the base DEF, as the pyramid ABCG to any folid which is Book XII. lefs than the pyramid DEFH. In the fame manner it may be demonftrated, that the base DEF is not to the bafe ABC, as the pyramid DEFH to any folid which is less than the pyramid ABCG. Nor can the bafe ABC be to the bafe DEF, as the pyramid ABCG to any folid which is greater than the py→ ramid DEFH. For, if it be poffible, let it be fo to a greater, viz. the folid Z. And because the bafe ABC is to the base DEF as the pyramid ABCG to the folid Z; by inverfion, as the bafe DEF to the bafe ABC, fo is the folid Z to the pyramid ABCG. But as the folid Z is to the pyramid ABCG, fo is the pyramid

DEFH to fome folid t, which must be lefs a than the pyramid a 14. s ABCG, because the folid Z is greater than the pyramid DEFH. And therefore, as the bafe DEF to the bafe ABC, fo is the pyramid-DEFH to a folid lefs than the pyramid ABCG; the con. trary to which has been proved. Therefore the bafe ABC is not to the bafe DEF, as the pyramid ABCG to any folid which is greater than the pyramid DEFH. And it has been proved, that neither is the bafe ABC to the bafe DEF, as the pyramid ABCG to any folid which is lets than the pyramid DEFH. Therefore, as the bafe ABC is to the bafe DEF, fo is the pyramid ABCG to the pyramid DEFH. Wherefore pyramids, &c. Q. E. D.

This may be explained the fame way as the like at the mark † in prop. 2.

Book XII.

PRO P. VI. THEOR.

See N. PYRAMIDS of the fame altitude which have polygons for their bafes, are to one another as their bafes.

Let the pyramids which have the polygons ABCDE, FGHKIS for their bases, and their vertices in the points M, N, be of the fame altitude : As the bafe ABCDE to the base FGHKL, so is the pyramid ABCDEM to the pyramid FGHKLN.

Divide the bafe ABCDE into the triangles ABC, ACD, ADE; and the bafe FGHKL into the triangles FGH, FHK, FKL: And upon the bases ABC, ACD, ADE let there be as many pyramids of which the common vertex is the point M, and upon the remaining bafes as many pyramids having their common vertex in the point N: Therefore, fince the triangle a 5. 12. ABC is to the triangle FGH, as a the pyramid ABCM to the pyramid FGHN; and the triangle ACD to the triangle FGH, as the pyramid ACDM to the pyramid FGHN; and alfo the

triangle ADE to the triangle FGH, as the pyramid ADEM té the pyramid FGHN; as all the firft antecedents to their comb 4. Cor. mon confequent; fo bare all the other antecents to their com24. 5. mon confequent; that is, as the bafe ABCDE to the base

FGH, fo is the pyramid ABCDEM to the pyramid FGHN : And, for the fame reafon, as the bafe FGHKL to the bafe FGH, fo is the pyramid FGHKLN to the pyramid FGHN: And, by inverfion, as the bafe FGH to the bate FGHKL, fo is the pyramid FGHN to the pyramid FGHKLN: Then, because as the bafe ABCDE to the bafe FGH, fo is the pyramid ABCDEM to the pyramid FOHN; and as the bafe FGH to the base FGHKL, fo is the pyramid FGHN to the pyramid FGHKLN; therefore,

therefore, ex æqualis, as the bafe ABCDE to the bafe FGHKL, Book XII. fo the pyramid ABCDEM to the pyramid FGHKLN. There- C 22. 5. fore pyramids, &c. Q. E. D.

PROP. VII. THEOR.

EVERY prifm having a triangular bafe may be divided into three pyramids that have triangular bases, and are equal to one another.

Let there be a prifm of which the bafe is the triangle ABC, and let DEF be the triangle oppofite to it: The prifm ABCDEF may be divided into three equal pyramids having triangular bafes.

