Note 648 — Chapter V

That is, so far as the facts are concerned. Geometry, considered in the most elevated manner, rests on ideas, and from that point of view is impregnable, unless the axioms can be overthrown. But if geometricians will insist on having definitions as well as axioms, they gain, no doubt, increased clearness, but they lose something in accuracy. I apprehend that, without definitions, geometry could not be a science of space, but would be a science of magnitudes, ideally conceived and consequently as pure as ratiocination could make it. This does not touch the question as to the empirical origin of the axioms.