contain the same straight line M, five, three, and four times respectively: the parallelopiped shall contain the cube of M, 5 x 3×4, or 60 times. In the straight line AB, take the five parts Ab, bb', &c. each equal to M, in AC the three parts Ac, cc', &c. each equal to M, and in AD the four parts Ad, dd', &c., each equal to the same M. Through the points b, c, and d, let planes be drawn parallel to the planes cAd, bAd, and 6 A c respectively, (11. Cor. 1.) and let them meet one another in the point k; then Ak is equal to the cube of M. Let the planes ck, dk be produced to meet the plane BN; and let them be cut by the planes b'k', &c. which are drawn through 5', &c. parallel to the plane bk, or the plane CAD. Then, because the lines Ab, bb', &c. are equal to one another, the bases of the parallelopipeds Ak, bk', &c. are equal to one another (I. 25.), and the parallelopipeds have the same altitude; therefore (24.), they are equal to one another, and the whole row AK is equal to five times the cube of M. Again, through the points c', &c., let the planes cl, &c. be drawn parallel to the plane c k, or (11. Cor. 2.) the plane BAD, to cut the plane dk produced. Then, because the lines A c, cc', &c. are equal to one another, the bases of the parallelopipeds A K, cl, &c. are equal to one another (I. 25.); and the parallelopipeds have the same altitude; therefore (24.) they are equal to one another, and the whole tier A L is equal to three times A K; that is, to 3 x 5, or fifteen times the cube of M. Lastly, through the points d', &c., let planes be drawn parallel to the plane Then, as before, because Ad, dd, &c. dk, or (11. Cor. 2.) the plane BA C. are equal, the parallelopipeds A L, dn, &c. are likewise equal to one another; and, therefore, the whole parallelopiped AN is equal to four times A L, that is to 4 × 3 × 5, or sixty times the cube of M. And, it is evident that a similar demonstration may be applied, whatever other numbers be taken in place of the numbers 5, 3, and 4. Therefore, &c. Scholium. Hence the solid content of a rectangular parallelopiped is said to be equal to the product of its three dimensions, that is to ABX ACX AD, if AB, A C, AD are the three edges; this expression being interpreted in the same sense with the product of the two dimensions or sides, which is said to constitute the area of a rectangle, viz., that the number of cubical units in the parallelopiped is equal to the product of the numbers which denote how often the corresponding linear unit is contained in the three edges. Thus, if the linear unit be a foot, and the edges 3, 4, and 5 feet respectively, the solid content will be 3 × 4 × 5, or 60 cubic feet. It is likewise said, in a similar sense, that the solid content of a rectangular parallelopiped is equal to the product of its base and altitude: thus, in the example just stated, the number of square feet in the base is 4 x 5, or 20; and this, being multiplied by 3, the number of linear feet in the altitude, gives 60 for the number of cubic feet in the parallelopiped, as before. The cube is also the unit in the mensuration of all other solids; their contents being the same with the contents of rectangular parallelopipeds equal to them. Thus, since every parallelopiped (24. Cor.) is equal to a rectangular parallelopiped of equal base and altitude, the solid content of every pa rallelopiped is equal to the product of its base and altitude. The prism and the pyramid will presently furnish new examples. PROP. 26. (Euc. xi. 32.) Parallelopipeds, which have equal altitudes, are to one another as their bases; and parallelopipeds, which have equal bases, are to one another as their altitudes; also, any two parallelopipeds are to one another in the ratio, which is compounded of the ratios of their bases and altitudes. First, let A B, A C be two rectangular parallelopipeds, having equal altitudes, and let AE, A F be their bases; the parallelopipeds being so placed that a solid angle of the one may coincide, as at A, with a solid angle of the other. Then, the parallelopipeds will have a common part AD, which is likewise a rectangular parallelopiped, having the same altitude AL with them, and the two adjoining edges AG, AH common to it with AB, AC respectively. Let the base AK of this parallelo piped be divided into any number f C F B E piped into the same number of equal parallelopipeds, by planes