B. XII.

See N.

a 2.6.

PROP. IV. THEOR.

IF there be two pyramids of the same altitude, upon triangular bases, and each of them be 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 shall all the prisms in one of them be 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 the like manner, and so on; as the base ABC is to the base 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 is equal to XC, and AL to LC, therefore XL is parallel a to AB, and the triangle ABC similar to the triangle LXC: for the same reason, the triangle DEF is similar to RVF: and because BC is double of CX, and EF double of FV, therefore BC is to CX, as EF to 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, as the trib 22.6. angle ABC is to the triangle LXC, sob is the triangle DEF to the triangle RVF, and, by permutation, as the triangle ABC to the triangle DEF, so is the triangle LXC to the triangle RVF: and because the planes ABC, OMN, as also the planes c 15. 11. DEF, STY, are parallel, the perpendiculars drawn from the points G, H to the bases ABC, DEF, which, by the hypothesis, are equal to one another, shall be cut each into two equal parts by the planes OMN, STY, because the straight lines GC, HF are cut into two equal parts in the points N, Y by the same planes: therefore the prisms LXCOMN, RVFSTÝ are of the same altitude; and, therefore, as the base LXC to

d 17. 11.

a

the base RVF; that is, as the triangle ABC to the triangle B. XII. DEF, so is the prism having the triangle LXC for its base, and OMN the triangle opposite to it, to the prism of which the a Cor. 32. base is the triangle RVF, and the opposite triangle STY: and 11. 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 is the parallelogram b 7. 5. 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 LXCOMN, so are the prisms PEVRTS, RVFSTY to the prish 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

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

PROP. V. THEOR. .

See N.

a 3. 12.

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 a 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 b 4. 12. ABC to the base DEF, sob 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 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

c 14. 5.

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

the base DEF, as the pyramid ABCG to any solid which is less B. XII. 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 the pyramid c 14. 5. ABCG, because the solid Z is greater than the pyramid DEFH. And, therefore, as the base DEF to the base APC, 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 mark† in prop. 2.

B. 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 triangle ABC is to the a 5. 12. 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 also the triangle ADE to

24. 5.

the triangle FGH, as the pyramid ADEM to the pyramid FGHN; 2. Cor. as all the first antecedents to their common consequent, so bare 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;