D

F

E

Join BD, EC, CD; and because ABED is a parallelogram of which BD is the diameter, the triangle ABD is equal a to a 34. 1. the triangle EBD; therefore the pyramid of which the bafe is the triangle ABD, and vertex the point C, is equal b to theb 5. 12. pyramid of which the bafe is the triangle EBD, and vertex the point C: But this pyramid is the fame with the pyramid the bale of which is the triangle EBC, and vertex the point D; for they are contained by the fame planes: Therefore the pyramid of which the bafe is the triangle ABD, and vertex the point C, is equal to the pyramid, the bafe of which is the triangle EBC, and vertex the point D: Again, because FCBE is a parallelogram of which the diameter is CE, the triangle ECF is equal a to the triangle ECB; therefore the pyramid of which the bafe is the triangle ECB, and vertex the point D, is equal to the pyramid, the bafe of which is the triangle ECF, and vertex the point D: But the pyramid of which the bafe is the triangle ECB, and vertex the point D has been proved equal to the pyramid of which the bafe is the triangle ABD, and vertex the point C. Therefore the prifm ABCDEF is divided into three equal pyramids having triangular bafes, viz. into the pyramids ABDC, EBDC, ECFD: And because the pyramid of which the bafe is the triangle ABD, and vertex the point C, is the fame with the pyramid of which the bafe is the triangle ABC, and vertex the point D, for they are contained by the fame planes; and that the pyramid of which the bafe is the triangle ABD, and vertex the point C, has been demonftrated

A

C

B

Book XII. demonftrated to be a third part of the prism the base of which is the triangle ABC, and to which DEF is the oppofite triangle; therefore the pyramid of which the bafe is the triangle ABC, and vertex the point D, is the third part of the prifm which has the fame bafe, viz. the triangle ABC, and DEF is the oppofite triangle. QE. D.

C 6. 12.

COR. I. From this it is manifeft, that every pyramid is the third part of a prifm which has the fame bafe, and is of an equal altitude with it; for if the bafe of the prifm be any other figure than a triangle, it may be divided into prifmns having triangular bafes.

COR. 2. Prifms of equal altitudes are to one another as their bafes; because the pyramids upon the fame bafes, and of the fame altitude, are c to one another as their bafes.

.PROP. VIII. THEOR.

SIMILAR pyramids having triangular bafes are one to another in the triplicate ratio of that of their homologous fides.

Let the pyramids having the triangles ABC, DEF for their bafes, and the points G, H for their vertices, be fimilar and fimilarly fituated; the pyramid ABCG has to the pyramid DEFH, the triplicate ratio of that which the fide BC has to the homologous fide EF.

Complete the parallelograms ABCM, GBCN, ABGK, and the folid parallelepiped BGML contained by these planes and

thofe oppofite to them: And, in like manner, complete the fo lid parailelepiped EHPO contained by the three parallelograms DEFP, HEFR, DEHX, and thofe oppofite to them: And be caufe the pyramid ABCG is fin.ilar to the pyramid DEFH, the

angle

a 11. def.

II.

ii.

angle ABC is equal a to the angle DEF, and the angle GBC Book XII. to the angle HEF, and ABG to DEH: And AB is b to BC, as DE to EF; that is, the fides about the equal angles are proportionals; wherefore the parallelogram BM is fimilar to EP : b 1. def. 6. For the fame reafon, the parallelogram BN is fimilar to ER, and BK to EX: Therefore the three parallelograms BM, BN, BK are fimilar to the three EP, ER, EX: But the three BM, BN, BK, are equal and fimilar e to the three which are oppo- c 24. fite to them, and the three EP, ER, EX equal and Gimilar to the three oppofite to them: Wherefore the folids BGML, EHPO are contained by the fame number of fimilar planes; and their folid angles are equal d; and therefore the folid d B. 11. BGML is fimilar a to the folid EHPO: But fimilar folid parallelepipeds have the triplicate e ratio of that which their homologous fides have: Therefore the folid BGML has to the folid EHPO the triplicate ratio of that which the fide BC has to the homologous fide LF: But as the folid BGML is to the folid EHPO, to is f the pyramid ABCG to the pyramid DEFH; f 15. 5. because the pyramids are the fixth part of the folids, fince the prifm, which is the half g of the folid parallelepiped, is triple h g 28. II, of the pyramid. Wherefore likewife the pyramid ABCG hash 7. 12. to the pyramid DEFH, the triplicate ratio of that which BC has to the homologous fide EF. Q. E. D.

33. 11.

COR. From this it is evident, that fimilar pyramids which See N. have multangular bafes, are likewife to one another in the triplicate ratio of their homologous fides: For they may be divided into fimilar pyramids having triangular bases, because the fimilar polygons, which are their bates, may be divided into the fime number of fimilar triangles homologous to the whole polygons; therefore as one of the triangular pyramids in the firft multangular pyramid is to one of the triangular pyramids in the other, fo are all the triangular pyramids in the firft to all the triangular pyramids in the other; that is, fo is the firft multangular pyramid to the other: But one triangular pyramid is to its fimilar triangular pyramid, in the triplicate ratio of their homologous fides; and therefore the firft multangular pyramid has to the other, the triplicate ratio of that which one of the fides of the firft has to the homologous fide of the other.

PROP.