Note 126 — Introductory material

In Grassmann's profound philosophical introduction to his Ausdehnungslehre of 1844, he suggested that Geometry, though improperly regarded as pure, was really a branch of applied mathematics, since it dealt with a subject-matter not created, like number, by the intellect, but given to it, and therefore not wholly subject to its laws alone. But it must be possible—so he contended—to construct a branch of pure mathematics, a science, that is, in which our object should be wholly a creature of the intellect, which should yet deal, as Geometry does, with extension—extension as conceived, however, not as empirically perceived in sensation or intuition.