Mathematical Philology in the Treatise on Double False Position in an Arabic Manuscript at Columbia University

This article examines an Arabic mathematical manuscript at Columbia University’s Rare Book and Manuscript Library (or. 45), focusing on a previously unpublished set of texts: the treatise on the mathematical method known as Double False Position, as supplemented by Jābir ibn Ibrāhīm al-Ṣābī (tenth century?), and the commentaries by Aḥmad ibn al-Sarī (d. 548/1153–4) and Saʿd al-Dīn Asʿad ibn Saʿīd al-Hamadhānī (12th/13th century?), the latter previously unnoticed. The article sketches the contents of the manuscript, then offers an editio princeps, translation, and analysis of the treatise. It then considers how the Swiss historian of mathematics Heinrich Suter (1848–1922) read Jābir’s treatise (as contained in a different manuscript) before concluding with my own proposal for how to go about reading this mathematical text: as a witness of multiple stages of a complex textual tradition of teaching, extending, and rethinking mathematics—that is, we should read it philologically.

The rules in this case are that the husband inherits one quarter of the estate, and that in dividing what remains, the son's share is twice each daughter's share. Suppose her estate is 100 dinars. How much does the son inherit?
You have never studied basic algebra. Or if you have, pretend for a moment that you have not. You know how to add, subtract, multiply, and divide integers and even integral fractions. So to answer this question, you begin guessing.
You guess that the son's share is 50. Since the husband's share is 100 4 25 = , and each daughter's share is half of the son's share, this would yield a total estate of 25 + 50 + 3 · 50 2 150 = . Too large. What about if the son's share is 20? Then the total estate would be 25 + 20 + 3 · 20 2 75 = . Too small. Maybe 40? But then the estate would be 25 + 40 + 3 · 40 2 125 = . Too large again. You could keep going, but you start to wonder if there is a better way to do this. Fortunately, you come upon a manuscript that includes a treatise that describes a method called "calculation by two errors" (ḥisāb al-khaṭa ʾayn). You've already made three errors, so this seems promising. You read on.
The present article is about just such a manuscript and just such a treatise. The manuscript is New York, Columbia University, or. 45 (ca. thirteenth century).2 The treatise is the Explication of the Demonstration of Calculation by Two Errors, Improved Edition (iṣlāḥ) by Abū Saʿd Jābir ibn Ibrāhīm al-Ṣābī. "Calculation by two errors," known in English as the method of Double False Position, appears as a minor chapter in the history of mathematics, especially when conceived as a linear history of progress from primitive problem-solving and limited understanding to sophisticated techniques and more complete theorems. Double False Position is a somewhat sophisticated technique but one that was at least at first glance entirely superseded by algebra.
To solve the above inheritance problem by Double False Position, we define x 1 and x 2 as the first and second guesses, y 1 and y 2 as the resulting outputs, where y = 100 is the desired output. The aim is to find x such that operating on x as stipulated in the problem yields the desired output y. Based on the above calculations, we can assign these terms the following values: x 1 = 50, y 1 = 150; x 2 = 20, y 2 = 75. We further define two "errors" e 1 = y 1 − y = 50 and e 2 = y 2 − y = −25. Finally, we plug these values into the formula And indeed, 25 30 3 30 2 55 45 100 + + ⋅ = + = , which was the deceased's total estate as stipulated in the problem. So the son's share is 30 dinars, making each daughter's share 15 dinars. Double False Position has allowed us to come to this conclusion without knowledge of algebra.
Once a systematic algebra of polynomials (albeit restricted to positive rational numbers and quadratic equations) had been developed by al-Khwārizmī in the ninth century,3 one might even expect Double False Position to have been abandoned altogether as superfluous.
But it was not. Part of the reason must be that Double False Position can come in handy even if one knows algebra.4 As Randy Schwartz has pointed out, not only was it an accessible method for a wide range of tradesmen with limited education to use; for some types of problems, it is in fact a quicker and simpler method than first expressing the problem as an algebraic equation and then solving for x. Schwartz's example is from a twelfth-century Latin treatise and can be summarized as follows: you carry some apples through three gates but at each gate must give up half of what you are carrying plus two apples to the gatekeeper; at the end you have one apple; how many did you start with? In the time that you will take to write out the algebraic expression corresponding to this problem, I can make two guesses, run them through the procedure, and have an answer from Double False Position.5 Indeed, the manuscript at the center of the present article contains not only a treatise on Double False Position with commentary but also Omar Khayyam's treatise on algebra, which built on al-Khwārizmī and his successors to produce a more systematic treatment that included cubic equations.6 This perhaps 3 Roshdi Rashed, The Development of Arabic Mathematics: Between Arithmetic and Algebra (Dordrecht: Kluwer, 1994), 8-21. 4 Indeed, it still has a place in modern mathematics as a standard way to approximate solutions to equations whose algebraic solutions are unknown. 5 Randy K. Schwartz, "Issues in the Origin and Development of Hisab al-Khata ʾayn (Calculation by Double False Position)," in Actes du huitième colloque maghrébin sur l'histoire des mathématiques arabes: Tunis, les 18-19-20 décembre 2004 (Tunis: Association Tunisienne des sciences mathematiques, 2004), 2-3. In this particular example, one might obtain the answer even faster by "working backwards": ([(1 + 2) · 2 + 2] · 2 + 2) · 2 = 36. But that is beside the point, which is that to solve this problem by algebra is exceedingly cumbersome compared to either of these numerical methods. 6 Rashed, Development of Arabic Mathematics, 43-50. Roberts philological encounters 5 (2020) 308-352 prove correct, it would place Jābir in the tenth century. (Otherwise all we have is the terminus ante quem provided by the text's commentators.) Interspersed with the original text is a commentary by the mathematician and philosopher Ibn al-Sarī (also known as Ibn al-Ṣalāḥ; from Hamadān; active in Baghdad;d. 1153-4),10 as well as a brief note by one Saʿd al-Dīn al-Hamadhānī. Works by both of these commentators appear elsewhere in the manuscript. As discussed in the following, Ibn al-Sarī's commentary points out a fatal flaw in Jābir's geometrical proof of the validity of the method of Double False Position; al-Hamadhānī explains a single aspect of Ibn al-Sarī's commentary.
In the early twentieth century, this treatise was studied by the Swiss teacher and historian of mathematics, Heinrich Suter (1848Suter ( -1922.11 Suter was primarily interested in the text as evidence that the method of Double False Position was known prior to the twelfth century, when it appears in Latin. His secondary interest was to evaluate the mathematical worth of the treatise; concurring with Ibn al-Sarī, he rated it quite low. (Suter subsequently gained access to a different Arabic treatise on Double False Position by the ninth/tenth-century Byzantine Christian scholar Qusṭā ibn Lūqā of Baʿlabakk. He concluded that it was of sufficient worth to merit being published in German translation.)12 Ibrāhīm, such as the well-known secretary of the Buyids Abū Isḥāq Ibrāhīm ibn Hilāl ibn Ibrāhīm ibn Hārūn (925-994) or his grandfather. See Alexandre M. Roberts, "Being a Sabian at Court in Tenth-Century Baghdad," Journal of the American Oriental Society 137, no. 2 (2017): 253-77; Sezgin, GAS, 5:314. Since the Sabians identified themselves as Abrahamic ḥunafāʾ, we might expect the name Ibrāhīm to be common among them. 10 Suter, Die Mathematiker und Astronomen, 120 = no. 287;Theodosius, Sphaerica: Arabic and Medieval Latin Translations, ed. and trans. Paul Kunitzsch and Richard Lorch (Stuttgart: Steiner, 2010), 2 (introduction), mentioned because one of the manuscripts of Theodosios's Sphairika says that it was copied from an exemplar that was copied from an exemplar in Ibn al-Sarī's own hand. 11 Heinrich Suter, "Einige geometrische Aufgaben bei arabischen Mathematikern," Bibliotheca Mathematica, 3rd ser., 8 (1907- Now, in the twenty-first century, I revisit this treatise as a locus for understanding the aims and approaches of premodern mathematical writers, readers of mathematical manuscripts, and twentieth-and twenty-first-century historians of mathematics. First, I sketch the contents of the Columbia manuscript containing Jābir's treatise ( §1). Then I present an editio princeps ( §2), translation ( §3), and analysis ( §4) of the treatise. After considering Suter's reading of this treatise ( §5), I offer my own historical, mathematical, and philological reading of the text, its manuscript context, and its commentators ( §6), arguing for the value of such a mathematical treatise-fatally flawed but nevertheless copied and read-for intellectual history, not only the history of mathematics, but also of how mathematical texts were read.

The Columbia Manuscript
New York, Columbia University, or. 45 is a medieval codex containing a significant collection of mathematical texts. It includes texts on astronomy and engineering, but the primary focus is geometry. Most of its texts are in Arabic; one is in Persian. A paper codex, most of it was written by a single scribe (Scribe 1) in a neat naskh (with some Persian features such as the shape of initial hāʾ), typically with 19 lines per page (texts no. 2-18, on fols. 15v-128v). Text no. 1 was written by a different scribe (Scribe 2), with no dots, also typically with 19 lines per page (fols. 1v-14v). The last portion of the manuscript, apparently added later, was written much more informally, in less regular scripts: one hand (Scribe 3) wrote items nos. 19 and 21 with no margins (fols. 129r-137v, 144v-146r), and another (Scribe 4) wrote no. 20 with slight margins (139v-143r). Two previous descriptions of the manuscript are known to me: a page devoted to the manuscript's contents in Gūrgīs ʿAwwād's catalog of Arabic manuscripts in American libraries,13 and the series of typewritten cards in the unpublished card catalog of Arabic and other Islamicate manuscripts housed at Columbia's Rare Book and Manuscript Library (RBML).14 ʿAwwād's list of the Islam," chap. 11 in Islamic Cultures, Islamic Contexts: Essays in Honor of Professor Patricia Crone, ed. Behnam Sadeghi et al. (Leiden: Brill, 2014), 306, citing Ibn Abī Uṣaybiʿah. 13 ʿAwwād, "Dūr al-kutub al-Amrīkiyyah," 262-63. 14 I am grateful to Jane Siegel, Rare Books Librarian at RBML, for introducing me to Columbia's Arabic manuscript collection and pointing me to this card catalog when I first arrived at Columbia in the Fall of 2015. Since then I have benefited from her knowledge of the collection's history. She kindly provided me with photographs of the manuscript so that I could continue to work on it after my departure from New York.

