(1) Given a cylinder inscribed in a rectangular parallelepiped on a square base in such a way that the two
bases of the cylinder are circles inscribed in the opposite square faces, suppose a plane drawn through one side of the square containing one base of the cylinder and through the parallel diameter of the opposite base of the cylinder. The plane cuts off a solid with a surface resembling that of a horse’s hoof. Archimedes proves that the volume of the solid so cut off is one sixth part of the volume of the parallelepiped.
(2) A cylinder is inscribed in a cube in such a way that the bases of the cylinder are circles inscribed in two opposite square faces. Another cylinder is inscribed which is similarly related to another pair of opposite faces. The two cylinders include between them a solid with all its angles rounded off; and Archimedes proves that the volume of this solid is two-thirds of that of the cube.
Having proved these facts by the mechanical method, Archimedes concluded the treatise with a rigorous geometrical proof of both
propositions by the method of exhaustion. The MS. is unfortunately somewhat mutilated at the end, so that a certain amount of restoration is necessary.
I shall now attempt to give a short account of the other treatises of Archimedes in the order in which they appear in the editions. The first is—
On the Sphere and Cylinder.
Book I. begins with a preface addressed to Dositheus (a pupil of Conon), which reminds him that on a former occasion he had communicated to him the treatise proving that any segment of a “section of a right-angled cone” (i.e. a parabola) is four-thirds of the triangle with the same base