inclination of the axis natural or rational, the work of the body's nature or of an angel? And second, are the axis of inclination and the magnetic axis that seeks the sun the same? And if they are different, how do they exist in the same globular planetary body? The two questions are interrelated.","1422":"I might almost have believed it to be natural, owing to the similarity between the natural power in the magnet and this one, were it not for the successive transposition of the nodes that had also been added, which clearly seems the work of a reasoning faculty, if not discursive, at least instinctual. For to maintain its equidistant position is less marvelous and more in accord with nature than previously in the matter of the eccentricity. There, we said that it is the sun that is sought by the axis of power, while here, it is the position beneath the far distant fixed stars. There, the axis was to have been turned about by the force of this magnetic power as the body is carried around, and would not remain equidistant to itself, if it were not restrained by a stronger directional force, or by an animate force, either unassisted or capable of reasoning in some manner. Here, the axis follows these equidistant positions by the force of our directional power itself, with no need of an animate power or of reasoning. Someone might, however, consider it the act of a mind, that the diameter that effects the latitude points directly towards the center of the sun when the planet is located at the limits, thus making a great circle of the planet's orbit, and causing the nodes to be at opposite positions with respect to the sun.","1423":"To this argument, in Chapter 39 above, I affirmed that the planet moves with respect to the sun. However, it is not just any kind of relationship with the sun that argues for the assistance of reason. It is of course true that he who first ordained the heavenly motions so directed this axis as to point at the sun when at the appointed position, and did so deliberately and with perfect rationality.","1424":"But this relationship with the sun can now be maintained without a mind, by the constancy of the magnetic faculty alone. For it is more like rest than motion, and hence is material, not mental.","1425":"Therefore, it is only the variation of this inclination which we call the translation of the nodes over the ages that still makes a case for a motive force that is more than natural, or physical, as are magnetic powers.","1426":"Nevertheless, I would prefer to think that the two must be conjoined, rather than to suppose that the rational faculty acts alone. Let the magnetic faculty be subordinate; let the rational be in charge, ruling over it, just as we said before in Chapter 57 concerning the power of seeking the sun.","1427":"Once this question is settled, there follows the other. If this directive power arises from magnetic, physical, natural [powers], its substrate will be a body. Could it therefore happen that the same diameter that seeks the sun or flees from it also governs the planet's deviation from the ecliptic, by being inclined to it? If the nodes were connected to the apsides and the limits to the middle elongations then the diameter would be the same in all respects, administering both the eccentricity and the latitude.","1428":" For it was said in Chapter 57 that the diameter that causes the eccentricity is directed towards the middle elongations, while it was just now said that the diameter that causes the latitude is directed towards the limits. Therefore, if the limits were connected to the middle elongations, both diameters would have the same direction, and, their positions thus being in agreement, nothing would prevent their being identical. However, the nodes, or intersections with the true ecliptic, do not coincide with the apsides. For Mars, the northern limit is 12 degrees before the aphelion, for Jupiter, the northern limit and the aphelion coincide exactly, for Saturn, the node follows the aphelion by 24 degrees, and for the moon, owing to its short orbit, everything becomes interchanged with everything else. For now the node is at apogee, now at the middle elongations, now at perihelion [sic]. So since these two powers differ in time and position, it follows that they are not identical.","1429":"There is, however, nothing to impede their residing in one and the same planetary body as a whole, except the motion or rotation of the globe. Thus if the planets are moved like the moon, which does not rotate, but always shows us the same face, nothing prevents our saying that the two are interwoven, as the weft is interwoven with the warp. For since the entire body of the planet would then stay in the same sidereal position as it is carried about the sun, any of its rectilinear parts, among which are numbered those two diameters, will stay in the same sidereal position. If, on the other hand, it is the earth's globe that is in question, which has a daily rotation in addition to its annual revolution, we are left in great doubt, no less than before, in Chapter 57. For if the body rotates, only one single diameter of power, that which is parallel to the axis of its rotational motion, remains constant and equidistant from itself. So if you were to say that in addition to that diameter, there is interwoven with it another altogether different one, a power of another sort, which causes the latitude, it will observe the same direction as the axis of rotation, since it circumscribes a cone about that axis, successively traversing each of its parts, and since it inclines, now to the right, now to the left side, it finally leads the body towards the middle position, whither the axis of rotation points.","1430":"Therefore, if the globe rotates, the subject of this declinational power is either not a body but something spiritual, or is not the same body. If it is something spiritual, how does it look to certain regions of the world, which are corporeal? And how does it impart this kind of motion (declination from the royal road) to the body? Is it perhaps that the body is more easily inclined, and departs from the royal road more easily (meanwhile receiving the cause of its translational motion extrinsically, from the sun), than it is carried from place to place by the force of its own proper mover? If, on the other hand, we prefer a corporeal subject, some mechanism will be brought into being for us, like those spherical oil lamps which, though thrown and spun around, do not spill any oil. For within is enclosed a little flask which, being drawn down by a weight in its belly, and held there, does not follow the convoluted motion of the surrounding sphere.","1431":"Is there then also some interior globe within this globe of the earth, to which the diurnal motion of the earth's exterior does not penetrate, but which is held in place by a very strong inclination towards certain sidereal positions, so as not to follow the revolving exterior of the body? For as we shall see in Chapter 68, this question pertains to the earth as well. We shall also see there whether, if some mean ecliptic be proposed for the six planets, that which we were requiring a little earlier can be accomplished, namely, that the nodes of each of the planets correspond to the apsides.","1432":"Or is it rather to be believed that there are some possible modes of celestial motion which, though physical like the magnetic [powers], cannot be comprehended by anyone on earth owing to the lack of examples? For if we had lacked the example of the magnet (which was indeed unknown at one time), we would have been ignorant of most of the causes of the celestial motions. ","1433":"Those who believe in solid orbs can easily set everything right, following what was said in Chapter 13. For they will attribute to the plane of Mars's eccentric FE an inclination to the plane of the ecliptic DC that does not librate, but is fixed and constant, above the diameter BA of intersection of these planes, drawn through the center of the world (the center of the sun, for Brahe), and they will say that over the ages it rotates about the center A beneath the ecliptic DC. ","1434":"And since the poles (F, G and B, C in the present diagram) of two great ","1435":"circles (ML and KH) are distant by an amount equal to their maximum declination MK, LH, the poles of Mars B, C will therefore describe small circles about the poles of the ecliptic F, G, with radius FB, GC, of 1\u00b0 50' 25\". These people will also say that the poles of the Martian sphere B, C revolve forward with a motion quantitatively the same as that which was expressed above in Chapter 17, and which will be corrected below in ch 69. ","1436":"Chapter 64","1437":"Examination of the parallax of Mars through the latitudes.","1438":"In Chapter 61, the two nodes were found to be at positions exactly opposed, a marvelous agreement and one which excludes all parallax.","1439":"Let it be the case that Mars's parallax is at least 1' and 2' [respectively] when at opposition to the sun (and nearer to the earth than the sun) in 1595 and 1589, and that on the former date Mars was about 38\u00b0 from the zenith, and on the latter about 66\u00b0. Accordingly, in 1589, when it was thought to be at the node, it would still have been nearly 2' to the north. Therefore, it would still have been one degree before the node. So the node would be, not 16\u00b0 46' Scorpio, but 17\u00b0 46' Scorpio. In 1595, on the other hand, it would have 1' of parallax. Therefore, on the day on which it was thought to be at the ascending node, it would now actually have had a latitude of 1', and it would thus now have been about 30' beyond the node. Therefore, the ascending node would be, not at 16\u00b0 46' Taurus, but at 16\u00b0 16' Taurus. You see that the descending node is at 17\u00be\u00b0 Scorpio, and the ascending at 16\u00bc\u00b0 Taurus, if you make use even of the least parallax. Let us therefore conclude, as in Chapter 11, that Mars's diurnal parallax is entirely imperceptible, if it is indeed true that the two observations of latitude are correct within 2'.","1440":"Another argument for no parallax, which is not dissimilar, will arise for us out of Chapter 62, premised upon the investigation of the inclination of the planes, perfectly true unless refraction will throw something off.","1441":"Let it be the case that in 1593, at an altitude of 22\u00b0, Mars had a parallax of at least 2', while in 1585, at an altitude of 53\u00b0, it had a parallax of one minute. The observed southern latitude would therefore be smaller then the northern, and so the inclination would also be smaller. But now, just above, without parallax, it appears somewhat smaller, by an amount attributable to a small error in observation or to a certain amount of refraction at an altitude of 23\u00b0. Therefore, when parallax is considered, the observation would be charged with a greater error, and conversely, if the observation stands, the parallax is entirely eliminated, if it is indeed true that the orbit of Mars is contained in a perfect plane that intersects the plane of the ecliptic at the very center of the sun.","1442":"But the same is proved much more certainly from the latitudes observed at other acronychal positions, especially those which the circumstances of observation or refraction did not render dubious. As I began to say in Chapter 15, this matter has so far been impossible to settle. For in 1587, when Mars was 55 degrees from the zenith, if it had had a parallax of 4' its latitude of 3\u00b0 37' would have been increased to 3\u00b0 41'. But in Chapter 62, it was found to be no greater than 3\u00b0 37'. And in 1589, when the nonagesimal was 64\u00b0 from the zenith, if Mars's parallax had been 5\u00bd (judging from the sun's horizontal parallax of 3'), then the northern latitude, instead of the observed 1\u00b0 7', would have been 1\u00b0 12\u00bd\u2019 freed of parallax. But we have computed no more than 1\u00b0 5\u2153\u2018, although a slight error of 2' could have occurred in the observation, such as if Mars at an altitude of 22\u00b0, still subject to refraction, had appeared 2' higher (to the north) than was correct, as was said in both Chapter 62 and Chapter 15. And in 1602, when with a parallax correction the observed latitude was found to be 4\u00b0 10', and without the correction, 4\u00b0 7\u00bd\u2019, we computed 4\u00b0 7 \u2156\u2019, very precisely. Similarly in 1604 we did not agree perfectly with the observed quantity of northern