passing through these lines parallel to the planes LAH (24.) Then, if one of the first parts be contained in the base A E any number of times, exactly, or with a remainder; one of the others will be contained in the parallelopiped A B the same number of times, exactly, or with a corresponding remainder (24.). Therefore (II. def. 7.) the parallelopiped A B is to AD as the base A E to the base A K. And in the same manner it may be shewn that the parallelopiped AD is to AC as the base AK to the base AF. Therefore, ex æquali (II. 24.), the parallelopiped A B is to the parallelopiped A C as the base A E to the base A F. Next, let AB, CD be two rectangular parallelopipeds, having equal bases; and let AE, CF be their altitudes. Then, if the altitude C F be divided into any number of equal parts; it may easily be shown that the parallelopiped CD will be divided into the same number of equal parallelopipeds, by planes bases AE, CF, and the altitudes AG, CH. Take A L equal to CH, and let AK be a third parallelopiped having the base AE and the altitude AL. Then, because the parallelopiped A B has to CD the ratio which is compounded of the ratios of A B to A K, and A K to CD (II. def. 12.); which ratios, by what has been just demonstrated, are the same with those of AG to AL or CH, and AE to CF; the parallelopiped A B has to the parallelopiped CD, the ratio which is compounded of the ratios of the altitude AG to the altitude GH, and of the base AE to the base C F. And what has here been demonstrated of rectangular parallelopipeds is true also with regard to any two parallelopipeds whatever, because these (24. Cor.) are equal to rectangular parallelopipeds having equal bases and altitudes with them. Therefore, &c. PROP. 27. Any two rectangular parallelopipeds are to one another in the ratio, which is compounded of the ratios of their edges. Let A E, A e be two rectangular parallelopipeds, the first having the edges AB, AC, AD, the other the edges Ab, Ac, Ad; the parallelopiped A E shall be to the parallelopiped AB in the ratio which is compounded of the A D E FG d હ For, the parallelo- b pipeds being placed so that a solid angle of the one may coincide with a solid angle of the other as at A, they will have a common part A F, which is likewise a rectangular parallelopiped; and the face of this parallelopiped, which is opposite to Db, being produced to meet the plane de, will cut off from the parallelopiped A e, the part A G, which is also a rectangular parallelopiped. Now, the parallelopipeds AE, AF have a common base DC; therefore (26.) they are to one another as their altitudes A B, Ab; and, for the like reason, the parallelopipeds A F, A G are to one another as their altitudes AD, Ad, and the parallelopipeds AG, A e are to one another as their altitudes AC, Ac. But the parallelopiped A E has to the parallelopiped Ae the ratio which is compounded of the ratios of AE to AF, AF to AG, and AG to Ae. Therefore, the parallelopiped A E has to the parallelopiped Ae, the ratio which is compounded of the ratios of AB to Ab, AD to Ad, and A C to Ac; or, which is the same thing, which is compounded of the ratios of AB to Ab, AC to A c, and AD to Ad. (II. 27.)* *This demonstration is analogous to that of II. 36. The proposition may, however, be otherwise, and more concisely established by the aid of Prop. 26.; for, since the parallelopipeds are to one another in the ratio which is compounded of the ratios of their bases and altitudes, and that the bases, which are rectangles under two of the conterminous edges of each, are to one another in the ratio which is compounded of the ratios of those edges (II. 36.), and the altitudes equal to the third edges, it follows that b C (26.), that is, as AB to Ab (II. 35. Cor.); and in like manner AF is to AG as AD to Ad, and AG to Ae as AC to Ac; therefore, as in the proposition, the parallelopiped AE is to the parallelopiped Ae, in the ratio which is compounded of the ratios of AB to Ab, AD to Ad, and AC to Ac; that is, in a ratio which is compounded of the ratio of the edges about the common angle A. Cor. 2. Cubes are to one another in the triplicate ratio of their edges (II. 27. Cor. 2.); therefore, the triplicate ratio of two straight lines A B, ab is the same with the ratio of their cubes. Cor. 3. If one straight line be to another as a third to a fourth, the cube of the first shall be to the cube of