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...dingly represent the Unlimited as an image. Thus, infinity exists, but not only for those things that proceed directly from the source, nor is it the principle of the primary Intelligible alone. It also belongs to those things which have proceeded from those first principles into a second rank. It generates, from the very beginning, the middle orders of beings and the variety found within them. Hence, while the bonds of these ratios original: logoi. In this context, these are the formal principles and mathematical proportions that structure reality. are encompassed by the Unlimited, they are held in check by the cause of the Limit.

For when number proceeds from the Monad original: monas. The fundamental unit of number and the principle of Unity., it possesses growth to infinity; yet every specific number is limited. Likewise, the division of parts progresses toward infinity, yet all distances are defined, and in no way are the parts of the whole limited in their potential for division.

It is a property of the Unlimited that all magnitudes would be commensurable, and nothing would be "inexpressible" or "irrational" original: arrhēton and alogon. In ancient Greek mathematics, these terms refer to magnitudes that cannot be expressed as a ratio of whole numbers, such as the square root of two. if it were not for these principles. In this way, the objects of geometry seem to differ from those of arithmetic; numbers reveal the simple, generative power of the Monad, yet they do not possess all the ratios found in actual beings, such as multiple ratios or super-particular ratios original: pollaplasious and epimerious. Technical terms for types of ratios: "multiple" refers to 2:1, 3:1, etc.; "super-particular" refers to ratios like 3:2 or 4:3 where the larger number contains the smaller plus one part of it.. For every number is named according to its ratio when examined in relation to the Monad and the number that came before it.

If the Limit were removed, then symmetry, the sharing of ratios, the likeness of character, equality, and all those things belonging to the "superior coordinate" original: systoichia. The Pythagoreans grouped concepts into two "coordinates" or columns: the better side (Limit, Light, Square, Good) and the lesser side (Unlimited, Darkness, Oblong, Evil). of the gods would vanish from the mathematical arts. For the gift of the geometer's science does not consist of ratio and accuracy alone; it is also necessary to provide a causal explanation of the principles, just as one does for the other classes of being.