be a triangular pyramid hav-ac ing the same altitude and an equivalent base with the pyramid S-ABCDE. The two bases may be regarded as situated in the same plane; in which case, the plane abcd, if produced, will form in the triangular pyramid a section fgh situated at the same distance above the common plane of the bases; and therefore the section fgh will be to the section abcde as the base FGH is to the base ABD (Prop. III.), and since the bases are equivalent, the sections will be so likewise. Hence the pyramids S-abcde, T-fgh are equivalent, for their altitude is the same and their bases are equivalent. The whole pyramids S-ABCDE, T-FGH are equivalent for the same reason; hence the frustums ABD-dab, FGH-hfg are equivalent; hence if the proposition can be proved in the single case of. the frustum of a triangular pyramid, it will be true of every other. Let FGH-hfg be the frustum of a triangular pyramid, having parallel bases: through the three points F, g, H, pass the plane FgH; it will cut off from the frustum the triangular pyramid g-FGH. This pyramid has for its base the lower base FGH of the frustum; its altitude likewise is that of the frustum, because F the vertex g lies in the plane of the upper base fgh. Pass the plane, This pyramid being cut off, there will remain the quadrangular pyramid g-fhHF, whose vertex is g, and base fhHF. fgH through the three points f, g, H; it will divide the quadrangular pyramid into two triangular pyramids g-FƒH, g-fhH. The latter has for its base the upper base gfh of the frustum; and for its altitude, the altitude of the frustum, because its vertex H lies in the lower base. Thus we already know two of the three pyramids which compose the frustum. It remains to examine the third g-FfH. Now, if gK be drawn parallel to fF, and if we conceive a new pyramid K-FfH, having K for its vertex and FfH for its base, these two pyramids will have the same base FfH; they will also have the same altitude, because their vertices g and K lie in the line gK, parallel to Ff, and consequently parallel to the plane of the base: hence these pyramids are equivalent. But the pyramid K-FƒH may be regarded as having its vertex in f, and thus its altitude will be the same as that of the frustum : as to its base FKH, we are now to show that this is a mean proportional between the bases FGH and fgh. Now, the triangles FHK, fgh, have each an equal angle F=ƒ; hence FHK: fgh :: FK × FH: fg× fh (Book IV. Prop. XXIV.); but because of the parallels, FK=fg, hence We have also, FHK fgh: FH : fh. FHG FHK: FG FK or fg. But the similar triangles FGH, fgh give hence, FG fg FH: fh; FGH: FHK: : FHK : fgh; or the base FHK is a mean proportional between the two bases FGH, fgh. Hence the frustum of a triangular pyramid is equivalent to three pyramids whose common altitude is that of the frustum and whose bases are the lower base of the frustum, the upper base, and a mean proportional between the two bases. PROPOSITION XIX. THEOREM. Similar triangular prisms are to each other as the cubes of their homologous sides. Let CBD-P, cbd-p, be two similar triangular prisms, of which BC, bc, are homologous sides: then will the prism CBD-P be to the prism cbd-p, as BC3 to bc3. For, since the prisms are similar, the planes which contain the homologous solid an gles B and b, are similar, like placed, and equally inclined to each other (Def. 17.): hence the solid angles B and b, are equal (Book VI. Prop. XXI. Sch.). If these solid angles be applied to each other, the angle cbd will coincide with CBD, the side ba with BA, and the prism cbd-p will take the position Bcd-p. From A draw AH perpendicular to the common base of the prisms: then will the plane BAH be perpendicular to the plane of the com P mon base (Book VI. Prop. XVI.). Through a, in the plane BAH, draw ah perpendicular to BH: then will ah also be perpendicular to the base BDC (Book VI. Prop. XVII.); and AH, ah will be the altitudes of the two prisms. Now, because of the similar triangles ABH,aBh, and of the similar parallelograms AC, ac, we have AH D a a ah: AB: ab :: BC: bc. But