He also assumes that, of pairs of lines (including broken lines) and pairs of surfaces, concave in the same direction and bounded by the same extremities, the outer is greater than the inner. These assumptions are fundamental to his investigation, which proceeds throughout by means of figures inscribed and circumscribed to the curved lines or surfaces that have to be measured.

After some preliminary propositions Archimedes finds (Props. 13, 14) the area of the surfaces (1) of a right cylinder, (2) of a right cone. Then, after quoting certain Euclidean propositions about cones and cylinders, he passes to the main business of the book, the measurement of the volume and surface of a sphere and a segment of a sphere. By circumscribing and inscribing to a great circle a regular polygon of an even number of sides and making it revolve about a diameter connecting two opposite angular points he obtains solids of revolution greater and less respectively than the sphere. In a series of propositions he finds expressions for (a) the surfaces, (b) the volumes, of the figures so inscribed and circumscribed to the sphere. Next he proves (Prop. 32) that, if the inscribed and circumscribed polygons which, by their revolution, generate the figures are similar, the surfaces of the figures are in the duplicate ratio, and their volumes in the triplicate ratio, of their sides. Then he proves that the surfaces and volumes of the inscribed and circumscribed figures respectively are less and greater than the surface and volume respectively to which the main propositions declare the surface and volume of the sphere to be equal (Props. 25, 27, 30, 31 Cor.). He has now all the material for applying the method of exhaustion and so proves the main propositions about the surface and volume of the sphere. The rest of the book applies the same procedure to a segment of the sphere. Surfaces of