Prologue I.

For the intelligible kinds, according to their own simplicity, participate primarily in the limit and the unlimited, being fulfilled by the limit through their unity, identity, and permanent, 5 stable existence, and enjoying the unlimited through their division into plurality, their generative abundance, and their divine otherness and progression. Mathematical entities are offspring of both limit and the unlimited, but not only of the most primary, intelligible, and hidden principles. They are also offspring of those principles which proceeded from the former into a second order, 10 and are sufficient to beget the middle realms of beings and the variety within them. Hence, in these mathematical entities, the logoi principles proceed toward the infinite, yet they are mastered by the cause of the limit. For number, starting from the monas unit, has an unceasing increase, 15 yet any given number is limited. The division of magnitudes also proceeds to infinity, but the things being divided are all defined, and the parts of the whole are limited in actuality. If the unlimited did not exist, all magnitudes would be commensurable, and there would be nothing incommensurable or irrational—20 in which things geometry is thought to differ from arithmetic—and numbers would not be able to demonstrate the generative power of the unit, nor would they contain within themselves all the logoi proportions of beings, such as the multiple or the superparticular original: "epimorios," a ratio where the greater contains the lesser plus a part of it, like 3/2.. For every number varies its ratio toward the unit and the one that came before it.