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11. 5.

*

Prism whofe Bafe is the Triangle RQF; and oppofite Base to that the Triangle STY. Therefore as the Triangle ABC is to the Triangle DEF, fo is the Prism whose Base is the Triangle LXC, and oppofite Bafe to that the Triangle OMN, to the Prifm whose Base is the Triangle RQF, and oppofite Base to that the Triangle STY; and because the two Prisms that are in the Pyramid ABCG are equal to one another, as also those two that are in the Pyramid DEF H; it fhall be as the Prifm whose Base is the Parallelogram KLXB, and oppofite Base to that the Right Line MO, is to the Prism whofe Bafe is the Triangle LXC; and opposite Base to that the Triangle OMN, fo is the Prifm whose Base is the Parallelogram EPRQ; and oppofite Bafe to that the Right Line ST, to the Prifm whofe Base is the Triangle RQF, and oppofite Bafe to that the Triangle STY. Therefore (by compounding) as the Prifms KBXLMO, LXCMNO, to the Prism LX CMNO, fo the Prifms PEQRST, RQFSTY, to the Prism RQFSTY. And (by Alternation) as the Prisms KBXLMO, LXCMNO, to the Prisms PEQRST, RQFSTY, fo the Prifm LX CMNO, to the Prifm RQFSTY; but as the Prifm LXC MNO is to the Prifm RQFSTY, fo has the Base LXC been proved to be to the Bafe RFQ; and fo the Base ABC to the Bafe DE F. Therefore alfo as the Triangle ABC is to the Triangle DEF, fo are the two Prisms that are in the Pyramid ABCG, to the two Prisms that are in the Pyramid DEFH. If in the fame Manner each of the Pyramids OMNG, STYH, made by the former Divifion, be divided, it fhall be as the Base OMN is to the Base STY, so the two Prisms that are in the Pyramid OMNG, to the two Prisms that are in the Pyramid STYH. But as the Bafe OMN is to the Bafe STY, fo is the Bafe ABC to the Base DEF. Therefore as the Base ABC is to the Base DEF, fo is the two Prisms that are in the Pyramid ABCG, to the two Prifms that are in the Pyramid DEFH; and fo the two Prisms that are in the Pyramid OMNG, to the two Prisms that are in the Pyramid STYH, and fo the four to the four. We demonftrate the fame of Prisms made by the Divifion of the Pyramids AKLO, DPRS,

and

and of all other Prisms, being equal in Multitude; which was to be demonftrated.

PROPOSITION V.

THE ORE M.

Pyramids of the fame Altitude, and having triangular Bafes, are to one another as their Bafes.

I ET there be two Pyramids of the fame Altitude,

having the triangular Bafes ABC, DEF, whofe Vertices are the Points G, H. I fay, as the Bafe ABC is to the Base DEF, fo is the Pyramid ABCG to the Pyramid DEF H.

For if it be not fo, then it fhall be as the Bafe ABC is to the Base DEF, fo is the Pyramid ABCG to fome Solid, greater or lefs than the Pyramid DEFH. First, let it be to a Solid lefs, which let be Z, and divide the Pyramid DEFH into two Pyramids equal to each other, and fimilar 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 Divifion, be divided after the fame 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 thefe, for Example, be the Pyramids DPRS, STYH; then the Prisms remaining in the Pyramid DEF H, are greater than the Solid Z. Alfo, let the Pyramid ABCG, be divided into the fame Number of fimilar Parts as the Pyramid DEFH is; and then as the Base ABC is to the Base DEF, fo* the Prisms that are in the* 4 of this. Pyramid ABCG, to the Prisms that are in the Pyramid DEFH. But as the Bafe ABC is to the Bafe DEF, fo is the Pyramid ABCG to to the Solid Z. And therefore as the Pyramid ABCG is to the Solid Z, fo 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 alfo the Solid Z, is greater than the Prisms that are in the Pyramid DEF H; bút

*From what it is lefs *alfo, which is abfurd.

has been al

Therefore the Bafe ready demon- ABC to the Bafe DEF, is not as the Pyramid ftrated ABCG to fome Solid less than the Pyramíd DEF H. After the fame Manner we demonftrate that the Base DEF to the Bafe ABC, is not as the Pyramid DEFH to fome Solid lefs than the Pyramid ABCG. Therefore, I fay, neither is the Bafe ABC to the Bafe DEF, as the Pyramid ABCG to fome Solid greater than the Pyramid DEF H. For if this is poffible, let it be to the Solid I, greater than the Pyramid DEFH. Then (by Inverfion) the Bafe DEF fhall be to the Base ABC, as the Solid I to the Pyramid ABCG: But fince the Solid I is greater than the Pyramid EDFH, it fhall be as the Solid I is to the Pyramid ABCG, fo is the Pyramid DEF H, to fome Solid lefs than the Pyramid ABCG, as just now has been proved. And fo as the Bafe DEF is to the Bafe ABC, fo is the Pyramid DEFH, to fome Solid lefs than the Pyramid ABCG, which is abfurd. Therefore the Base A B C to the Base DEF, is not as the Pyramid ABCG to fome Solid greater than the Pyramid DEF H. But it has been alfo proved, that the Base ABC to the Base DEF, is not as the Pyramid ABCG to fome Solid lefs than the Pyramid DEF H. Wherefore as the Bafe ABC is to the Bafe DEF, fo is the Pyramid ABCG to the Pyramid DEF H. Therefore, Pyramids of the fame Altitude, and having triangular Bafes, are to one another as their Bafes; which was to be demonftrated.

