ARCHIMEDES

ON THE SPHERE AND CYLINDER

BOOK ONE.

A poorly corrupted passage: I read, "These things existed by nature in the aforementioned figures; yet they had not been observed by those who had examined geometry before us, as he will understand who compares the proposed theorems concerning these figures with their demonstrations." — For "and" (kai) I read "as" (hosper).

Previously, indeed, I sent those things which we discovered by contemplation, writing down their demonstrations; just as that every section contained by a straight line and a section of a right-angled cone is one and one-third times the triangle having the same base as the section and an equal height. Now, we have elaborated the demonstrations of certain incident theorems, which are of this kind. First, that the surface of a sphere is four times the greatest circle of those which are in it. Second, that for every portion of a sphere, a circle is equal to its surface, the radius of which is equal to the straight line drawn from the vertex of the portion to the periphery of the circle which is the base of the portion. Furthermore, that for every cylinder of a sphere, having the base the same as the greatest circle of those in the sphere, and a height equal to the diameter of the sphere, the cylinder is one and one-half times the sphere, and its surface is one and one-half times the surface of the sphere. These things had preceded by nature to be demonstrated concerning the said figures, yet they had not been observed by those who before us had applied themselves to geometry; as anyone will understand who compares the proposed theorems concerning these figures with their demonstrations. Just as also many of those things have been received, which Eudoxus speculated concerning solids; such as that every pyramid is the third part of a prism having the same base as the pyramid,