latitude. Therefore, we shall complain that it is much less when it is increased through the removal of parallax.","1443":"By these three procedures, we have overcome our uncertainty about Mars's parallax. However, we have not completely proved that it is utterly imperceptible, since the matter of refraction eludes us, and besides, the observations do not descend to within 2 or 3 minutes. So if anyone wishes to attribute to Mars a maximum latitudinal parallax of 2 or 2\u00bd minutes, these Brahean observations do not significantly disagree with him. For the inclination, too, will be accommodated to this view, becoming 1\u00b0 51' 0\".","1444":"Chapter 65","1445":"Investigation of the maximum latitude in both regions: in conjunction as well as opposition with the sun.","1446":" Once the inclination is established, it is easy to define the maximum latitude, and this can be done in two ways. For one can find the maximum for all time, or how great it could be in our time. Today the two hardly differ, since the limits are the midpoints between the apsides of Mars and of the sun or earth, and they are no more than 54 degrees from one another, and the eccentricity of the sun or earth is not great. Nonetheless, let it be the case (as it once was) that the apsides of Mars and the sun coincide, along with the limits of Mars's latitudes. And let the ecliptic maintain its sidereal position. Now since, in the diagram of Chapter 13, Mars's greatest distance AC is 166,465, the sun's least distance AB is 98,200, and BAC is 1\u00b0 50\u00bd\u2019, the maximum northern latitude at opposition to the sun computed from these data is 4\u00b0 29' 10\". At conjunction with the sun, when the sun's distance from the earth is 101,800, this is decreased to 1\u00b0 8' 34\". But the southern latitude, from Mars's distance of 138,234, and the sun's of 101,800, is computed to be 6\u00b0 58' 24\", a little less than 7\u00b0. At conjunction with the sun, when the sun's distance is 98,200, this is decreased to 1\u00b0 4' 36\". If, however, one considers the contrary case, in which the sun's apogee coincides with Mars's perihelion, the maximum northern latitude at opposition comes out to be 4\u00b0 44' 12\", and at conjunction 1\u00b0 9' 32\", while the southern latitude at opposition is 6\u00b0 20' 50\", and at conjunction 1\u00b0 3' 32\".","1447":"And this is how it would be if the apsides and the limits were to coincide at some time, but whether this is going to happen before the whole fabric comes to ruin is uncertain. It is certain that Ptolemy attributed equal motions to the apsides and nodes, and if this were so, that conjunction would never happen. But even though today they appear to undergo different motions, the observations of the ancients are not sufficiently reliable, and the difference of these motions even in modern astronomy is not sufficiently great, for us to conclude how many myriads of years apart these conjunctions of the apsides and the limits occur.","1448":"Therefore, let us return to our era, that which extends between us and Ptolemy. And here, one who is looking for geometrically precise determinations is presented with a manifold obstacle to computation. ","1449":"First, the apsides of the sun and Mars are not in conjunction, and second, the planets' orbits are not perfect circles. So even if we project a new line of apsides through the centers of the circles of Mars and the earth (through B, C in the diagram of Chapter 52), it will still be possible for the nearest approach of the celestial bodies to occur elsewhere than on this line.","1450":"Finally, even if the position of the nearest approach were established, the position of the northern and southern limit is different. For example, the limit is at 16\u00b0 50' Leo. But the straight line BC projected through the centers of the circles is directed towards 24\u00bd Leo and Aquarius, approximately; in the same direction, that is, in which Brahe put his line of apsides HF, to which our line BC runs parallel, both eccentricities being bisected, AF at C, and AH at B.","1451":" And I was now about to choose the mean between 17\u00b0 Leo and 25\u00b0 Leo, namely, 21\u00b0 Leo, but the year 1585 gave me pause, since in that year the latitude observed at 21\u00b0 36' Leo was clearly not a maximum. For while the opposition was on the night following January 30, the latitude observed on the 24th, preceding the opposition, was 4\u00b0 31', still increasing, while on January 31, 16 hours past opposition, the observed latitude was again 4\u00b0 31'. It is therefore evident that on the 24th, if the opposition had occurred at that eccentric position, a latitude greater than 4\u00b0 31' would have been observed, for two reasons: first, because the celestial body was nearer the earth than when it was not at its acronychal position, and second, because Mars was farther from apogee, and was lower.","1452":" Therefore, let the maximum latitude occur about 19\u00b0 Leo and Aquarius, where Mars was on January 24. And since the supplement of the equated anomaly was 10\u00b0, the distance of Mars will be 166,200, and of the sun, 98,670. And so the maximum northern latitude will be about 4\u00b0 31\u00be\u2019. At conjunction with the sun, since its distance is 101,280, this appears as 1\u00b0 8' 30\". ","1453":"For the maximum southern latitude, Mars's equated anomaly of 170\u00b0 shows us a distance of about 138,420, and the sun at 19\u00b0 Leo has a distance of 101,280. Hence it is concluded that the maximum southern latitude will be about 6\u00b0 52' 20\", and at conjunction it appears as 2\u00b0 4' 20\". ","1454":"Chapter 66","1455":"The maximum excursions in latitude do not always occur at opposition to the sun.","1456":"Concerning the maximum latitude that can occur in any particular period of Mars, however, it is a much more complicated business to define its exact positions geometrically, and also involves this great paradox, which I found emphasized among the observations of 1593 in Tycho Brahe's hand, in the following words:1","1457":"\"It is worthy of consideration that on about the tenth day of August Mars had its maximum southern latitude, and that it decreased afterwards, so that at opposition on the 24th it was about one fourth of a degree nearer to the ecliptic. However, the Canons2 do not show this at 18 Aquarius, even when the position of maximum latitude is corrected, no matter how that maximum latitude is derived there. The cause of this needs to be looked into carefully.\"","1458":"When I later had come to him in Bohemia, and frequently inquired about how the latitudes are arranged, he answered that the nodes are at opposite positions, and the line of intersection passes through the point of the sun's mean position, or through the center of its epicycle (for which see Chapter 67 below), and enumerated many other things. Reminded by this mention, he said, regarding the present matter, \"this is remarkable, that the latitudes reach their maximum before or after opposition to the sun.\" Mention was also made of this above in Chapter 15.","1459":"The cause of this occurrence is in fact contained in the true hypothesis of the latitude established in this fifth part; however, you would have almost as much trouble finding the boundaries of the maximum latitudes geometrically, as Apollonius of Perga had in finding the boundaries of the stations.","1460":"For in the business of the stations, a certain condition can be described through which the position of the stations may be known (and that condition is this, that the line of vision of Mars, the earth being in motion, remains parallel to itself). But the position of the stations cannot be demonstrated a priori from this condition without multiple calculations, owing to the confluence of many causes. And matters stand just the same with the maximum latitude for any given occurrence. For the latitude is greatest when the distance of Mars from the earth is increasing or decreasing in the same ratio in which the lines of Mars's inclinations increase or decrease. And the latitude is increasing when the ratio of the distance decreases more than the ratio of the lines of inclination, or when the former is decreasing while the latter, on the contrary, is increasing. And, in turn, the latitude is decreasing either when the distance of Mars from the earth increases more than the lines of inclination, each in its own proportion, or when the distance is increasing while the lines are decreasing.","1461":"These conditions are satisfied indiscriminately, now at opposition, now before, now after, depending on whether the opposition falls at the limit, or before, or after the limit.","1462":"That these results follow from the hypothesis of this work, my ephemerides prove. In 1604, about Feb. 25 or March 6, the northern latitude was a maximum, while opposition followed by an entire month. On Sept. 27 or October 7, in turn, the southern latitude was a maximum, while Mars was between its quintile and sextile aspects to the sun. Again, at the end of 1605 the northern latitude was maximum, while the sun was moving from quintile to quadrature with Mars. And, in turn, at the end of July of 1606, the southern latitude was maximum when the sun was trine with Mars. But in 1607, the maximum northern latitude occurred a little after the conjunction of Mars with the sun.","1463":"The reason why these things would appear remarkable in ancient astronomy is chiefly that Ptolemy and his imitators had fabricated the extremely intricate motions of inclination, deviation and reflection. For since Ptolemy clung to his invention of the epicycle, as soon as he saw that when the planet was at opposition to the sun (and was thus visible) the epicycle went out to one side, he immediately indulged in conjecture, asserting that at conjunction with the sun, when it is not visible, the epicycle goes out in the other direction, and generally, that at conjunction the epicycle does the opposite of what he observed it to do at opposition. This is done in order that there be some compensation and equality of return and coherence with the sun. However, this is not discovering the true by observing, but fabricating the observations by a falsely conceived fancy. Nevertheless, it should be condoned in him, since he had few observations. On this subject, see Chapter 14 also. ","1464":"But come, let us see whether our calculation gives the observed latitude on August 10. For we are sure of July 21 and August 25 of that year, since the calculation yields the observations upon which it is based. ","1465":" So, on August 10 at 13h 45m, Mars's eccentric position on the ecliptic is computed to be 2\u00b0 41' 18\" Pisces, the sun was 27\u00b0 37' 49\" Leo, the angle at the sun 5\u00b0 3' 29\", the angle at the earth 18\u00b0 25', and from the calculation Mars was at 16\u00b0 3' Pisces, while it was observed at 16\u00b0 7' Pisces. And since 2\u00b0 40\u2019 48\" Pisces, the position on the orbit, is distant from 16\u00b0 43' Taurus by 74\u00b0 2', the inclination will therefore be 1\u00b0 46' 10\". From this and the two angles mentioned, using the method given in Chapter 62, the observed latitude is found to be 6\u00b0 21' 14\", still two minutes more than the observation has. But lest the angle's small magnitude trip us up, let us use the true distances of Mars from the earth and the sun (as the method given above requires), or in their place, the true angles. In the diagram of Chapter 20 you see that CB, BA differ from CL, LA. And our method did not say that the sine of the angle LAB is to the sine of the angle LCB as CB is to BA, but as CL is to LA. Let the ecliptic position be 2\u00b0 41' 18\" Pisces, Mars standing beneath the point \u03bb, and \u03ba be the position opposite the sun, 27\u00b0 37' 49\" Aquarius. Therefore, \u03ba\u03b2 is 5\u00b0 3' 29\", and \u03b2\u03bb is 1\u00b0 46' 10\". From this and the right angle \u03bb\u03b2\u03ba, \u03ba\u03bb or CAL is given as 5\u00b0 21' 36\", to which corresponds the true distance of Mars L from the sun A. So in triangle CAL, from the sides CA, 101,077, and AL, 138,261, and from the angle just found, let LCA be sought, which is found to be 160\u00b0 33'. Its supplement is 19\u00b0 27', to which corresponds the true distance of Mars L from the earth C.","1466":" So now, using these angles of the operation, I find the apparent latitude LCB to be 6\u00b0 19' 10\", very nearly the same as the observed value. ","1467":"Thus the hypothesis established in this work shows this