the second as the cube of the third to the cube of the fourth (II. 27.). PROP. 28. Every triangular prism is equal to a rectangular parallelopiped, which has an equal base and the same altitude. Let Abc be a triangular prism, and let E F be a rectangular parallelopiped, a A having its base E G equal to the base A B C of the prism, and its altitude F G the same with the altitude of the prism: the prism Abc shall be equal to the parallelopiped E F. Let the bases A B C, a b c of the prism be completed into the parallelograms A D, ad, and let D d be joined. Then it is evident that the solid A d is a parallelopiped (def. 11.). But the prism Abc is equal to half the parallelopiped A d (21.); and the parallelopiped EF is also equal to half the parallelopiped Ad, because the base EG is equal to A B C, that is, to half of the base A D, and the altitude FG is the same with the altitude A a (26.). Therefore (I. ax. 5.) the prism Abc is equal to the parallelopiped E F Therefore, &c. PROP. 29. Every prism is equal to a rectangular the parallelopipeds are to one another in the ratio parallelopiped, which has an equal base which is compounded of the ratios of their edges. And the first corollary (analogous to II. 36. Cor. 1.) admits of a demonstration little different. and the same altitude. Let AdC be a prism, having the polygonal bases ABCDE, abcde, and B C FL N G then, because A a and Cc are parallel, they lie in the same plane (I. def. 12.) and the solid Abc is a triangular prism upon the base ABC (def. 14.). In the same manner it may be shewn, that Acd, Ade, are triangular prisms upon the bases ACD, ADE. Now, because the rectangle FH is equal to the polygon A B C D, it may be divided (I. 57.) into rectangles FK, LM, NH, severally equal to the triangles ABC, ACD, ADE, which together make up the polygon; and the parallelopiped F H may be divided into as many rectangular parallelopipeds, having the same altitude GH, and these rectangles for their bases. And, because these bases are severally equal to the bases of the triangular prisms Abc, A cd, Ade, and that their common altitude G H is the same with the common altitude of the prisms, the parallelopipeds, which stand upon them, are severally equal to the prisms (28.). Therefore their sum is equal to the sum of the prisms; that is, the parallelopiped FG is equal to the prism Ad C. Therefore, &c. Cor. 1. The solid content of every prism is equal to the product of its base and altitude (25. Scholium). Cor. 2. (Euc. xii. 7. Cor. 2.) Prisms which have equal altitudes are to one another as their bases; and prisms which have equal bases are to one another as their altitudes: also, any two prisms are to one another in the ratio which is compounded of the ratios of their bases and altitudes. (26.). Scholium. With regard to the lateral surface of a prism, if it be a right prism, it is measured by the product of the principal edge, and the perimeter of the base; if oblique, by the product of the principal edge and the perimeter of a section, which is made by a plane per pendicular to the principal edge. For, in the case of the right prism, the sides are rectangles, which have a common altitude, and for their bases the sides of the base of the prism; therefore (I. 30.) their sum is equal to a rectangle of the same altitude, and having its base equal to the sum of those sides. And, in the case of the oblique prism, the sides of the perpendicular section, being perpendicular to the principal edges (5.), are the altitudes respectively of the the prism and have for their bases each parallelograms, which are the sides of the sum of those parallelograms is equal a principal edge of the prism; therefore to a rectangle, having likewise for its base altitude the sum of their altitudes, that is, a principal edge of the prism, and for its the perimeter of the perpendicular section. In the latter case it is easy also to perceive that the content of the oblique prism is measured by the product of the principal edge, and the area of the perpendicular section above mentioned: for, if the lateral perpendicular planes passing through surface be produced, and cut by two such the extremities of any principal edge, as in the figure, the solids included between these planes and the bases of the oblique prism, may be made to coincide, and are therefore equal to one another: therefore the whole oblique prism is