Geometry of Indivisibles (Geometria Indivisibilibus Continuorum)

...having, for example, a ratio based on numbers arranged sequentially from one. The circles themselves will not maintain that same ratio of their radii. Instead, they will follow the ratio that their g squares have to one another.

g Corollary 1 to Proposition 11, Book 3

h Luca Valerio, Book 1, Proposition 39

i Luca Valerio, Book 1, Proposition 19

k Luca Valerio, Book 2, Proposition 41

l Archimedes, On the Equilibrium of Planes, Proposition 8

Having observed these things, I then turned my attention to the centers of gravity of both plane and solid figures. When I encountered a similar variety there, my wonder increased even further. In a cone, the center of gravity h is located on the axis at a distance of one quarter of the way from the base. However, in the triangle that generates that cone, the center of gravity is on that same axis but at a distance of one third of the axis from the base. Similarly, in a parabolic conoid A three dimensional shape formed by rotating a parabola, now called a paraboloid., the center of gravity is on the axis at a distance of one third from the base. In the parabola that generates it, however, the center is removed from the base by l two thirds of that same axis.

I meditated often on this kind of variation in many other figures. Previously, I had imagined a cylinder as being composed of an indefinite number of parallelograms, and a cone on the same base and axis as being compacted from an indefinite number of triangles passing through that axis. I used to think that by finding the ratio of these planes to one another, the ratio of the solids generated by them would immediately appear. But it became clearly established that the ratio of the solids did not agree at all with the ratio of the planes that generated them. It seemed to me then that anyone seeking to measure figures in such a way was wasting both oil and effort original: "oleum, & operam perdere," a Latin idiom for wasting time and resources on a fruitless task. I felt they would be like someone trying to thresh grain from empty husks.

However, after contemplating the matter a little more deeply, I finally arrived at this opinion: for our purposes, we should not assume that the lines and planes coincide with one another, but rather that they are parallel. By investigating the logic of many cases in this way, I found that the ratio of the planes exactly matches the proportion of the solids, and the ratio of the lines matches the proportion of the planes. This holds true in all cases, provided they are taken in the manner explained in m Book 2.

m Definitions 1 and 2, Book 2

n Definitions 1 and 2, Book 2

o Definitions 1 and 2, Book 2

Therefore, I no longer looked at the cylinder and cone as being cut through the axis. Instead, I viewed them as if they were cut into sections parallel to the base. I discovered they truly have the same ratio that I have called in Book 2 n "all the planes" of a cylinder compared to "all the planes" of a cone, o according to the rule of their common base. This refers to the collection of circles which, inside the cylinder and cone, are like cloth...

cylindrus: cylinder
conus: cone
parabola: a specific type of curved line or plane figure
gravitatum centra: centers of gravity, the balance points of shapes
plana: planes, two dimensional surfaces
lineæ: lines
proportio: ratio or proportion
Archimedes: a famous Greek mathematician
Luca Valerio: a contemporary mathematician known for his work on centers of gravity
conoide parabolico: a paraboloid
sectio: a cross section or cut