very thing whose cause Brahe advised was diligently to be sought, and which ancient astronomy, for all its apparatus, cannot show. And, I would add, it shows this in all its simplicity, in that the plane of the eccentric is given a constant inclination or obliquity, and this is variously increased or diminished, not in reality, but optically only, insofar as our sighting approaches it or recedes from it, or (for Brahe and Ptolemy) it approaches or recedes from our sighting.","1468":"Chapter 67","1469":"From the positions of the nodes and the inclination of the planes of Mars and the ecliptic, it is demonstrated that the eccentricity of Mars takes its origin, not from the point of the sun's mean position (or, for Brahe, the center of the sun's epicycle), but from the very center of the sun.","1470":" The end is a reply to the beginning. In Chapter 6, I argued on physical grounds that when solid orbs are denied, the eccentricities of the planets cannot take their origin from any point other than the very center of the sun. I postponed part of the geometrical proof of this, based upon the observations, to Chapters 22, 23, and 52, in which places I think I have satisfied even the sharpest- eyed critic. The other part I shall now expound. This is done first through the positions of the nodes. It was proven in Chapter 61 that when Mars's eccentricity is constructed from the very center of the sun, or, what is the same, using acronychal observations taken when the planet is at opposition to the sun's apparent position, the nodes fall at positions that are very precisely opposite in relation to the sun's center; that is, that the diameter of the apsides and the diameter of the intersection of the planes of the ecliptic and of Mars coincide, or intersect one another, at the same center from which the eccentricity is computed, namely, at the center of the sun. The question now is, if we use the sun's mean motion instead of its apparent motion, will the nodes as a result still be at opposite positions about the point whence the eccentricity is computed? Not at all. Consider again the Copernican diagram in Chapter 6. In it let \u03ba\u03b4 now be the line of the limits, at 16\u00be\u00b0 Leo and Aquarius (not, as in Chapter 6, the line of apsides at 29\u00b0 Leo). Therefore, the line [NN'] drawn through \u03ba perpendicular to \u03ba\u03b4 will be the diameter of the nodes. But if we use the mean sun instead of the apparent sun, then we are given \u03b2 instead of \u03ba as the point from which the eccentricity is reckoned. So from \u03b2 let \u03b2\u03c2 be drawn perpendicular to \u03ba\u03b4 [and extended to PP']. This will fall at positions exactly opposite about \u03b2, but will not fall at the positions of the nodes, because the former perpendicular [NN'] through \u03ba, falling at the positions of the nodes, is above \u03b2\u03c2 by the distance \u03ba\u03c2. It is desirable to enquire into the magnitude of the angles at the circumference of the eccentric when the point \u03ba is connected to the points of intersection of the line \u03b2\u03c2 with the circumference of the eccentric. Since, by supposition, \u03c2\u03ba is at 16\u00b0 45' Leo, and \u03b2\u03ba is at about 5\u00b0 45' Cancer, the angle \u03b2\u03ba\u03c2 will be 41\u00b0; and since \u03b2\u03c2\u03ba is right, \u03ba\u03b2\u03c2 will be 49\u00b0. And since \u03ba\u03b2 is the sun's eccentricity, 3600, where the orb of the earth or the sun is 100,000, therefore, as the whole sine of the angle \u03c2 is to \u03b2\u03ba, 3600, so is the sine of angle \u03b2 to \u03ba\u03c2, 2717. And in the same units (where the semidiameter of the earth\u2019s orb is 100,000), the semidiameter of Mars's orb, from Chapter 54, is 152,350. Therefore, where the semidiameter of Mars's orb is 100,000, \u03ba\u03c2 will be 1790, showing an angle of 1\u00b0 1' 33\" in the [table of] sines. ","1471":"The ascending node [P'] should therefore have been moved backward, and the descending node [P] moved forward, by that number of degrees, minutes, and seconds, if I had been mistaken in taking the sun's center \u03ba instead of the Ptolemaic, Copernican, and Brahean point \u03b2. But in turn, where the observations are referred to the mean sun, and thus the point \u03b2 is taken, if this is done in error, and \u03ba should have been chosen instead, the ascending node [\u039d'] found from \u03b2 should be in a place farther forward, and the descending node [N] farther back, so as to shorten the northern semicircle [N\u2019\u03b7N] by an arc of 2\u00b0 3' 6\".","1472":"Let us see whether it happens in this way. In the observations of Chapter 12, considered approximately, on 1595 October 28 Mars was considered to have been at the node. From the Brahean equations, which rely on the point \u03b2, its eccentric position was found to be 16\u00b0 48' Taurus. And on the morning of 1589 May 9 we supposed Mars to have been at the other, descending, node. Using the same Brahean equations, we computed Mars's eccentric position to be 15\u00b0 44\u00bd\u2019 Scorpio at that time. So what I said should happen, does happen: there are 1 degree and 3\u00bd minutes less in the northern semicircle. If the observations are treated more accurately, as in Chapter 61, Mars arrives at the ascending node one day and 15 hours late. Therefore, about 50 minutes are added to the eccentric position, so that the planet falls at 17\u00b0 38' Taurus, in its eccentric motion. Accordingly, the abbreviation of the upper semicircle is 1\u00b0 53\u00bd\u2019, virtually equal to the computed value of 2\u00b0 3'.","1473":"Therefore, the point \u03ba is entirely confirmed, and \u03b2 is rejected. For why will the diameter of the intersection of the planes not intersect the diameter of the apsides in the center from which the eccentricity originates, as above? What would be the cause of such a thing?","1474":" The same is also demonstrated through the inclination of the planes demonstrated in Chapter 62, using the diagram of Chapter 20. The inclination, that is, the angle LAB under which the digression of the northern limit appears when seen from the sun A, was there found to be 1\u00b0 50' 45\". But the angle MAD, under which the southern limit's digression from the ecliptic appears when viewed from the sun A, was found to be nearly equal to it, namely, 1\u00b0 50' 8\". So the angles at A, above and below, are equal, and the line drawn from A to the ecliptic positions of the limits B, D, is one line (since it is in the one plane of the ecliptic). It was therefore concluded from this that the other line, drawn from A to the limits themselves L, M, is also one line; and further, that what is enclosed within the orbit of Mars is a single plane. Furthermore, if the common intersection of the planes were not at \u03ba in the former diagram (which is A in the present one), but at \u03b2\u03c2 (that is, below A in the present diagram), when the limits L, M are connected with some point on the line BD below A, the angle under which LB appears from that point would be smaller, and the angle under which MD appears would be larger, by about two minutes.","1475":"It is true that if we are allowed the liberty of making the parallax as great as we please, the arguments of this chapter are easily weakened. But it is a well- documented certainty that it is impossible to allow a parallax great enough to fully enervate this demonstration.","1476":"Also, since the point of this chapter was demonstrated most soundly in Chapter 52, I could take another tack, and instead of demonstrating this point by denying parallax, I could deny parallax, as in Chapter 64, by maintaining this point, which has its own demonstration in Chapter 52.","1477":"It does not matter which way you do it. For both points have other demonstrations. The present way occurred to me first, and suited my purpose of showing how everything is in agreement.","1478":"Chapter 68","1479":"Whether the inclinations of the planes of Mars and the ecliptic are the same in our time and in Ptolemy\u2019s. Also, on the latitudes of the ecliptic and the nonuniform circuit of the nodes.","1480":"It was said in Chapter 14 that in any one period of Mars whatever, the obliquity\nor inclination of Mars's plane to the plane of the ecliptic remains fixed. There\nis, however, some doubt whether this obliquity is the same, and fixed, for all\nages. The reason for the doubt is this.\nIn the first volume of the Progymnastica, p. 233, Brahe demonstrated that the latitudes of the fixed stars are different today than at the time of Ptolemy, the difference being this: that in the region of the summer solstice, the latitudes of the northern stars increased and those of the southern stars decreased; and, in turn, in the region of the winter solstice, the latitudes of the northern stars decreased and those of the southern stars increased. As one goes from these boundaries towards the equinox points, the alteration of the latitudes diminishes, until near the equinox points there is none at all. This observation of our time we shall accommodate to our principles laid down in Chapter 63, thus:","1481":"It is established that the sphere of the fixed stars is raised above the planets by an immense interval, and it is accordingly not affected by those motions that in the planets. Copernicus puts the matter very simply: the fixed stars is raised above the planets by an immense interval, and it is accordingly not affected by those motions that in the planets. Copernicus puts the matter very simply: the fixed stars are not subject to any motion from place to place, and thus are truly fixed forever in the same places. \n The ecliptic, in turn, is the great circle in the sphere of the fixed stars beneath which, for us on earth, the sun ever appears, and which it is seen to traverse annually. And whether this motion belongs to the sun or the earth, in either case it belongs to one of the planets. Therefore, the fixed stars do not themselves contain the cause of the ecliptic: it only results from the annual motion of the earth or of the sun about the center of the world.\nThus, since the ecliptic is found to have changed its position with respect to the fixed stars, it is not the fixed stars that have moved away from the ecliptic, but the latter that has moved away from the fixed stars.\nThe reason for this translation is shown beyond doubt by our principles of Chapter 63, if, indeed, they are sound. Since the sun, through its most rapid rotation in its space which, for Copernicus, is the center of the world, sets the planets in motion through an emitted species, this rotation will have determinate poles. In the last diagram of Chapter 63, let the body of the sun be IO, and the poles of rotation be A, E, above which stand the points F, G on the sphere of the fixed stars. The great circle IO of the rotating solar body will thus be arranged beneath some great circle of the fixed stars: let this be ML. This is doubtless one and the same circle beneath the fixed stars, the poles F and G remaining constant, and the dignity of its body declaring that it first instils motion into the others. Nevertheless, the planets are found to move on various circles that are inclined to one another, owing to the natural principles explained in Chapter 63. Therefore, beyond doubt, the various circles of all the planets depend upon this \"royal circle\" ML, described by the rotation of the solar body about its axis AE, and each of them will keep its inclination to this circle constant in quantity, though having a translational motion, since we know by experience that the nodes are transposed.","1482":" Since the ecliptic, too, is one of the planetary circles, either the sun's or the earth's, it is consistent for it, too, to have some inclination to the royal circle ML, described among the fixed stars by the great circle IO of the solar body. For what would be the reason why the other planets would decline from one another, while the ecliptic alone, standing above the solar or the terrestrial path, coincides exactly with this royal circle ML?","1483":" Let this therefore be granted: that the ecliptic, properly so called, is inclined to the royal solar circle. Let it be represented to us by the circle KH drawn among the fixed stars, and let its poles be BC. Under these conditions, we easily discover the occasion of the alteration of the fixed stars' latitudes: as the name suggests, these are computed from the true ecliptic and not from that royal solar circle, hitherto unknown. For the intersections or common nodes of the ecliptic, truly and properly so called (as a result of eclipses' occurring only beneath that line along which the sun proceeds), with the circle ML, which we might call the \"mean ecliptic\", will be carried along no less than the nodes of the other planets. Nevertheless, the maximum obliquity MK or LH, which is measured by the distance of the poles FB, GC, remains fixed and constant, as in the rest of the planets. That is, if about centers F, G, with constant radii FB, GC, small circles be described upon which we suppose the poles of the ecliptic B, C, to revolve, then the circle KH as well will depart from its original position on the sphere of the fixed stars FMG, and over the ages will make the southern limit come to be near the same fixed stars where the northern limit once was. Over a shorter period, however, it will be as follows. Since the limits K, H have not moved far from their fixed stars, their latitudes will be changed by some imperceptible quantity. However, since the nodes have progressed by the same amount from their fixed stars, the latitudes of their fixed stars will be altered more evidently. This is because at the end of the quadrant, near the limit, the sines of the inclinations increase by imperceptible increments, while at the beginning, near the nodes, these increments are quite perceptible.