II.

the base RVF; that is, as the triangle ABC to the triangle Book XII. DEF, fo a is the prifm having the triangle LXC for its bafe, a Cor. 32. and OMN the triangle opposite to it, to the prism of which the base is the triangle RVF, and the oppofite triangle STY: And because the two prifms 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 oppofite fide MO, to the prifin having the triangle LXC for its bafe, and OMN the triangle oppofite to it; fo is the prifm of which the bafe bis the parallelogram, b7. 5. PEVR, and opposite fide TS, to the prism of which the base is the triangle RVF, and oppofite triangle STY. Therefore, componendo, as the prifms KBXLMO LXCOMN together

are unto the prism LXCOMN; so are the prisms PEVRTS, RVFSTY to the prifm RFSTY: 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 base DEF: Therefore, as the base ABC to the bafe DEF, fo are the two prifms 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 fame manner; as the base OMN is to the bafe STY, fo fhall the two prisms in the pyramid OMNG be to the two prifms in the pyramid STYH: But the base OMN is to the bafe STY, as the bafe ABC to the bafe DEF; therefore, as the bafe ABC to the base DEF, fo are

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 prisms made by dividing the pyramids AKLO and DPRS, and of all made by the fame number of divifions. Q. E. D.

See N.

PYRAMIDS of the fame altitude which have tri

angular 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 fame altitude: As the bafe ABC to the base DEF, so is the pyramid ABCG to the pyramid DEFH.

For, if it be not fo, the base ABC must be to the base DEF, as the pyramid ABCG to a folid either lefs than the pyramid DEFH, or greater than it *. First, let it be to a folid less than it, viz. to the folid Q: And divide the pyramid DEFH into two equal pyramids, fimilar to the whole, and into two equal 3. 12. prifms: Therefore these two prifms are greater a than 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, lefs 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 rest of the pyramid DETH, 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 base 4. 12. ABC to the bafe DEF, fob are the prifms in the pyramid ABCG to the prifms in the pyramid DEFH: But as the base ABC to the base 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 prisms contained in it; wherefore also the folid Qis greater than the prisms in the pyramid DEFH. But is it alfo lefs, which is impoffible. Therefore the bafe ABC is not to

€ 14. 5.

the

*This may be explained the fame way as at the note † in propofition 2. in the like cafe.

the base DEF, as the pyramid ABCG to any folid which is Book XII. less than the pyramid DEFH. In the fame manner it may be demonftrated, that the base DEF is not to the base ABC, as the pyramid DEFH to any folid which is less than the pyramid ABCG. Nor can the bafe 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 bafe ABC is to the bafe DEF as the pyramid ABCG to the folid Z; by inverfion, as the base DEF to the base 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 a than the pyramid ABCG, because the folid Z is greater than the pyramid DEFH. And therefore, as the base DEF to the base ABC, fo is the pyramid DEFH to a folid less than the pyramid ABCG; the con trary to which has been proved. Therefore the bafe ABC is not to the bafe 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 base DEF, as the pyramid ABCG to any folid which is lefs than the pyramid DEFH. Therefore, as the bafe ABC is to the base 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.

a 14. 5.

Book XII.

See N.

PROP. VI. THEOR.

PYRAMIDS of the fame altitude which have poly gons for their bafes, are to one another as their

bafes.

Let the pyramids which have the polygons ABCDE, FGHKL for their bases, and their vertices in the points M, N, be of the fame altitude: As the bafe ABCDE to the base FGHKL, fo 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 bafes as many pyramids having their common vertex in the point N: Therefore, fince the triangle 5. 12. ABC is to the 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 the triangle FGH, as the pyramid ADEM to the pyramid FGHN; as all the firft antecedents to their comb 2. Cor. mon confequent; fob are all the other antecedents to their com24.5. mon confequent; that is, as the base ABCDE to the bafe FGH, fo is the pyramid ABCDEM to the pyramid FGHN: And, for the fame reafon, as the base FGHKL to the bafe FGH, fo is the pyramid FGHKLN to the pyramid FGHN: And, by inverfion, as the base FGH to the base FGHKL, fo is the pyramid FGHN to the pyramid FGHKLN: Then, because as the base ABCDE to the base FGH, fo is the pyramid ABCDEM to the pyramid FGHN; and as the bafe FGH to the base FGHKL, fo is the pyramid FGHN to the pyramid FGHKLN; therefore,

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therefore, ex æqualis, as the bafe ABCDE to the bafe FGHKL, Book XII. fo the pyramid ABCDEM to the pyramid FGHKLN. There- c 22. 5. fore pyramids, &c. Q. E. D.

E

PROP. VII. THEOR.

VERY 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 prifm of which the base is the triangle ABC, and let DEF be the triangle oppofite to it: The prism ABCDEF may be divided into three equal pyramids having triangular bafes.

Join BD, EC, CD; and because ABED is a parallelogram of which BD is the diameter, the triangle ABD is equal a to a 34. T. the triangle EBD; therefore the pyramid of which the base is the triangle ABD, and vertex the point C, is equal b to the b5. 12. pyramid of which the base is the triangle EBD, and vertex the point C: But this pyramid is the fame with the pyramid the bafe of which is the triangle EBC, and vertex the point D; for they are contained by the fame planes: Therefore the pyramid of which the base is the triangle ABD, and vertex the point C, is equal to the pyramid, the bafe 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 a 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

D

F

E

B

bafe is the triangle ABD, and vertex the point C. Therefore the prifm 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 fame with the pyramid of which the base is the triangle ABC, and vertex the point D, for they are contained by the fame planes; and that the pyramid of which the base is the triangle ABD, and vertex the point C, has been demonstrated

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