The Complete Works of Archimedes

...and let the point be [that] at which $zh$ and $mn$, being produced, meet one another and the line $ag$. But the sides $bt$ and $mn$ will be carried along a conical surface, the base of which is the circle about the diameter $bd$, standing at right angles to the circle $abcd$. Its vertex is the point at which $bh$ and $dm$, being produced, meet one another and the line $ag$. In like manner, the sides [formed] by the other semicircle will also be carried along conical surfaces, again in like manner to these. Thus, the figure inscribed in the sphere will be contained by conical surfaces, of which all [the lines] are

A circular diagram representing a sphere with a vertical axis labeled α and γ, and a horizontal diameter labeled β and δ. Several horizontal lines are drawn across the circle (labeled with Greek letters like ζ, η, θ, μ, ν), representing the parallel bases of conical sections within the sphere. Diagonal lines from these points suggest inscribed figures.

sides. The surface [of the figure] will be less than the surface of the sphere. For, the sphere being divided by the plane about $bd$ [which is] at right angles to the circle $abcd$, the surface of the one hemisphere and the surface of the figure inscribed in it have the same boundaries in one plane. For the boundary of both surfaces is the circle—the circumference—about the diameter $bd$ at right angles to the circle $abcd$. And both are concave in the same direction, and one of the surfaces is enclosed by the other and the plane having the same boundaries as it. Similarly, the surface of the figure in the other hemisphere is also less than the surface of the hemisphere.

A second circular diagram showing a sphere with an inscribed regular polygon. Parallel horizontal chords connect the vertices of the polygon across the circle. Points are labeled with Greek letters including α, β, γ, δ, ε, ζ, η, θ, κ, λ, μ, ν. To the right of the diagram, several straight line segments are drawn horizontally, representing lengths used in the proof.

Therefore, the whole surface of the figure inscribed in the sphere is also less than the surface of the sphere.

24 The surface of the figure inscribed in the sphere is equal to a circle, the square on the radius of which is equal to the rectangle contained by the side of the figure and a line equal to all the straight lines (those that are parallel to one another) joining the sides of the polygon, being equal to the straight line subtending two sides of the polygon. Let $abcd$ be a great circle in a sphere, and let there be inscribed in it an equilateral polygon whose sides are measured by four. And let a certain figure be conceived as inscribed in the sphere from the inscribed polygon. And let $ez, ht, gd, kl, mn$ be joined, [which are] equal to one another [and] to the [line] subtending two