Prologue

Let it be so for mathematical objects as well. The lowest things—those carried in matter and shaped by nature—appear from afar to participate in both The "both" refers to the principles of the Limit (order/definition) and the Unlimited (potential/indeterminacy)., though they do not perceive the "infinite" in terms of the indeterminacy underlying them. Instead, they perceive what is prior: the principles, the shapes, and the forms.

On the mathematical sciences

It is clear that these are the all-encompassing principles of mathematics, even if one sets aside the higher levels of true being. Just as we speak of the common theoretical principles that permeate all classes of mathematics,

Common mathematical theorems

let us also speak of those general theorems, simple and born of a single science, which encompass all mathematical knowledge together. Let us agree on how these theorems apply to all things—to shapes, to numbers, to magnitudes, and to motions. These are the theorems concerning proportions original: analogiai., and those concerning compositions, divisions, conversions, and alternations. Furthermore, there are those concerning all types of ratios, such as multiple, super-particular original: epimorios. A ratio where the larger number contains the smaller number once, plus one part of it (e.g., 3:2 or 4:3)., and super-partient original: epimerēs. A ratio where the larger number contains the smaller number once, plus several parts of it (e.g., 5:3)., as well as their opposites. Simply put, these are the general and common theorems regarding equality and inequality, considered not just in points, numbers, or motions, but according to the common nature that each possesses in itself, containing a simpler form of knowledge within it.

Furthermore, the method and the order are common to all mathematical sciences: the path from those things more familiar to us toward the objects of inquiry, and the transition made through correct deductions. These methods are what they call analyses and syntheses.

A decorative flourish or scrollwork marks the conclusion of this section of the text.