Roberts
philological encounters 5 (2020) 308-352 manuscript's contents is more complete but still quite limited. In what follows, I present a much improved and elaborated description.
My aim here, I should note, is not simply to improve our bibliographic knowledge by supplementing Sezgin's handbook on the evidence of this manuscript (that is a happy side-effect). Instead, I include a description of the manuscript as an integral part of the project to construct a philology that treats manuscripts not merely as warehouses to be mined for the texts they contain but also as evidence for the intellectual milieux that produced and studied them. Like the anonymous treatise on Double False Position, Jābir's attempt to improve it, Ibn al-Sarī's critique of that attempt, and al-Hamadhānī's brief gloss on that critique (and, we might be tempted to add, Suter's account and critique of the whole assemblage), the Columbia manuscript represents one of many historical layers of interest in and engagement with a particular mathematical problem.15 To interpret the manuscript as such, first we must read it.
ʿAwwād dated the script to the seventh Hijrī century (thirteenth century ce). Modern bibliographical notes added to the manuscript itself place it in the thirteenth or fourteenth century ce. The manuscript includes Arabic translations of ancient Greek texts, but many of its texts date from the eleventh and especially twelfth centuries.
Two brief notes, one on the treatise on Double False Position, are ascribed to one Saʿd al-Dīn Asʿad ibn Saʿīd al-Hamadhānī. This may be someone associated with the production of this manuscript or of the collection it contains. The manuscript refers to him as one refers to an acquaintance: "the wise judge (al-qāḍi al-ḥakīm) Saʿd al-Dīn Asʿad ibn Saʿīd al-Hamadhānī, may God preserve his high rank." The Persian treatise (no. 8) and features of Scribe 1's handwriting would tend to situate the manuscript's production in Iran. Its apparently close connection to al-Hamadhānī, as well as the prominence of another author from the same city, Ibn al-Sarī, points to Hamadān as a possible place of the manuscript's production and subsequent use.
[ 125, no. 9, and further 156-57 = §10; Sezgin, GAS, 6:246-248. 23 Al-Nasawī begins his preface (fol. 49r) by dividing astral scientists (ʿulamāʾ al-nujūm) into four cumulative levels (ṭabaqāt): those who know (1) calendars and the astrolabe; (2) basic astrology like planetary and zodiacal attributes and the astrological verdicts (aḥkām) that result from their combination-these he calls the "verdicticians" (aḥkāmiyyūn); (3) basic calculations of astral positions and use of astronomical tables and calendars; (4) mathematical astronomy (hayʾah) and geometrical proofs of the validity of such calculations-the province of the "complete astronomer" (al-munajjim al-tāmm). Most people of "our time," continues al-Nasawī, only reach the first two levels. 24 The beginning of the text is indeed about calendars and conversions between them, but this is presumably because al-Zīj al-Jāmiʿ began with this topic. Al-Nasawī's text soon proceeds to discussing the geometry and mathematical astronomy necessary for the construction of zījes. Al-Nasawī does not always give an example; for "part 4, chapter 1," he writes, "You need no example because it is obvious" (lā taḥtāju ilā mithālin li-annahu ẓāhirun; fol. 55v 6 ). 25 At one point (fol. 50v 4 ), al-Nasawī gives an example of how to convert a Hijrī date to other formats; the example he gives is of his own present day, which he gives as 12 Ṣafar 439 (8 August 1047 ce). A later hand pointed out that the text was composed in 439 ah, in a note in the top margin on the first page of the text (fol. 49r). The creator of the card catalog entry housed at Columbia's RBML seems to have misread this note as 429 ah (rather than 439 ah) and misinterpreted it as the year when the text was copied rather than when it was written. 26 Indeed, an anonymous reviewer has informed me that a recent study (Abū l- . This is the same person who added a brief note to Ibn al-Sarī's commentary on Jābir's "improved edition" (iṣlāḥ) of the treatise on Double False Position appearing earlier in this manuscript (no. 10). 16. a problem from Abū Naṣr al-Fārābī (d. ca. 950), Music fann 1, maqālah 1 (Masʾalah dhakarahā Abū Naṣr al-Fārābī fī l-maqālah al-ūlā min al-fann al-awwal fī l-mūsīqā, 122v-125r 2 ). 17. Abū l-Futūḥ ibn al-Sarī (d. 548/1153-4), Problem, on constructing "a triangle whose sides are equal to the diameter" of a given circle (Masʾalah min kalām Abī l-Futūḥ ibn al-Sarī, 125r 3 -126r 1 ).35 This is the same Ibn al-Sarī who commented on the treatise on Double False Position (no. 10 above). 30 Sezgin,GAS,5:333,no. 26, citing only ʿAwwād's description. ʿAwwād only mentioned the author's nisbah, so Sezgin treats this as a work of Aḥmad ibn Muḥammad ibn ʿAbd al-Jalīl al-Sijzī (explicitly named as the author of the next text in the Columbia manuscript), but the scribe here seems deliberately to have named someone else-whom is not clear to me. Someone by this name studied with ʿAbd al-Qāhir al-Jurjānī (11th century): Yāqūt al-Rūmī, Muʿjam al-udabāʾ, ed. Iḥsān ʿAbbās, 6 vols., continuous pagination (Beirut, 1993), 187, no. 53. This is probably a coincidence. 31 Sometimes called al-Saḥarī, but here (on fol. 119r 6 ) the name is marked with diacritics, as al-Sijzī. 32 Sezgin, GAS, 5:333, no. 25, citing only ʿAwwād's description of the Columbia manuscript. 33 Sezgin, Geschichte des arabischen Schrifttums, 5:336. Following Suter, Sezgin considers al-Qummī a "younger contemporary" of Aḥmad ibn Muḥammad ibn ʿAbd al-Jalīl al-Sijzī. 34 This "ibn" is a mistaken addition that should be suppressed to make the name correspond with that of the same personage in no. 10. 35 See n. 10 above.

Text
In what follows, I present an editio princeps of the treatise on Double False Position as it appears in New York, Columbia University, or. 45 (no. 10; fols. 109v-113r). As with the sketch of the manuscript's contents in the previous section, I carry out this work, philological in the narrower sense (textual criticism), in the spirit and in the service of a more ambitious philology that shows curiosity for and invests resources in texts preserved by a textual tradition that may at first sight appear irrelevant to modern critics.
Within the text, diagrams (see Figure 1) illustrate the geometrical definitions and proofs on fol.  of text, as if for a diagram that was never added; at the top of the next page a modern hand pencilled in the heading ḥisāb al-khaṭa ʾayn, but the original text simply began at the top of the page (with the beginning of Ibn al-Sarī commentary) with no new heading.
The commentaries by Ibn al-Sarī and al-Hamadhānī were originally copied as a single block of text visually undifferentiated from the main text of the treatise. Marginal and interlinear labels were subsequently added to distinguish commentary from focus text ( Figure 2). This suggests that an ancestor of the Columbia manuscript had the commentaries (or at least Ibn al-Sarī's commentary) in the margin; that a more proximate ancestor that descended from the first then incorporated the marginalia into the body of the text, probably rubricated or visually differentiated from the focus text some other way; and

Roberts
philological encounters 5 (2020) 308-352 that a subsequent scribe, perhaps the scribe of the Columbia manuscript, then copied this ancestor without rubrication.
In the Arabic text, I supply hamzas where necessary to suit modern orthography. I do not emend the consonantal skeleton without indicating it (e.g., with angle brackets), except that wherever the manuscript says ‫خاائين‬ ‫,ال‬ I write ‫خاأين‬ ‫.ال‬ When the letter jīm is used as a mathematical symbol, I print , even when that is not precisely the shape used in the manuscript.
In the translation, I render the Arabic letters used to represent geometric points with the English letter corresponding to the abjad order, i.e.,

Explication of the Demonstration of Calculation by Two Errors, Improved Edition by Abū Saʿd Jābir ibn Ibrāhīm al-Ṣābī
1. If you wish to calculate this sort of thing, then you will come up with an amount of the kind about which you inquire, whatever amount it may be, such as number, line, surface, or other things that can be calculated. That amount becomes the first estate.38 Then operate upon it as you were instructed in the question. If you happen to be right, then that is the answer. Getting it right this way is unreliable. If it errs from what you were seeking, find the amount by which you erred, and call it the first error. If the operation yielded an excess by that amount above what the question requires, then call it the excessive error; if it yielded a deficit, then call it the deficient error. Then after that come up with another amount different from the first, and call it the second estate. Do the same to it as you did to the first estate. If it errs, find the amount of the error and call it the second error. If it too is excessive, call it excessive; if deficient, call it deficient. Then look, and if the two errors are both excessive or both deficient, displace [i.e., subtract] the lesser from the greater; but if they are different, one excessive and the other deficient, then take their sum. Call the result of either operation the part; this will be the divisor.39 Then multiply the first estate by the second error, and the second estate by the first error. Look, and if you summed the two errors, then sum these two [products] too; and if you subtracted the lesser of the two errors from the greater of the two, then subtract the lesser of these two [products] from the greater of the two. Either way, divide the result by the part. The result is the answer.
2. The justification of this method: Each line is divisible into three segments.40 That line in its entirety multiplied by its middle segment,41 plus <…?>42 the product of multiplying the middle segment plus one of the two adjacent segments joined into a line by the middle segment itself plus the other segment joined into a single line.43 For example, the line AB is divided into three segments, namely AG, GD, DB. I'm saying that AB in its entirety multiplied by GD and AG multiplied by DB, summed together, are congruent with the two lines GD, AG summed together, which is AD, multiplied by the two lines GD, DB summed together, which is GB.44 The proof of this is for us to draw, upon the line GB, an equilateral right quadrangle45 GBED [read: GBEZ]; determine its diagonal, which is GE; extend from point D a vertical line upon side GB such that it intersects the square's diagonal at point T and meets side ZE at point H; place line YTK on T parallel to side GB; complete the oblong46 surface AE and extend line YTK straight until it meets the side of oblong surface AE at point L.
[See Figures 1 and 3.] It is clear that the surface GT is an equilateral right quadrangle [a square], and so is the surface TE. The two surfaces DY and TZ are equal because they are the two complements that are on either side of the diagonal of the square GBEZ.47 If that is so, then we say that the two surfaces 39 Literally, "that over which the division" will take place. 40 Call them a, b, c.
The text here appears corrupt; for maʿ perhaps read mithl, "is like," i.e., equal to. 43 (b + c)(b + a) = b2 + bc + ab + ac. As the text stands, it is not a complete sentence. If we emend maʿ to mithl, it still isn't quite right because the first half of the equation would be missing a term: ac. 44 I.e., AB × GD   57 b(a − c).

