and of all other Prisms, being equal in Multitude ; PROPOSITION V. THÉOR E M. gular Bases, are to one another as their Bases. having the triangular Bases ABC, DEF, whofe For if it be not so, then it shall be as the Base ABC is to the Base DEF, so is the Pyramid ABCG to fome Solid, greater or less than the Pyramid DEFH. First, let it be to a Solid less, which let be Z, and divide the Pyramid DEFH into two Pyramids equal to each other, and similar to the Whole, and into two equal Prisms; then these two Prisms are greater than the half of the whole Pyramid. And again, let the Pyramids made by the former Division, be divided after the same manner, and let this be done continually, until the Pyramids in the Pyramid DEFH, are less than the Excess by which the Pyramid DEFH exceeds the Solid Z. Let these, for Example, be the Pyramids DPRS, STYH; then the Prisms remaining in the Pyramid DEF H, are greater than the Solid Ž. Also, let the Pyramid ABCG, be divided into the fame Number of similar Parts as the Pyramid DEF H is; and then as the Base ABC is to the Base DEF, so * the Prisms that are in the * 4 of thiso Pyramid ABCG, to the Prisms that are in the Pyramid DEFH. But as the Base ABC is to the Base DEF, fo is the Pyramid ABCG to to the Solid Z. And therefore as the Pyramid ABCG is to the Solid Z, so are the Prisms that are in the Pyramid ABCG, to the Prisms that are in the Pyramid DEFH; but the Pyramid ABCG, is greater than the Prisms that are in it. Wherefore also the Solid Z, is greater than the Prisms that are in the Pyramid DEFH; but * From wbat it is less * also, which is absurd. Therefore the Bafe bas been al ABC to the Base DEF, is not as the Pyramid ready demonfrated ABCG to some Solid less than the Pyramíd DEFH. After the same Manner we demonstrate that the Base DEF to the Base ABC, is not as the Pyramid DEFH to some Solid less than the Pyramid ABCG. Therefore, I say, neither is the Base ABC to the Bafe DEF, as the Pyramid ABCG to some Solid greater than the Pyramid DEFH. For if this is poffible, let it be to the Solid I, greater than the Pyramid DEFH. Then (by Inverfion) the Base D'EF shall be to the Base ABC, as the Solid I to the Pyramid ABCG: But since the Solid I is greater than the Pyramid EDFH, it shall be as the Solid I is to the Pyramid ABCG, so is the Pyramid DEFH, to fome Solid less than the Pyramid ABCG, as just now has been proved. And so as the Base DEĚ is to the Base ABC, fo is the Pyramid DEFH, to fome Solid less than the Pyramid ABCG, which is absurd. Therefore the Base AB C to the Base DEF, is not as the Pyramid ABCG to fome Solid greater than the Pyramid DEFH. But it has been also proved, that the Base ABC to the Base DEF, is not as the Pyramid ABCG to some Solid less than the Pyramid DEFH. Wherefore as the Base ABC is to the Base DEF, fo is the Pyramid ABCG to the Pyramid DEFH. Therefore, Pyramids of the fame Altitude, and baving triangular Bases, are to one another as their Bases; which was to be demonstrated. PROPOSITION VI. THEOR M. nous Bases, are to one another as their Bases. LE ET there be Pyramids of the fame Altitude, which have the polygonous Bases ABCDE, FGHKL, and let their Vertices be the Points M, N, I fay, as the Base ABCDE is to the Base FGHKL, so is the Pyramid ABCDEM to the Pysamid F GHKLN, For For let the Base ABCDE be divided into the Triangles ABC, ACD, ADE; and the Bafe F GHKL into the Triangles FGH, FHK, FKL; and let Pyramids be conceived upon every one of those Triangles of the fame Altitude with the Pyramids ABC, DEM, FGHKLN. Then because the Triangle ABC is to the Triangle ACD, as * the Pyramid * 5 of thiso ABCM is to the Pyramid ACDM: And (by compounding) as the Trapezium ABCD is to the Triangle A CD, so is the Pyramid ABCDM to the Pyramid ACDM; but as the Triangle A CD is to the Triangle ADE, so is * the Pyramid ACDM to the Pyramid ADEM. Wherefore (by Equality of Proportion) as the Base ABCD is to the Base ADE, fo is the Pyramid ABCD'M to the Pyramid ADEM. And again (by composition of Proportion) as the Base ABCDE is to the Base A DE, so is the Pyramid ABCDEM to the Pyramid ADEM. For the fame Reason, as the Base FGHKL is to the Base FKL, so is the Pyramid FGHKLN to the Pyramid FKLN. And since