ON THE HARMONY OF THE FIGURES:

Preface.

Proclus on the intellectual essence of geometric things.

Since we must seek the causes of harmonic proportions from the divisions of a circle into a certain number of equal parts—divisions which are performed geometrically and scientifically (that is, from demonstrable regular plane figures)—I have deemed it necessary to point out at the start that the conceptual original: "mentales," referring to distinctions existing in the mind or as abstract categories differences of geometric things are today, so far as appears from published books, entirely ignored. Thus, one encounters no one, even among the ancients, who indicated that they had 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 and the ancients whom he follows is suitable enough for explaining the mental habits arising around each part of the geometric subject; yet that distribution is both brief in its wording and applied to practice, making no mention of theory. And truly, unless we occupy ourselves with our whole mind in the theory of this matter, we shall never be able to attain the harmonic ratios. Proclus Diadochus, in his four books published on the first book of Euclid, acted expressly as a theoretical philosopher regarding the mathematical subject. If only he had also left us his commentaries on the tenth book of Euclid! Euclid’s Book X is notoriously difficult and deals with irrational magnitudes; Kepler considers this essential for distinguishing which proportions are "harmonic." He would have freed our geometers from a neglected ignorance and would have entirely relieved my labor in explaining these differences of geometric things. For it is easily apparent from his very introduction that those distinctions between "Mental Beings" were sufficiently known to him, since he established the same principles for the whole of mathematical essence which also run through all Beings and beget all things from themselves: namely, the Limit and the Infinite (or the Bound and the Unbound), recognizing the Limit or the circumscription as the Form and the Infinite as the Matter of geometric things.

For Shape and Proportion are the proper characteristics of Quantities; shape of individual things, proportion of things joined together. Shape is perfected by boundaries; for a straight line is bounded by points, a plane surface by lines, and a body by surfaces; it is by these that it is circumscribed and shaped. Therefore, those things which are finite, circumscribed, and shaped can also be grasped by the mind; but infinite and indeterminate things, insofar as they are such, can be narrowed by no boundaries of demonstration and compared to no knowledge that relies on definitions. Furthermore, shapes exist in the Archetype In Kepler's philosophy, the "Archetype" is the supreme internal blueprint in the mind of God used to create the universe. before they exist in the Work, and in the divine mind before they exist in creatures—different indeed in the manner of their subject, but the same in the form of their essence. Therefore, for quantities, "Shaping" becomes a kind of Mental Essence, or intellection: their essential difference. This is much more clear from proportions. For since a shape is perfected by several boundaries, it happens that because of this plurality, the shaping makes use of proportions. Truly, what a proportion could be without the action of the mind can in no way be understood. And so, even on this account—that he gives boundaries to quantities as an essential principle—he posits that shaped quantities have an intellectual essence. But there is no need for argument; let the whole book of Proclus be read, and it will be sufficiently apparent that the intellectual differences of geometric things were rightly known to him. Even if he does not place this affirmation separately on its own in the open and conspicuous view so as to remind even the half-asleep reader; for his speech flows like a full riverbed, strewn on all sides with the most abundant maxims of the more recondite Platonic philosophy, among which is this singular argument of this book.