without multiplication or division, since DB is known, and the known DA is subtracted from it, leaving the known AB …
[Jābir, ¶5 continued.] … by the first error, which is DA. Since the two errors are different, one of them excessive and the other deficient, he summed these two products of multiplication (maḍrūbayn). The sum was congruent to the intermediate, AB, multiplied by the excess of the greatest over the least, namely GD, [by which] he meant DG, which is the sum of the two errors. Then he performed the division over it-I mean the division of the sum of the two products (maḍrūbayn). From the division came out the amount AB, which is the desired result.

[Comment, part 2.] … and for that [i.e., finding AB] he needs no other operation. Thus what this man is in fact doing with this operation64 is computing the amount stipulated for him in the question, not the desired result. Suppose for example we ask him for knowledge of what estate is such that if we increase it by half of itself plus a third of itself, it becomes eleven.65 Then it is as if he stipulated that eleven is AB, then stipulated [as his first guess] a number, say nine, and added to it half of itself plus a third of itself, resulting in sixteen and a half.66 Next he stipulates that this is the number DB67 then compares DB to eleven, which is AB, and finds [it] in excess over eleven by five and a half [which he assigns] to
67 I.e., let DB = 16 1 2 .
70 The phrase "in the second place" (ʿalā l-wajhi l-thānī) straddles the intervening comment; this block of focus text begins with al-thānī. GB (as appears in the next line). 72 I.e., GD − AB = GA and DB − AB = AD. 73 murtafaʿayn, lit., "raised [numbers]." 74 This rather interventionist emendation is meant to make the sentence make sense both grammatically and mathematically. On mathematical grounds, this is clearly the meaning that the text originally conveyed; I base the wording of this emendation on how the text describes similar terms in other equations. 75 I.e., DB × GA + AB × DG = BG × AD.

Roberts
philological encounters 5 (2020) 308-352 is to multiply, in place of BD, nine, which is related to it, by AG; and in place of BG, four, which is related to it, by AD, so that the sum of the two is 55.76 Then he divides it by the part, which is nine and a sixth, and the desired result emerges.
[Jābir, ¶6 continued.]77 … above AB by the amount DA, and he called DA the first, excessive error. Then he went back and tried another amount, coming up with GB, which he called the second estate. It too was found to be in excess above what was sought, namely AB. So he took the amount by which he had erred, which is AG, and it was called the second error, also excessive. Then he multiplied the second estate, GB, by the first error, AD, and he subtracted from that the product of the first estate, DB, multiplied by the second error, GA, for the errors in this case are both excessive.78 So he is left with two amounts equal to the product of AB multiplied by GD [read: GB?]. He carries out the division of this remainder by GD [read: GB?], which is the surplus between the first and second errors, and so he obtains from the division the amount AB, the desired result that he sought to know. 7. Also, in the third place, let DB, GB each be less than AB, such that the lines or numbers AB, DB, GB are also all three different, where AB is the greatest, DB is the intermediate, and GB is the least. Then DG is the excess of the intermediate over the least; AD is the excess of the greatest over the intermediate; and GA is the excess of the greatest over the least. [diagram 4] According to what we have demonstrated in the foregoing, //fol. 113r// AB × DG + GB × DA summed together is congruent to DB × DA. Let that [amount] be remembered. At this point too, the author of Calculation by Two Errors, since he was seeking to find AB, first took DB and called it the first estate. Using it, he erred by DA, so he called DA the first, deficient error. Then he went back and tried a second estate, and he happened to get GB, in which he erred by GA, so he called GA the second error, also deficient. Then he multiplied DB, the second estate, by DA, the first error, and he was left with an amount equal to AB × DG. When he divided this remainder by DG, which is the surplus between the first error and the second error, he obtained AB from the division, being the result which he was seeking. Q.E.D.
The end. Praise be to God, lord of the worlds, and his blessings upon our sayyid Muḥammad and all of his family. 76 The manuscript expresses this number using decimal 'Arabic' numerals. 77 The manuscript indicates the continuity between this text and the previous portion of the matn with a small circle that appears above both the last word of that portion (zāʾidan) and the first word of this continuation (ʿalā). 78 I.e., this is why he subtracted here instead of adding.

