/Introduction/

Archimedes to Dositheus: greetings

Earlier you sent me a request to write the proofs of the problems, whose proposals I had myself sent to Conon; and for the most part they happen to be proved through the theorems whose proofs I had sent you earlier: <namely, through the theorem> that the surface of every sphere is four times the greatest circle of the <circles> in it, and through <the theorem> 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 base, and through <the theorem> that, in every sphere, the cylinder having, <as> base, the greatest circle of the <circles> in the sphere, and a height equal to the diameter of the sphere, is both: itself, in magnitude, half as large again as the sphere; and, its surface, half as large again as the surface of the sphere, and through <the theorem> that every solid sector is equal to the cone having, <as> base, the circle equal to the surface of the segment of the sphere <contained> in the sector, and a height equal to the radius of the sphere. Now, I have sent you those theorems and problems that are proved through these theorems <above>, having proved them in this book. And as for those that are found through some other theory, <namely:> those concerning spirals, and those concerning conoids, I shall try to send quickly.