Cor. 3.—In any two pyramids, sections parallel to their bases, which divide their altitudes in the same ratio, are proportional to their bases. PROP. V.—Theorem. Pyramids (P–ABCD, p–abc), having equal altitudes (PO, po) and bases (ABCD, abc) of equal areas, have equal volumes. Dem.—If they be not equal in volume, let abc be the base of the greater; and let ox be the altitude of a prism, with an equal base, and whose volume is equal to their difference; then let the equal altitudes PO, po be divided into such a number of equal parts, by planes parallel to the bases of the pyramids, that each part shall be less than ox. Then [iv. Cor. 3] the sections made by these planes will be equal each to each. Now let prisms be constructed on these sections as bases and with the equal parts of the altitudes of the pyramids as altitudes, and let the prisms in P–ABCD be constructed below the sections, and in p–abc, above. Then it is evident that the sum of the prisms in P–ABCD is less than that pyramid, and the sum of those on the sections of p–abc greater than p–abc. Therefore the difference between the pyramids is less than the difference between the sums of the prisms, that is, less than the lower prism in the pyramid p–abc; but the altitude of this prism is less than ox (const.). Hence the difference between the pyramids is less than the prism whose base is equal to one of the equal bases, and whose altitude is equal to ox, and the difference is equal to this prism (hyp.), which is impossible. Therefore the volumes of the pyramids are equal. Cor. 1.—The volume of a triangular pyramid E–ABC is one third the volume of the prism ABC–DEF, having the same base and altitude. For, draw the plane EAF, then the pyramids E–AFC, E–AFD are equal, having equal bases AFC, AFD, and a common altitude; and the pyramids E–ABC, F–ABC are equal, having a common base and equal altitudes. Hence the pyramid E–ABC is one of three equal pyramids into which the prism is divided. Therefore it is one third of the prism. Cor. 2.—The volume of every pyramid is one-third of the volume of a prism having an equal base and altitude. Because it may be divided into triangular pyramids by planes through the vertex and the diagonals of the base. PROP. VI.—Theorem. The volume of a cylinder is equal to the product of the area of its base by its altitude . Dem.—Let O be the centre of its circular base; and take the angle AOB indefinitely small, so that the arc AB may be regarded as a right line. Then planes