Commentary on the First Book of Euclid's Elements

For intelligible referring to the realm of pure forms kinds, by virtue of their own simplicity, partake primarily of Limit and the Infinite; through unity, identity, and their permanent and 5 stable existence, they are filled with Limit, while through their division into multiplicity, their generative abundance, and their divine otherness and procession, they enjoy the Infinite. Mathematical objects are indeed offspring of Limit and the Infinite, but not of those primary ones alone, nor of the intelligible and hidden 10 principles; rather, they are offspring of those principles that have proceeded from the former into a second rank, and which are sufficient to generate, in conjunction with one another, the intermediate systems of existing things and the variety within them. Hence, it is that within these 15 mathematical objects, the logoi rational principles proceed toward the infinite, yet are mastered by the cause of Limit. For the number, starting from the Unit, possesses unceasing growth—yet every number that is grasped is limited—and the division of magnitudes proceeds toward the infinite, yet the things being divided are all defined, and the parts of the whole are limited 20 in actuality. If the Infinite did not exist, all magnitudes would be commensurable, and there would be nothing incommensurable or irrational—the very things by which those in geometry are thought to differ from those in arithmetic—and the numbers would not be able to demonstrate the generative power of the Unit, nor would they contain within themselves all the logoi rational principles of existing things, such as the multiple or epimoric superparticular ratios. For every number, when examined, alters its ratio in relation to the Unit and the one that came before it.