\nHence, because no change in the latitudes of the fixed stars is perceived near the equinoxes, while it is noticeable enough near the solstices, we correctly conclude that the limits of the ecliptic's latitudes are near the equinoxes, and the nodes are near the solstices. Therefore, the points K, H will be near the equinoxes. We likewise conclude this: that since the northern part of the true ecliptic flees from the north, in that the northern latitudes are increasing in Gemini and Cancer, the ecliptic's northern limit is therefore either in Libra, if the nodes progress, or in Aries, if (as is more probable) they retrogress. For the moon's nodes also retrogress, traversing the zodiac in 19 years, while the apogee progresses, traversing it in 8\u00bd years.","1484":"Now the sun's apogee, or the earth's perihelion, is at 5\u00bd\u00b0 Cancer, and thus by Chapter 57, the diameter of power, causing the eccentricity, points at the sun when the earth is at 5\u00bd\u00b0 Aries. But also, by Chapter 63, the diameter of power that causes the latitude points at the sun when the earth is at the limit, which is in Aries by the present Chapter 68. Therefore, by the same Chapter 63, both powers can be effected by the same diameter of the earth's body. Hence one may argue plausibly that this invisible circle or mean ecliptic and the true one known to us coincide at 5\u00bd\u00b0 Cancer and Capricorn.","1485":"If the aphelia of all the planets were arranged on a single great circle, we could say that this is what we are seeking. For then it could be true of all planets, as it is here in the earth's circuits, that the nodes coincide with the apsides, and thus both variations\u2014that of the eccentricity (in height) and that of the the obliquity (in latitude)\u2014are effected by the same diameter of power. This would free us from the great difficulties with which we were left in Chapter 63.","1486":"And in fact the apogees of the sun, Mars, Jupiter, and Saturn fit approximately. For the aphelia of all three superior planets are in the same semicircle, and at the same time in the same northern direction. Therefore, the southern limit of the true ecliptic would be in Libra, and the northern in Aries, which agrees with the above.","1487":"A full consideration of this question must, however, be deferred until the motions of all the planets are examined with reference to the true ecliptic, the one known to us.","1488":"Further confirmation of this opinion of a hidden royal circle, projected from the sun among the fixed stars, is provided by the obliquity of the ecliptic that is in common use, which is computed from the equator, but which we might more correctly call the equator's latitude from the ecliptic. Now the equator is the great circle of the earth's body that is intermediate between the poles of the earth's daily rotation on its axis. And the same name of \"equator\" or \"equinoctial\" is given to that region of the sphere of the fixed stars that stands above the terrestrial equator in any era. The same name of \"poles\" is given to the points of the fixed stars that stand above the earth's poles in any era. Hence this axis, and this great circle, are inclined to the ecliptic differently in different eras. For to the extent that the northern latitude of the fixed stars in Cancer, and the southern latitude in Capricorn, is greater today, the equator's latitude from the ecliptic is smaller than it was once, since this obliquity is greatest in Cancer and Capricorn. It was once 23\u00b0 51\u00bd\u2019, while today it is 23\u00b0 31\u00bd\u2019, the difference of 20' being the change in latitude of the fixed stars.","1489":"It is, however, reasonable to suppose that the circle of the equator with its axis and poles would forever decline from the poles of this ecliptic HK by an equal and fixed distance, if the true ecliptic were the world's primary circle. But the ecliptic has changed, and the inclination of this axis to the ecliptic (and with it the inclination of the equator, to which this axis belongs) has been altered, so that to the extent that the ecliptic has receded from the fixed stars in Cancer, it has approached the equator. Therefore, the equator appears to maintain a constant inclination to some other circle. So a great cause, and a great dignity, ought to belong to this hidden circle. And thus from all these plausible arguments there arises a royal circle LOM, middle among the circles of the planets, to which all the planets, and Mars with them, maintain a constant inclination.\nThe example of the moon should not trouble us, whose inclination to the ecliptic, but not to any other great circle, is a constant 5\u00b0, both in the past and today, even though the ecliptic has been moved. For there is an enormous difference between the moon and the other planets. The orbs of the others encircle the center of the world. The moon's orb alone (roughly speaking) is outside the center and is transported from place to place. The others in common circle the sun, while the moon circles the earth. The eccentricities of the others and the whole theory of longitude and latitude originate from the sun, while those of the moon originate from the moving earth. The sun sweeps the others around in a circle, while the earth so moves the moon. What wonder, then, if the moon keeps the limits of its latitude constant with respect to the changeable ecliptic HK, beneath which lies the terrestrial circle, while the other planets do so with respect to some other invariable circle, such as LOIM? So the moon should not prevent us from giving credence to this theory.","1490":"It is therefore granted that Mars's orbit is inclined at a constant angle to some circle that maintains its position beneath the same fixed stars, such as LOIM. It follows that this same orbit of Mars has different inclinations to the ecliptic HK in different ages, since in certain of its parts it leaves the fixed stars it originally lay beneath and moves on to others. This only follows, however, if we grant that the nodes of Mars and the nodes of the earth, that is, the intersections of these orbits with the invisible circle LOIM, are not always carried over the same intervals in the heavens, some being faster than others. An authentic example of this was just given. For since the equator maintains a constant inclination to this invisible circle LOIM, while the ecliptic is meanwhile moved, the declination of the equator from the ecliptic is consequently perceived to be changeable.","1491":"Let A be the pole of the mean ecliptic, or the point upon which the straight line falls that is drawn from the center of the sun through the pole of the solar body. About center A with radius AB of 23\u00b0 42' (or thereabouts) let a smaller circle be described, and let B, C be the positions of the north pole of the world, or the points upon which falls a line from the center of the earth's body through the pole of the daily rotation on the same body, B at the time of Ptolemy, and C at our time. If the nodes of the ecliptic also retrogress, the northern limit must be placed near the fixed stars in the region of Aries and Pisces. For the northern latitude of the fixed stars in Gemini and Cancer has increased, as was previously said. Let the midpoint D between B and C be taken, marking the position of the pole of the equator at an intermediate time, and let AD be joined. Thus the circle AD extended will pass through the solstice of the intermediate time. From A at right angles to AD let AE be drawn, which, being extended, will pass through the vernal equinox of the intermediate time. Therefore, close to the line AE there would be the pole of the circle beneath which the orbit and circuit of the earth was once arranged. And because the northern limit is in Aries, let EA be extended in the direction of A, and let the point I be taken on the extension below A. Thus the Ptolemaic pole of the ecliptic would be at I. About center A with radius AI let a small circle be described, on which let another point O be taken, nearer to C than I is to B. And let O be the present pole of the ecliptic, 23\u00b0 31\u00bd\u2019 from C, while I, the Ptolemaic pole of the ecliptic, is 23\u00b0 51\u00bd\u2019 from B. This will be the theory of the ecliptic's altered obliquity and of the altered latitude of the fixed stars, except that the size of the small circle OI is not known to us. For the ecliptic's 20' alteration of obliquity can be produced in various ways.","1492":" And because O is today's pole of the ecliptic, and OC points towards the beginning of Cancer, let CP be the eighth part of the circle, and P the middle of Leo, where the northern limit of Mars is today. Let PO be extended beyond O, and let GI be drawn through I nearly parallel to PO, but slanting somewhat forward in longitude (for the sidereal position of Mars's limit was once a little farther forward than today), and let it be extended beyond I. And about A let a small circle be described, intersecting PO at F and GI at H. Let the size of the circle be sufficient to make OF greater than IH. And let the pole of the circle beneath which the circuit of Mars is arranged, be placed at F today, and at H in the past. Today\u2019s obliquity OF, or the inclination of the plane of Mars to the ecliptic, will be greater, and the Ptolemaic obliquity HI will be smaller. Nevertheless the pole of Mars\u2019s orbit H, F would have moved from H to F keeping at a constant distance AH, AF from A.