there are two Pyramids ADEM, FKLN, having triangular Bases, and the same Altitude, the Base ADE Ihall be * to the Base FKL, as the Pyramid ADEM to the Pyramid FKLN. And since the Base ABCDE is to the Base ADE, as the Pyramid ABCDEM is to the Pyramid ADEM; and as the Base ADE is to the Bafe FKL, so is the Pyramid ADEM to the Pyramid FKLN; it shall be (by Equality of Proportion) as the Base ABCDE to the Base F KL, so is the Pyramid ABCDEM to the Pyramid FKLN; but as the Base FKL is to the Base FGHKL, so was the Pyramid FKLN to the Pyramid FGHKLN. Wherefore, again, (by Equality of Proportion) as the Bafe ABCDE is to the Base FGHKL, fo is the Pyramid ABCDEM to the Pyramid F GH KLN. Therefore, Pyramids of the fame Altitude, and having polygonous Bases, are to one another as their Bases : which was to be demonstrated, PROPOSITION VII. THEOREM. vided into three Pyramids equal to one another, A B C, and opposite Base to that the Triangle DEF. I say, the Prifm ABCDEF may be divided into the three equal Pyramids that have triangular Bases. For join BD, EC, CD. Then because ABED is a Parallelogram, whose Diameter is BD, the Tri34. 1. angle ABD shall be * equal to the Triangle EBD. Therefore the Pyramid whose Base is the Triangle +6 of tbls. ABD, and Vertex the Point C, is equal to the Py ramid whose Bafe is the Triangle EDB, and Vertex the Point C. But the Pyramid whose Base is the Triangle EDB, and Vertex the Point c, is the fame as the Pyramid whose Base is the Triangle EBC, and Vertex the Point D, for they are contained under the same Planes. Therefore the Pyramid, whose Base is the Triangle ABD, and Vertex the Point C, is equal to the Pyramid whose Base is the Triangle E BC, and Vertex the Point D. Again, because FCBE is a Parallelogram, whose Diameter is CE, the Triangle ECF shall be * equal to the Triangle CBE. And so the Pyramid whose Base is the Triangle BEC, and Vertex the Point D, is t equal to the Pyramid whose Base is the Triangle ECF, and Vertex the Point D: But the Pyramid whose Base is the Triangle BCE, and Vertex the Point D, has been proved equal to the Pyramid whose Base is the Triangle ABD, and Vertex the Point C. Wherefore also the Pyramid, whose Base is the Triangle CEF, and Vertex the Point D, is equal to the Pyramid, whose Base is the Triangle ABD, and Vertex the Point C. Therefore the Prism ABCDEF is divided into three Pyramids equal to one another, and having triangular Bases. And because the Pyramid, whose Base is the Triangle ABD, and Vertex the Point C, is the fame with the Pyramid whose Base is the Triangle CA B, and Vertex the Point D; for they 22 1 they are contained under the same Planes ; and the Pyramid, whose Base is the Triangle A BD, and Vertex the Point C, has been proved to be a third Part of the Prism, whose Base is the Triangle ABC, and opposite Base to that the Triangle DEF. Therefore also the Pyramid, whose Base is the Triangle A B C, and Vertex the Point D, is a third Part of the Prism having the fame Base, viz. the Triangle A B C, Hand the opposite Base the Triangle DEF; which was to be demonstrated. Coroll. 1. It is manifest from hence, that every Pyra mid is a third Part of a Prism, having the fame Base and an equal Altitude; because if the Base of a Prism, as also the opposite Base, be of any other Figure, it may be divided into Prisms having tri angular Bases: 2. Prisms of the fame Altitude are to one another as their Bases. PROPOSITION VIII. THEOREM. ; ate, having the triangular Bases ABC, DEF, and let their Vertices be the Points G, H. I say, the Pyramid ABCG to the Pyramid DEFH has a Proportion triplicate of that which B C has to EF. For complete the folid Parallelepipedons BGML, EHPO; then because the Pyramid ABCG is similar to the Pyramid DEFH, the Angle ABC shall be * equal to the Angle DEF, the Angle GBC* Def. 9. 11. equal to the Angle HEF, and the Angle ABG equal to the Angle DEH. And A B is to DE as BC is to EF, and so is BG to EH. Therefore because AB is to DE, as BC is to EF; and the Sides about the equal Angles are proportional, the Parallelogram BM shall be similar to the Parallelo- + 6.6. gram EP. For the same Reason, the Parallelogram BN is similar to the Parallelogram ER, and the Pa rallelogram |