PREFACE

Euclid, Elements, Book 12, Proposition 10

Euclid, Elements, Book 1, Proposition 41

Archimedes, On the Sphere and Cylinder, Book 1, Corollary to Proposition 34

Archimedes, On the Sphere and Cylinder, Book 1, Corollary to Proposition 31

Archimedes, On the Measurement of a Circle, Proposition 1

Archimedes, On Conoids and Spheroids, Proposition 4

I believe that no one has truly touched the sweetness of mathematical proofs with even the tip of their lips who does not then strive with all their might to enjoy them fully. Once a person has tasted this sweetness, they are like a bear original: "ursum" that has discovered honey hidden in a tree. Even if a swarm of bees attacks him with countless stings, he can hardly be kept away from the prize. In the same way, a mathematician is not easily driven back by the many difficulties that accompany these studies, even when they strike like frequent blows. To you, dear reader, who are accustomed to feeding on these honeyed fruits, I offer these delights to taste. They were born from a wonder of geometry that occurred to me during my meditations.

While I was reflecting on how solid shapes are created by rotating a flat shape around an axis, I compared the ratios of the original flat shapes to the resulting solids. I was truly amazed that the resulting figure seemed to change so much from the nature of its "parents" the flat shapes that generate the solid. They appeared to follow an entirely different mathematical logic.

For example, a cylinder and a cone that share the same base and axis have a specific relationship: the cylinder is three times the size of the cone. However, the rectangle that creates the cylinder is only double the size of the triangle that creates the cone.

Similarly, consider a hemisphere, a hemispheroid a solid shaped like half an egg, a parabolic conoid a shape formed by rotating a parabola, and a cylinder, all standing on the same base and axis. In this case, the cylinder is one and a half times the size of the hemisphere or hemispheroid. It is double the size of the conoid. Yet, the rectangle that generates the cylinder has a ratio of fourteen to eleven to the circle or ellipse that generates the other shapes. Against the parabola, that same rectangle is only one and a half times larger.

Indeed, you can find this same variety in flat figures created by spinning straight lines around a central point, such as circles. If we imagine several concentric circles with their radii laid out...