since the bases are similar, we have : base BCD base bcd BC2: bc2 (Book IV. Prop. XXV.); hence, base BCD: base bcd :: AH2: ah2. Multiplying the antecedents by AH, and the consequents by ah, and we have base BCD × AH : base bcd×ah :: AH3 ah3. But the solidity of a prism is equal to the base multiplied by the altitude (Prop. XIV.); hence, the prism BCD-P: prism bcd-p :: AH3 : ah3 : : BC3 : bc3, or as the cubes of any other of their homologous sides. Cor. Whatever be the bases of similar prisms, the prisms will be to each other as the cubes of their homologous sides. For, since the prisms are similar, their bases will be similar polygons (Def. 17.); and these similar polygons may be di vided into an equal number of similar triangles, similarly placed (Book IV. Prop. XXVI.): therefore the two prisms may be divided into an equal number of triangular prisms, having their faces similar and like placed; and therefore, equally inclined (Book VI. Prop. XXI.); hence the prisms will be similar. But these triangular prisms will be to each other as the cubes of their homologous sides, which sides being proportional, the sums of the triangular prisms, that is, the polygonal prisms, will be to each other as the cubes of their homologous sides. PROPOSITION XX. THEOREM. Two similar pyramids are to each other as the cubes of their homologous sides. But the bases ABCDE, abcde, being similar figures, we have ABCDE: abcde: AB2: ab2 (Book IV. Prop. XXVII.). Multiply the corresponding terms of these two proportions; there results the proportion, ABCDEXSO: abcdex So :: AB3: ab3. Now ABCDEX SO is the solidity of the pyramid S-ABCDE, and abcdexSo is that of the pyramid S-abcde (Prop. XVII.); hence two similar pyramids are to each other as the cubes of their homologous sides. The chief propositions of this Book relating to the solidity of polyedrons, may be exhibited in algebraical terms, and so recapitulated in the briefest manner possible. Let B represent the base of a prism; H its altitude: the solidity of the prism will be Bx H, or BH. Let B represent the base of a pyramid; H its altitude: the solidity of the pyramid will be B× H, or H×B, or BH. Let H represent the altitude of the frustum of a pyramid, having parallel bases A and B ; AB will be the mean proportional between those bases; and the solidity of the frustum will be Hx (A+B+ √AB). In fine, let P and p represent the solidities of two similar prisms or pyramids; A and a, two homologous edges: then we shall have Pp A3: a3 BOOK VIII. THE THREE ROUND BODIES. Definitions. 1. A cylinder is the solid generated by the revolution of a rectangle ABCD, conceived to turn about the immoveable side AB. In this movement, the sides AD, BC, continuing always perpendicular to AB, describe equal circles DHP, CGQ, which are called the bases of the cylinder, the side CD at the same time describing the convex surface. The immoveable line AB is called the axis of the cylinder. E M P N B C Every section KLM, made in the cylinder, at right angles to the axis, is a circle equal to F either of the bases; for, whilst the rectangle ABCD turns about AB, the line KI, perpendicular to AB, describes a circle, equal to the base, and this circle is nothing else than the section made perpendicular to the axis at the point I. Every section PQG, made through the axis, is a rectangle double of the generating rectangle ABCD. 2. A cone is the solid generated by the revolution of a rightangled triangle SAB, conceived to turn about the immoveable side SA. In this movement, the side AB describes a circle BDCE, named the base of the cone; the hypothenuse SB describes the convex surface of the cone. The point S is named the vertex of the cone, SA the axis or the altitude, and SB the side or the apothem. Every section HKFI, at right angles to the axis, is a circle; every section SDE, through the axis, is an isosceles triangle double of the generating triangle SAB. S F H E D 3. If from the cone S-CDB, the cone S-FKH be cut off by a plane parallel to the base, the remaining solid CBHF is called a truncated cone, or the frustum of a cono |