AEGH, HKLD is similar to the whole pyramid ABCD. Again, because BF is equal to FC, the parallelogram EBFG is double (I. 41.) of the triangle GFC. But (XI. 40.) when there are two prisms of the same altitude, of which one has a parallelogram for its base, and the other a triangle that is half of the parallelogram, these prisms are equal: therefore the prism having the parallelogram EBFG for its base, and the straight line KH opposite to it, is equal to the prism having the triangles GFC, HKL for its bases; for they are of the same altitude, because they are between the parallel planes (XI. 15.) ABC, HKL. It is also manifest that each of these prisms is greater than either of the pyramids AEGH, HKLD, because, if EF be joined, the prism having the parallelogram EBFG for its base, and KH the line opposite to it, is greater than the pyramid EBFK: but this pyramid (XI. C.) is equal to the pyramid AEGH, because they are contained by equal and similar planes. Therefore the prism having EBFG for its base, and opposite side KH, is greater than the pyramid AEGH: and the prism of which the base is EBFG, and opposite side KH, is equal to the prism having for its bases the triangles GFC, HKL: and the pyramids AEGH, HKLD are equal; therefore the two prisms before-mentioned are greater than these two pyramids. Therefore the whole pyramid ABCD is divided into two equal pyramids similar to one another, and to the whole; and into two equal prisms; and the two prisms are together greater than half of the whole pyramid. A triangular pyramid, therefore, &c.

PROP. IV. THEOR.

IF two triangular pyramids of the same altitude be each divided into two equal pyramids, similar to the whole pyramid, and also into two equal prisms; and if each of these pyramids be divided in the same manner as the first two, and so on: as the base of one of the first two pyramids is to the base of the other, so are all the prisms in one of them to all the prisms in the other, that are produced by the same number of divisions.

Let there be two pyramids of the same altitude upon the triangular bases ABC, DEF, and having their vertices in the points G, H; and let each of them be divided into two equal pyramids similar to the whole, and into two equal prisms; and let each of the pyramids thus made be conceived to be divided in like manner, and so on: as ABC is to DEF, so are all the prisms in the pyramid ABCG to all the prisms in the pyramid DEFH, made by the same number of divisions.

Make the same construction as in the foregoing proposition: and because BX, XC are equal, and also AL, LC, therefore

(VI. 2.) XL is parallel to AB, and the triangle ABC is similar to LXC. For the same reason, DEF is similar to RVF. Now, because BC is double of CX, and EF of FV, therefore BC: CX:: EF: FV; and upon BC, CX are described the similar and similarly situated rectilineal figures ABC, LXC; and upon EF, FV, in like manner, are described the similar figures DEF, RVF: therefore (VI. 22.) as the triangle ABC is to LXC, so is the triangle DEF to RVF, and, alternately, as the triangle ABC to DEF, so is the triangle LXC to RVF. Again, because (XI. 15.) the planes ABC, OMN, as also DEF, STY are parallel, the perpendiculars drawn from G, H to the bases ABC, DEF, which (hyp.) are equal to

B

G

M

Y

T

L

P

R

X

C

E

V

F

one another, will be each bisected (XI. 17.) by the planes OMN, STY, because the straight lines GC, HF are bisected in the points N, Y, by the same planes. Therefore the prisms LXCOMN, RVFSTY are of the same altitude; and therefore (XI. 32. cor.) as the base LXC to the base RVF, that is, (V. 11.) as the triangle ABC to DEF, so is the prism having the triangles LXC, OMN for its bases, to the prism of which the bases are the triangles RVF, STY. And because the two prisms in the pyramid ABCG are equal to one another, as also the two in the pyramid DEFH; as the prism of which the base is the parallelogram KBXL, and opposite side MO, to the prism having the triangles LXC, OMN for its bases; so (V. 7.) is the prism of which the base is the parallelogram PEVR, and opposite side TS, to the prism of which the bases are the triangles RVF, STY. Therefore, by composition, as the prisms KBXLMO, LXCOMN together are to the prism LXCOMN; so are the prisms PEVRTS, RVFSTY to the prism RVFSTY: and, alternately, 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, ABC to DEF: therefore (V. 11.) as ABC to DEF, so are the two prisms in the pyramid ABCG to the two in the pyramid DEFH. In like manner, if the pyramids now made, for example, the two OMNG, STYH, be divided similarly; as the base OMN is to the base STY, so will the two prisms in the pyramid OMNG be to the two in the pyramid STYH. But OMN: STY:: ABC: DEF; therefore (V. 11.) as ABC to DEF, so are the two prisms in the pyramid ABCG to the two in the pyramid DEFH; and so are the two prisms in the pyamid OMNG to the two in the pyramid STYH; and so are all four to all four and the same may be shown of the prisms made by dividing the

pyramids AKLO and DPRS, and of all made by the same number of divisions. Therefore, if two triangular pyramids, &c.

