40. 11.

tude, of which one has a parallelogram for its bafe, and the other Book XII. a triangle that is half of the parallelogram, these prifms are equal * to one another; therefore the prifm having the parallelograma EBFG for its bafe, and the ftraight line KH oppofite to it, is equal to the prifm having the triangle GFC for its bafe, and the triangle HKL oppofite to it; for they are of the fame altitude, because they are between the parallel planes ABC, b 15. 11. HKL: And it is manifeft that each of these prifms is greater than either of the pyramids of which the triangles AEG, HKL are the bafes, and the vertices the points H, D; becaufe, if EF be joined, the prifm having the parallelogram EBFG for its bafe, and KH the ftraight line oppofite to it, is greater than the pyramid of which the bafe is the triangle EBF, and vertex the point K; but this pyramid is equal to the pyramid the c C. II. bafe of which is the triangle AEG, and vertex the point H; because they are contained by equal and fimilar planes: Where fore the prifm having the parallelogram EBFG for its base, and oppofite fide KH, is greater than the pyramid of which the base is the triangle AEG, and vertex the point H: And the prifm of which the bafe is the parallelogram EBFG, and oppofite fide KH is equal to the prifm having the triangle GFC for its bafe, and HKL the triangle oppofite to it; and the pyramid of which the bafe is the triangle AEG, and vertex H, is equal to the pyramid of which the bafe is the triangle HKL, and vertex D: Therefore the two prifms before mentioned are greater than the two pyramids of which the bafes are the triangles AEG, HKL, and vertices the points H, D. Therefore the whole pyramid of which the bafe is the triangle ABC, and vertex the point D, is divided into two equal pyramids fimilar to one another, and to the whole pyramid; and into two equal priims; and the two prifms are together greater than half of the whole pyramid. Q. E. D.

Book XII.

See N.

a a. 6.

b 22. 6.

I

there be two pyramids of the fame altitude, upon triangular bafes, and each of them be divided into two equal pyramids fimilar to the whole pyramid, and alfo into two equal prifms; and if each of these pyramids be divided in the fame manner as the first two, and fo on : As the bafe of one of the first two pyramids is to the base of the other, fo fhall all the prifms in one of them be to all the prifms in the other, that are produced by the fame number of divifions.

Let there be two pyramids of the fame altitude upon the triangular bafes ABC, DEF, and having their vertices in the points G, H; and let each of them be divided into two equal pyramids fimilar to the whole, and into two equal prifms; 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, fo are all the prifms in the pyramid ABCG to all the prifms in the pyramid DEFH made by the fame number of divifions.

Make the fame conflruction as in the foregoing propofition: and because BX is equal to XC, and AL to LC, therefore XL is parallel to AB, and the triangle ABC fimilar to the triangle LXC: For the fame reason, the triangle DEF is fimilar 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 fimilar and fimilarly fituated rectilineal figures ABC, LXC; and upon EF, FV, in like manner, are defcribed the fimilar figures DEF, RVF: Therefore, as the triangle ABC is to the triangle LXC, fob is the triangle DEF to the triangle RVF, and, by permutation, as the triangle ABC to the triangle DEF, fo is the triangle LXC to the triangle RVF: And because the planes ABC, OMN, as alfo the planes 15. 11. DEF, STY are parallel, the perpendiculars drawn from the points G, H to the bafes ABC, DEF, which, by the hypothefis, are equal to one another, fhall be cut each into two equal 17. 11. 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 fame planes: Therefore the prifms LXCOMN, RVFSTÝ are of the fame altitude; and therefore, as the bafe LXC to

d

the

a Cor. 32.

II.

the bafe RVF; that is, as the triangle ABC to the triangle Book XII. DEF, fo is the prifm having the triangle LXC for its base, and OMN the triangle oppofite to it, to the prifm of which the bafe is the triangle RVF, and the oppofite triangle STY: And because the two prifms in the pyramid ABCG are equal to one another, and alfo the two prifms in the pyramid DEFH equal to one another, as the prifm of which the bafe is the parallelogram KBXL and oppofite fide MO, to the prifm having the triangle LXC for its bafe, and OMN the triangle oppofite to it; fo is the prifm of which the bafe is the parallelogram b 7. 5PEVR, and oppofite fide TS, to the prifm of which the base is the triangle RVF, and oppofite triangle STY. Therefore, componendo, as the prifms KBXLMO, LXCOMN together

