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what bas

the Prisms that are in the Pyramid DEF H. But it is lefs alfo ; which is abfurd; therefore the Bafe ABC to the Bafe DEF, is not as the Pyramid A B C G to been before fome Solid less than the Pyramid DEFH. After the proved. fame manner we demonftrate, that the Bafe DEF to the Bafe A B C, is not as the Pyramid D E F H to fome Solid less than the Pyramid A BCG: 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 be poffible, let it be to the Solid I, greater than the Pyramid D E FH; then (by Inverfion) the Bafe D E F fhall be to the Bafe A B C, as the Solid I to the Pyramid A BCG: But fince the Solid I is greater than the Pyramid DEFH, it fhall be as the Solid I is to the Pyramid ABCG, fo is the Pyramid D E F H to fome Solid lefs than the Pyramid A BCG; and fo as the Bafe DEF is to the Base A B C, fo is the Pyramid DEFH to fome Solid lefs than the Pyramid A BCG, which is abfurd, as juft now has been proved: Therefore the Bafe ABC to the Bafe DEF, is not as the Pyramid A BCG to fome Solid greater than the Pyramid D E F H. But it has been alfo proved, that the Base A B C to the Bafe DE F, is not as the Pyramid A BCG to fome Solid lefs than the Pyramid DEFH; wherefore, as the Bafe A B C is to the Base D E F, fo is the Pyramid ABCG to the Pyramid DE FH. Therefore, Pyramids of the fame Altitude, and having triangular Bafes, are to one another as their Bafes; which was to be demonstrated.

PROPOSITION VI.
THEOREM.

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

LET there be Pyramids of the fame Altitude, which have the polygonous Bafes A B C DE, FGH KL; and let the Vertices be the Points M, N. I fay, as the Bafe A B C D E is to the Bafe F G H KL, fo is the Pyramid ABCDEM to the Pyramid F G H K L N.

For,

For, let the Bafe A B C D E be divided into the Triangles A B C, ACD, A DE; and the Bafe FGHKL into the Triangles FGH, FH K, FKL; and let the Pyramids be conceived upon every one of thofe Triangles, of the fame Altitude with the Pyramids ABCDEM, FGHKLN; Then, because the Triangle ABC is to the Triangle ACD, •s of this, as the Pyramid ABCM is to the Pyramid ACDM; and (by compounding) as the Trapezium ABCD is to the Triangle A CD, fo is the Pyramid ABCDM to the Pyramid AC DM: But as the Triangle ACD is to the Triangle A DE, fo is * the Pyramid A CDM to the Pyramid A D E M. Wherefore, (by Equality of Proportion), as the Base ABCD is to the Bafe ADE, fo is the Pyramid ABCDM to the Pyramid ADE M: And again (by Compofition of Proportion), as the Bafe A B C DÉ is to the Bafe ADE, fo is the Pyramid ABC DEM to the Pyramid ADEM. For the fame Reason, as the Bafe F G H K L is to the Bafe F K L, fo is the Pyramid F G H K L N to the Pyramid F K L N: And fince there are two Pyramids ADEM, FK LN, having triangular Bafes, and the fame Altitude; the Bafe A D E fhall be * to the Bafe F K L, as the Pyramid ADEM to the Pyramid FKLN: And fince the Base ABCDE is to the Bafe ADE, as the Pyramid A BCDEM is to the Pyramids ADEM; and as the Bafe ADE is to the Bafe FK L, fo is the Pyramid ADEM to the Pyramid F KLN; it shall be (by Equality of Proportion), as the Bafe A B C D E is to the Bafe FKL, fo is the Pyramid A B C DEM to the Pyramid F K L N: But as the Bafe F KL is to the Bafe F GHK L, fo was the Pyramid F KLN to the Pyramid F G H KLN. Wherefore, again, (by Equality of Propertion), as the Bafe ABCDE is to the Bafe F GHK L, fo is the Pyramid A B C D E M 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.

If the Bafes had not confifted of equal Numbers of Sides, the Demonftration had been the fame.

PRO

PROPOSITION XII.
THEOREM.

Every Prifm having a triangular Bafe, may be
divided into three Pyramids, equal to one ano-
ther, and having triangular Bafes.

LET there be a Prifm, whofe Bafe is the Triangle

A B C, and the oppofite Bafe to that the Triangle.. DEF. Ifay, the Prism ABCDEF may be divided into three equal Pyramids, that have triangular Bases.

34. 2.

