Note 343 — APPENDIX. (continued 2)

It is plain that Plato, by Hypotheses, in this place, means the usual foundations of Arithmetic and Geometry; namely, Definitions and Postulates. He says that "the arithmeticians and geometers take as hypotheses (hυποθεμενοι) odd and even, and the three kinds of angles (right, acute, and obtuse); and figures, (as a triangle, a square,) and the like." I say his "hypotheses" are the Definitions and Postulates, not the Axioms: for the Axioms of Arithmetic and Geometry belong to the Higher Faculty, which ascends to First Principles. But this Faculty operates rather in using these axioms than in enunciating them. It knows them implicitly rather than expresses them explicitly.