Analysis
The text does not explicitly distinguish between the original (anonymous) text and Jābir's revisions and additions. Nevertheless, ¶4 begins by indicating that the foregoing text ( ¶1-3) may require further elaboration, and then ¶5 explicitly refers to the author of the Calculation by Two Errors, which seems to be a short version of the title of the work that Jābir undertook to revise and improve. This strongly suggests that Jābir's contribution begins there, probably with ¶4 and definitely with ¶5. As for the comments by Ibn al-Sarī and Saʿd al-Dīn al-Hamadhānī, these are clearly labeled in the text.
In the present section, I will provide a rather detailed mathematical paraphrase and analysis of the treatise. Though mathematicians and historians of mathematics may find it excessive to spell out every step, my hope is that this will make explicit more of my interpretive reasoning. In other words, I aim to foreground the philology-the self-critical interpretation of a text and a textual tradition-involved in rendering a medieval mathematical text into modern mathematical modes of expression, rather than elide it.
Find x such that f (x) = y for some y ∈ R+.
Let x 1 be the first guess (māl, "estate"). Let y 1 = f (x 1 ). If y 1 = y, then x = x 1 . Lucky guess. If not, then let e 1 = y 1 − y be the first error. If y 1 > y, then e 1 is an excessive error (i.e., e 1 > 0). If y 1 < y, then e 1 is a deficient error (i.e., e 1 < 0).79 Let x 2 be the second guess. Let y 2 = f (x 2 ). Assuming y 2 ≠ y (so that the answer x is not simply x 2 ), let e 2 = y 2 − y be the second error, excessive if y 2 > y and deficient if y 2 < y.
Now if e 1 < 0 and e 2 < 0 or e 1 > 0 and e 2 > 0 (i.e., if e 1 e 2 > 0), then let the 'part' ( juzʾ) be j = |e 2 − e 1 |. Otherwise (if e 1 e 2 < 0), j = |e 1 | + |e 2 |. (In either case, we can express this as j = |e 2 − e 1 |, since when e 1 e 2 < 0, |e 2 − e 1 | = |e 1 | + |e 2 |.) Compute x 1 e 2 and x 2 e 1 . If e 1 e 2 < 0, then sum them together: |x 1 e 2 | + |x 2 e 1 |. (Since x 1 > 0 and x 2 > 0 by assumption, for the text does not employ negative numbers, this is just |x 1 e 2 − x 2 e 1 |.) If e 1 e 2 > 0, then subtract: |x 1 e 2 − x 2 e 1 |. (Thus, either way we are finding |x 1 e 2 − x 2 e 1 |.) Now divide by j to obtain (2) Using modern algebraic computation (including the concept of negative numbers), it is trivial to justify this method by expressing x in terms of x 1 , x 2 , f (x 1 ), f (x 2 ) then expressing f (x) as ax + b (i.e., reducing it to linear and constant terms) and reducing the result to which is the algebraic solution to the equation Perhaps more intuitively, working in the other direction, the method of Double False Position can be derived from a basic result of linear algebra, namely that two points define a line, whose slope is thus known (see Figure 4). Once we know (x 1 , y 1 ) and (x 2 , y 2 )-by choosing guesses x 1 and x 2 arbitrarily then calculating the corresponding outputs y 1 and y 2 )-we can express them in terms of the two (possibly negative) errors e 1 = y 1 − y and e 2 = y 2 − y: (x 1 , y + e 1 ) and (x 2 , y + e 2 ). The line's slope is then But of course this is not how the text proceeds.

Geometrical Proof of a Relation between Line Segments ( ¶2)
Let AB be a line (segment) subdivided by two points along it, G and D: A-G-D-B. The resulting line segments are related as follows: Proof: Construct diagram 1 (see Figures 1 and 3). The rectangles DY and TZ are equal because they are complements about the diagonal of the square GBEZ [Euclid, Elements 1.43]. This lets us equate the gnomon BAMZKY [supposing we label the lower-right corner of the diagram M]-which the text calls AY plus LZ-with rectangle AH, since the only difference between the two is that AH contains TZ rather than DY, but as we just saw, DY = TZ. The rectangle AY can be expressed as AB × YB = AB × GD, and rectangle LZ is Suter says that this step is flawed because Jābir unnecessarily restricts his result by using a square rather than a rectangle in the proof's geometrical construction. But this part of the text is not purporting to be the entire proof; it is simply proving a geometrical relation between lines and the numbers corresponding to their lengths. It is only when we arrive at ¶5 that Suter's critique hits home. Indeed, it is there that Ibn al-Sarī critiques Jābir-a critique that I believe amounts to the same one that Suter makes.

Generalize This Result to Any Three Numbers ( ¶3)
In today's algebra, this is trivial to prove. The text, however, offers a geometric proof that rests on the proof in ¶2: Assign the three numbers to segments of the original line in Figure 1: Furthermore, we can define all the differences more simply as their own line segments: DB − AB = DA, AB − GB = AG. (Also, though the text doesn't mention this relation explicitly, DB − GB = GD.) Substituting in these simpler expressions, we obtain

Relating This Result to Double False Position ( ¶5)
According to Jābir, the method of Calculation by Two Errors (Double False Position) can be mapped onto Equation 6.
In the first case (one error excessive, the other deficient): [This step, if I have understood it correctly, is Jābir's misstep: by defining the two errors as differences between the unknown, x, and, respectively, the two guesses x 1 and x 2 , Jābir has entirely changed their definition as it appears in the method of Double False Position, namely e 1 = y 1 − y and e 2 = y 2 − y.] Thus, continues Jābir, x 1 e 2 = DB × AG, and x 2 e 1 = GB × DA. Since one error was excessive and the other deficient, the method says to sum them: x 1 e 2 + x 2 e 1 = DB × AG + GB × DA. But (by Equation 6) we know that this equals AB × GD. Since GD = DG = DB − GB, we can write GD = DB − GB = (DB − AB) + (AB − GB) = e 1 + e 2 . Thus AB × GD = x(e 1 + e 2 ). Therefore, x 1 e 2 + x 2 e 1 = x(e 1 + e 2 ).
Then, Jābir tells us, the author of Calculation by Two Errors divided the left side of this equation by e 1 + e 2 to obtain the result x e x e e e AB x

The Same, for the Case Where Both Errors Are Excessive ( ¶6) [Instead of continuing in the order of the text and translation, I will skip the comments of Ibn al-Sarī and al-Hamadhānī for now and get back to them after finishing the analysis of Jābir.]
In the second case, both guesses produce an output that is greater than the target: DB > GB > AB, so that we can again return to Equation 4

