...I might bring forth and indicate the reasons why certain parts of the divisions were omitted by Euclid: the ancient Greek "Father of Geometry" whose "Elements" remains the foundation of the field; then, finally, the figures themselves had to be addressed. In cases where they were demonstrated most clearly by Euclid, I was satisfied with a simple citation of the propositions. However, many things that were demonstrated by Euclid in another way were repeated here for my proposed goal—namely, for the comparison of knowable and unknowable In this context, "knowable" (scibilium) refers to figures that can be constructed using only a compass and straightedge, which Kepler links to the "knowable" proportions of the divine mind. figures—or things that were separate had to be joined, or the order had to be changed. I have organized the series of Definitions, Propositions, and Theorems with continuous numbering, as I did in my Dioptrics: Kepler's 1611 work on the theory of lenses and optics, for the convenience of citations. Even in the lemmas: a "helping theorem" used as a stepping stone to a larger proof themselves, I was not overly precise, nor was I too worried about specific vocabulary, being more intent on the things themselves; for I am no longer acting as a Geometer in the realm of Philosophy, but as a Philosopher in this part of Geometry. And I truly wish I could have discussed these geometric matters even more popularly, provided it were also more clear and tangible. But I hope that fair-minded readers will think well of my effort in both respects: both that I hand down these Geometrics in a popular style, and that I could not entirely overcome the obscurity of the material through my diligence. To those readers, I give this final piece of advice: if they are entirely unskilled in Mathematical matters, they should skip over my narrations and read only the propositions, from XXX until the end; and, having placed faith in the propositions themselves without the demonstration, let them proceed to the other books, especially to the last; lest, terrified by the difficulty of the Geometric arguments, they deprive themselves of the most delightful fruit of Harmonic contemplation. Now, with God's help, let us approach the matter.

On the Demonstrations of

Regular Figures.

I. Definition.

A plane figure is called regular which has all sides and all angles, facing outward, equal to one another.

II. Definition.

Of these figures, some are primary and radical, which do not exceed their own boundaries, and to which the given definition properly applies. Some are "augmented," which, as it were, exceed their own sides; this happens by continuing the non-adjacent sides of some radical figure until they meet. These are called Stars: Kepler was the first to mathematically define and name these "star polygons," such as the pentagram,