DE FIGURARUM HARMON:
original: "DE FIGURARUM HARMONIA" — On the Harmony of Figures

Preface.

Proclus on the intellectual essence of geometric things.

Decorative woodcut initial 'C' featuring ornate foliate designs and scrolled patterns in a square frame.Since we must seek the causes of Harmonic Proportions from the divisions of a circle into equal parts—divisions which are performed geometrically and scientifically (that is, from demonstrable regular plane figures)—I thought it necessary to point out at the start that the mental differences Kepler means the conceptual or "intellectual" distinctions that exist in the mind rather than just physical properties. of geometric things are, today, as far as can be seen from published books, entirely ignored. Thus, among the ancients, no one occurs to me who indicated that they exactly understood these specific differences of geometric things, except for Euclid and his commentator Proclus. To be sure, the distribution of problems into Plane, Solid, and Linear by Pappus of Alexandria Pappus (c. 290–350 AD) was one of the last great Greek mathematicians; "Plane" problems used only a compass and straightedge, "Solid" used conic sections, and "Linear" used more complex curves., and the ancients whom he follows, is appropriate enough for explaining the mental habits arising around each part of the geometric subject; yet that distribution is both brief in words and applied to practice. No mention is made of theory. Truly, unless we occupy our whole mind with the theory of this matter, we will never be able to attain the harmonic ratios. Proclus the Successor Diadochus, meaning "the Successor" to Plato’s Academy., in his four books on the first book of Euclid, acted professedly as a Theoretical Philosopher regarding mathematical subjects. If only he had also left us his commentaries on the tenth book of Euclid! Book X of Euclid’s Elements deals with "irrational" lines, a notoriously difficult topic that Kepler found essential for his theories of harmony. It would have freed our geometers from a neglected ignorance and relieved me of the heavy labor of explaining the differences of geometric things from the ground up. For it is easily apparent from his very introduction that those distinctions of Mental Beings original: "Entium Mentalium" — things that exist as concepts in the mind. were well known to him, since he establishes that the principles of the whole mathematical essence are the same ones that permeate all Beings and give birth to all things from themselves: namely, the Limit that is and the Infinite; or the Bound and the Unbounded. He recognizes the bound or the "circumscription" as the Form, and the "unbounded" as the Matter of geometric things.

For Configuration and Proportion are proper to quantities: configuration belongs to individual things, while proportion belongs to things joined together. Configuration is perfected by boundaries: for a line is bounded by points, a plane surface by lines, and a body is bounded, circumscribed, and shaped by surfaces. Therefore, those things which are finite, circumscribed, and shaped can also be grasped by the mind; but the infinite and indeterminate, insofar as they are such, can be narrowed down by no knowledge (which is compared to definitions) and by no barriers of demonstration. Furthermore, figures exist in the Archetype The "Archetype" is the original divine blueprint or "idea" in the mind of God. before they exist in the Work; they are in the divine mind before they are in creatures; though different in the manner of their subject, they are nonetheless the same in the form of their essence. Therefore, configuration is a certain Mental essence for quantities, or a "thought-process," which is their essential difference. This is much clearer from proportions. For since configuration is perfected by several boundaries, it happens that because of this plurality, configuration makes use of proportions. But what a proportion might be without the action of the mind can in no way be understood. And for this reason, he who gives boundaries to quantities as an essential principle posits that shaped quantities have an intellectual essence. But there is no need for argument; let the whole book of Proclus be read. It will be sufficiently apparent that the intellectual differences of geometric things were well known to him; even if he does not place this affirmation so separately and alone in the open as to warn even a drowsy reader. For his speech flows like a full river-bed, paved on all sides with the most abundant sentiments of the more abstruse Platonic philosophy, among which is also this unique argument of this book: