parts in the points N, Y by the same planes: therefore the prisms LXCOMN, RVFSTY are of the same altitude; and therefore as the base LXC to the base RVF; that is, as the triangle ABC to the triangle DEF, so (Cor. 32. 11.) is the prism having the triangle LXC for its base, and OMN the triangle opposite to it, to the prism of which the base is the triangle RVF, and the opposite triangle STY: and because the two prisms in the pyramid ABCG are equal to one another, and also the two prisms in the pyramid DEFH equal to one another, as the prism of which the base is the parallelogram KBXL and opposite side MO, to the prism having the triangle LXC for its base, and OMN the triangle opposite to it, so is the prism of which the base (7. 5.) is the parallelogram PEVR, and opposite side TS, to the prism of which the base is the triangle RVF, and opposite triangle STY. Therefore componendo, as the prisms KBXLMO, LXCOMN together are unto the prism LXOMN, so are the prisms PEVRTS, RVFSTY,

to the prism RVFSTY; and permutando, as the prisms KBXLMO, LXCOMN are to the prisms PEVRTS, RVFSTY, so is the prism LXCOMN to the prism RVFSTY: but as the prism LXCOMN to the prism RVFSTY, so is, as has been proved, the base ABC to the base DEF: therefore, as the base ABC to the base DEF, so are the two prisms in the pyramid ABCG to the two prisms in the pyramid DEFH: and likewise if the pyramids now made, for example, the two OMNG, STYH, be divided in the same manner; as the base OMN is to the base STY, so shall the two prisms in the pyramid OMNG be to the two prisms in the pyramid STYH: but the base OMN is to the base STY, as the base ABC to the base DEF; therefore, as the base ABC to the base DEF, so are the two prisms in the pyramid ABCG to the two prisms in the pyramid DEFH; and so are the two prisms in the pyramid OMNG to the two prisms in the pyramid STYH; and so are all four to all four: and the same thing may be shown of the prisms made by dividing the pyramids AKLO and DPRS, and of all made by the same number of divisions. Q. E. D.

PROP. V. THEOR.

PYRAMIDS of the same altitude, which have triangular bases, are to one another as their bases.*

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 pyramid 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 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

+ This may be explained the same way as at the note † in proposition 2 in the like case.

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 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 (14. 5.) than 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.

PROP. VI. THEOR.

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: thereM

N

*This may be explained the same way as the like at the mark † in prop. 2. † See Note.

fore, since the triangle ABC is to the triangle FGH, as (5. 12.) the pyramid ABCM to the pyramid FGHN; and the triangle ACD 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 FGHKLN: then, because as the base ABCDE to the base FGH, so is the pyramid ABCDEM to the pyramid FGHN; 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.

PROP. VII. THEOR.

EVERY prism having a triangular base, may be divided 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 (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 D
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 trian-
gle ABD, and vertex the point C. Therefore A

F

E

B

the prism ABCDEF is divided into three equal pyramids having triangular bases, viz. into the pyramids ABDC, EBDC, ECFD: 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 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 of the same altitude, are (6. 12.) 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 opposite to them: and, because the pyramid ABCG is similar to the pyramid DEFH, the angle ABC is equal (11. def. 11.) to the angle DEF and the angle GBC to the angle HEF, and ABG to DEH: and AB is