PROPOSITION VI.

THEOREM.

Pyramids of the fame Altitude, and having polygonous Bafes, are to one another as their Bafes.

LEwhich have the polygonous Bafes ABCDE,

ET there be Pyramids of the fame Altitude,

FGHKL, and let their Vertices be the Points M,
N. I fay, as the Bafe ABCDE is to the Bafe
FGHKL, fo is the Pyramid ABCDE M to the Py-
ramid F GHKLN. BES

For

*

For let the Bafe ABCDE be divided into the Triangles ABC, ACD, ADE; and the Bafe F GHKL into the Triangles F GH, FHK, FKL; and let Pyramids be conceived upon every one of thofe Triangles of the fame Altitude with the Pyramids ABC, DEM, FGH KLN. Then because the Triangle ABC is to the Triangle A CD, as * the Pyramid * 5 of this. AB CM is to the Pyramid ACDM: And (by compounding) as the Trapezium ABCD is to the Triangle ACD, fo is the Pyramid ABCDM to the Pyramid A CDM; but as the Triangle ACD is to the Triangle ADE, fo is the Pyramid ACDM to the Pyramid ADEM. Wherefore (by Equality of Proportion) as the Base ABCD is to the Bafe ADE, fo is the Pyramid ABCDM to the Pyramid ADEM. And again (by compofition of Proportion) as the Bafe ABCDE is to the Bafe ADE, fo is the Pyramid ABCDE M to the Pyramid ADEM. For the fame Reason, as the Bafe F GHKL is to the Bafe F KL, fo is the Pyramid FGH KLN to the Pyramid F KLN. And fince there are two Pyramids ADEM, FKLN, having triangular Bafes, and the fame Altitude, the Bafe ADE fhall be to the Bafe FKL, as the Pyramid ADEM to the Pyramid FKLN. And fince the Base ABCDE is to the Base ADE, as the Pyramid ABCDEM is to the Pyramid ADEM; and as the Bafe ADE is to the Bafe FKL, fo is the Pyramid ADEM to the Pyramid FKLN; it fhall be (by Equality of Proportion) as the Base ABCDE to the Bafe F KL, fo is the Pyramid ABCDEM to the Pyramid F KLN; but as the Bafe FKL is to the Bafe FGHKL, fo was the Pyramid FKLN to the Pyramid F GHKLN, Wherefore, again, (by Equality of Proportion) as the Bafe ABCDE is to the Bafe FGHKL, fo is the Pyramid ABCDEM to the Pyramid F G H KLN. Therefore, Pyramids of the fame Altitude, and having polygonous Bafes, are to one another as their Bafes which was to be demonftrated.

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PROPOSITION VII.
THEORE M.

Every Prifm having a triangular Bafe, may be di-
vided into three Pyramids equal to one another,
and having triangular Bafes.

L

ET there be a Prism whose Base is the Triangle ABC, and oppofite Bafe to that the Triangle DEF. I fay, the Prism 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 Tri* 34. I. angle ABD fhall be equal to the Triangle EBD. Therefore the Pyramid whofe Bafe is the Triangle +6 of this, ABD, and Vertex the Point C, is † equal to the Pyramid whose Base is the Triangle ED B, 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 whofe Bafe is the Triangle EB C, and Vertex the Point D, for they are contained under the fame Planes. Therefore the Pyramid, whofe Base is the Triangle ABD, and Vertex the Point C, is equal to the Pyramid whose Base is the Triangle EBC, and Vertex the Point D. Again, because FCBE is a Parallelogram, whofe Diameter is CE, the Triangle ECF fhall be equal to the Triangle CBE. And fo the Pyramid whofe Base is the Triangle BEC, and Vertex the Point D, is † equal to the Pyramid whose Base is the Triangle ECF, and Vertex the Point D: But the Pyramid whofe Bafe is the Triangle BCE, and Vertex the Point D, has been proved equal to the Pyramid whofe Bafe is the Triangle ABD, and Vertex the Point C. Wherefore alfo the Pyramid, whose Base is the Triangle CEF, and Vertex the Point D, is equal to the Pyramid, whofe Bafe is the Triangle ABD, and Vertex the Point C. Therefore the Prifm ABCDEF is divided into three Pyramids equal to one another, and having triangular Bases. And because the Pyramid, whofe Bafe 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

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