square of FH, as the circle ABCD to the square of FH, so is the circ! any space less than the circle EFGH: ABCD to the circle EFGH. Circles Wherefore as the square of BD to therefore are, &c. Q. E. D. PROP. III. THEOREM. I Every pyramid having a triangular base, may be divided into two equal and similar pyramids having triangular bases, and which are similar to the whole pyramid ; and into two equal prisms, which together are greater than half of the whole pyramid. Let there be a pyramid of which not in the same plane with them, they the base is the triangle ABC and its contain equal (10. 11.) angles; theresertex the point D: The pyramid fore the angle EHG is equal to the ABCD may be divided into two equal angle KDL. Again, because EH, and similar pyramids D HG are equal to KD, DL, each to having triangular bases, each, and the angle EHG equal to the and similar to the whole; angle KDL; therefore the base EG and into two equal is equal to the base KL: And the prisms which together K L triangle EHG equal (4. 1.) and simiare greater than half of lar to the triangle KDL: For the same the whole pyramid. reason, the triangle AEG is also eDivide AB, BC, CA, qual and similar to the triangle HKL. AD, DB, DC, each into B Therefore the pyramid, of which the two equal parts in the base is the triangle AEG, and of points E, F, G, H, K, L, and join which the vertex is the point H, is EH, EG, GH, HK, KL, LH, ÉK, equal (C. 11 ) and si D KF, FG. Because AE is equal to milar to the pyramid EB, and AH to HD, HE is parallel the base of which is the (2. 6.) to DB : For the same reason, triangle KHL, and ver, HK 18 parallel to AB: Therefore tex the point D; And K HEBK is a parallelogram, and HK because HK is parallel equal (34. 1. to EB : But EB is equal to AB a side of the trito AE; therefore also AE is equal to angle ADB, the trianHK : And AH is equal to HD; gle ADB is equiangu- BF wherefore EA, AH are equal to KH, lar to the triangle HDK, HD, each to each; and the angle and their sides are proportionals: (4. EAH is equal (29. 1.) to the angle 6.) Therefore the triangle ADB is siKHD; therefore the base EH is equal milar to tbe triangle HDK. And for to the base KD, and the tria::gle AEH the same reason, the triangle DBC equal (1. 1.) and similar to the trian- is similar to the triangle DÅL: and gle HKD: For the same reason, the the triangle ADC to the triangle triangle AGH is equal and similar to HDL; and also the triangle ABC to the triangle HLD. And because the the triangle AEG : But the triangle two straight lines EH, HG, which AEG is similar to the triangle HKL, meet one another, are parallel to KD, as before was proved; therefore the DL that meet one another, and are triangle ABC is similar (21. 6.) to the • Because as a fourth proportional to the squares of BD, FH, and the circle ABCD, is possible, and that it can neither be less nor greater than the circle EFGH, it must be equal to it. AL 日 triangle HKL. And the pyramid of and the vertices the points H, D: bewhich the base is the triangle ABC, cause, if EF be joined, the prism har. and vertex the point D, is therefore ing the parallelogram EBFG for its similar (B. 11. & 11. def. 11.) to the base, and KH the straight line oppopyramid of which the base is the tri- site to it, is greater than the pyramid angle HKL, and vertex the same of which the base is the triangle EBF, point D): But the pyramid of which and vertex the point K; but this pythe base is the triangle HKL, and ramid is equal (C. 11.) to the pyramid vertex the point D), is similar, as has the base of which is the triangle AEG, been proved, to the pyramid the base and vertex the point H: because they of which is the triangle AEG, and are contained by equal and similar vertex the point H: Wherefore the planes: Wherefore the prism having pyramid, the base of which is the tri- the parallelogram EBFG for its base, angle ABC, and vertex the point D, and opposite side KH, is greater than is similar to the pyramid of which the the pyramid of which the base is the base is the triangle AEG and vertex triangle AEG, and vertex the point H: Therefore each of the pyramids H: And the prism of which the base AEGH, HKLD is similar to the is the parallelogram EBFG, and opwhole pyramid ABCD: And because posite side KH is eqnal to the prism BF is equal to FC, the parallelogram haring the triangle GFC for its base, EBFG is double (41. 1.) of the trian- and HKL the triangle opposite to it; gle GFC: But when there are two and the pyramid of which the base is prisms of the same altitude, of which the triangle AEG, and vertex H, is one has a parallelogram for its base, equal to the pyramid of which the and the other a triangle that is half base is the triangle HKL, and vertex of the parallelogram, these prisms D: Therefore the two prisms before are equal (40. 