4.8
Ibn al-Sarī's Critique of ¶5-7 [The critique focuses on ¶5, but as Ibn al-Sarī points out, it applies just as much to ¶6-7.] Ibn al-Sarī begins by pointing out that Jābir has misrepresented what the method of Double False Position entails. Jābir, Ibn al-Sarī explains, has defined x = AB, x 1 = DB, e 1 = DA = DB − AB = x 1 − x, and so on. Thus x = AB is the unknown quantity sought but it is used to calculate e 1 ; thus it has "become known without multiplication or division," since x 1 (our first guess) is known and, apparently, e 1 is known as well. Thus all one has to do to find x, as Jābir has defined it, is to calculate x = x 1 − e 1 . Therefore, Ibn al-Sarī continues, in effect Jābir is simply "computing the amount stipulated for him in the question" (y), "not the desired result" (x).
Ibn al-Sarī is not saying that Jābir does not know how to use the method of Double False Position in practice, but rather that Jābir's purported proof implies the faulty method Ibn al-Sarī describes.
Ibn al-Sarī proceeds to explain his critique by means of an example: suppose we want to know x such that The method implied by Jābir's proof would be to say AB[= x] = 11 (even though in fact y, not x, is supposed to equal 11), then make a guess, 9, then plug it into the left side of Equation 9 to obtain 9 9 2 9 3 16 1 2 + + = .
But instead of defining DB (which is supposed to be x 1 ) as 9, the first guess, now Jābir would have us define DB as 16 1 2 (which is actually the output of the first guess, y 1 ) and then compares DB to AB, finding that as he has defined them DB AB DA − = = − = 16 1 2 11 5 1 2 .
This result is then called GB (or BG), so BG = 7 1 3 , and then compared to 11; the difference between them, defined as GA, is GA = − = 11 7 1 3 3 2 3 , which is then called the second error.
[Ibn al-Sarī skips the next step, presumably because it is obvious, namely taking the other product, DB × AG, where AG is GA, which is 16 1 2 3 2 3 60 1 2 ⋅ = .] Then we sum these two products to obtain 40 1 3 60 1 2 100 5 6 + = , and divide that sum by GD, where GD is defined as "the sum of the two errors," namely DA GA + = + = 5 1 2 3 2 3 9 1 6 . And so this means that we calculate 100 5 6 9 1 6 605 6 6 55 11 ÷ = × = . This, observes Ibn al-Sarī, is nothing but the desired output initially stipulated in the question (y), not the unknown that was to produce it (x). Thus, Ibn al-Sarī concludes, Jābir's proof is not about Double False Position at all because "the first number" (i.e., x 1 ) should be 9, not 16 1 2 (the first output y 1 ); and "the second number" (i.e., x 2 ) should be 4, not 7 1 3 (the second output y 2 ). These numbers stand in a relation of proportionality to each other. In any case, this is not the main point he is trying to make.) If we had used Double False Position properly and calculated x 1 e 2 + x 2 e 1 = x e x e The next two parts of Jābir's proof (for the cases where the two errors have the same sign, positive or negative) follow the example of the first part, so Ibn al-Sarī doesn't deal with them individually; instead, he dismisses Jābir's proof as insufficient to tell us anything about Double False Position.

4.9
Al-Hamadhānī's Comment Here a brief comment by one Saʿd al-Dīn Asʿad ibn Saʿīd al-Hamadhānī appears, spelling out the calculation implied by Ibn al-Sarī's statement (indeed, al-Hamadhānī does precisely what I just did in my paraphrase of Ibn al-Sarī). He says that in the formula BD × AG + BG × AD one should replace BD (as Jābir had defined it) with the proportional number 9 and BG (as Jābir had defined it) with the proportional number 4, in order to obtain 9 3 2 3 4 5 1 2 55 ⋅ + ⋅ = . Then divide that by the part, which is 9 1 6 , to get "the desired result." (Al-Hamadhānī doesn't spell out what that result is, presumably leaving it to the reader to perform the calculation.)

Suter as a Reader of the Treatise
Suter did not have a high opinion of Jābir's treatise. He consulted the text contained in Leiden,Univ. Library,or. 14,.80 To judge from his description of the text, it was very similar to the version contained in the Columbia manuscript, including the intermingled commentary of (Ibn) al-Sarī.81 Suter did not deign to publish the text or a translation: "Since [the 80 Suter,"Einige geometrische Aufgaben,[23][24] Item no. 4 of the Leiden manuscript, Suter describes, "contains not only … the commentary but also the text's continuation mixed together with glosses" ("Nr. 4 enthält nämlich nicht nur … den Kommentar, sondern die Fortsetzung des Textes mit Glossen text's] proof itself is a bit flawed, it would be a waste of effort to wish to provide a complete word-by-word translation of it …"82 Instead, he summarizes the proof and points out its flaw. In the process, he says, "I … avail myself as often as possible of our present-day manner of representation," that is, modern mathematical notation.83 Jābir's first step is correct, Suter remarks, namely his statement and geometrical proof of a relation between three arbitrary, consecutive segments of a line: given the line AB and two points G and D between A and B, in the order A-G-D-B, Jābir shows ( ¶2) that Roberts philological encounters 5 (2020) 308-352 Jābir seems not to have recognized, however, that this proof is valid only for a very special case, namely for the case where e 1 + e 2 is exactly equal to BG = x 2 − x 1 . This was also recognized by the glossator Aḥmad ibn al-Surrī [i.e., Ibn al-Sarī] when he remarks that for calculating the unknown here there would of course be no need at all for any multiplication or division, since x would of course be simply = AG + DG = x 1 + e 1 , or = AB − BD = x 2 − e 2 .85 Indeed, this is precisely the point that Ibn al-Sarī makes.
But Suter's next remark seems to misread the rest of Ibn al-Sarī's commentary. This suggests that Suter did not realize that Ibn al-Sarī's entire commentary (assuming it is the same in the Leiden and Columbia manuscripts) is devoted to addressing the same fatal flaw in Jābir's proof that Suter identified. As described in §4 above, Ibn al-Sarī begins by noting this fatal flaw then devotes the rest of his note to illustrating that flaw with a numerical example. So Ibn al-Sarī is not pointing out "another error" at all, as Suter thought, but simply seeking to make clear to his reader why Jābir's proof fails. Suter's explanation of Ibn al-Sarī's purported error indicates that Suter must have read Ibn al-Sarī's commentary very cursorily, since he imagines that Ibn al-Sarī was confused by Jābir's repeated redefinition of the line segments corresponding to the underlying quantities in question ( ¶4-5, 6, and 7). 85 Suter, "Einige geometrische Aufgaben," 25: "Ǧâbir scheint aber nicht erkannt zu haben, daß dieser Beweis nur für einen ganz speziellen Fall zutrifft, nämlich für den Fall, wo f (α 1 ) + f (α 2 ) genau gleich bg = α 2 − α 1 ist. Das hat auch der Glossator Aḥmed b. el-Surrî eingesehen, indem er bemerkt, daß es hier zur Berechnung der Unbekannten ja gar keiner Multiplikation und Division bedürfe, denn x wäre ja einfach = ag + dg = α 1 + f (α 1 ), oder = ab − bd = α 2 − f (α 2 )." 86 Suter, "Einige geometrische Aufgaben," 25: "Ein anderer Fehler, den der Glossator dem Verfasser vorwirft, ist aber unbegründet, er scheint übersehen zu haben, daß Ǧâbir el-Ṣâbî bei der Anwendung seines geometrischen Satzes auf die Regel der beiden Fehler andere Buchstaben annimmt als in der Beweisfigur …" Suter's subsequent remark confirms his cursory reading not only of Ibn al-Sarī but of Jābir's text as well. According to Suter, Ibn al-Sarī "also seems not to have correctly construed the sense of some admittedly obscure passages." Here Suter opens the possibility that he himself has overlooked something, continuing, we at least found no other error than the one already discussed, including in the continuation of the treatise, where the author [Jābir] gives the proofs for the cases where the errors e 1 and e 2 have the same sign, so that x 1 and x 2 are either both greater or both smaller than AD [= x].87 It is not clear which passages of Jābir's text Suter found "obscure," since he correctly understood that the rest of Jābir's treatise repeats the proof for the other two cases ( ¶6-7). It is even less clear, then, which part of Ibn al-Sarī he thought might be misinterpreting those obscurities. To his credit, Suter does not claim here to have a full understanding of either Jābir's or Ibn al-Sarī's text. In spite of this, he is nonetheless inclined to view Ibn al-Sarī's commentary as flawed.
Suter's subsequent discussion embraces the assumption that he, Suter, has understood the texts in question sufficiently to be able to evaluate Jābir's (and presumably also Ibn al-Sarī's) worth as a mathematician. He introduces his own corrections to Jābir's proof with the words Jābir certainly cannot have been much of a mathematical mind; otherwise, he would have recognized his own error and would easily have figured out how to come to his own aid: he could have generalized his proof in the following way …88 What then follows is Suter's revised version of the geometrical proof that omits certain constraints. In particular, he constructs the same diagram with the same labels (see Figure 3), but without requiring the rectangles BZ (= BGZE), 87 Suter, "Einige geometrische Aufgaben," 25: "… und scheint auch den Sinn einiger allerdings undeutlicher Stellen nicht richtig aufgefaßt zu haben; wir wenigstens haben keinen andern Fehler als den eben besprochenen gefunden, auch nicht in der Fortsetzung der Abhandlung, wo der Verfasser die Beweise für die Fälle gibt, wo die Fehler f (α 1 ) und f (α 2 ) beide gleiches Zeichen haben, also α 1 und α 2 entweder beide größer oder beide kleiner als ad sind; auf diese Beweise treten wir hier aber nicht mehr ein, sie sind leicht aus dem ersten abzuleiten." 88 Suter, "Einige geometrische Aufgaben," 25: "Ein bedeutender mathematischer Kopf kann Ǧâbir allerdings nicht gewesen sein, sonst hätte er seinen Fehler erkannt und sich leicht zu helfen gewußt, er hätte seinen Beweis in folgender Weise verallgemeinern können …"

