CASE II. When the bases ABCD, CEFG, are equiangular, having the ZDCB the ZGCE, place DC, CE, in one, then GC, CB, are also in one . Also let their sides be produced to meet in the points H, K. Since thes AC, CF, are equal, they are

complements of the AF, and the HCK is its diagonal. Upon the base AF let a parallelopiped be erected, of the same altitude with those upon AC, CF, and let it be cut by planes | its sides, and touching the lines DCE, GCB; these planes divide the whole parallelopiped into four other parallelopipeds, upon the bases DG, AC, BE, CF. But the diagonal plane touching HK divides each of the parallelopipeds upon AF, DG, BE, into two equal prisms, (Prop. 98)... the prisms upon HGC, CEK, are together = the prisms upon HDC, CBK; and these being taken away from the= prisms upon HFK, HAK, the remainders, the parallelopipeds upon AC, CF, are equal. Consequently any two parallelopipeds upon these bases, and between the same parallel planes, are equal, (Prop. 99).

CASE III. When the bases are neither equiangular, nor have one side common, a parallelogram can be described on the base of the one equal to it, (Prop. 26), and equiangular to the other, by which it is reduced to the second Therefore, universally, parallelopipeds upon equal bases, and between the same parallel planes, are equal. Cor. 1. Parallelopipeds of equal bases and equal altitudes are equal.

case.

Cor. 2. Prisms, upon equal bases, which are either both triangles, or both parallelograms, and of equal altitudes, are equal.

Cor. 3. Two prisms, upon equal bases, the one a triangle, and the other a parallelogram, and having the same altitude, are equal.

Cor. 4. A prism upon any rectilineal base is equal to a parallelopiped having an equal base and the same altitude. PROPOSITION CI.-THEOREM.

Parallelopipeds EK, AL, of equal altitudes, are to one another as their bases EFGH, ABCD.

Produce EF both ways to N and R, and make the line FQ such that the altitude of the EG is to the altitude of the AC as AB is to FQ, and complete the OFW and the parallelopiped FS. Hence the OFW is the

AC, (Prop. 64); and the parallelopipeds FS and AL

having equal bases and altitudes, are equal. Take FR any multiple of FQ, and FN any multiple of FE, and complete the □s WR, EM, and MÑ, and the solids QT, EP, and PN; it is manifest, that since FQ, QR, are equal, the

FW, WR, are equal, (Prop. 27); and that, since FW, WR, are equal, the solids FS, QT, are also equal, (Prop. 100); ..what multiple soever the base GR is of FW, the same multiple is the solid FT of the solid FS. In the same manner it may be shown, that what multiple soever the base GN is of the base EG, the same multiple is the solid KN of the solid EK. Now, if the base NG be the base GR, the solid NK will be the solid FT; if equal, equal; and if less, less. .. EK: FS = the base EG: the base FW; and since the base FW=AC, and the solid FS=AL; the solid EK: the solid AL➡ the base EG: the base AC.

Cor. 1. Prisms standing on their ends, and of equal altitudes, are to one another as their bases.

Cor. 2. Parallelopipeds, upon the same or equal bases, are to one another as their altitudes.

For the parallelopipeds KN, KR, on the same base KF, are to one another as NG: GR, that is, as FN: FR.

PROPOSITION CII.-THEOREM.

Two parallelopipeds DE, KL, which have a solid angle B, of the one equal to a solid angle G of the other, are to one another in the ratio compounded of the ratios of the linear sides BA, BC, BE of the one, to the linear sides GF, GH, GL of the other, each to each, about these solid angles.

For, let AB, CB, EB be produced to M, N, O, so that BM=GF, BN=GH and BO D =GL. With the lines EB, BM, and BN, let the parallelopiped EP be completed; and with the lines OB, BM, and BN, the parallelopiped OP. Then the ratio DE: OP is

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 Ls 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 half the whole pyramid, and two equal pyramids, which are similar to the whole and to one another,

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,

= a

AC, of the ABC, are bisected in EF, the EF is = and || CH, or HB, (Prop. 59), half the remaining side BC; and so of the other Is that join the points of section. Hence EC, CG, and consequently EL, ares, 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 BÍK-E, are triangular pyramids. They are equal and similar to one another, and also similar to the whole, because the ▲s which bound the one, are equal and similar to the ▲s which bound the other, and also similar to those which bound the whole, each to each, and alike situated. But either of these pyramids 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 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 = (+16+87+128+ &c.) of a prism, having the same base and altitude as the pyramid. But +++11+ &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.