Prologue I.

but as each of these possesses a certain common nature in itself, providing a simpler knowledge of itself. Furthermore, beauty and order are common to all mathematical subjects, as is the path from the more familiar to the sought-after, and the transition from these to those, which they call analyses and syntheses. Similarity and dissimilarity of ratios do not leave any mathematical genus at all. 10 For we speak of shapes as similar or dissimilar, and likewise of numbers as similar or dissimilar. Whatever emerges according to the powers original: "dynameis," referring to roots or squares in mathematical contexts. applies to all mathematics alike, both the powers and the things powered. These are what Socrates in the Republic dedicated to the high-speaking Muses, 15 encompassing the common principles of all mathematical logoi propositions within defined limits, and setting them before us in the aforementioned numbers, from which the measures of both good birth and its opposite, sterile birth, become apparent.

21 Therefore, we must not think that these common principles exist primarily in the many divided forms, nor that they have their generation later and from the many, 25 but we must posit them as standing before them, distinguished by their simplicity and accuracy. For this reason, knowledge of them precedes the many knowledges and gives