Archimedis Syracusani Opera (1544 Basel Editio Princeps)

...polygon circumscribed about circle $β$, so that the surface of the prism circumscribed about the cylinder is equal to the polygon circumscribed about circle $β$. And since the polygon circumscribed about circle $β$ has a lesser ratio to the polygon inscribed in circle $β$ than the surface of the cylinder has to circle $β$, the surface of the prism circumscribed about the cylinder will have a lesser ratio to the polygon inscribed in circle $β$ than the surface of the cylinder has to circle $β$, and inversely. This is impossible. For the surface of the prism circumscribed about the cylinder has been shown to be greater than the surface of the cylinder. And the inscribed polygon in circle $β$ is less than circle $β$. Therefore, circle $β$ is not less than the surface of the cylinder. Let it be greater, if possible. Again, let a polygon be inscribed in circle $β$ and another circumscribed, such that the ratio of the circumscribed to the inscribed is less than the ratio of circle $β$ to the surface of the cylinder. And let a polygon be inscribed in circle $α$, similar to the one inscribed in circle $β$. And let a pyramid be constructed from the polygon inscribed in the circle. And again, let $κδ$ be equal to the perimeter of the polygon inscribed in circle $α$. And let $κη$ be equal to the radius original: α' ζ'. The triangle $κτη$ is greater than the polygon inscribed in circle $α$, because it has a base equal to its perimeter and a height greater than the perpendicular drawn from the center to one of the sides of the polygon. The parallelogram $λρ$ is equal to the surface of the prism, composed of parallelograms, because it is contained by the side of the cylinder and the line equal to the perimeter of the polygon, which is the base of the prism. Thus, triangle $ρλζ$ is equal to the surface of the prism. And since the polygons inscribed in circles $α$ and $β$ are similar, they have the same ratio to each other as the squares of their radii. Triangles $κτη$ and $ζρλ$ also have a ratio to each other as the squares of the radii of the circles. Therefore, the polygon inscribed in circle $α$ has the same ratio to the polygon inscribed in circle $β$ as triangle $κτη$ has to triangle $λζρ$. The polygon inscribed in circle $α$ is less than triangle $κτη$. Therefore, the polygon inscribed in circle $β$ is also less than triangle $ζρλ$, and so is the surface of the prism inscribed in the cylinder. This is impossible. For since the polygon circumscribed about circle $β$ has a lesser ratio to the inscribed polygon than circle $β$ has to the surface of the cylinder, then inversely, the circumscribed polygon about circle $β$ is greater than circle $β$. Therefore, the inscribed polygon in circle $β$ is greater than the surface of the cylinder. Thus, the surface of the prism is also greater. Therefore, circle $β$ is not greater than the surface of the cylinder. It was shown that it is not less either. Therefore, it is equal.

The surface of every isosceles having two equal sides cone, excluding the base, is equal to a circle whose radius has a mean proportional between the side of the cone and the radius of the circle which is the base of the cone. 14 Let there be an isosceles cone whose base is circle $α$. Let $γ$ be its radius. Let $α'$ be equal to the side of the cone. Let $ε$ be the mean proportional between $γ$ and $α'$. Let circle $β$ have a radius equal to $ε$. I say that circle $β$ is equal to the surface of the cone, excluding the base. If it is not equal, it is either greater or smaller. Let it first be smaller. There are two unequal magnitudes, the surface of the cone and circle $β$, and the surface of the cone is greater. It is possible to inscribe an isopleuron equilateral polygon in circle $β$, and to circumscribe another similar to the inscribed one, such that the ratio of the circumscribed to the inscribed is less than the ratio of the surface of the cone to circle $β$. Let a polygon be circumscribed about circle $α$, similar to the one circumscribed about circle $β$. And let a pyramid be constructed from the polygon circumscribed about circle $α$, having the same apex as the cone. Since the polygons circumscribed about circles $α$ and $β$ are similar, they have the same ratio to each other as the squares of their radii; that is, the ratio that $γ$ has to...

A geometrical diagram on the right side of the page consists of a vertical line with markings at intervals, several horizontal line segments below it labeled with the Greek letters α, γ, η, ε. There is a right-angled structure with vertices labeled κ, τ, η and another triangle or related structure with vertices λ, ζ, ρ. The diagram illustrates the geometric proofs discussed in the text regarding the surface areas of cylinders and cones.