equal to the right prism, which has for its bases the two perpendicular sections. The convex surface of a prism has this obvious but remarkable property, that planes are similar and equal figures (12. the sections made by any two parallel and 15. Cor.): the convex surface, also, similar, but they are not equal to one of a pyramid has its parallel sections another. altitude KL. Let K L be divided into any number of equal parts; and let the pyramids be cut by planes, parallel to the plane of the bases, passing through the points of division. Let bed and fgh, bed and f'g' h', &c. be the sections made by these planes. Then, because the triangles bcd, BCD lie in parallel planes, which cut the dihedral angles at the edges A B, A C, and AD of the pyramid ABCD, the angles b, c, d of the one are (15. Cor.) severally equal to the angles B, C, D of the other, and therefore (II. 31. Cor. 1.) the two triangles are similar. Therefore the triangle bed is to the triangle B CD as bc2 to B C2 (II. 42. Cor.), that is, because bc is (12.) parallel to BC, as A b2 to A B2 (II. 30. Cor. 2. and II. 37. Cor. 4.) And, in the same manner it may be shewn, that the triangle f g h is to the triangle F G H as Ef2 to E F2, that is (14.) as Ab2 to A B2. Therefore the triangle bed is to the triangle B CD as the triangle f g h to the triangle FGH: and, because BCD is equal to FGH (II. 12.), bcd is equal to ƒgh (II. 18. Cor.). In the same manner it may be shewn, that b'c'd' is equal to f'g'h'; and so of the rest. Now, if the triangular prisms with the bases b cd, b' c' d', &c., and the edges bB, b' b, &c., be completed, and also the triangular prisms with the bases B CD, bed, &c., and the edges B b, bb', &c.; the former will, together, constitute a series of prisms inscribed in the pyramid A B C D, and the latter a series of prisms circumscribed about the same pyramid, which, excepting that upon the base BCD, are equal to the former, each to each, because they have equal altitudes towards opposite parts of the same base (29. Cor. 2.). And, because the pyramid exceeds the inscribed series, but is less than the circumscribed; it differs from the former, by less than the difference of the two series, that is, by less than the prism upon the base BCD. But, by increasing the number of parts into which the altitude is divided, the altitude of this prism may be diminished without limit, and so the prism itself be made less than any given solid. Therefore, the series of inscribed prisms may be made to approach to the pyramid ABCD, by less than any given difference. And in the same manner it may be shewn, that if prisms be similarly inscribed in the other pyramid EFGH, this series will approach to the pyramid E F G H, by less than the prism upon the base FGH, which is equal to that before-mentioned upon the equal base B CD; by less, that is, than the same given difference. But of the prisms inscribed in the two pyramids, those upon the bases b cd, and fgh, b'c' d' and f'g'h', &c. are equal to one another, because their bases are equal to one another, and they have the same altitude (29. Cor. 2.). Therefore, the series inscribed in the pyramid A B C D is equal to the series inscribed in the pyramid E F G H (I. ax. 2.). And because this equality always subsists, and that the two series may be made to approach to the two pyramids respectively within any the same given difference, the pyramid ABCD is equal to the pyramid E F G H. (II. 28.). Therefore, &c. PROP. 31. (EUc. xii. 7. Cor. 1.) Every triangular pyramid is equal to the third part of a prism, having the same base and the same altitude with the pyramid. Let A B C D be a triangular pyramid, having the base BCD; let the sides ABC, ABD be com- A pleted into the parallelograms A B C E, A B D F, and join EF; then it is evident that the triangle AEF is the upper base B of a prism B EF, standing upon the same base BCD with the pyramid, and having the same altitude: the pyramid A B C D shall be equal to a third part of the prism B E F. The Join D E. Then the prism BEF is made up of three pyramids, viz. AB CD, ACDE, and ADEF. first of these is the original pyramid; the second is equal (30.) to a pyramid BCDE, upon the same base CDE, and having its vertex B in the same line AB parallel (10.) to the base; and the third is equal to a pyramid BDEF, upon the same base D E F, and having its vertex B in the same line AB paral |