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 D B : a 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 bcdxah :: 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 divided 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. For, since the pyramids are similar, the solid angles at the vertices will be contained by the same number of similar planes, like placed, and equally inclined to each other (Def. 17.). Hence, the solid angles at the vertices may be made to coincide, or the two pyramids may be so placed as to have the solid angle S common. B In that position, the bases ABCDE, abcde, A 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 (Book VI. Prop. XIII.). This being proved, 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, we shall have SO: So: SA: Sa :: AB: ab (Prop. III.); and consequently, ISO: 80: AB : ab. But the bases ABCDE, abcde, being similar figures, we have ABCDE: abcde: AB2: ab (Book IV. Prop. XXVII.). Multiply the corresponding terms of these two proportions; there results the proportion, ABCDE×1SO: abcdex So :: AB3 : ab3. Now ABCDE× 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. General Scholium. 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× 4H, or H× 4B, 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 E 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. M 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, perpen P G dicular 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 buse 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. F H E B 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 cone. We may conceive it to be generated by the revolution of a trapezoid ABHG, whose angles A and G are right angles, about the side AG. The immoveable line AG is called the axis or altitude of the frustum, the circles BDC, HFK, are its bases, and BH is its side. 4. Two cylinders, or two cones, are similar, when their axes are to each other as the diameters of their bases. 5. If in the circle ACD, which forms the base of a cylinder, a polygon ABCDE be inscribed, a right prism, constructed on this F base ABCDE, and equal in altitude to the cylinder, is said to be inscribed in the cylinder, or the cylinder to be circumscribed about the prism. The edges AF, BG, CH, &c. of the prism, being perpendicular to the plane of the base, are evidently included in the convex surface of the cylinder; hence the prism and the cylinder touch one another along these edges. 6. In like manner, if ABCD is a polygon, circumscribed about the base of a cylinder, a right prism, constructed on this base ABCD, and equal in altitude to the cylinder, is said to be circumscribed about the cylinder, or the cylinder to be inscribed in the prism, mi Let M, N, &c. be the points of contact in the sides AB, BC, &c.; and through the points M, N, &c. let MX, NY, &c. be drawn A perpendicular to the plane of the base: these perpendiculars will evidently lie both in the surface of the cylinder, and in that of the circumscribed prism; hence they will be their lines of contact. 7. If in the circle ABCDE, which forms the base of a cone, any polygon ABCDE be inscribed, and from the vertices A, B, C, D, E, lines be drawn to S, the vertex of the cone, these lines may be regarded as the sides of a pyramid whose base is the polygon ABCDE and vertex S. The sides of this pyramid are in the convex A surface of the cone, and the pyramid is said to be inscribed in the cone. 8. The sphere is a solid terminated by a curved surface, all the points of which are equally distant from a point within, called the centre. The sphere may be conceived to be generated by the revolution of a semicircle DAE about its diameter DE: for the surface described in H this movement, by the curve DAE, will have all its pointsequally distant from its centre C. F 9. Whilst the semicircle DAE revolving round its diameter DE, describes the sphere; any circular sector, as DCF or FCH, describes a solid, which is named a spherical sector. D E 10. The radius of a sphere is a straight line drawn from the centre to any point of the surface; the diameter or axis is a line passing through this centre, and terminated on both sides by the surface. All the radii of a sphere are equal; all the diameters are equal, and each double of the radius. 11. It will be shown (Prop. VII.) that every section of the sphere, made by a plane, is a circle: this granted, a great circle is a section which passes through the centre; a small circle, is one which does not pass through the centre. 12. A plane is tangent to a sphere, when their surfaces have but one point in common. 13. A zone is a portion of the surface of the sphere included between two parallel planes, which form its bases. One of these planes may be tangent to the sphere; in which case, the zone has only a single base. 14. A spherical segment is the portion of the solid sphere, included between two parallel planes which form its bases. One of these planes may be tangent to the sphere; in which case, the segment has only a single base. 15. The altitude of a zone or of a segment is the distance between the two parallel planes, which form the bases of the zone or segment. Note. The Cylinder, the Cone, and the Sphere, are the three round bodies treated of in the Elements of Geometry. |