\nAnd since the pole of the Martian orbit has traversed a fairly large arc from H to F, whether forward or backward, but at the same time the pole of the ecliptic has gone from I to O about the same point A, the pole of Mars would appear to be nearly at rest, since IH and OF are nearly parallel.","1493":"A great inequality in the motion of the nodes must indeed follow if it is true that the poles of the individual planets circle some common pole in different times.","1494":"For there is an anomaly originating from this source in the precession of the equinoxes, whose treatment is very much like the present ones. ","1495":"I have stated what is in agreement with the principles established in this work, and by what hypotheses this can be brought about, so that the inclinations of the planes can be made different in different ages. Let us now examine the observations of Ptolemy. For since the northern latitude of Mars is with Cor Leonis, a northern star, while the southern latitude is with the southern stars of Capricorn, it is reasonable that the same happened to Mars's maximum latitudes as happened to those stars: they both increased. For the stars' latitudes did increase, the northern ones around the summer solstice, and the southern ones around the winter solstice. Ptolemy therefore said that Mars's maximum observed northern latitude was 4\u00b0 20', while today it is 4\u00b0 32'. This confirms our opinion, since he shows a maximum latitude that is 12' smaller than today's, while the nodes stayed at approximately the same distance from aphelion as they are today. On the other hand, he makes the southern latitude about 7\u00b0, while today it could also be that much, namely, 6\u00b0 52\u2153\u2018. We are therefore left undecided by his observations. For concerning those 12' in the northern latitude, it should be noted that the smallest graduations on his instruments were 10 minutes, and that he usually supposed an error amounting to one of these parts. Also, the difference between the Greek symbols for 20' and 40' is very small and slippery, often neglected by translators. Nevertheless, the Arabic says 20' here.","1496":"There is nothing besides this in Ptolemy that can lead us to a judgement of how these matters stood in antiquity. For the observation examined in the following Chapter 69 is shown to be in error. Therefore, as long as we are wanting suitable observations from antiquity, circumstances compel us to leave this discussion of the motion of the nodes, along with many other matters, to posterity, if, indeed, it should please God to vouchsafe the human race a length of time in this world sufficient to work through such remaining questions thoroughly.","1497":"Chapter 69","1498":"A consideration of three Ptolemaic observations, and the correction of the mean motion and of the motion of the aphelion and nodes.","1499":"1","1500":"From all antiquity, there have survived no more than five written observations of the star Mars, as well as one of extreme antiquity noted by Aristotle, who saw Mars occulted by the dark part of the half moon. However, neither the year nor the time of day were given. Nevertheless, I have discovered, using a very lengthy process of induction extending over the 50 years from Aristotle's fifteenth year to the end of his life, that this could not have happened on any other day than the evening of April 4 in the 357th year before the commonly accepted epoch of Christ, when the 21-year-old Aristotle was, as we know from Diogenes Laertius, a student of Eudoxus. The second observation, obtained from the Chaldeans, was preserved for us by Ptolemy. This was made on the morning of January 18, 272 BC, when Mars occulted the northern star in the head of Scorpius. Here again, no particular time was given. The other four were by Ptolemy himself, using an astrolabe to measure Mars's distance from fixed stars. However, he reports only the zodiacal position at the exact moment of Mars's opposition to the sun's mean motion. ","1501":"Upon these few observations, arguments of the greatest moment are to be founded; or, if this is not possible, astronomy must remain incomplete. First, through the four Ptolemaic observations, the epoch of the mean motions, related to the fixed stars, belonging to Ptolemy's time, is to be found, and by a comparison of this with the modern data the mean motion itself is to be determined. Next, it seems possible to use the Chaldean observation to inquire whether the solar eccentricity was really once greater than it is today. And then, using both this observation and the Aristotelian one, if the time were known, one could hazard a guess at Mars's latitude at those times.","1502":"But by God immortal, what is this path upon which we shall be treading? For we have hardly anything from Ptolemy that we could not with good reason call into question prior to its becoming of use to us in arriving at the requisite degree of accuracy.\nI","1503":" First, at the times set out, he records the mean position of the sun by a calculation that depends upon observation of the equinoxes and solstices. The sun reveals the beginning of Aries, not by pointing a finger at the place, but by a blind conjecture of the time. For we call the beginning of Aries that point that the sun occupies when the day is observed to equal the nights. What if Ptolemy had been in error as to the time? We are not wanting conjectures. First, he does not give his method of observing it. My hope is that he observed the meridian altitudes, for by induction from these the moment of the sun's entering the northern hemisphere is obtained without error. But what if he had observed it using Alexandrian armillaries, where refraction could have put him off? He himself clearly suggests that he did so, when he says that the equinox was measured twice on the same day using these armillaries. He ascribes this to instrumental error; I, however, suspect that the error arose from refraction. ","1504":" Let it stand, however: let him have observed it through meridian altitudes. There is another suspicion that unwelcomely but most forcefully insinuates itself: the moments of the equinoxes recorded by Ptolemy do not agree within a day and a half when compared with the previous observations of Hipparchus and the later ones of Albategnius6 and Brahe, all of which conspire in a single equality. The Ptolemaic equinoxes alone deviate. This fact has given rise to many extremely complicated opinions on the heavens, and gave birth to the motions of trepidation and libration, all of which are brought to ruin once one sees that the observations after Ptolemy always agree to the point of equality with the most ancient ones of Hipparchus.","1505":" Ptolemy nevertheless supports himself by comparing the vernal equinoxes with the autumnal. For if, as a result of instrumental error, he had pronounced the true equinox to be on the following day, when it had occurred on the previous one, the autumnal equinox would have been pronounced to be on the previous day when it belonged on the following one. If two days were thus subtracted from the length of the summer, a great alteration in the sun's eccentricity would have resulted. Nevertheless, following his observations, he left this at the same quantity found by Hipparchus. So no alternative is left us but to trust Ptolemy in believing him to have observed correctly the time at which the sun stood at the beginning of Aries.","1506":"II","1507":"Once a beginning is made, and the obliquity of the ecliptic is found through observation, it is a trivial matter to use the sun's declinations on each day to report its elongation from the point occupied by the sun at the time stated to be the moment of the equinox, whatever it might be or in whichever sphere it might be established. For various authors thought up a variety of spheres for this purpose: after the eighth and ninth spheres established by Ptolemy, others have set up a tenth, and most recent authors an eleventh and a twelfth, all through the merest of speculations, against which overabundance of mechanisms7 Brahe vehemently inveighed. He never, however, told me what he intended to substitute in their place, nor did he leave it in writing anywhere. Copernicus, on the other hand, acted ingeniously and wittily (in the common opinion), and wisely (in mine), in removing his eyes from the heavens and seeking that point in the globe of the earth, above which, in the sphere of the fixed stars, there stands a point, determinate for any age you please, as was said in Chapter 68. However, this is not the place to discuss this matter further.","1508":"III","1509":"There follows a demonstration of the equation, which depends upon the sun's observed entry into the beginnings of the cardinal signs. For when the equation is subtracted from or added to the sun's apparent position, the sun's mean motion is established with respect to the point which the sun is observed to occupy at the time of the equinox. Here too, as to the quantity of the equation, there is greater uncertainty than before over the equinox or the beginning of the zodiac. For today that equation appears to be 20' less than the quantity which Hipparchus appears to have demonstrated for himself, and which Ptolemy retained. Nor is there sufficiently good reason why we should say that the ratio of the orbs is different today from what it once was. For affirmations of the greatest moment require the most solid evidence, and this we are lacking. And those observations cannot be that accurate, especially concerning the entry into Cancer and Capricorn. If we substitute the modern equations for Ptolemy's, we shall not change his observations as much as Ptolemy himself says he can discern in observing, nor as much as the Ptolemaic observations can be vitiated by the matter of refraction. For we can with certainty name the day of the Ptolemaic equinox observation, though meanwhile remaining uncertain about the time of day. And here, the partnership of the vernal and autumnal equinoxes is no such defense against the small error we are considering here as it was before against a large one.","1510":"That the equations of Ptolemy's day were really equal to ours, the constancy of the modern ones argues. For those found today by Brahe, and those found several centuries ago by Albategnius and Arzachel, are nearly the same.","1511":"There is therefore suspicion that the equation of the sun used by Ptolemy is in error, since it is deduced from erroneous apparent positions of the sun. Consequently, Ptolemy did not relate Mars to either the sun's mean or its apparent position, without the chance of error.","1512":"Nevertheless, there is this consolation, that we need to make use of the sun's apparent position, the comprehensive argument for which was given above.","1513":" We can, however, proceed on a twofold path: either we believe Ptolemy on the equinoxes, or, using the modern equations, we apply a correction to the Ptolemaic equinoxes, making the vernal equinox three hours later than that noted by Ptolemy, and the autumnal earlier by the same amount, the result being an error of 8' in the sun's declination at each place. Ptolemy's instruments were doubtless not calibrated with divisions any finer than 10'. And Hipparchus assigns an uncertainty of one such division. For this reason, the times, too, that the sun takes to traverse the quadrants of the zodiac, were not expressed more precisely than to within a quarter of a day. So much for the true length of summer and winter.","1514":"IV","1515":" But what shall we now say of the sun's entry into Cancer and Capricorn, upon which the apogee and the setting out of the equations depends? And of how easily a quarter of a day can diminish the vernal quadrant of the zodiac, and increase the autumnal? For the sun's entry into Cancer is quite imperceptible. Nor can I be persuaded that Hipparchus and Ptolemy examined the very moment of this entry, ignoring the intermediate points. I find it more credible that throughout the summer they were carefully noting the sun's declinations, and were matching up the equal ones on both sides of the solstice, taking as the true entry of the sun into Cancer the time intermediate between the moments of equal declinations. In this way, if a comparison had been made of positions near to the solstice, there would indeed be little error, but there still could have occurred as much as a quarter of a day, during which time 15' of the sun's motion elapse. Therefore, even if the equinoxes were perfectly certain, it is nonetheless possible that there be an excess or defect of a quarter of a degree in the sun's position in the two segments about the solstice, and that the apogee should then fall eight degrees farther forward or farther back. So much for the sun's motion.","1516":"V","1517":"Now, as for the observations of Mars itself, even if Ptolemy in fact managed to line the astrolabe up accurately on the fixed stars, there is still clearly no more certainty about Mars's zodiacal position (just as in the previous consideration of the sun's position) than there is about the positions of our fixed stars. If Ptolemy committed an error in assigning a fixed star its degree of elongation from the point of the equinox, the same error will be committed in proclaiming the position of Mars. Furthermore, not even the elongation of the fixed stars from the sun (and thus from the point of Aries from which the sun's elongation is known through its declination) is free from suspicion of error. For consider both the manner in which it is found and the argument of error. In the year 2 of Antonine Ptolemy sought it through the half-illuminated moon. Using the astrolabe, he found the moon's elongation from the sun, and that of Cor Leonis from the moon. Therefore, when the sun's elongation from the point of the equinox is given, the elongation of the fixed star from the same point is also given*. Now, in measuring the elongation of the moon from the sun, an error of half a degree appears to have been committed. For the measurement was made at sunset. But the sun when setting appears through refraction higher than it should by about half a degree. Therefore, the moon's elongation appears less than it should, and so also that of Cor Leonis from the sun, and likewise from the equinoctial. Thus it appears that half a degree must be added to the positions of the fixed stars at the time of Ptolemy.