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 and height, and adds that he is now sending the proofs of certain theorems which he has since discovered, and which seem to him to be worthy of comparison with Eudoxus’s propositions about the volumes of a pyramid and a cone. The theorems are (1) that the surface of a sphere is equal to four times its greatest circle (i.e. what we call a “great circle” of the sphere); (2) that the surface of any segment of a sphere is equal to a circle with radius equal to the straight line drawn from the vertex of the segment to a point on the circle which is the base of the segment; (3) that, if we have a cylinder circumscribed to a sphere and with height equal to the diameter, then (a) the volume of the cylinder is 1½ times that of the sphere and (b) the surface of the cylinder, including its bases, is 1½ times the surface of the sphere.
Next come a few definitions, followed by certain Assumptions, two of which are well known, namely:—
1. Of all lines which have the same extremities the straight line is the least (this has been made the basis of an alternative definition of a straight line).
2. Of unequal lines, unequal surfaces and unequal solids the greater exceeds the less by such a magnitude as, when (continually) added to itself, can be made to exceed any assigned magnitude among those which are comparable [with it and] with one another (i.e. are of the same kind). This is the Postulate of Archimedes.
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 revolution are inscribed and circumscribed to a segment less than a hemisphere, and the theorem about the surface of the segment is finally proved in Prop. 42. Prop. 43 deduces the surface of a segment greater than a hemisphere. Prop. 44 gives the volume of the sector of the sphere which includes any segment.
Book II begins with the problem of finding a sphere equal in volume to a given cone or cylinder; this requires the solution of the problem of the two mean proportionals, which is accordingly assumed. Prop. 2 deduces, by means of 1., 44, an expression for the volume of a segment of a sphere, and Props. 3, 4 solve the important problems of cutting a given sphere by a plane so that (a) the surfaces, (b) the volumes, of the segments may have to one another a given ratio. The solution of the second problem (Prop. 4) is difficult. Archimedes reduces it to the problem of dividing a straight line AB into two parts at a point M such that
MB : (a given length) = (a given area) : AM².
The solution of this problem with a determination of the limits of possibility are given in a fragment by Archimedes, discovered and preserved for us by Eutocius in his commentary on the book; they are effected by means of the points of intersection of two conics, a parabola and a rectangular hyperbola. Three problems of construction follow, the first two of which are to construct a segment of a sphere similar to one given segment, and having (a) its volume, (b) its surface, equal to that of another given segment of a sphere. The last two propositions are interesting. Prop. 8 proves that, if V, V′ be the volumes, and S, S′ the surfaces, of two segments into which a sphere is divided by a plane, V and S belonging to the greater segment, then