1 compounded of the ratios DE: EP, and EP: PO, of which the former is = AC: MN, and the latter is = EM: MO, (Prop. 101), or EB : BO, (Prop. 101, cor. 2); the ratio of DE: OP is the same with that which is compounded of AC: MN, and EB : BO, or of AB:BM, CB: BN, and EB: BO. But OP is = KL, (Prop. 98, cor. 5); these parallelopipeds have the plane Ls and linear sides about their solid Z8 B and G respectively equal and alike situated. Consequently, the ratio DE: KL is the same with that which is compounded of AB: BM, CB: BN, and EB: BO; or of AB: FG, CB : HG, and EB: LG. Cor, 1. Two rectangular parallelopipeds are to one another in the ratio compounded of the ratios of the linear sides of the one to the linear sides of the other, each to each. And any ratio compounded of three ratios, (whose terms are straight lines), is the same with the ratio of the rectangular parallelopipeds under their homologous terms. Cor. 2. Two cubes, or, in general, two similar parallelopipeds are to one another in the triplicate ratio of their homologous linear sides. Cor. 3. Similar parallelopipeds are to one another as the cubes of their homologous linear sides. Cor. 4. If four straight lines be in continued proportion, the first is to the fourth as the cube of the first to the cube of the second. Cor. 5. The rectangular parallelopipeds, under the corresponding terms of three analogies, are proportional. Cor. 6. If four straight lines be proportional, their cubes are also proportional; and conversely. Cor. 7. Rectangular parallelopipeds, and consequently any other parallelopipeds, are to one another in the ratio compounded of the ratios of their bases and altitudes. Cor. 8. Parallelopipeds whose bases and altitudes are reciprocally proportional are equal, and conversely. Cor. 9. Prisms are to one another in the ratio compounded of the ratios of their bases and altitudes. Therefore the 2d cor., prop. 101, and 8th Cor. of this, may be applied to prisms. PROPOSITION CIII.-THEOREM. Every triangular pyramid may be divided into two equal prisms, which are together greater than balf the whole pyramid, and two equal pyramids, which are similar to the whole and to one another, G H D = a Let BCD-A be any triangular pyramid. Let its linear sides, AB, AC, AD, be bisected in E, F, G, and the points of section joined; and let the sides of the base CB, BD, DC, be bisected in H, K, L, and the points of section joined. Also let EH, EK, and LG, be drawn. ::: the two sides, AB, AC, of the AABC, are bisected in EF, the EF is = and || CH, or HB, (Prop. 59), half the remaining side BC; and so of the other |s that join the points of section. Hence EC, CG, and consequently EL, are Os, and the planes EFG, BCD, are parallel ... the solid EFG-HCL, upon the triangular base HCL, is a prism. And the solid EHKGLD is a triangular prism = (Prop. 98, cor. 7) a prism on the base, (HKL=HCL), and having the same altitude as the former prism ; .. the two prisms EFG-HCL, and EHK-GLD, are equal to one another, and together prism on the base HCL, and having the same altitude as the pyramid BCD-A. :. since the base HCL is the fourth part (Prop. 70) of the base BCD, the whole solid HCDKEFG is the fourth part of a prism on the base BCD, and having the same altitude as the pyramid. The two remaining solids EFG-A, and BHK-E, are triangular pyramids. They are equal and similar to one another, and also similar to the whole, because the As which bound the one, are equal and similar to the As which bound the other, and also similar to those which bound the whole, each to each, and alike situated. But either of these ramids is, for the same reason, = the pyramid KLD-G, and .. < either of the two prisms. Consequently, the solid HCLK-EFG is the two pyramids EFG-A, BHK-E, and 7 half the whole pyramid. Cor. 1. By taking from the whole pyramid the two equal prisms, and from each of the remaining pyramids two equal prisms, formed in like manner, there will remain at length a magnitude less than any proposed magnitude, (Prop. 73). Cor. 2. Since the magnitude taken away from each of the remaining pyramids is equal to a prism on a fourth part of its base, and having the same altitude as the pyramid; the solids thus taken from both will be equal to a prism on the fourth part of their base, or the sixteenth part of the original base, and its altitude equal to that of the original pyramid. Cor. 3. For the same reason the solids taken from the four remaining pyramids will be equal to a prism having its base a sixty-fourth part of the original base, and its altitude equal to the altitude of the original pyramid. Cor. 4. Therefore all the solids thus taken away, that is, the whole pyramid, is = (t+18 tot 18+ &c.) of a prism, having the same base and altitude as the pyramid. But (1+++ido + &c., to infin.) = } (Alg. 96). Cor. 5. A triangular pyramid is equal to the third part of a prism on the same base, and having the same altitude. Cor. 6. Since (Alg. 109) quantities are proportional to their equimultiples, whatever has been proved of prisms, in regard to their ratios, will also be true of the pyramids on the same base, and having the same altitude. Cor. 7. A polygonal pyramid is equal to the third part of a prism on the same base, and having the same altitude. For it can be divided into as many triangular pyramids as there are sides in the polygon, and each of these pyramids will be the third part of a prism on the same base, and having the same altitude; the sum of all the pyramids, that is, the polygonal pyramid, will be the third part of a prism upon the sum of all the triangles, that is, on the whole polygon. PROPOSITION CIV.