\n Therefore, when Ptolemy considered Mars (when observed in relation to the fixed stars) to be at opposition to the sun's mean position, it would now really have been half a degree beyond this opposition. So when these four observed positions of Mars are presented by Ptolemy: 21\u00b0 0' Gemini, 28\u00b0 50' Leo, 2\u00b0 34' Sagittarius, 1\u00b0 36' Sagittarius; we should take these: 21\u00b0 30' Gemini, 29\u00b0 20' Leo, 3\u00b0 4' Sagittarius, 2\u00b0 6' Sagittarius. Now Ptolemy actually defended himself against such presumption, affirming that he frequently had sought out this one thing, namely, the elongation of the fixed stars from the moon, of the moon from the sun, and hence the distance of the fixed stars from the sun and from the equinox point, and had always found it to be the same. So although he produced only one observation in order to demonstrate the method, it is nonetheless credible that he consulted several observations, at both the rising and the setting of the sun or moon, finally choosing the one which he saw to be intermediate among many operations that produced various positions.","1518":"Now this argument over 30' seems to be irrelevant to Mars's mean motion, if indeed on these four occasions Mars, as it was observed with respect to the fixed stars, can be referred to them, without consideration of the equinox point, whose distance is uncertain. This is the method I used in Chapter 17 above to investigate the position of the aphelion in Ptolemy's times. We are nevertheless hindered in this respect, that the observed positions of Mars are to be referred to a point opposite the sun's apparent position. This task can never proceed correctly unless the distances of both Mars and the sun from the common equinox point is previously known, since the arc of the true elongations of Mars from the sun cannot be deduced otherwise than through these components (so to speak).","1519":"If, at the moment taken as the true opposition of the bodies, the planet should appear to be 30' beyond the sun's true positions, the planet still has some involvement with the second inequality, and is not yet ready for an enquiry into the first inequality. And at apogee, these 30' in the equation of the orb occupy a large arc on the eccentric, to which corresponds an even larger portion of the time or the mean motion. At perigee the opposite takes place. For this equation of the center occupies a small arc on the eccentric, to which corresponds an even smaller portion of the mean motion. Therefore, anyone who says that on these four occasions Mars was observed 30' farther along on the zodiac, says in effect that at the equinoctial point Mars's mean motion was many minutes farther back at apogee, and a few minutes back at perigee. And since the arc on the eccentric is smaller than this 30' arc resulting from faulty observation, neither Mars's eccentric position nor even its sidereal position are as far forward as Mars itself appears to have moved with respect to the fixed stars, the difference being that quantity by which the arc on the eccentric differs from the 30' arc in the observation. And since this arc is large at aphelion, differing but little from the 30' arc in the observation, and the opposite at perigee, it therefore finally will follow that at aphelion a small amount, and at perihelion more, must be subtracted from Mars's mean motion with respect to the fixed stars, if we accept that the fixed stars are 30' farther forward on the zodiac. Thus not only is the mean motion made smaller (although by a much smaller quantity than the 30' resulting from faulty observation), but also the arrangement of the three acronychal observations used by Ptolemy is disturbed, whence must arise another aphelion and another eccentricity. However, this will cause us no trouble later. For we may neglect it, even if the observations introduce something large, as long as there is no suspicion of error in the fixed stars, since it is certain that they do not have the same precision as the Brahean ones. So we shall use the form of equation found through the Brahean observations, as if they remain the same throughout the ages.","1520":"Since we have encountered three forks in the road, one concerning the sun's eccentricity, another concerning the position of the sun's apogee, and the third concerning the zodiacal positions of the fixed stars and of Mars, there are therefore eight ways of establishing the mean motions and the aphelion at the moments of observation, even if, ignoring the zodiac, we compute only with reference to the fixed stars.","1521":"Let the first investigation retain all the Ptolemaic data concerning the sun and the fixed stars.","1522":" Since the positions of the sun's mean motion were 21\u00b0 0' Sagittarius, 28\u00b0 50' Aquarius, and 2\u00b0 34' Gemini, and the sun's apogee was 5\u00b0 30' Gemini, the apparent positions of the sun were 21\u00b0 40' Sagittarius, 1\u00b0 13' Pisces, and 2\u00b0 41' Gemini, all three beyond opposition. The true opposition therefore precede them. And since the diurnal motion at 21\u00b0 Gemini (Cancer, today) is about 23', and that of the sun, 61', and the sum is 1\u00b0 24', those 41' therefore require 8 hours, at which time Mars was visible at 21\u00b0 8' Gemini, opposite the sun's apparent position. Likewise, at 29\u00b0 Leo (Virgo, today) Mars's diurnal motion is usually taken to be 24', the sun's diurnal motion 59', and the sum 1\u00b0 23'. Therefore, a difference of 2\u00b0 23' requires 1 day 17 hours 21 minutes, at which time Mars was visible at 29\u00b0 31' Leo. Finally, at 3\u00b0 Gemini (Cancer, today) Mars's diurnal motion is 23', the sun's 57', the sum 1\u00b0 20', by which it is shown that for 7' there are required 2h 6m, at which time Mars was visible at 2\u00b0 36' Sagittarius.","1523":"12 13 14","1524":"To the first interval there corresponds a mean sidereal motion of 80\u00b0 57' 14\" beyond the complete cycles, and to the second, 96\u00b0 16' 24\". But in the former instance the apparent motion of Mars was 68\u00b0 21' 20\" beyond the complete cycles, with precession over the interval subtracted, in the amount that it was at that time. In the latter instance, the apparent motion was 93\u00b0 2' 20\".","1525":"Now let the hypothesis previously under investigation, established on the basis of the most recent observations, be brought in, and let the question be raised, at what anomalistic position do the apparent motions on the eccentric correspond to the mean motions, as I have just now given them? After a few trials, this is found: If for the last time the aphelion of Mars is placed at 0\u00b0 41' Leo, and for the remaining times somewhat before that, owing to the precession of the equinoxes, while at the first time the mean anomaly is 46\u00b0 37', at the second, 34\u00b0 21', and the third, 130\u00b0 37\u00bd\u2019; and thus the elongation from the equinox at the middle time was 5s 4\u00b0 59' 20\", then by the modern hypothesis of the equations the star Mars is placed at 21\u00b0 7' Gemini for the first time, 29\u00b0 [31']16 Leo for the second, and 2\u00b0 37\u00bd' Sagittarius for the third, fortuitously precise. For the foundations are not such as to allow one to hope for such precision. Had Ptolemy made note of more oppositions of his day, we would doubtless be experiencing greater difficulty. For with three solar oppositions it is handled easily. Compare this aphelion with Chapter 17.","1526":"Second, the Ptolemaic equation and apogee being retained, let 30 minutes be added to the fixed stars.","1527":"The result will be slightly different. For since Mars is half a degree beyond opposition to the sun, the corrected opposition will follow. The sums of the diurnal motions were 1\u00b0 24', 1\u00b0 23', and 1\u00b0 20'. Therefore, for the extra 30' the corresponding times come out almost exactly the same, to be added to all three, namely, about 8 hours 40 minutes. To this there corresponds 8\u00bd\u2019 of Mars's apparent motion, which is to be subtracted from these 30'. The remaining 21\u00bd\u2019 are to be added to the planet's positions, placing it at 21\u00b0 29\u00bd\u2019 Gemini, 29\u00b0 52\u00bd\u2019 Leo, and 2\u00b0 57\u00bd' Sagittarius. The intervals both of time and of zodiacal positions will remain almost exactly the same. Thus the distribution of the mean anomaly among these observations, which was just now found, will also be the same. Only the aphelion will be transposed by the same number of minutes, so that on the last date it is at 1\u00b0 2\u00bd\u2019 Leo. It therefore has to be moved 8\u00bd\u2019 back among the fixed stars. And the mean motion from the equinox will be increased by the above mentioned 21\u00bd\u2019, but it will be 8 hours 40 minutes longer. And to these hours there correspond 11' 24\" of mean motion. Therefore, at the proposed time, the mean motion from the equinox will be only 10' greater than before. But the positions of the fixed stars are 30' farther removed from the equinox. Therefore, Mars's mean motion with respect to the fixed stars has proceeded 20' less than before.","1528":"Third, the sun's apogee being transposed by 11 or 12 degrees, while the longitude and equation of the fixed stars remains the same.","1529":"Then on the first date the sun will be 20' back in position;19 on the middle date practically nothing will be changed; and on the last date it will be 21\u2019 farther along in position, owing to the altered equations of the sun. Therefore, the first opposition will be 4 hours later, and Mars will be the same number of minutes farther back in position, while the last opposition will occur 4\u2153 hours earlier, with Mars the same number of minutes farther along in position.\n22 23","1530":"The first interval of time becomes smaller. So also the corresponding mean motion is 5' 15\" smaller, so as to be 80\u00b0 53'. The second interval of time again becomes smaller. Therefore, the mean motion corresponding to it is 5' 40\" smaller, namely, 96\u00b0 10' 48\". So, since to the two mean motions, both smaller, there corresponds a greater apparent motion than before, and on the supposition that the mean anomaly is the same as before for both, the apparent motion is greater by about 9', it therefore appears that Mars has to move down from aphelion. However, the first interval is unchanged unless a great descent is made, while the second [is corrected] by a descent of about 36\u2019. Therefore, if we were to indulge in our enquiry, and not take the modern hypothesis as given, we would arrive at a completely different hypothesis with a new eccentricity. And if, on the other hand, these three observations of Ptolemy were perfectly certain, this would constitute the basis for an argument that he established the sun's apogee correctly.","1531":"But when 36' are subtracted from Mars's aphelion, placing it at 0\u00b0 3' Leo for the last time, and when its mean motion is so adjusted that for the middle time the [mean] anomaly is 34\u00b0 58\u00bd\u2019 with longitude from the equinoctial of 5s 5\u00b0 0' 50\", the following observations result:27","1532":"Again, an accurate enough approach. For we cannot hope that the observations were of such certainty. So, whether the sun's apogee is known correctly or not, the distance of the mean motion from the equinoctial is certain within 1\u00bd\u2019.","1533":"Fourth, the same things will be changed in the computed positions of the second case, and in establishing the mean longitude, that is, by transposing the apogee and the fixed stars. ","1534":"Fifth, the sun's apogee and the Ptolemaic longitude of the fixed stars remaining the same, the modern solar eccentricity is used.","1535":"Thus, while the first and last positions of the sun remain almost exactly the same, the sun's apparent position will be changed in the middle observation by 20'. For the former fall near the sun's apsides, where the equation is small, while the latter is near the middle longitude, where the equation caused by the eccentricity is maximum. And since in Aquarius the equation is additive, when 20' are taken away from the equation, the sun will be moved back through the same number of minutes, and will be, not in 29\u00b0 31' Aquarius, but in 29\u00b0 11' Aquarius. The correct and truest opposition therefore follows by 4 hours. The planet will then be at 29\u00b0 27' Leo. The earlier time interval and its mean motion is increased, and the apparent motion is decreased, while the later time interval is decreased, and the apparent motion is increased. So once again, more evidently than before, the application of this correction calls upon us to change the hypothesis, unless we had sworn allegiance to the words and numbers of the hypothesis of this era by our best judgement. For to move the planet forward a smaller distance in a greater time near apogee, and a greater distance in a smaller time near perigee, nothing can suffice but to increase the eccentricity. If everything were kept the same, as in the first case, the results for the first and third times would indeed again be the same as then, namely 21\u00b0 7' Gemini and 2\u00b0 37\u00bd' Sagittarius. But in the middle position, it would come out to be 29\u00b0 36\u00bd\u2019 Leo, where it ought to have been 29\u00b0 27' Leo, a difference of 9\u00bd'. To eliminate this, the aphelion ought to remain in about the same place, but the mean motion should omit 3\u00bd\u2019. Then the results will be:32 ","1536":"Sixth, the same change of the second case will occur, if we at once change both the sun's eccentricity and the longitude of the fixed stars.","1537":"Seventh, if on the other hand we at once change both the eccentricity and apogee of the sun, combining the third and fifth cases, the fundamentals will be these. 