are unto the prifm LXCOMN; fo are the prifms PEVRTS, RVFSTY to the prifm RVFSTY: And, permutando, as the prifms KBXLMO, LXCOMN are to the prifms PEVRTS, RVFSTY; fo is the prifm LXCOMN to the prifm RVFSTY: But as the prifm LXCOMN to the prifm RVFSTY, fo is, as has been proved, the base ABC to the bafe DEF: Therefore, as the base ABC to the bafe DEF, so are the two prifms in the pyramid ABCG to the two prifms in the pyramid DEFH; And likewife if the pyramids now made, for example, the two OMNG, STYH be divided in the fame manner; as the base OMN is to the bafe STY, fo fhall the two prifms in the pyramid OMNG be to the two prifms in the pyramid STYH: But the bafe OMN is to the bafe STY, as the base ABC to the bafe DEF; therefore, as the base ABC to the base DEF, so are R 3 the

Book XII. the two prifms in the pyramid ABCG to the two prifms in the pyramid DEFH; and fo are the two prifms in the pyramid OMNG to the two prifms in the pyramid STYH; and fo are all four to all four: And the fame thing may be fhewn of the prifms made by dividing the pyramids AKLO and DPRS, and of all made by the fame number of divifions. Q. E. D,

See N.

a 3. 12.

b 4. 12.

c 14.5.

YRAMIDS of the fame altitude which have triangular bases, are to one another as their bases.

PR

Let the pyramids of which the triangles ABC, DEF are the bafes, and of which the vertices are the points G, H, be of the fame altitude: As the bafe ABC to the bafe DEF, fo is the pyramid ABCG to the pyramid DEFH.

For, if it be not fo, the bafe ABC must be to the bafe DEF, as the pyramid ABCG to a folid either lefs than the pyramid DEFH, or greater than it t. First, let it be to a folid lefs than it, viz. to the folid Q: And divide the pyramid DEFH into two equal pyramids, fimilar to the whole, and into two equal prifms: Therefore these two prifms are greater then the half of the whole pyramid. And, again, let the pyramids made by this divifion be in like manner divided, and fo 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 folid Q: Let thefe, for example, be the pyramids DPRS, STYH: Therefore the prifms, which make the reft of the pyramid DEFH, are greater than the folid Q Divide likewife the pyramid ABCG in the fame manner, and into as many parts, as the pyramid DEFH: Therefore, as the bafe ABC to the bafe DEF, fob are the prifms in the pyramid ABCG to the prifms in the pyramid DEFH: But as the bafe ABC to the bafe DEF, fo, by hypothefis, is the pyramid ABCG to the folid Q; and therefore, as the pyramid ABCG to the folid Q, fo are the prifms in the pyramid ABCG to the prifms in the pyramid DEFH: But the pyramid ABCG is greater than the prifms contained in it; wherefore alfo the folid Qis greater than the prifms in the pyramid DEFH. But it is alfo lefs, which is impoffible. Therefore the bafe ABC is not to

the

This may be explained the fame way as at the note in propofition 2. in

the like cafe,

も

the bafe DEF, as the pyramid ABCG to any, folid which is Book XII. lefs than the pyramid DEFH. In the fame manner it may be demonftrated, that the bafe DEF is not to the bafe ABC, as the pyramid DEFH to any folid which is lefs than the pyramid ABCG. Nor can the base ABC be to the base DEF, as the pyramid ABCG to any folid which is greater than the pyramid DEFH. For if it be poffible, let it be fo to a greater, viz. the folid Z. And because the base ABC is to the base DEF as the pyramid ABCG to the folid Z; by inverfion, as the base DEF to the bafe ABC, fo is the folid Z to the pyramid ABCG. But as the folid Z is to the pyramid ABCG, fo is the pyramid

DEFH to fome folid, which must be less than the pyramid a 14. 5. ABCG, because the folid Z is greater than the pyramid DEFH. And therefore, as the bafe DEF to the bafe ABC, fo is the pyramid DEFH to a folid lefs than the pyramid ABCG; the contrary to which has been proved. Therefore the bafe ABC-is not to the base DEF, as the pyramid ABCG to any folid which is greater than the pyramid DEFH. And it has been proved, that neither is the bafe ABC to the bafe DEF, as the pyramid ABCG to any folid which is less than the pyramid DEFH. Therefore, as the bafe ABC is to the bafe DEF, fo is the pyramid ABCG to the pyramid DEFH. Wherefore pyramids, &c. Q. E. D.

• This may be explained the fame way as the like at the mark † in prop. 2.