11.) to one another; mentioned are greater than the two therefore the prism having the paral- pyramids of which the bases are the lelogram CBFG for its base, and the triangles AEG, HKL, and vertices straight line KH opposite to it, is e- the points H, D. Therefore the whols qual to the prism having the triangle pyramid of which the base is the tri GFC for its base, and the triangle angle ABC, and vertex the point D, HKL opposite to it; for they are is divided into two equal pyramids siof the same altitude, because they milar to one another, and to the whole are between the parallel(15.11.) planes pyramid ; and into two equal prisms; ABC, HKL: And it is manifest that and the two prisms are together each of these prisms is greater than greater than half of the whole pyraeither of the pyramids of which the mid. Q. E. D. triangles AEG, HKL are the bases, PROP. IV. THEOREM. If there be two pyramids of the same altitude, upon triangular bases, and each of them be divided into two equal pyramids similar to the whole pyramid, and also into two equal prisms; and if each of these pyramids be divided in the same manner as the first two, and so on : As the base of one of the first two pyramids is to the base of the other, so shall all the prisms in one of them be to all the prisms in the other that are produced by the same number of divisions. Let there be two pyramids of the bases, ABC, DEF, and having their same altitude: upon the triangular vertices in the points G, H; and let G each of them be divided into two equal in the pyramid ABCG are equa. tn pyramids similar to the whole, and one another, and also the two prisms into two equal prisms; and let each in the pyramid DEFH equal to one of the pyramids thus made be con- `another, as the prism of which the ceived to be divided in the like man- base is the parallelogram KBXL and ner, and so on : As the base ABC is opposite side MO, to the prisın having to the base DEF, so are all the prisms the triangle LXC for its base, and in the pyramid ABCG to all the prisms OMN the triangle opposite to it; so in the pyramid DEFH made by the is the prism of which the base (7. 5.) same number of divisions. is the parallelogram PEVR, and op • Make the same construction as in posite side TS, to the prism of which the foregoing proposition: And be the base is the triangle RVF, and op cause BX is equal to XC, and AL to posite triangle STY. Therefore, come LC, therefore XL is parallel (9. 6.) to ponendo, as the prisms KBXLMO, AB, and the triangle ABC similar to LXCOMN together are unto the the triangle LXC: For the same rea Ir MANY TKY BC, CX are described the similar and similarly situated rectilineal figures ABC, LXC; and upon EF, FV, in like manner, are described the similar figures DEF, RVF: Therefore, as в Х. the triangle ABC is to the triangle prism LXCOMN: so are the prisms LXC, so (22. 6.) is the triangle DÈF PEVRTS, RVFSTY to the prison to the triangle RVF, and, by permu- RVFSTY: And permutando, as the tation, as the triangle À BC to the tri- prisms KBXLMO, IXCOMN are angle DEF, so is the triangle LXC to to the prisms PEVRTS, RVFSTY; the triangle RVF: And because the so is the prism LXCOMN to the planes ABC, OMN, as also the planes prism RVI STY: But as the prism DEF, STY, are parallel, the per- LXCOMX to the prism RVFSTY, pendiculars drawn from the points so is, as has been proved, the base G, H to the bases ABC, DEF, which ABC to the base DEF: Therefore, by the hypothesis, are equal to one as the base ABC to the base DEF, another, shall be cut each into two so are the two prisms in the pyramid equal (17. 11.) parts by the planes ABCG to the two prisms in the pyraOMN, STY, because the straight mid DEFH: And likewise if the pye lines GC, HF are cut into two equal ramids now made, for example, the parts in the points N, Y by the same two OMNG, STYH be divided in planes : Therefore the prisms LXC- the same manner; as the base OMN OMN, RVFSTY are of the same al- is to the base STY, so shall the two titude; and therefore 'as the base prisms in the pyramid OMNG be to LXC to the base RVF; that is, as the two prisms in the pyramid STYH: the triangle ABC to the triangle DEF, But the base OMN is to the base STY. so (Cor. 32. 11.) is the prism having as the base ABC to the base DEF: the triangle LXC for its base, and therefore, as the base ABC to the OMN the triangle opposite to it, to base DEF, so are the two prisms in the prism of which the base is the tri. the pyramid ABCG to the two prisma angle AVF, and the opposite triangle in the pyramid DEFH; and so are STY: And because the two prisms the two prisms in the pyramid OMNG to the two prisms in the pyramid the pyramids AKLO, and DPRS, and STYH; and so are all four to all of all made by the same number of four: And the same thing may be divisions. Q. E. D. shewn of the prisms made by dividing PROP. V. THEOREM. Pyramids of the same altitude which have triangular bases, are to one another as their bases. Let the pyramids of which the tri- pyramid ABCG in the same manner, angles ABC, DEF are the base?, and and into as many parts, as the pyraof which the vertices are the points mid DEFH: Therefore as the base, G, H be of the same altitude: As the ABC to the base DEF, so (4. 