It is appropriate to share with those of one's own kin those things that have been learned. I am sending you the proofs I have recorded, regarding which those involved in mathematical study will be able to inquire. Farewell.

ON THE SPHERE

AND CYLINDER OF ARCHIMEDES, BOOK I.

FIRST, the axioms and the assumptions taken for their proofs are written down. There are certain finite curved lines in a plane which, regarding the straight lines connecting their ends, either entirely lie on the same side, or contain nothing on the other side. I call such a line "concave on the same side" if, whenever any two points are taken, the straight lines between the points either all fall on the same side of the line, or some on the same side and some on the line itself, but none on the other side. Similarly, there are certain finite surfaces. These are in a plane, or they have their boundaries in a plane. Either the planes in which the boundaries lie, or some of them entirely lie on the same side, or contain nothing on the other side. I call such surfaces "concave on the same side" if, whenever two points are taken, the straight lines between the points either all fall on the same side of the surface, or some on the same side and some on the surface itself, but none on the other side.

I call a "solid sector" the figure contained by the surface of a cone and the surface of the sphere within the cone, when a cone cuts a sphere, having its vertex at the center of the sphere. I call a "solid rhombus" the solid figure composed of two cones when two cones have the same base and their vertices lie on opposite sides of the plane of the base, such that their axes lie on the same straight line.

I assume that of all lines having the same endpoints, the straight line is the smallest. Regarding other lines in a plane that have the same endpoints, I assume they are unequal when both are concave on the same side, and either one is entirely enclosed by the other and by the straight line having the same endpoints, or part is enclosed and part is common, and the enclosed part is smaller. Similarly, of surfaces having the same endpoints, if their endpoints are in a plane, the plane surface is smaller. Of other surfaces having the same endpoints, if their endpoints are in a plane, I assume they are unequal when both are concave on the same side, and either one is entirely enclosed by the other surface and the plane containing the same endpoints, or part is enclosed and part is common, and the enclosed part is smaller. Further, of unequal lines, unequal surfaces, and unequal solids, the difference by which the greater exceeds the smaller is such that, when added to itself, it is possible for it to exceed any of those described in relation to one another. Given these premises, if a polygon is inscribed in a circle, it is evident that the perimeter of the inscribed polygon is less than the circumference of the circle. For each of the sides of the polygon is less than the circumference of the circle, as it is cut off by the same referring to the arc spanning the chord.

IF a polygon is circumscribed about a circle, the perimeter of the circumscribed polygon is greater than the perimeter of the circle. For let a polygon be circumscribed about the given circle. I say that the perimeter of the polygon is greater than the circumference of the circle. Since the sum of $α λ$ and $λ β$ is greater than the arc $β λ$ because it encompasses the arc having the same endpoints, similarly the sum of $α η γ$ and $γ β$ is greater than the arc $α β$. And the sum of $λ κ θ$ and $θ λ$ is greater than the arc—$θ$. And the sum of $ζ η θ$ is greater than the arc $ζ θ$. And further, the sum of $δ ε$ and $ε ζ$ is greater than the arc $δ ζ$. Therefore, the whole perimeter of the polygon is greater than the circumference of the circle.

A geometric diagram shows a circle with a circumscribed polygon, a pentagon. Vertices and points are labeled with Greek letters: $α$ at the top left, $β$ at the top right, $γ$ at the far right vertex, $δ$ at the bottom right vertex, $ε$ at the bottom vertex, $ζ$ at the bottom left vertex, and $η$ at the far left vertex. Inside the top arcs are labels $θ$, $κ$, and $λ$. A horizontal line segment $α β$ is also indicated.

Given two unequal magnitudes, it is possible to find two unequal straight lines such that the ratio of the greater straight line to the smaller is less than the ratio of the greater magnitude to the smaller. Let two unequal magnitudes be $α β$ and $δ$.