to the two prisms in the pyramid STYH; and so are all four to all four: And the same thing may be shewn of the prisms made by dividing

the pyramids AKLO, and DPRS, and of all made by the same number of divisions. Q. E. D.

Let the pyramids of which the triangles ABC, DEF are the bases, and of which the vertices are the points G, H be of the same altitude: As the base ABC to the base DEF, so is the pyramid ABCG to the pyramid DEFH. For, if it be not so, the base ABC must be to the base DEF, as the pyramid ABCG to a solid either less than the pyrainid DEFH, or greater than it. First, let it be to a solid less than it, viz. to the solid Q: And divide the pyramid DEFH into two equal pyramids, similar to the whole, and into two equal prisms: Therefore these two prisms are greater (3. 12.) than the half of the whole pyramid. And again, let the pyramids made by this division be in like manner divided, and so on, until the pyramids which remain undivided in the pyramid DEFH be, all of them together, less than the excess of the pyramid DEFH above the solid Q: Let these, for example, be the pyramids DPRS, STYH: Therefore the prisms, which make the rest of the pyramid DEFH, are greater than the solid Q: Divide likewise the

G

pyramid ABCG in the same manner, and into as many parts, as the pyramid DEFH: Therefore as the base, ABC to the base DEF, so (4. 12.) are the prisms in the pyramid ABCG to the prisms in the pyramid DEFH: But as the base ABC to the base DEF, so by hypothesis is the pyramid ABCG to the solid Q; and therefore, as the pyramid ABCG to the solid Q, so are the prisms in the pyramid ABCG to the prisms in the pyramid DEFH: But the pyramid ABCG is greater than the prisms contained in it; wherefore (14. 5.) also the solid Q is greater than the prisms in the pyramid DEFH. But it is also less, which is impossible. Therefore the base 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 Á BCG. Nor can the base ABC be to the base DEF, as the pyramid ABCG to any solid which is greater than the pyra

Y

MN

T

like ease.

mid 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 ABG to the solid Z: by inversion, as the base DEF to the base ABC, so is the solid Z to the pyramid A5CG. But as the solid Z is to the pyramid ABCG, so is the pyramid DEFH to some solid,* which must be less (14. 5.) than the pyramids ABCG, because the solid Z is greater than the pyramid DEFH. And therefore, as the base DEF to the base ABC, so is the py

ramid 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.

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 triangle ABC is to the triangle FGH, as (5. 12.) the pyramid ABCM to the pyramid FGHN; and the triangle ACD

M

T D

to the triangle FGH, 'as' the pyramid ACDM to the pyramid FGHN; and also the triangle ADE to the triangle FGH, as the pyramid ADEM to the pyramid FGHN: as all the first antecedents to their common consequent; so (2. Cor. 24. 5.) are all 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 FGÖKLN: Then, because as the base ABCDE to the base FGH, so is the pyramid ABCDEM to the pyramid FGIN; and as the base FGH to the base FGHKL, so is the pyramid FGHN to the pyramid FGHKLN: therefore, ex æquali, (22. 5.) as the base ABCDE to the base FGHKL, so the pyramid ABCDEM to the pyramid FGHKLN. Therefore pyramids, &c. Q. E. D.

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 (34 1.) to the triangle EBD; therefore the pyramid of which the base is the triangle ABD, and vertex the point C, is equal (5. 12.) to the pyramid of which the base is the triangle 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 (34. 1.) to the triangle ECB; therefore the pyramid of which the base 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

F

B

E

point C. Therefore the prism ABCDEF is divided into three equal pyramids having triangular bases, viz. into the pyramids ABDC, EBDC, ECFD: And because the pyramid of A 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 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; be. cause the pyramids upon the same bases, and of the same altitude, are (6. 12.) to one another as their bases.

Let the pyramids having the triangles ABC, DEF for their bases; and the points G, H for their vertices,

be similar, and similarly situated; and the pyramid ABCG has to the pyramid DEFH, the triplicate ratio

posite to them: And because the pyramid ABCG is equal to the pyramid DEFH, and that the solid BGML is sextuple of the pyramid ABCG, and the solid EHPO sextuple of the pyramid DEFH; therefore the solid BGML is equal (1. Ax. 5.) to the solid EHPO but the bases and altitudes of equal solid parallelopipeds are reciprocally proportional; (34. 11.) therefore as the base BM to the base EP, so is the altitude of the solid EHPO to the altitude of the solid BGML: But as the base BM to the base EP, so is (15. 5.) the triangle ABC to the triangle DEF, therefore as the triangle ABC to the triangle DEF, so is the altitude of the sold EHPO to the altitude of the solid BGML: But the altitude of the solid EHPO is the same with the altitude of the pyramid DEFH; and the altitude of the solid BGML is the same with altitude of the pyramid ABCG: Therefore, as

the base ABC to the base DEF, so is the altitude of the pyramid DEFH to the altitude of the pyramid ABCG: Wherefore the bases and altitudes of the pyramids ABCG, DEFH are reciprocally proportional.

Again, let the bases and altitudes of ciprocally proportional, viz. the base the pyramids ABCG, DEFH be reABC to the base DEF, as the altitude of the pyramid ABCG: The pyramid of the pyramid DEFH to the altitude ABCG is equal to the pyramid DEFH.

The same construction being made, because as the base ABC to the base DEF, so is the altitude of the pyramid DEFH to the altitude of the pyramid ABCG: And as the base ABC to the base DEF, so is the parallelogram BM to the parallelogram EP; therefore the parallelogram BM is to EP, as the altitude of the pyramid DEFH to the altitude of the pyramid ABCG: But the altitude of the pyramid DEFII is the same with the altitude of the solid parallelopiped EHPO: and the altitude of the pyramid ABCG is the same with the altitude of the solid parallelopiped BGML: As, therefore, the base BM to the base EP, so is the altitude of the solid parallelopiped EHFO to the altitude of the solid paralelopiped BGML. parallelopipeds having their bases But solid and altitudes reciprocally proportional, are equal (31. 11.) to one another. Therefore the solid parallelopiped BGML is equal to the solid parallelopiped EHPO.

ABCG is the sixth part of the solid And the pyramid BGML, and the pyramid DFFH is the sixth part of the solid EHPO. Therefore the pyramid ABCG is equal to the pyramid DEFH. Therefore the bases, &c. Q. E. D.