Mathematical Philology
Suter was a prolific historian and philologist of Arabic mathematics and did much to advance the field. The son of a farmer and postmaster in a village outside of Zurich, he was remembered as an "unpretentious man," a hardworking and humble scholar who resolved to learn Arabic at the age of forty out of : e 2 = (x − x 1 ) : e 1 , i.e., that there is a fixed proportion between how far off the guess is (where we are dealing with the case in which x 2 > x and x 1 < x) and how far off the resulting output is. Since we are dealing only with linear functions, this constraint poses no problem; the inverse of that fixed proportion is (the magnitude of) the line's slope. 90 This exploits Euclid, Elements 1.43, to make a point that is less obvious than the one Jābir had used that theorem to make; see n. 47 above. 91 Suter, "Einige geometrische Aufgaben," 26: "Dies hat auch der Glossator Aḥmed b. el-Surrî nicht erkannt." a fascination with the Islamic world, and a "free thinker" who believed in the similarity of world religions and the common humanity of all.92 He worked at a time when even less (much less) of the relevant source material was available outside of manuscripts and when photographs of distant manuscripts were much harder to come by. He read many mathematical texts attentively and with great discernment. His publications on the topic are a vast repository of information and astute analysis and remain key references today. In the case of Jābir's text and Ibn al-Sarī's response, he does not pretend to have dwelt on it at length or captured every nuance of the text. For all these reasons, it would be rash, unproductive, and entirely unfair to hurl back Suter's insults at him, calling him not much of a philological mind, just as he called Jābir not much of a mathematical mind, and thus generalize about Suter based on a single section of a single scholarly article. Nevertheless, the rapidity of Suter's reading of the treatise was driven by the overarching priorities and methodological principles embraced by Suter and his fellow historians of mathematics. For this reason-combined with his warm and conscientious attitude toward Arabic mathematical texts, which rules out any facile dismissal of his work-it is perhaps worth dwelling for a moment on the characteristics of Suter's reading of the treatise before considering what alternative mode might best suit a different set of priorities.
Suter was working within the tradition that approaches the history of science and mathematics by asking who first discovered things that we now know to be true and when. Information not pertaining to this line of inquiry was accordingly unimportant-hence his decision to refrain from publishing Jābir's faulty proof verbatim.
Tellingly, after he mentions the treatise's (correct) proof of Equation 11 in ¶2, Suter continues by regretting that he could not answer the question that presumably he could expect his reader to be asking: who, and in particular which nation, first came up with that correct proof? Suter writes: "Whether this theorem belongs properly to the Greeks or the Arabs, we cannot decide; it is not to be found in Euclid to our knowledge."93 Likewise, to conclude his discussion of Jābir's treatise and Ibn al-Sarī's commentary, Suter writes that he cannot help but mention that the existence of this treatise and Ibn al-Nadīm's references to other works on Double False Position refute the view expressed by some of his 92 Ruska, "Heinrich Suter," esp. 409 ("diesem anspruchslosen Manne"), 411 ("ein freier Denker"), 411-412. 93 Suter, "Einige geometrische Aufgaben," 25: "Ob dieser Satz griechisches oder arabisches Eigentum sei, können wir nicht entscheiden, bei Euklid findet er sich unseres Wissens nicht." contemporaries that the method of Double False Position was first discovered in the twelfth century by European mathematicians.94 To avoid any misunderstanding, it is worth emphasizing here that the response to Suter that I propose is not a critique of "Orientalist thought," a vindication of Arabic or Islamic mathematics in the face of European bias or ignorance. That vindication is precisely what Suter was eager to carry out. Instead, Suter's blind spot is connected to the approach to the history of science and mathematics that he embraced, one in which the wheat must be separated from the chaff-according to simple scientific or mathematical criteria, not hermeneutically recursive historical or textual criteria-so that the historian could avoid wasting too much effort on the chaff. In other words, it is a historical approach in which the historian adopts the criteria of his own contemporaries in the natural and mathematical sciences and uses them as historical criteria.95 The question of most interest to historians embracing this approach is when each aspect of modern science or mathematics was first "discovered." In addressing an individual scholar of the past, the questions then become how much he knew and understood of (modern) science or mathematics, and how much credit he deserves for uncovering some part of that modern body of knowledge.96 While quite powerful in its own way, this approach tends to downplay or omit altogether an account of how mathematicians of the past thought about, discussed, arrived at, and communicated their results. With an emphasis on what they knew and when they knew it, in other words, it tends to skip over false starts, flawed proofs, and critiques of such errors, thus suppressing valuable evidence for the aims of mathematicians and the conceptual frameworks that conditioned those aims and how they were pursued and that were in turn shaped by all aspects of mathematical production, not only the statements and proofs admired by modern mathematicians. 94 Suter, "Einige geometrische Aufgaben," 26-27: "Man entschuldige uns, wenn wir hier folgende Bemerkung nicht unterlassen können …" 95 This should not be confused with using one's contemporary scientific criteria as scientific criteria for assessing the past, which entails using what we know or think we know today in order to gain a perspective on past scientific work that might not have been available to past scientists themselves. The difference is crucial: when these scientific criteria are used as a substitute for historical criteria, we allow present-day scientific concerns to warp our understanding of how and why ideas developed. A philology of mathematics that takes such evidence into account will be the best equipped to produce the kind of deeper history of mathematics that Roshdi Rashed has advocated, a history not only of methods available and theorems proven (or at least exploited) but also of conceptual framing, modes of understanding, and notions of the possible directions available to a given field of mathematics in a given time and place.97 In the case of Jābir's treatise on Double False Position and the Columbia manuscript, such a mode of reading, applicable to flawed and flawless mathematical texts alike, allows us to return to the juxtaposition with which this article began. What was a flawed proof of a numerical method that today's mathematicians would regard as hopelessly elementary doing in the same manuscript, copied by the same scribe, as Omar Khayyam's pathbreaking treatise on algebra and, indeed, coming right after it?
