Commentary on the First Book of Euclid's Elements

they exist in shapes, numbers, or motions, but insofar as each of these possesses a common nature and provides a simpler knowledge of itself. Furthermore, beauty and order are common to all mathematical subjects, as is the path from things better known to those being sought, and the transition from these to those—which they call analysis and synthesis. The likeness and unlikeness of ratios never depart from any of the mathematical kinds. 10 For we speak of shapes as alike or unalike, and likewise of numbers as alike or unalike. And whatever is revealed according to the powers mathematical faculties or potentials belongs to all mathematical subjects alike, both to the things that have power and the things that are empowered. These are what Socrates in the Politeia Republic Plato's Republic, 546-547 dedicated to the high-speaking Muses, having encompassed the common logoi rational principles of all mathematical things within defined limits and having established them in the aforementioned numbers, from which the measures of both eugonia noble birth/good offspring and the contrary state of agonia sterility/infertile offspring are revealed.

It must not be thought, then, that these common things exist primarily in the many and divided forms, nor that they have their generation subsequently and from the many; rather, they must be posited as standing before them, differing in simplicity and precision. For this reason, the knowledge of them precedes the knowledge of the many and grants 25 the principles to them; the many stand around it and are referred back to it.