Geometry of Indivisibles (Geometria Indivisibilibus Continuorum)

Kepler considers these shapes to be similar to straight horns, some of which are acute and others obtuse, just as in cattle when they first begin to butt with their horns, he says. These are indeed twenty solids in number, to which can also be added the narrow elliptical ring g Corollary 28, section 1, book 1. that is wider on one side, and the wide elliptical ring h Corollary 29, section 4, book 1. that is narrower on the other. Kepler thinks these are similar to a Tiara or a Turkish Globe The "Turkish Globe" refers to a turban, a shape Kepler used to describe specific geometric rotations., as well as those solids that arise from opposite sections i Corollaries 21, 30, book 5., which seem to accompany the previously mentioned ones.

These, I say, are the shapes we have selected from his list to be examined. We have taken nothing else from him except for some of the names, as will be clear to anyone who looks closely. Let the reader know, however, that we also contemplate many other solids besides these, which do not belong to the group listed above. Above all, I will not keep silent about the great universality of this method of demonstration. For while others demonstrate properties concerning only one or a few types of solids, we demonstrate them continuously for infinite types.

For example, we do not only show here that a cylinder k 10, Elements book 12; l Corollary 7, Elements book 12. is triple the volume of a cone, or a prism is triple a pyramid, when they share the same base and height. Rather, even if the base is changed in any way, without being limited by any assigned number, the solid standing upon it, which we call "cylindrical" m Definition 3, book 1; n Scholium 9, Corollary 4, 34, book 1., is triple o Definition 4, book 1. the volume of that which we call "conical," provided it is established on the same base and height. It is clearly apparent that the types of such solids are infinite in number. From this single example, as if "recognizing the lion by its claw" original: "ex ungue Leonem". A proverb suggesting that one can judge the scale of a great work or the talent of a creator from a single small sample., the studious reader will discern how much the geometric field becomes more fertile and wider through this method. Indeed, we shall constantly pursue this universality concerning almost all solids considered here.

In the first and second Books, for the most part, lemmas Lemmata: Preliminary propositions used to prove later theorems. are proposed which seem necessary for understanding the doctrine of the following books, although many of them are also demonstrated simply for their own sake. In the 3rd, 4th, and 5th Books, we examine solids that acknowledge their origin from conic sections. In the 6th, we treat the spaces of the helix This refers to the Archimedean spiral and the three-dimensional shapes generated by rotating it. and the solids generated from them, and we construct problems concerning the things previously demonstrated. Finally, in the seventh Book, our ship having traversed the Ocean of the infinity of indivisibles Cavalieri famously viewed volumes as being composed of an infinite number of parallel planes, which he called "indivisibles.", we establish another method...