-THEOREM. A cone is the third part of a cylinder on the same base, and having the same altitude. For, if a series of polygons be inscribed in the circular base of the cone or cylinder, each having its number of sides double of the former, and on each of these a pyramid and prism, having the same altitude as the cylinder, be erected; the polygon will ultimately be = the circle, (Prop. 75, cor. 3), while at the same time the prism will be equal to the cylinder, and the pyramid to the cone; but the pyramid is the third part of the prism ; .. also the cone is the third part of the cylinder. PROPOSITION CV.-THEOREM. A sphere is two-thirds of a cylinder, having its altitude and the diameter of its base each equal to the diameter of the sphere. D E K M м T N B В Let CDLB be a quadrant of a circle, DCBE a square described about it, and DCE a right angled triangle, having the side DE=BC, and consequently any line, as KL || DE, will be =KC, and GN will be =GC. If the whole figure thus formed revolve about DC, as a fixed axis, the figure DCBE will generate a cylinder, (Def. 7), the ACDE will generate a cone, (Def. 8), and the quadrant will generate a hemisphere; now these figures may be conceived to be made up of an infinite number of lamina, represented in thickness by the line GI or the line KM, the solids generated by the several parts of the line GI will be as the squares of their generating lines; but the generating lines are in the cone GN, in the circle GH, and in the cylinder GI. Now the square of GI is = the sum of the squares of GH and GN, for GN is equal to GC, and the squares of CG and GH are equal to the square of the radius = the square of GI. .. the lamina thus added to the cone and sphere are together equal to the lamina added to the cylinder, and the same is evidently true at any other point. Hence the cone and hemisphere together are equal to the cylinder; but the cone was shown (Prop. 104) to be one-third of the cylinder, therefore the hemisphere is two-thirds of the cylinder; ... the whole sphere will be equal to two-thirds of a cylinder, having its altitude double of the line DC, that is, equal to the diameter of the sphere; and it is evident, that the diameter of the base being the double of CD, is also equal to the diameter of the sphere. Cor. 1. The portion of the sphere, together with the portion of the cone lying between the lines CB and GI, are together equal to the portion of the cylinder lying between the same lines. Cor. 2. Any portion of the sphere, together with the corresponding portion of the cone, is equal to the corresponding portion of the cylinder. END OF SOLID GEOMETRY. GEOMETRICAL EXERCISES. 1. If a straight line bisect another at right angles, every point of the first line will be equally distant from the two extremities of the second line. 2. If straight lines be drawn, bisecting two sides of a triangle at right angles, and from the point of their intersection a perpendicular be drawn to the third side, it will bisect the third side. 3. If two angles of a triangle be bisected, and from the point where the bisecting lines cut one another, a straight line be drawn to the third angle, it will bisect the third angle. 4. The difference of any two sides of a triangle is less than the third side. 5. The sum of two sides of a triangle is greater than twice the straight line drawn from the vertox to the middle of the base. 6. If the opposite sides of a quadrilateral figure be equal, the figure is a parallelogram. 7. If the opposite angles of a quadrilateral figure be equal, the figure is a parallelogram. 8. If a straight line bisect the diagonal of a parallelogram, it will bisect the parallelogram. 9. The diagonals of a rhombus bisect one another at right angles. 10. If a straight line bisect two sides of a triangle it will be parallel to the third side, and equal to the half of it, and will cut off a triangle equal to one-fourth part of the original triangle. 11. The diagonals of a right angled parallelogram are equal. 12. From a given point between two indefinite straight lines given in position, but not parallel, to draw a line which shall be terminated by the given lines, and bisected in the given point. 13. If the sides of a quadrilateral figure be bisected, and the adjacent points of bisection joined, the figure so formed will be a parallelogram, equal to half of the quadrilateral figure. 14. If a point be taken either within or without a rectangle, and straight lines drawn from it to the angular points, the sum of the squares of those drawn to the extremities of one diagonal will be equal to the sum of the squares of those drawn to the extremities of the other. 15. In any quadrilateral figure, the sum of the squares of the diagonals, together with four times the square of the line joining their middle points, is equal to the sum of the squares of the sides. 16. If the vertical angle of a triangle be two-thirds of two right angles, the square of the base will be equal to the sum of the squares of the side, increased by the rectangle contained by the sides; and if the vertical angle be two-thirds of one right angle, the square of the base will be equal to the sum of the squares of the sides, diminished by the rectangle contained by the sides. 17. To bisect a triangle by a line drawn from a given point in one of its sides. 18. A perpendicular drawn from an angle of an equilateral triangle to the opposite side, is equal to three times the radius of the inscribed circle. |