Commentary on the First Book of Euclid's Elements

If the Limit were removed, symmetry, the communion of ratios, the identity of forms, equality, and all that belongs to the superior series would never appear in mathematics, nor would there be sciences of such things, nor stable and precise grasps of them. 5 Therefore, both of these principles are required for mathematical objects, just as they are for the other kinds of existing things. The lowest things, those carried in matter and fashioned by nature, are clearly seen to partake of both from their very nature: of the Infinite, according to the underlying 10 seat of the forms; and of the Limit, according to the logoi rational principles, shapes, and forms.

That these are the principles presiding over mathematics—just as they are over all existing things—is evident. And just as we have contemplated their common principles and seen them running through all mathematical kinds, so let us also reckon their common theorems—simple and offspring of the one science that encompasses all mathematical knowledge in one—and let us examine how this applies to all and can be studied in numbers, magnitudes, and motions. Such are the theorems of proportions, compositions, divisions, inversions, and alternations; furthermore, those regarding all ratios, such as the multiple, epimoric superparticular, and epimeres super-partient ratios, as well as those opposed to them; and, simply, those regarding the equal and the unequal, considered generally and commonly, not insofar as 25