As already mentioned, Double False Position could be very useful in practice. But this manuscript was not a manual for traders; clearly this collection was produced by and for mathematicians, focused on theoretical texts and demonstration of theorems, not practical numerical methods and their applications. Why, then, include the treatise on Double False Position?
The answer, I propose, lies in precisely what Suter found unsatisfactory about the text: the faulty proof that Jābir added to the basic description of the method of Double False Position, along with Ibn al-Sarī's critique of that proof. This may seem like an odd proposal: why would working mathematicians wish to preserve and even study a misguided, incorrect proof? But the Columbia manuscript is evidence of just that wish: mathematicians and students of mathematics in the medieval Islamic world-in particular Iran, probably Hamadān-were interested in understanding what was wrong with Jābir's proof.98 This would have offered them a lesson in how to catch a proof's fault while preserving an episode in the history of their discipline.
Nor was this episode necessarily lodged exclusively in the past from the perspective of the scholars who used this manuscript. After all, there were plenty of other treatises on Double False Position. As already mentioned, that 97 Rashed, Development of Arabic Mathematics, ch. 1, esp. 14-16. Such an approach is related more broadly to the methods practiced and advocated, for example, by Kuhn and the historians and sociologists who have taken inspiration from aspects of his approach; see Barry Barnes, T.S. Kuhn and Social Science (London: Macmillan, 1982). 98 A single scribe (Scribe 1) copied texts no. 2-18 in the manuscript, including the treatise on Double False Position (no. 10). Even if the compilation represented by this subset of the manuscript had already been compiled piecemeal over time (such that Scribe 1 would not be the compiler of this compilation, only its copyist), nevertheless it was still the scribe's choice to copy a pre-existing compilation in its entirety-a choice that suggests an interest in studying the text on Double False Position alongside the other texts.
of the Arabophone Byzantine Christian scholar Qusṭā ibn Lūqā (d. ca. 912-13) sparked Suter's interest because it contained a more nearly valid geometric proof of why Double False Position works.99 Various other treatises on the topic are known today only by their titles. Sezgin lists treatises entitled (Ḥisāb) al-Khaṭa ʾayn, or (Calculation by) Two Errors, by Abū Kāmil (whom Sezgin tentatively places in the second half of the ninth century), Abū Yūsuf al-Rāzī and Abū Yūsuf al-Miṣṣīṣī (both probably active in the first half of the tenth century according to Sezgin), al-Karajī (active ca. tenth/eleventh century), and Ibn al-Haytham (965-1039).100 (There is no significance to the fact that Ibn al-Haytham is the latest author in this list; Sezgin's multivolume biobibliographical reference work stops at ca. 430 ah/1038 ce, so it would automatically have excluded any treatises on Double False Position that might have been composed after the mid-eleventh century.) In other words, there seems to have been enduring interest in this algorithm and its mathematical justification.
Further research into such treatises-especially if any of them should turn up in the vast number of uncatalogued and undercatalogued Arabic manuscripts around the world-might help us understand the context of Jābir's treatise. For example, if indeed he was working later than Qusṭā, as Suter thought, we might imagine that Jābir was seeking to produce a simpler proof, or else that he sought to reproduce Qusṭā's proof from memory and ended up getting it wrong without realizing his mistake. Similarly, if indeed Jābir did not have much of a head for math, as Suter claimed, it would be interesting to know what social and cultural incentives impelled him to take up the task of proving Double False Position nevertheless. Or, if other works by the same Jābir turn up showing him to be more of a mathematical mind than Suter thought, we might ask what led him astray in this one treatise-or we might reconsider what he was 99 See n. 12 above. As Suter points out (Suter, "Die Abhandlung Qosṭā ben Lūqās," 119-21), Qusṭā's treatise (at least as translated by Suter) sets up the correspondence between the line segments in its geometric proof and Double False Position's parameters in such a way as to assume implicitly that the equation in question is of the form ax = y, i.e., that the y-intercept is zero. Suter is puzzled that a mathematician like Qusṭā would have missed this and suggests that the attribution may be false. But if the correspondence is tweaked, the proof is successful; indeed Suter also suggests that an error of transmission could have introduced the error into the text. To be sure of what is going on, it will be necessary to consult the original Arabic of Qusṭā's treatise anew. 100 Abū Kāmil: Sezgin,GAS,esp. 277 (date)  trying to do in this treatise and ask why subsequent readers from Ibn al-Sarī to Suter to the present author misunderstood his aims. 101 In any case, we must still contend with the widespread interest in proofs of Double False Position. Jābir's purported proof was clearly something that Ibn al-Sarī considered worth his time to refute in the twelfth century, and his refutation was still being studied closely when Saʿd al-Dīn al-Hamadhānī subsequently explained it (presumably to students) and when the Columbia manuscript was produced. This concern for refuting a bad proof of Double False Position might have stemmed in part from the numerical method's widespread use, but ultimately it must have been part of medieval Arabic mathematicians' broader project. Perhaps it was precisely because Double False Position was clearly applicable to many of the same problems that the new algebra subsumed that it was important to study it not simply as a handy numerical method but as a theorem to be demonstrated by a satisfactory and revealing geometric proof and thus properly integrated into the new mathematics. 102 Suter's observation that Jābir's treatise attests to the existence of the method of Double False Position already in early Arabic mathematics, then, is only the beginning of the historian's task. Rather than stop there and dismiss the Roberts philological encounters 5 (2020) 308-352 treatise as otherwise useless because mathematically incorrect, philologicallyminded historians of mathematics might ask how the treatise, its commentary, its subsequent study, and other treatises like it on Double False Position can be reconciled and integrated into the picture of medieval mathematics that continues to emerge, one newly edited mathematical text at a time.