/Introduction: general/

Archimedes to Dositheus: greetings.

Earlier, I have sent you some of what we had already investigated then, writing it with a proof: that every segment contained by a straight line and by a section of the right-angled cone is a third again as much as a triangle having the same base as the segment and an equal height. Later, theorems worthy of mention suggested themselves to us, and we took the trouble of preparing their proofs. They are these: first, that the surface of every sphere is four times the greatest circle of the <circles> in it. Further, that the surface of every segment of a sphere is equal to a circle whose radius is equal to the line drawn from the vertex of the segment to the circumference of the circle which is the base of the segment. Next to these, that, in every sphere, the cylinder having a base equal to the greatest circle of the <circles> in the sphere, and a height equal to the diameter of the sphere, is, itself, half as large again as the sphere; and its surface is <half as large again> as the surface of the sphere.

In nature, these properties always held for the figures mentioned above. But these <properties> were unknown to those who have engaged in geometry before us – none of them realizing that there is a common measure to those figures.