But it is also less, which is impossible. Therefore the base Book XII. ABC is not to the base DEF, as the pyramid ABCG to any solid which is less than the pyramid DEFH. In the same manner it may be demonstrated, that the base DEF is not to the base ABC, as the pyramid DEFH to any solid which is less than the pyramid ABCG. Nor can the base ABC be to the base DEF, as the pyramid ABCG to any solid which is greater than the pyramid DEFH. For, if it be possible, let it be so to a greater, viz. the solid Z. And because the base ABC is to the base DEF as the pyramid ABCG to the solid Z; by inversion, as the base DEF to

the base ABC, so is the solid Z to the pyramid ABCG. But as the solid Z is to the pyramid ABCG, so is the pyramid DEFH to some solid*, which must be less than 14. 5. the pyramid ABCG, because the solid Z is greater than the pyramid DEFH. And therefore, as the base DEF to the base ABC, so is the pyramid DEFH to a solid less than the pyramid ABCG; the contrary to which has been proved. Therefore the base ABC is not to the base DEF, as the pyramid ABCG to any solid which is greater than the pyramid DEFH. And it has been proved that neither is the base ABC to the base DEF, as the pyramid ABCG to any solid which is less than the pyramid DEFH. Therefore, as the base ABC is to the base DEF, so is the pyramid ABCG to the pyramid DEFH. Wherefore pyramids, &c.. Q. E. D.

This may be explained the same way as the like at the markt in Prop. 2.

BOOK XII.

PROP. VI. THEOR.

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

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

Divide the base ABCDE into the triangles ABC, ACD, ADE; and the base 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 bases as many pyramids having their common vertex in the point N: Therefore since the 5. 12. triangle ABC is to the triangle FGH, as the pyramid ABCM to the pyramid FGHÑ; and the triangle ACD to the triangle FGH, as the pyramid ACDM to the pyramid

B

FGHN; and also the triangle ADE to the triangle FGI, as the pyramid ADEM to the pyramid FGHN; as all the are all

2 Cor. first antecedents to their common consequent, so b 24. 5. the other antecedents to their common consequent, that is, as the base ABCDE to the base FGH, so is the pyramid ABCDEM to the pyramid FGHN: And for the same reason, as the base FGHKL to the base FGH, so is the pyramid FGHKLN to the pyramid FGHN: And, by inversion, as the base FGH to the base FGHKL, so is the pyramid FGHN to the pyramid FGHKLN: Then, because, as the base ABCDE to the base FGH, so is the pyramid ABCDEM to the pyramid FGHN; and as the base FGII to the base FGHKL, so is the pyramid FGHN to the py

ramid FGHKLN; therefore, ex æqualis, as the base Book XII. ABCDE to the base FGHKL, so the pyramid ABCDEM to the pyramid FGHKLN. Therefore pyramids, &c. Q. E.D.

€ 22. 5.

.

PROP. VII. THEOR.

EVERY prism having a triangular base
may be di-
vided into three pyramids that have triangular
bases, and are equal to one another.

Let there be a prism of which the base is the triangle ABC, and let DEF be the triangle opposite to it: The prism ABCDEF may be divided into three equal pyramids having triangular bases.

Join BD, EC, CD; and because ABED is a parallelogram of which BD is the diameter, the triangle ABD is equal a to the triangle EBD; therefore the pyramid of 34. 1. which the base is the triangle ABD, and vertex the point

a

F

C, is equal to the pyramid of which the base is the triangle * 5. 12. EBD, and vertex the point C: But this pyramid is the same with the pyramid the base of which is the triangle EBC, and vertex the point D; for they are contained by the same planes: Therefore the pyramid of which the base is the triangle ABD, and vertex the point C, is equal to the pyramid, the base 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 to the triangle ECB; therefore the pyramid of which the base D. is the triangle ECB, and vertex the point D, is equal to the pyramid the base of which is the triangle ECF, and vertex the point D: But the pyramid of which the base is the triangle ECB, and vertex the point D, has been proved equal to the pyramid of which the base is the triangle ABD, and vertex the point C. Therefore the prism ABCDEF is divided into three equal pyramids having triangular bases, viz. into the pyramids ABDC, EBDC, ECFĎ: And because the pyramid of which the base is the triangle ABD, and vertex the point C, is the same with the pyramid of which the base is the triangle ABC, and vertex the point D, for they are contained by the same planes; and that the pyramid of which the base is the triangle ABD, and vertex S