For, join BD, EC, CD: Then, because ABED is a Parallelogram, whofe Diameter is BD, the Triangle ABD fhall be equal to the Triangle E BD. Therefore the Pyramid whofe Bafe is the Triangle AB D, and Vertex the Point C, is + equal to the Pyramid + 6 of this. whose Base is the Triangle E D B, and Vertex the Point C. But the Pyramid, whofe Bafe is the Triangle ED B, and Vertex the Point C, is the fame as the Pyramid whofe Bafe is the Triangle E B C, and Vertex the Point D; for they are contained under the fame Planes: Therefore the Pyramid, whose Base is the Triangle A B D, and Vertex the Point C, is equal to the Pyramid whofe Bafe is the Triangle E B C, 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 Bafe is the Triangle B EC, and Vertex the Point D, is equal to the Pyramid whose Bafe is the Triangle E CF, 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 A B D, 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 Bafes. And because the Pyramid, whofe Bafe is the Triangle A B D, and Vertex the Point C, is the fame with the Pyramid whofe Bafe is the Triangle C A B, and Ver

tex

tex the Point D; for they are contained under the
fame Planes; and the Pyramid, whose Base is the
Triangle ABD, and Vertex the Point C, has been
proved to be a third Part of the Prifm, whose Base is ́.
the Triangle AB C, and the oppofite Base to that the
Triangle DEF: Therefore, alfo, the Pyramid, whofe
Bafe is the Triangle A B C, and Vertex the Point D, is
a third Part of the Prifm, having the fame Bafe, viz. the
Triangle ABC, and the oppofite Bafe the Triangle DEF;
which was to be demonftrated.

Coroll. 1. It is manifeft from hence, that every Pyramid is a third Part of a Prifm, having the fame Bafe, and an equal Altitude; becaufe, if the Base of a Prifm, as alfo the oppofite Bafe, be of any other Figure, it may be divided into Prifms having triangular Bases.

2. Prifms of the fame Altitude are to one another as their Bafes.

Def. 9. 11.

+6. 6.

PROPOSITION VIII.

THEOREM.

Similar Pyramids, baving triangular Bafes, are in a triplicate Proportion of their homologous Sides.

L

ET there be two Pyramids fimilar and alike fituate, having the triangula Baies ABC, DEF; and let their Vertices be the Points G, H. I fay, the Pyramid A BCG to the Pyramid DE FH, has a Proportion triplicate of that which BC has to EF.

*

For, compleat the folid Parallelopipedon B G ML, EHPO; then, because the Pyramid ABCG is fimilar to the Pyramid D E F H, the Angle A B C fhall be equal to the, Angle DE F, the Angle G BC. equal to the Angle HEF, and the Angle ABG equal to the Angle DE H. And A B is to DE, as BC is to EF; and fo is B G to EH. Therefore, because the Angle A B C is equal to the Angle DEF; and the Sides about the equal Angles are proportional; the Parallelogram B M fhall be + fimilar to the Parallelogram E P. For the fame Reason the Pa

ral

rallelogram B N is fimilar to the Parallelogram ER, and the Parallelogram BK to the Parallelogram EX. Therefore three Parallelograms B M, B K, BN, are fimilar to three Parallelograms EP, EX, ER. But the three Parallelograms B M, B K, BN, are equal and fimilar to the three oppofite ones; as alfo the three Parallelograms EP, EX, ER: Therefore the Solids BGML, EHP O, are contained under equal Numbers of fimilar and equal Planes; and confequently, the Solid B G M L is fimilar to the Solid EHPO. But fimilar folid Parallelopipedons are to each other 33. 11. in a triplicate Proportion of their homologous Sides; therefore the Solid B G M L to the Solid EH PO, has a Proportion triplicate of that which the homologous Side B C has to the homologous Side E F. But as the Solid B GML is to the Solid EHPO, fo is + the † 15.5. Pyramid A BCG to the Pyramid DEFH; for the Pyramid is the one fixth Part of that Solid, fince the Prifm, which is the Half of the folid Parallelopipedon, is triple of the Pyramid. Wherefore, the Pyramid, ABCG to the Pyramid D E F H, fhall have a triplicate Proportion to that which BC has to EF; which was to be demonftrated.

Coroll. From hence it is manifeft, that fimilar Pyramids having polygonous Bafes, are to one another in a triplicate Proportion of their homologous Sides. For, if they be divided into Pyramids, having triangular Bafes, because their fimilar polygonous Bafes are divided into fimilar Triangles equal in Number, and homologous to the Wholes; it shall be, as one Pyramid, having a triangular Bafe in one of the Pyramids, is to a Pyramid having a triangular Base in the other Pyramid; fo are all the Pyramids, having triangular Bafes in one Pyramid, to all the Pyramids having triangular Bafes in the other Pyramid; that is, fo is one of the Pyramids, having the polygonous Bafe, to the other: But a Pyramid having a triangular Bafe, to a Pyramid having a triangular Bafe, is in a triplicate Proportion of the homologous Sides. Therefore one Pyramid, having a polygonous Bafe, to another Pyramid having a fimilar Bafe, is in a triplicate Proportion of their homologous Sides.

PRO.