34","1538":"The first interval remains the same as in the first case, while the last is much altered. And because more of the path is traversed in a smaller time, it must descend farther down towards perigee. To 8 hours of mean motion correspond 10' 30\", to which add the extra 8 minutes of travel. The total is thus 18\u00bd, which we shall make up if we move the aphelion back by 1\u00b0 12', putting it at 29\u00b0 29' Cancer for the last time, with a mean anomaly of 131\u00b0 45'. Therefore, [the position of] its mean motion is 11\u00b0 4\u2019 Sagittarius, which, in the first case, was 11\u00b0 18\u00bd\u2019 Sagittarius. From this we compute:38","1539":"Finally, with alterations in all three data that we have taken from Ptolemy, the result will be the combined effect of the seventh and second cases.","1540":"It is therefore apparent that the epoch of the mean motion with respect to the equinox and the fixed stars is not much changed by an alteration in the sun's eccentricity, or in the apogee, or in both at once, but that it is only changed when the positions of the fixed stars are altered. For the third case adds 1' 30\", the fifth subtracts 3' 30\", and the seventh subtracts 4\u2019 30\". The second case alone subtracts 10\u201939 from the mean motion measured from the equinoctial, and 20' measured from the fixed stars. ","1541":"As a result of this, two epochs of motion at the time of Ptolemy are established.","1542":"But what if we make something suitable by combining the second and fifth cases, by which we can hold simply to the Ptolemaic longitude of the fixed stars, eliminating any need for us to suspect that there might be two epochs of Mars's mean motion? For Ptolemy explicitly affirms that in his observation he found the moon's distance from the sun to be 92\u00b0 8\u2019, the same amount he computed from his hypothesis of the moon's motions. Ptolemy would have spoken truly: he would have been skillful enough in observing, and would have plainly seen this distance on his instrument to be the same as that prescribed by his hypothesis of the moon's motions, which was not in error near the quadratures. From this, I argue thus. If the sun had been at 3\u00b0 5' Pisces, where Ptolemy placed it using his eccentricity, the moon could not have appeared 92\u00b0 8' from it, the measure that is just and computed from the hypothesis, for the reason that the setting sun reaches the eye by refraction, and appears higher than it actually is (and thus 30\u2019 farther to the east). But because the arc from the moon to the sun was observed to be 92\u00b0 8\u2019, and because of refraction, this was in actual fact 92\u00b0 38\u2019, the sun was therefore not at 3\u00b0 3' Pisces, but at 2\u00b0 33' Pisces. And this is in agreement with the fifth case, where we said that Ptolemy's maximum additive equation (which occurs at 5\u00b0 Pisces) becomes 20' smaller when today's eccentricity is used, thus putting the sun at 2\u00b0 43' Pisces instead of 3\u00b0 3' Pisces. And so, on the supposition that refraction is universal throughout all places and times, as is discussed in the Optics, and supposing this observation to stand, we arrive at the conclusion that the sun's eccentricity is less than that reckoned by Ptolemy.","1543":"It should not trouble you that I spoke of a refraction of 30', while this diminution is but 20'. For if you consider well, since 30\u00b0 Taurus had culminated, 1\u00b0 Pisces was then setting at Alexandria, and the sun, being at 3\u00b0 Pisces, consequently had an altitude of two degrees, or perhaps even more, and therefore the refraction was less than 30'; nor was all the refraction simply longitudinal. Thus these two causes43 were almost exactly the same in quantity, and cancelled one another.","1544":"However, anyone who knows anything of the Ptolemaic reckoning of the fixed stars will not consider this difference of ten minutes worth mentioning. For example, Ptolemy comes up with an interval of 54\u00b0 10' between Cor Leonis and Spica Virginis, although in the heavens themselves it is not more than 53\u00b0 59'.","1545":" Let us therefore follow whither our inclinations and our arguments lead: as in the first case, in the second year of Antonine, on the 12th day of Epiphi, at the 8th hour, at Alexandria in Egypt, let Mars's mean motion from the equinoctial be 11\u00b0 18' 30\" Sagittarius. This time corresponds to the common year of Christ 139 May 27. The difference of meridians between Hven and Alexandria is nearly two hours, from the most recent geographical tables. Therefore, at Hven in the year of Christ 139 May 27 at 6h the mean motion was 8s 11\u00b0 18' 30\". But in that year, Cor Leonis had a longitude of 2\u00b0 30' Leo, that is, 4s 2\u00b0 30' 0\". Therefore Mars's mean motion was 4s 8\u00b0 48' 30\" from Cor Leonis. But on 159944 May 27 at 6h Mars's mean motion was 0s 0\u00b0 47' 30\" from the equinoctial, while the distance of Cor Leonis from that point, as demonstrated by Brahe, was 4s 24\u00b0 15' 45\". Therefore Mars was 7s 6\u00b0 31' 45\" from Cor Leonis.","1546":"For each year nearly one second must be subtracted. Therefore, at noon on 1 January in the first year of Christ, at Hven, it is elongated in its mean motion by 5s 8\u00b0 52' 45\" from Cor Leonis. ","1547":"And so much for Mars's mean motion with respect to the fixed stars.","1548":"The motion of the aphelion will come out a little different from what it was above in Chapter 17. For in the year of Christ 139 May 27 it was at 0\u00b0 41' Leo, while Cor Leonis was at 2\u00b0 30' Leo. It therefore preceded the latter by 1\u00b0 49'. But today, in 1599 May 27, it is at 28\u00b0 58' 50\" Leo, while Cor Leonis is at 24\u00b0 15' 45\" Leo.","1549":"over an interval of 1460 Julian years, which makes an annual motion of a little greater than 16\". Therefore, the root of the Christian era, at noon on January 1, has this aphelion 2\u00b0 27' before Cor Leonis.","1550":"On the mean motion of the sun with respect to the fixed stars, treated in passing, for future reference.","1551":"In the year of Christ 139 Pharmouthi 9, which is February 23, at sunset at 5h 30m, 3h 30m at Hven, the apparent position of the sun was computed as 3\u00b0 3' Pisces; therefore, the mean position was 0\u00b0 43' Pisces. But the longitude of Cor Leonis was found to be 2\u00b0 30' Leo. Therefore, the mean motion of the sun preceded Cor Leonis by 5s 1\u00b0 47' 0\". But on 1599 February 23 at 3h 30m at Hven the mean motion of the sun was at 12\u00b0 47' 41\" Pisces, and Cor Leonis 24\u00b0 15\u2019 30\" Leo. So the mean motion of the sun preceded Cor Leonis by 5s 11\u00b0 27' 49\".","1552":"Over 1460 Egyptian years, 9\u00b0 40' 49\" are removed. ","1553":"Our conclusion is 2' 42\" less than that from the Prutenics over the same","1554":"number of years, and the epoch will be 5s 7\u00b0 14\u2019 36\" from Cor Leonis at the root of the Christian era on January 1. ","1555":"Similarly, the progression of the sun's apogee is found to be 8\u00b0 23\u2019, and at the root of the Christian era it was 1s 27\u00b0 48\u2019 0\" before Cor Leonis. ","1556":"Chapter 70","1557":"Consideration of the remaining two Ptolemaic observations, in order to investigate the latitude and ratio of the orbs at the time of Ptolemy","1558":" It is true, as I have more than once remarked, that Ptolemy had at his disposal many more observations than were presented in his Opus. This may be seen in the presentation of the method for investigating the ratio of the orbs, where he uses a single observation that is within three days of the opposition. For it was said in Chapter 53 that observations that are so close result in a very large error if they are off by even one minute. Nevertheless, let us follow his footprints, and upon the hypothesis just established, erected upon the foundation of the first case, let us compute this fourth position1 as well. 2","1559":"The sun's true position on the 12th was 2\u00b0 36' Gemini. Add the motion of the three days and one hour, near apogee, which is 2\u00b0 53' 40\" from modern experience. This makes it 5\u00b0 29' 40\" Gemini, and let the present distance of the apogee be used, 101,800. Therefore, the point opposite the sun and Mars's eccentric position differ by 1\u00b0 5' 6\". This arc appears to be 3\u00b0 43' 14\", so that Mars would appear at 1\u00b0 46' 26\" Sagittarius.","1560":"If, however, we use the Ptolemaic eccentricity of the sun, the sun's motion over the three days will be 1' smaller, and the sun will be at 5\u00b0 28' 40\" Gemini. The difference is thus 1\u00b0 4' 6\". The apparent magnitude of this arc, using the Ptolemaic distance of the sun and earth, 102,100, will be 3\u00b0 45' 45\", so the planet will fall at 1\u00b0 43' Sagittarius. But Ptolemy said that it was seen at 1\u00b0 36' Sagittarius. We have therefore come out 7' to 10' beyond the correct figure. But the least division of the Ptolemaic instrument, which can always be considered as the uncertainty, has the value 10'.","1561":"You should also note that if we have erred by two minutes in the eccentric position, we shall now err by seven minutes in the observed position. For let Mars be moved back to 4\u00b0 22' Sagittarius on the eccentric: it will now appear at 1\u00b0 36' Sagittarius.","1562":"Above, on Epiphi 12, there was also an excess of 1\u00bd'. So these results are in agreement.","1563":"And because at such proximity to opposition a difference in the eccentricity has little effect, let us also consult the more ancient observation. Between the morning of January 18 during the year 272 before Christ, and noon on January 1 of the year 1 of Christ, there are 272 Egyptian years, 51 days, and several hours. For since at Alexandria, the sun at 25\u00b0 Capricorn rises at 7h, and the morning observation of Mars was made one hour earlier, as dawn was breaking, it was therefore made at the sixth hour, which is the fourth hour at Hven, from which time there are eight hours until noon. From this time interval, by the principles laid down above, the sun's mean motion is found to have gone 5s 25\u00b0 32' 50\" beyond Cor Leonis, with an anomaly of 234\u00b0 54' 34\". The corresponding equation from Ptolemy is 2\u00b0 0' 30\", and from Brahe, 1\u00b0 42' 54\", additive; and for the former, the sun's distance from the earth is 98,790, and for the latter, 98,976. But Mars's mean motion was then 2s 6\u00b0 7' 12\" beyond Cor Leonis. Also, since the aphelion is 3\u00b0 40' 20\" before Cor, Mars's anomaly will be 69\u00b0 47' 32\", the equated anomaly 60\u00b0 15' 27\", and the distance 158,320.","1564":"Here we shall follow a twofold path to the end of the calculation. The first is through the Ptolemaic eccentricity and equation. Then the sun's elongation from Cor Leonis is 5s 27\u00b0 33' 20\", differing from Mars's eccentric longitude of 1s 26\u00b0 35' 7\" by 4s 0\u00b0 58' 13\". This arc length, and the distances of the earth and Mars from the sun, show an apparent elongation from the sun of 82\u00b0 43' 46\". Thus the apparent elongation