12.) are base ABC to the base DEF, so is the the prisms in the pyramid ABCG to pyramid ABCG to the pyramid DEFH. the prisms in the pyramid DEFH: For, if it be not so, the base ABC But as the base ABC to the base DEF, must be to the base DEF, as the py- so by hypothesis is the pyramid ABCG ramid ABCG to a solid either less to the solid Q; and therefore, as the than the pyrarnid DEFH, or greater pyramid ABCG to the solid Q, so are than it.* *First, let it be to a solid the prisms in the pyramid ABCG to less than it, viz. to the solid Q: And the prisms in the pyramid DETH: divide the pyramid DEFH into two But the pyramid ABCG is greater equal pyramids, similar to the whole, than the prisms contained in it; and into two equal prisms : Therefore wherefore (14. 5.) also the solid Q is these two prisms are greater (3. 12.) greater than the prisms in the pyramid than the half of the whole pyramid. DEFH. But it is also less, which is And again, let the pyramids made by impossible. Therefore the base ABC this division be in like manner divided, is not to the base DEF, as the pyramid and so on, until the pyramids which ABCG to any solid which is less than remain undivided in the pyramid the pyramid DEFH. In the same DEFH be, all of them together, less manner it may be demonstrated, that than the excess of the pyramid DEFH the base DEF is not to the base ABC, above the solid Q : Let these, for ex- as the pyramid DEFH to any solid ample, be the pyramids DPRS, STYH: which is less than the pyramid ABCG. Therefore the prisms, which make the Nor can the base ABC be to the base rest of the pyramid DEFH, are greater DEF, as the pyramid ABCG to any than the solid Q : Divide likewise the solid which is greater than the pyra G н This may be explained she same way us at the Doce in Proportion 2, in the like ease. mid DEFH. For if it be possible, ramid DEFH to a solid less than the let it be so to a greater, viz. the solid pyramid ABCG; the contrary to Z. And because the base ABC is to which has been proved. Therefore the base DEF as the pyramid ABG the base ABC is not to the base DEF, to the solid Z: by inversion, as the as the pyramid ABCG to any solid base DEF to the base ABC, so is the which is greater than the pyramid solid Z to the pyramid A5CG. But DEFH. And it has been proved, that as the solid Z is to the pyramid ABCG, neither is the base ABC the base 80 is the pyramid DEFH to some so- DEF, as the pyramid ABCG to any lid," which must be less (14.5.) than solid which is less than the pyramid the pyramids ABCG, because the 80- DEFE. Therefore, as the base ABC lid Z is greater than the pyramid is to the base DEF, so is the pyramid DEFH. And therefore, as the base ABCG to the pyramid DEFH. DEF to the base ABC, so is the py- Wherefore pyramids, &c. Q. E. D. PROP. VI. THEOREM. Pyramids of the same altitude which have polygons for their bases, are to one another as their bases. Let the pyramids which have the to the triangle FGH, 'as' the pyramid polygons ABCDE, FGHKL for their ACDM to the pyramid FGHN ; and bases, and their vertices in the points also the triangle ADF to the triangle M, N be of the same altitude: As the FGH, as the pyramid ADEM to the base ABCDE to the base FGHKL, fyramid FGHŇ: as all the first ante80 is the pyramid ABCDEM to the cedents to their common cousequent ; pyramid FGHKLN. so (2. Cor. 24. 5.) are all the other an. Divide the base ABCDE into the tecedents to their common consequent; triangles ABC, ACD, ADE; and the that is, as the base ABCDE to the base FGHKL into the triangles FGH, base FGH,so is the pyramid ABCDEM FHK, FKL: And upon the basts to the pyramid FGHN. And, for the ABC, ACD, ADE let there be as same reason, as the base FGHKL to many pyramids of which the common the base FGH, so is the pyramid vertex is the point M, and upon the FGHKLN to the pyramid FGHN: remaining bases as many pyramids And, by inversion, as the base FGH to having their common vertex in the the base FGHKL, so is the pyramid point N: Therefore, since the triangle FGHN to the pyramid FGHKLN : ABC is to the triangle FGH, as (6. Then, because as the base ABCDE 12.) the pyramid ABCM to the pyra. to the base FGH, so is the pyramid mid FGHN; and the triangle ACD ABCDEM to the pyramid FGIN; and as the base FGH to the base FGHKL, so is the pyramid FGHN to the pyramid FGHKLN: therefore, ex æquali, (22. 5.) as the base ABCDE I. AS K to the base FGHKL, so the pyramid ABCDEM to the pyramid FGHKLN. G Therefore pyramids, &c. Q. E. D. M D This may be explained the same way as the like at the mark + in Prop. 2. |