less than this. Hence the difference between the sum of all the exterior prisms and the sum of all the interior ones, must be greater than the difference between the two pyramids themselves. Now, beginning with the bases ABC, abc, the second exterior prism DEFG is equivalent to the first interior prism defa, because they have the same altitude k, and their bases DEF, def are equivalent: for like reasons, the third exterior prism GHIK, and the second interior prism ghid, are equivalent; the fourth exterior, and the third interior; and so on, to the last in each series. Hence all the exterior prisms of the pyramid S - ABC, excepting the first prism DABC, have equivalent corresponding ones in the interior prisms of the pyramid s-abc: hence the prism DABC is the difference between the sum of all the exterior prisms of the pyramid S — ABC, and the sum of all the interior prisms of the pyramid s — abc. But the difference between these two sets of prisms has already been proved to be greater than that of the two pyramids, which latter difference we supposed to be equal to the prism aABC: hence the prism DABC must be greater than the prism aABC; but in reality it is less, for they have the same base ABC, and the altitude Ax of the first is less than Aa the altitude of the second. Hence the supposed inequality between the two pyramids cannot exist; hence the two pyramids S — ABC, s ABC, s — abc, having equal altitudes and equivalent bases, are themselves equivalent. THEOREM XVII. Every triangular pyramid is the third of the triangular prism having the same base and altitude. Let F-ABC be a triangular pyramid, ABCDEF a triangular prism of the same base and altitude; the pyramid will be equal to one third of the prism. A E B D C Conceive the pyramid F-ABC to be cut off from the prism by a section made along the plane FAC, and there will remain the solid FACDE, which may be considered as a quadrangular pyramid whose vertex is F, and base the parallelogram ACDE. Draw the diagonal AD, and extend the plane FAD, which will cut the quadrangular pyramid into two triangular ones F ACD, F-ADE. These two triangular pyramids have for their common altitude the perpendicular drawn from F to the plane ACDE; they have equal bases, the triangles ACD, ADE being halves of the same parallelogram; hence the two pyramids F-ACD, F-ADE are equivalent. But the pyramid F-ADE and the pyramid F - ABC have equal bases, ABC, DEF; they have also the same altitude, namely, the distance of the parallel planes ABC, DEF: hence the two pyramids are equivalent. Now the pyramid FADE has already been proved equivalent to FACD; hence the three pyramids F- ABC, FADE, FACD, which compose the prism ABCD, are all equivalent. Hence the pyramid F― ABC is the third part of the prism ABCD, which has the same base and the same altitude. Cor. The volume of a triangular pyramid is equal to a third part of the product of its base by its altitude. THEOREM XVIII. The volume of any pyramid has for its measure the area of its base multiplied into one third of its altitude. Let S - ABCDE be a pyramid, having the altitude SO; then will it be measured by the base ABCDE into one third of the altitude SO. A S E D C B For, extending the planes SEB, SEC through the diagonals EB, EC, the polygonal pyramid S—ABCDE will be divided into several triangular pyramids, all having the same altitude SO. But (T. XVII., C.) each of these pyramids is measured by multiplying its base ABE, BCE, or CDE by the third part of its altitude SO; hence the sum of these triangular pyramids, or the polygonal pyramid S - ABCDE will be measured by the sum of the triangles ABE, BCE, CDE, or the polygon ABCDE, multiplied by SO. Hence every pyramid is measured by a third part of the product of its base by its altitude. Cor. I. Every pyramid is the third part of the prism which has the same base and the same altitude. Cor. II. Two pyramids having the same altitude, are to each other as their bases. Scholium. The volume of any polyedral body may be computed, by dividing the body into pyramids; and this division may be accomplished in various ways. One of the simplest is to make all the planes of division pass through the vertex of one solid angle; in that case, there will be formed as many partial pyramids as the polyedron has faces, minus those faces which form the polyedral angle whence the planes of division proceed. THEOREM XIX. The volume of a frustum of a pyramid is equivalent to three pyramids having the common altitude of the frustum, and for bases, the lower base of the frustum, the upper base, and a mean proportional between them. We will represent the lower base ABCDE of the frustum by A, and the upper base abcde. by a; also, we will denote the altitude oO of the frustum by h. If we denote the altitude SO of the pyramid, whose base is ABCDE by x, we shall have x h for the altitude So of the pyramid whose base is abcde. α S E D A B C Since the two bases of the frustum are similar polygons (T. XIV.), their areas are to each other as the squares of their homologous sides AB and ab (B. III., T. XXVIII.), which sides are to each other as SO to So; consequently we have A: a :: x2: (x − h)2. Extracting the square root of each term, we have Hence the volume of the larger pyramid is A × 3x= If we denote the volume of the frustum, which is the differ ence of these pyramids by V, we shall have V 1 which becomes V=(A+a+A2 a3) × 1 h = A × 1 h +.a × } h + √ A × a × 3 h, which establishes the Theorem. THEOREM XX. Two similar pyramids are to each other as the cubes of their homologous sides. a C E D For, two pyramids being similar, the smaller may be placed within the greater, so that the angle S shall be common to both. In that position the bases ABCDE, abcde will be parallel; because, since the homologous faces are similar, the angle Sab is equal to SAB, and Sbc to SBC; hence the plane ABC is parallel to the plane abc. This granted, let SO be the perpendicular drawn from the vertex S to the plane ABC, and o the point where this perpendicular meets the plane abc: from what has already been shown (T. XIV.), we shall have SO: So::SA: Sa:: AB: ab; and, consequently, SO: So:: AB: ab. A B C But the bases ABCDE, abcde being similar figures, we have ABCDE: abcde:: AB2: ab2. Multiply the corresponding terms of these two proportions; there results the proportion, ABCDE × SO: abcde × So :: AB3: ab3. Now ABCDE × SO is the volume of the pyramid S-ABCDE, and abcde So is that of the pyramid S-abcde (T. XVII., C.); hence two similar pyramids are to each other as the cubes of their homologous sides. SEVENTH BOOK. THE THREE ROUND BODIES. DEFINITIONS. I. A cylinder is a solid, which may be produced or generated by the revolution of a rectangle ABCD, conceived to revolve about the side AB. In this rotation, the sides AD, BC, continuing always perpendicular to AB, describe equal circular planes DHP, CGQ, which are called the bases of the cylinder; the side CD at the same time describing the convex surface. The immovable line AB is called the axis of the cylinder. Every section KLM made in the cylinder, at right angles to the axis, is a circle equal to either of the bases; for, while the rectangle ABCD revolves about AB, the line KI, perpendicular to AB, describes a circular plane, equal to the base, which is a section made perpendicular to the axis at the point I. Every section PQGH passing through the axis is a rectangle, and is double of the generating rectangle ABCD. II. A cone is a solid, which may be produced or generated by the revolution of a right-angled triangle SAB, conceived to revolve about the side SA. In this rotation, the side AB describes a circular plane BDCE, named the base of the cone; and the hypotenuse SB describes its convex surface. H S G K B A D Ε C The point S is named the vertex of the cone; SA its axis, or altitude. |