of Mars from Cor Leonis is 3s 4\u00b0 49' 34\". And the second path is through the Brahean eccentricity and equations, if they are assumed to have been the same then as well. The sun's apparent position will be 17' 36\" farther back, or 5s 27\u00b0 15' 44\". Thus the angle of commutation is 4s 0\u00b0 40' 37\". Through this, together with our value for the sun's distance from earth, as if it too were the same then, Mars's apparent elongation from Cor Leonis is shown as 3s 4\u00b0 51' 28\". The difference between the two calculations is very small and of no significance. Is it then true that","1565":"Mars was observed as if placed upon or fitted to the northern star in the brow of Scorpius,","1566":"as the description of the observation proclaims? Let us see. For Ptolemy, Cor Leonis is at 2\u00b0 30' Leo, and the Northern Bright Star in the Brow of Scorpius is at 6\u00b0 20' Scorpio, at an elongation of 3s 3\u00b0 50' 0\". For Brahe, Cor Leonis is at 24\u00b0 17' Leo. The brow of Scorpius is at 27\u00b0 36' Scorpio. The elongation is 3s 3\u00b0 20' 0\". But Mars's elongation was just computed to be 3s 4\u00b0 51' 28\". The difference is a degree and a half.","1567":" Since he had confidence in this observation, it being the most ancient of those upon which he could have depended, Ptolemy doubtless established that ratio of the orbs which we have hitherto discovered in his numbers, and which this observation appeared to require. For in the mean motion computed for this time, he differs from me by no more than 20 minutes. The remaining discrepancy therefore comes from the ratio of the orbs. Now because he pretends to investigate this ratio using an observation three days from opposition, he made different things seem to be deduced from different phenomena. And so since the ancient observation was to be reserved for investigating the mean motions, he substituted the other one for finding the ratio of the orbs, which had already been found using the ancient one. As was just said, it is absurd to test the ratio of the orbs using an observation as close to opposition as the one by which Ptolemy pretends to have demonstrated this ratio.","1568":" Let no one therefore wonder at our differing by a degree and a half from the observation that Ptolemy summoned from antiquity: let him rather examine Ptolemy's ratio of the orbs, so different from those proven by present day observations, and consider that in order to keep this observation, Ptolemy corrupted the ratio of his orbs.","1569":" As for the observation itself, of which this is the verbal description: \u1f11\u1ff3\u03bf\u03c2 \u1f41 \u03c4\u03bf\u1fe6 \u1f08\u03c1\u03ad\u03c9\u03c2 \u1f10\u03b4\u03cc\u03ba\u03b5\u03b9 \u03c0\u03c1\u03bf\u03c3\u03c4\u03b5\u03b8\u03b5\u03b9\u03ba\u03ad\u03bd\u03b1\u03b9 \u03c4\u1ff7 \u03b2\u03bf\u03c1\u03b5\u03af\u1ff3 \u03bc\u03b5\u03c4\u03ce\u03c0\u1ff3 \u03c4\u03bf\u1fe6 \u03c3\u03ba\u03bf\u03c1\u03c0\u03af\u03bf\u03c5\u00b7 (In the morning, the [star] of Mars appeared to have just come upon the northern brow of Scorpius), I believe that an error was committed by Ptolemy, who understood the first star of Scorpius, while the observer nodded towards the fifth. This is proven from the words themselves. For the brow of Scorpius has six bright stars. Of these, there are three prominent ones, of third magnitude, or better, second. The remaining three are of fourth, or, by my estimate, third magnitude, and one is higher than the three bright ones, and farther north. Now if the observer called the \"Bright Star in the Brow\" (which Brahe correctly pronounces to be of second magnitude, and which Ptolemy understood to be the intended star) the \"Northern Brow\", did he not speak ambiguously in saying simply \"northern\" rather than \"brightest of the northern\", since the star was not the northernmost? Thus I, much more prudently, will take it to be the northernmost, the fifth in number, that was described by the observer.","1570":"Furthermore, my computed longitude of Mars agrees with this, and not with the Bright Star of the Brow; and this while the hypothesis remains valid which the modern Brahean observations have generated. For Brahe places the northernmost star at 29\u00b0 3\u00bd\u2019 Scorpio. Subtract Cor Leonis at 24\u00b0 17' Leo. The difference will be its elongation from Cor, 94\u00b0 46\u00bd\u2019. But our calculation puts Mars at an elongation of 94\u00b0 49\u00bd' or 94\u00b0 51\u00bd' from Cor Leonis. The difference is 3 or 5 minutes, not greater.","1571":"I do not deny that the latitude presents a difficulty for me, in that I interpret the words, \u1f10\u03b4\u03cc\u03ba\u03b5\u03b9 \u03c0\u03c1\u03bf\u03c3\u03c4\u03b5\u03b8\u03b5\u03b9\u03ba\u03ad\u03bd\u03b1\u03b9, as if he said, \"It appeared to approach so near that the two stars could be taken as if they were one, that they appeared to touch one another.\" The Arabic, however, translates it [with a word signifying] \"to have covered up,\" as if the Greek had read, \u1f10\u03c0\u03b9\u03c0\u03c1\u03bf\u03c3\u03c4\u03b5\u03b8\u03b5\u03b9\u03ba\u03ad\u03bd\u03b1\u03b9. Accordingly, in the Optics, p. 304, I used the word, 'superimposed'. The best word in German is \"drangesetzt.\" From this I reasoned as follows. Whether Mars ran beneath it centrally, or grazed its northern or southern margin, it could not have been removed from the star latitudinally by any great distance. For indeed, the latitudes are less uncertain than the longitudes, since the manner of their variation is simpler and more consistent, as is proven in this book. We now know that the node retrogresses with respect to the fixed stars, by 4\u00b0 15' during one \"year of the Dog,\"18 as was proven in Chapter 17. For Ptolemy, the northern limit was considered to precede Cor Leonis by 3\u00bd degrees. For us, over the intervening 410 years, it had retrogressed one degree, so that at the time of observation it would be 2\u00bd\u00b0 before Cor Leonis. Therefore, the node is 87\u00bd\u00b0 past Cor Leonis. But Mars is 56\u00b0 35' past Cor Leonis. Therefore, it is 31\u00b0 from the node, making the inclination 57\u00bd\u2019, which, by the parallax of the orb, results in a true latitude of 1\u00b0 7'. But now it is clear from Brahe that the latitude of the Bright Star of the Brow is 1\u00b0 5', while that of the Northernmost Star of the Brow is 1\u00b0 42'. So the latitude appeared to refute me concerning the Bright Star of the Brow, leading me to believe that this star was occulted by Mars, and not the other.","1572":"But this collusion of numbers is fortuitous. For in the latitude of the Northernmost Star of the Brow Brahe and Ptolemy are in agreement, the former pronouncing it to be 1\u00b0 46', and the latter, 1\u00b0 42'. In the latitude of the Bright one they differ. Ptolemy has 1\u00b0 20'; Brahe, 1\u00b0 5'. But the former numerical equality results from an error, and the latter difference is really more like an agreement. For the latitudes of the northern stars in Scorpius, Sagittarius, Capricorn, and Aquarius are smaller today than they once were by about 16' 20\", and those of the southern stars are greater by the same amount, since the ecliptic has been transposed and the declinations of the degrees of the ecliptic have been altered by the same amount, as Brahe proved and as we have said in Chapter 68. Thus if it is true\u2014and it is very true\u2014that the latitude of the Bright Star in the Brow of Scorpius is 1\u00b0 5' today, at the time of Ptolemy and Hipparchus it was no less than 1\u00b0 20', probably greater. And so Mars has a smaller northern latitude than either of the stars mentioned, and passed beneath both. For it is certain that even if we are too high by a whole degree in [the position of] the node, the latitude in the calculation was wrong by no more than three minutes. Also, it has now been shown in Chapter 64 to be entirely uncertain whether the northern latitude for Mars was also once greater in the southern signs. Therefore, my clever interpretation of the word \u201c\u03c0\u03c1\u03bf\u03c3\u03c4\u03b5\u03b8\u03b5\u03b9\u03ba\u03ad\u03bd\u03b1\u03b9 was in vain. It can only be explained as denoting the stars' being placed side by side in the same longitude; and on this ground, the one that I favor is just as good a candidate as the Bright, its greater latitude notwithstanding.","1573":" Consider whether the meaning could be this: that since in the northern part of the brow there are three stars in the form of a triangle, Mars was sighted in the middle of them, and was thus \"placed upon the northern brow\" of Scorpius, it having simply been made one of that number of stars that are in the northern part of the brow of Scorpius.","1574":"This interpretation is furthered by the observer's having said \"northern brow\" rather than \"northern star of the brow\", since he is denoting, not one single star, but an entire part of the constellation.","1575":"So these two ancient observations are of no use to us in estimating either the latitude or the ratio of the orbs at that time. Therefore, since there are no observations to the contrary to impede us, while the extreme likelihood of our position confirms us in it, let us conclude that the ratio of the orbs is also the same today as it was once, while the maximum latitudes today are somewhat altered.","1576":"Appendix B","1577":"On the Table of Oppositions in Chapter 15","1578":"by Yaakov Zik Table i:","1579":"Initial positions of Mars in Chapter 15 computed with Guide 9 using JPL DE430.","1580":"General notes:","1581":"The time is measured from midnight; Gregorian dates are obtained by adding 10 days to Kepler's dates. The dates are given in old style.","1582":"Geographical coordinates: Uraniborg 12 41 46 E, 55 54 30 N [50m]; Wandesburg (Wandsbek-Hamburg) 10 05 53 E, 53 35 10 N [15m]; Prague 14 26 E, 53 35 10 N [250m].","1583":"The longitudes are given in geocentric ecliptic coordinates.","1584":"1 2345","1585":"Table 2:","1586":"Table of oppositions of Mars, 1580-1604","1587":"COMPUTED WITH GUIDE 9 USING JPL DE43O.","1588":"Compare with the table at the end of Chapter 15 (reproduced below). Longitudes are given in geocentric ecliptic coordinates; dates are in old style.","1589":"Kepler's table, for comparison ","1590":"Mars from Earth to be a little greater than half of the distance of the Sun from Earth. Kepler computed the diurnal latitudinal parallax of Mars to be 3' 28\", from which he determined the latitude of Mars without parallax to be 2\u00b0 25'. Since Mars's distance from the nonagesimal was 56\u00b0 and its ecliptic latitude 2\u00b0 25', Kepler computed the value of the diurnal longitudinal parallax (i.e., the difference between Mars's distance from the nonagesimal and its ecliptic longitude) to be 3' 32\". Kepler's computations for the latitudinal and longitudinal parallax resulted in positions for Mars's ecliptic longitude of 198\u00b0 21' 30\" (18\u00b0 21' 30\" Libra), and its latitude 2\u00b0 21' 30\". The longitude of the true Sun was at 19\u00b0 20' 8' Aries (Figure 2A). The deviation of elongation between Mars and the Sun was 58' 38\". Kepler determined the diurnal motions of the Sun and Mars to be 58' 38\" and 22' 36\" respectively, and the sum of the diurnal motions was 1\u00b0 21' 14\". With the sum of the diurnal motions of Mars and the Sun (1\u00b0 21' 14\"), Kepler computed the time it takes for Mars and the Sun to move 58' 38\", that is, about 17 hours 20 minutes.","1591":" As shown in Figure 2B, the Sun is assumed to be at the center of the planetary system. The longitudes of Mars and Earth's in reference to the center of the Sun are 198\u00b0 21' 30\" and 199\u00b0 20' 8\" respectively. The deviation of elongation between Mars and the Earth remain 58' 38\". To find the position of Mars at opposition, Kepler computed the angular distance that Mars and Earth\u2014now substituting the place of the Sun\u2014moved during 17 hours 20 minutes; Mars moved eastward about 16' 20\" and the Sun westward about 42' 18\". Accordingly, Kepler determined the longitude of Mars at opposition to be 198\u00b0 37' 50\" from which he subtracted about 39\" in order to correct Mars's orbit; he got 198\u00b0 37' 10\" (18\u00b0 37' 10\" Libra).","1592":"The Sun moved westward and its longitude decreased from the time of observation to its position opposite to Mars. Therefore, the time of opposition is 17 hours 20 minutes before March 29, at 21:43, the time when the observation was made. Kepler determined the time of opposition on March 28, 4h 23m AM, old style."}}