A

B

BOOK XII. the point C, has been demonstrated to be a third part of the prism, the base of which is the triangle ABC, and to which DEF is the opposite triangle; therefore the pyramid of which the base is the triangle ABC, and vertex the point D, is the third part of the prism which has the same base, viz. the triangle ABC, and DEF is the opposite triangle. Q. E. D.

COR. 1. From this it is manifest, that every pyramid is the third part of a prism which has the same base, and is of an equal altitude with it: for if the base of the prism be any other figure than a triangle, it may be divided into prisms having triangular bases.

COR. 2. Prisms of equal altitudes are to one another as their bases; because the pyramids upon the same bases, and 6. 12. of the same altitude, are to one another as their bases.

PROP. VIII. THEOR.

SIMILAR pyramids, having triangular bases, are one to another in the triplicate ratio of that of their homologous sides.

Let the pyramids having the triangles ABC, DEF, for their bases, and the points G, H for their vertices, be similar, and similarly situated; the pyramid ABCG has to the pyramid DEFH, the triplicate ratio of that which the side BC has to the homologous side EF.

Complete the parallelograms ABCM, GBCN, ABGK, and the solid parallelopiped BGML contained by these

planes and those opposite to them: And, in like manner, complete the solid parallelopiped EHPO contained by the three parallelograms DEFP, HEFR, DEHX, and those

с

11 Def. 11.

opposite to them: And because the pyramid ABCG is si- Book XII. milar to the pyramid DEFH, the angle ABC is equal to the angle DEF, and the angle GBC to the angle HEF, and ABG to DEH: And AB is to BC, as DE to EF:1 Def. 6. that is, the sides about the equal angles are proportionals; wherefore the parallelogram BM is similar to EP: For the same reason, the parallelogram BN is similar to ER, and BK to EX: Therefore the three parallelograms BM, BN, BK, are similar to the three EP, ER, EX: But the three BM, BN, BK, are equal and similar to the three which 24. 11. are opposite to them, and the three EP, ER, EX, equal and similar to the three opposite to them: Wherefore the solids BGML, EHPO are contained by the same number of similar planes and their solid angles are equal; and therefore the solid BGML is similar to the solid EHPO: But similar solid parallelopipeds have the triplicate ratio of that which their homologous sides have: Therefore the solid BGML has to the solid EHPO the triplicate ratio of that which the side BC has to the homologous side EF: But as the solid BGML is to the solid EHPO, so is f the pyramid f 15. 5. ABCG to the pyramid DEFH; because the pyramids are the sixth part of the solids, since the prism, which is the halfs of the solid parallelopiped, is tripleh of the pyramid. 28. 11. Wherefore likewise the pyramid ABCG has to the pyramid 7. 12. DEFH, the triplicate ratio of that which BC has to the homologous side EF. Q. E. D.

g

B. 11.

33. 11.

COR. From this it is evident, that similar pyramids which See N. have multangular bases, are likewise to one another in the triplicate ratio of their homologous sides: For they may be divided into similar pyramids having triangular bases, because the similar polygons, which are their bases, may be divided into the same number of similar triangles homologous to the whole polygons; therefore as one of the triangular pyramids in the first multangular pyramid is to one of the triangular pyramids in the other, so are all the triangular pyramids in the first to all the triangular pyramids in the other; that is, so is the first multangular pyramid to the other: But one triangular pyramid is to its similar triangular pyramid, in the triplicate ratio of their homologous sides; and therefore the first multangular pyramid has to the other, the triplicate ratio of that which one of the sides of the first has to the homologous side of the other.