A new branch of mathematics: Grassmann's Ausdehnungslehre

Grassmann's Ausdehnungslehre is an algebra that directly refers to and combines geometric figures and their incidence relations, with arbitrary Cartesian coordinates relegated to their properly auxiliary role, and equally arbitrary symbol strings relegated to their secondary role.

Grassmann's philosophical introduction is essentially one of the first expositions of the rudimentary principles of what today might be called universal algebra.  The content appears to be a simple and clear natural scientist's version of the basic principles of dialectical materialism, as applied to the formal sciences.

Grassmann insists that his reason for including the philosophical introduction is an attempt to provide an orientation to help the student form for himself the proper estimation of the relation between general and particular at every stage of the learning process. His formulation that philosophy moves from general to particular, and mathematics from particular to general, can be traced to Kant.