Geometry of Indivisibles (Geometria Indivisibilibus Continuorum)

...I had discovered a new and, if I may say so, very concise method for [measuring] these figures. I considered myself fortunate that these solids, beyond those described by Archimedes, were provided to me. I could then test the power and energy of my method upon them. However, let no one think that I have attained the measurement of all the solids mentioned. Even Kepler could not achieve this except for a few, and not entirely successfully, as will be clear to anyone reading his Stereometria Stereometria: Referring to Kepler's 1615 work "Nova Stereometria Doliorum Vinariorum," which investigated the volumes of wine barrels and its supplement. It was enough for me to investigate some of them by a more certain reasoning, if I am not mistaken. These can be counted as more than twenty in number, especially if the Archimedean solids are included: specifically, five for each of the four conic sections, along with others to be listed further down. For a revolution occurs around the axis of the said sections, and thus the Archimedean solids are formed.

Namely, a Sphere comes from a circle, a parabolic conoid Conoides: a solid shaped somewhat like a cone, formed by rotating a parabola or hyperbola around its axis from a parabola, a hyperbolic conoid r from a hyperbola, and an oblong or prolate original: "prolatum", meaning elongated like a football spheroid from an ellipse. Or, a revolution occurs around a line parallel to the axis, outside the figure but not touching it. In this way, a circle produces q a wide circular ring. A parabola produces s a wide parabolic semi-ring. A hyperbola produces t a hyperbolic ring (Kepler calls these "Craters," as they are similar to the cavity of Mount Etna). From an ellipse comes x a wide elliptical ring, which Kepler calls a "steep ring" y original: "Anulum arduum", comparing it to the flower garlands worn by country girls. Or, a revolution occurs around a line parallel to the axis that touches the figure. Then, from a circle comes u a narrow circular ring. From a parabola comes z a narrow parabolic semi-ring. From a hyperbola comes a a hyperbolic ring. Finally, from an ellipse comes what is likewise called b a narrow elliptical ring. Finally, when a revolution occurs around a line parallel to the axis that cuts the figure into two unequal portions, different shapes are formed. From the larger portion of a circle comes a Rose Apple. From the smaller portion comes c a Citron Apple. In an ellipse, the larger portion produces d a Quince Apple original: "Malum cotoneū", referring to the Cydonian apple or quince, and the smaller portion produces b an Olive. From the larger portion of a parabola comes e a larger parabolic Heap original: "Acervus", a term used by Kepler to describe the volume of a segment of a solid. From the smaller portion comes d a smaller Heap. From the larger portion of a hyperbola comes a larger hyperbolic Heap. From the smaller portion comes f a smaller Heap. These smaller parabolic and hyperbolic Heaps, the same...