Geometry of Indivisibles (Geometria Indivisibilibus Continuorum)

...the planes flowing continuously from the base to the opposite base and always parallel to it, are understood to be left in a certain way to that ratio which the cylinder has to the cone. I therefore realized that the best method for preserving the measurement of figures was first to investigate the ratios of lines for planes, and of planes for solids, so that I might then better compare the measurement of the figures themselves. The matter, I believe, succeeded according to my wishes, as will be clear to the reader.

I have used a device similar to that which practitioners of algebra Algebrici: early modern mathematicians who used symbolic algebra to solve problems are accustomed to use for solving proposed questions. These mathematicians, although they deal with roots of numbers that are inexpressible, irrational original: "surdas", literally "deaf" numbers. This was a common historical term for irrational numbers like the square root of two., and unknown, nevertheless persist in adding, subtracting, multiplying, and dividing them. Provided they can extract the desired knowledge of the matter at hand, they are convinced they have fulfilled their duties well enough. In the same way, I have used a collection of indivisibles indivisibilium: the infinitely small parts, such as lines making up a plane or planes making up a solid, that form the basis of Cavalieri's method, whether of lines or of planes (assuming the same principles explained in Book 2). Although this collection is unnameable, irrational, and unknown in terms of the number of its parts, it is nevertheless enclosed within visible limits regarding its total magnitude. I have used this to investigate the measurement of continuous quantities, as will become apparent to the reader in the course of the work.

p Kepler's Measurement of the Volume of Barrels

It is my purpose as a geometer in these seven books to discover the dimensions of many figures, both plane and solid. Some of these were treated by others, especially by Euclid and Archimedes. Others, however, have been attempted by no one else so far as I know, with the single exception of Kepler Johannes Kepler (1571 to 1630), the astronomer who developed new geometric methods to calculate the volume of wine barrels.. On the occasion of measuring the Austrian wine barrel Dolij Austriaci: a practical problem regarding the capacity of wine casks that led Kepler to develop new geometric techniques using a measuring rod, Kepler summarized those discoveries of Archimedes that were useful to him in his Archimedean Stereometry. After adding some new reasons of his own, whatever they may be, he eventually added that part which he called the Supplement to Archimedean Stereometry. In that work, he considered the various conic sections Sectionum conicarum: curves obtained by cutting a cone, such as circles, parabolas, hyperbolas, and ellipses, namely the circle, parabola, hyperbola, and ellipsis, as well as the revolution of their parts around various axes. He announced to geometers, with a very elegant introduction, eighty-seven solids in total. These were in addition to the five Archimedean solids: the sphere, the parabolic conoid Conoides: a shape like a bowl formed by rotating a parabola, the hyperbolic conoid, and the oblong and prolate spheroids Sphæroides: a sphere-like shape that is either stretched like a football or flattened like a disk. Since, therefore, the already explained method of measuring...