PROP. V. THEOR.

TRIANGULAR Pyramids, of the same altitude, are to one another as their bases.

Let the pyramids ABCG, DEFH, of which the vertices are 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, ABC must be to 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 Q less than it: and divide DEFH into two equal pyramids similar to the whole, and into two equal prisms: these two prisms (XII. 3.) are greater than the half of the whole pyamid. Again, let the pyramids made by this division be in like manner divided, and so on, until the pyramids which remain undivided in DEFH, are all of them together, less than the excess of DEFH above Q; which (XII. lem. 1.) it is possible to do. Let these, for example, be the pyramids DPRS, STYH. Therefore the prisms, which make the rest of the pyramid DEFH, are greater than Q. Divide likewise the pyramid ABCG in the same manner, and into as many parts as DEFH. Therefore (XII. 4.) as the base ABC is to the base DEF, so are the prisms in ABCG to those in DEFH. But as ABC to DEF, so, by hypothesis, is the pyramid ABCG to Q; and therefore as ABCG is to Q, so are the prisms in ABCG to those in DEFG. But ABCG is greater than the prisms contained in it; wherefore also (V. 14.) Q is greater than the prisms in DEFH. But it is also less, which is impossible. Therefore ABC is not to DEF, as ABCG to any solid which is less than DEFH. In the same manner, it may be demonstrated, that DEF is not to ABC, as DEFH to any solid which is less than ABCG. Nor can ABC be to DEF, as ABCG to any solid which is greater than DEFH. For, if it be possible, let it be so

to a greater, Z. Then, because ABC is to DEF, as ABCG to Z; by inversion, as DEF to ABC, so is Z to ABCG. But as Z is to ABCG, so is DEFH to some solid, which (V. 14.) must be less than ABCG, because Z is greater than DEFH. And, therefore, as DEF to ABC, so is DEFH to a solid less than ABCG;

the contrary to which has been proved. Therefore ABC is not to DEF, as ABCG to any solid which is greater than DEFH. And it has been proved, that neither is ABC to DEF, as ABCG to any solid which is less than DEFH. Therefore, as ABC is to DEF, so is ABCG to DEFH. Wherefore, triangular pyramids, &c.

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.

A

M

K

Divide the base ABCDE into the triangles ABC, ACD, ADE; and the other base into the triangles FGH, FHK, FKL: and upon the bases ABC, ACD, ADE let there be as many pyramids having the common vertex M, and upon the remaining bases as many pyramids having the common vertex N. Then, since (XII. 5.) the triangle ABC is to FGH, as the pyramid ABCM to FGHN; and the triangle ACD to FGH, as the pyramid ACDM to FGHN; and also the triangle ADE to FGH, as the pyramid ADEM to FGHN; as all the first antecedents to their common consequent, so (V. 24. cor. 2.) 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

B

D

F

G

K

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 the base ABCDE is to the base FGH, as the pyramid ABCDEM to the pyramid FGHN; and the base FGH to the base FGHKL, as the pyramid FGHN to the pyramid FGHKLN; therefore, ex æquo, as the base ABCDE to the base FGHKL, so is the pyramid ABCDEM to the pyramid FGHKLN. Therefore, pyramids of the same altitude, &c.

PROP. VII. THEOR.

EVERY triangular prism may be divided into three triangular pyramids, equal to one another.

Let there be a prism of which the bases are the triangles ABC, DEF: the prism may be divided into three equal triangular pyramids.

It is

D

F

Join BD, EC, CD; and because ABED is a parallelogram, and BD its diagonal, the triangles ABD, EBD are (I. 34.) equal; therefore (XII. 5.) the pyramid of which the base is ABD, and vertex C, is equal to the pyramid of which the base is EBD and vertex C. But EBC may be taken as the base of this pyramid, and D as its vertex. therefore equal (XII. 5.) to the pyramid of which ECF is the base, and D the vertex; for they have the same altitude, and (I. 34.) equal bases ECF, ECB and it has been already proved to be equal to the pyramid ABDC. Therefore the prism ABCDEF is divided into three equal pyramids having triangular bases, viz., into the pyramids ABDC, EBDC, ECFD. Therefore, every triangular prism, &c.

A

B

Cor. 1. From this it is manifest, that every pyramid is the third part of a prism which has the same base, and an equal altitude; for if the base of the prism be any other rectilineal 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 (XII. 6.) to one another as their bases.

PROP. VIII. THEOR.

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

Let the pyramids of which ABC, DEF are the bases, and G, H,