Supplement Take S, so that AC may be to KM as KO to S, and it will be fhewn, as was done above, that the folid AG is to COR. In the fame manner, may it be demonftrated, that equal prifms have their bases and altitudes reciprocally proportional, and converfely. a 9. def. 3. Sup. PROP. XI. THEOR. SIMILAR folid parallelepipeds are to one another in the triplicate ratio of their homologous fides. Let AG, KQ be two fimilar parallelepipeds, of which AB and KL are two homologous fides; the ratio of the solid AG to the folid KQ is triplicate of the ratio of AB to KL. Because the folids are fimilar, the parallelograms AF, KP are fimilar a, as also the parallelograms AH, KR; therefore, the ratios of AB to KO. Now, the ratio of AC to KM, because they are c14. 6. equiangular parallelograms, is compounded of the ratios of AB to KL, and of AD to KN. Wherefore, the ratio of AG to KQ is compounded of the three ratios of AB to KL, AD to KN, and AE to KO; and these three ratios have already been proved to be equal; therefore, the ratio that is compounded compounded of them, viz. the ratio of the folid AG to the Book III. folid KQ, is triplicate of any of them d; it is therefore tripli- d 12. def. 5. cate of the ratio of AB to KL. Therefore, fimilar solid pa rallele pepids, &c. Q. E. D. COR. I. If as AB to KL, fo KL to m, and as KL to m, fo is m to n, then AB is to n as the folid AG to the folid KQ. For the ratio of AB to n is triplicate of the ratio of AB to KLe, and is therefore equal to that of the folid AG to the fo- e 12. def. 5. lid KQ. COR. 2. As cubes are fimilar folids, therefore the cube on AB is to the cube on KL in the triplicate ratio of AB to KL, that is in the same ratio with the folid AG to the folid KQ. Similar folid parallelepipeds are therefore to one another as the cubes on their homologous fides. COR. 3. In the fame manner it is proved, that fimilar prisms are to one another in the triplicate ratio, or in the ratio of the cubes, of their homologous fides.. Ifes and plitudes, be cut by planes that are paral F two triangular pyramids, which have equal ba lel to the bafes, and at equal distances from them, the fections are equal to one another. Let ABCD and EFGH be two pyramids, having equal bafes BDC and FGH, and equal altitudes, viz. the perpendi A A culars AQ, and ES, drawn from A and E upon the planes T BDC Supplement BDC and FGH: and let them be cut by planes parallel to BDC and FGH, and at equal altitudes QR and ST above thofe planes, and let the fections be the triangles KLM, NOP; KLM and NOP are equal to one another. Because the plane ABD cuts the parallel planes BDC, a 14.2. Sap. KLM, the common fections BD and KM are parallel 2. For the fame reason, DC and ML are parallel. Since therefore KM and ML are parallel to BD and DC, each to each, though not in the fame plane with them, the angle KML is equal to the 9. 1. Sup. angle BDC b. In like manner the other angles of these triangles are proved to be equal; therefore, the triangles are equiangular, and confequently fimilar; and the fame is true of the triangles NOP, FGH. A A G Now, fince the ftraight lines ARQ, AKB meet the paralc16. 2.Sup. lel planes BDC and KML, they are cut by them propord 18. 5. tionally, or QR: RA:: BK: KA; and AQ:AR:: AB : AK d, for the fame reafon, ES: ET:: EF EN; therefore, AB: AK:: EF: EN, because AQ is equal to ES, and AR to ET. Again, because the triangles ABC, AKL are fimilar, AB: AK:: BC: KL; and for the fame reason EF EN:: FG: NO; therefore, BC KL:: FG: NO. : And, when four ftraight lines e 22. 6. are proportionals, the fimilar figures defcribed on them áre also proportionals e; therefore the triangle BCD is to the triangle KLM as the triangle FGH to the triangle NOP; but the f14. 5. triangles BDC, FGH are equal; therefore, the triangle KLM is alfo equal to the triangle NOPf. Therefore, &c. Q. E. D. COR. I. Because it has been fhewn that the triangle KLM is fimilar to the base BCD; therefore, any fection of a triangu larpyramid parallel to the bafe, is a triangle fimilar to the bafe. And 1 And in the fame manner it is fhewn, that the sections parallel Book III. to the base of a polygonal pyramid are similar to the base. COR. 2. Hence alfo, in polygonal pyramids of equal bafes and altitudes, the fections parallel to the bafes, and at equal distances from them, are equal to one another. A SERIES of prifms of the fame altitude may be circumfcribed about any pyramid, fuch that the fum of the prifms fhall exceed the pyramid by a folid lefs than any given folid. Let ABCD be a pyramid, and Za given folid; a feries of prifms having all the fame altitude, may be circumfcribed about the pyramid ABCD, fo that their fum fhall exceed ABCD by a folid less than Z. W N T U H I M. R r G Let Z be equal to a prism standing on the fame bafe with the pyramid, viz. the triangle BCD, and having for its altitude the perpendicular drawn from a certain point E in the line AC upon the plané BCD. It is evident, that CE multiplied by a certain number m will be greater than AC; divide CA into as many equal parts as there are units in m, and let thefe be CF, FG, GH, HA, each of which will be lefs than CE. K Through each of the points F, G, H let planes be made to pafs parallel to the plane BCD, making with the fides of the pyra- B mid the fections FPQ, GRS, HTU, which will be all fimilar to one another, and to the bafe P h 00 D BCD a. From the point B draw C a i. cor. 12. in the plane of the triangle ABC the, straight line BK parallel 3. Sup *The folid Z is not represented in the figure of this, or the following Piopofition. dr. cor 8. 3. Sup. N W V Supplement to CF meeting FP produced in K. In like manner, from D draw DL parallel to CF, meeting FQ in L: Join KL, and b 11. def. 7. it is plain, that the folid KBCDLF is a prifm b. By the fame conftruction, let the prifms PM, RO, TV be defcribed. Alfo, let the ftraight line IP, which is in the plane of the triangle ABC be produced till it meet BC in h; and let the line MQ be produced till it meet DC in g: Join hg; then hCgQFP is a prifm, and is equal to the prifm PM d. In the fame manner is defcribed the prifm MS equal to the prifm RO, and the prifm qU equal to the prifm TV. The fum, therefore, of all the infcri- K, bed prifms hQ, mS, and qU is equal to the fum of the prifms PM, RO and TV, that is, to the fum of all the circumfcribed B prifms except the prifm BL; wherefore, BL is the excefs of the prifms circumfcribed about the pyramid ABCD above the I T U H M R S r G L P m n F h C prifms infcribed within it. But the prifm BL is less than the prifm which has the triangle BCD for its bafe, and for its altitude the perpendicular from E upon the plane BCD; and the prifm which has BCD for its bafe, and the perpendicular from E for its altitude, is by hypothefis equal to the given folid Z; therefore, the excefs of the circumfcribed, above the infcribed prifms, is lefs than the given folid Z. But the excess of the circumfcribed prifms above the infcribed is greater than their excefs above the pyramid ABCD, becaufe ABCD is greater than the fum of the infcribed prifms. Much more, therefore, is the excefs of the circumfcribed prifms above the pyramid, lefs than the folid Z. A feries of prifms of the fame altitude has therefore been circumfcribed about the pyramid ABCD exceeding it by a folid lefs than the given folid Z. Q. E. D. PROP. |