Can We Identify the Theorem in Metaphysics 9, 1051a24-27 with Euclid’s Proposition 32? Geometric Deductions for the Discovery of Mathematical Knowledge

Abstract

This paper has two specific goals. The first is to demonstrate that the theorem in Metaphysics Θ 9, 1051a24-27 is not equivalent to Euclid’s Proposition 32 of book I (which contradicts some Aristotelian commentators, such as W. D. Ross, J. L. Heiberg, and T. L. Heith). Agreeing with Henry Mendell’s analysis, I argue that the two theorems are not equivalent, but I offer different reasons for such divergence: I propose a pedagogical-philosophical reason for the Aristotelian theorem being shorter than the Euclidean one (and the previous Aristotelian versions). Aristotle wants to emphasize the deductive procedure as a satisfactory method to discover scientific knowledge. The second objective, opposing some consensus about geometrical deductions/theorems in Aristotle, is to briefly propose that the theorem, exactly as we found it in Metaphysics and without any emendation to the text (therefore opposing Henry Mendell’s suggested amendments), allows the ancient philosopher to demonstrate that universal mathematical knowledge is in potence in geometrical figures. This tentatively proves that Aristotle emphasizes that geometrical deduction is sufficient to actualize mathematical knowledge.