d 4.

II.

Book XII. AEH equal d and fimilar to the triangle HKD: For the fame reafon, the triangle AGH is equal and fimilar to the triangle HLD: And because the two straight lines EH, HG which meet one another are parallel to KD, DL that meet one another, and are not in the fame plane with them, they contain 10. 11. equal angles; therefore the angle EHG is equal to the angle KDL. Again, becaufe EH, HG are equal to KD, DL, each to each, and the angle EHG equal to the angle KDL; therefore the bafe EG is equal to the bafe KL: And the triangle EHG equal d and fimilar to the triangle KDL: For the fame reafon, the triangle AEG is alfo equal and fimilar to the triangle HKL. Therefore the pyramid of which the bale is the triangle AEG, and of which the vertex is the point H, is eFC. 11. qual f and fimilar to the pyramid the

KA

D

bafe of which is the triangle KHL, and vertex the point D: And because HK is parallel to AB a fide of the triangle ADB, the triangle ADB is cquiangular to the triangle HDK, and their 4. 6. fides are proportionals 8: Therefore the triangle ADB is fimilar to the triangle HDK: And for the fame reason, the triangle DBC is fimilar to the triangle DKL; and the triangle ADC to the triangle HDL; and alfo the triangle. ABC to the triangle AEG: But the triangle AEG is fimilar to the triangle HKL, as before was proved; therefore h21. 6. the_triangle ABC is fimilar to the triangle HKL, And the pyramid of which the bafe is the triangle ABC, and vertex the point D, i B. 11. is therefore fimilar i to the pyramid of which the bafe is the triangle HKL, and vertex the fame point D: But the pyramid of which the bafe is the triangle KL, and vertex the point D, is fimilar, as has been proved, to the pyramid the bafe of which is the triangle AEG, and vertex the point H: Wherefore the pyramid the bafe of which is the triangle ABC, and vertex the point D, is fimilar to the pyramid of which the bafe is the triangle AEG and vertex H: Therefore each of the pyramids AEGH, HKLD is fimilar to the whole pyramid AECD: And becaufe 41. 1. BF is equal to FC, the parallelogram EBFG is double k of the triangle GFC: But when there are two prifms of the fame alti

&

Def. 11.

B

F

C

tude,

tude, of which one has a parallelogram for its bafe, and the other Book XII. a triangle that is half of the parallelogram, thefe prifms are equal to one another; therefore the prifm having the parallelogram a 40. 11. 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, becaufe they are between the parallel b planes ABC, b 15. 11. HKL: And it is manifeft that each of thefe prifms is greater than either of the pyramids of which the triangles AEG, HKL are the bafes, and the vertices the points H, D; because, if EF be joined, the prifm having the parallelogram EBFG for its bafe, and KH the ftraight line oppolite 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. 11. bafe of which is the triangle AEG, and vertex the point H; because they are contained by equal and fimilar planes: Wherefore the prifm having the parallelogram EBFG for its base, and oppofite fide KH, is greater than the pyramid of which the bafe is the triangle AEG, and vertex the point H: And the prifm of which the bafe is the parallelogram EBFG, and oppofire 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 prifms; and the two prifms are together greater than half of the whole pyramid. QE. D.

Book XII.

Sce N.

6.

PROP. IV. THEOR.

IF 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 prifins; and if each of thefe pyramids be divided in the fame manner as the firft 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 prifmis 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 bates 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 fo on : As the bafe ABC is to the bafe DEF, fo are all the prifms in the pyramid ABCG to all the prifms in the pyramid DEFH made by the fame number of di vifions.

Make the fame conftruction as in the foregoing propofition : and because BX is equal to XC, and AL to LC, therefore XL is parallel a to AB, and the triangle ABC fimilar to the triangle LXC: For the fame reafon, the triangle DEF is fimilar to RVF: And becaufe BC is double of CX, and EF double of FV, therefore BC is to CX, as EF to FV: And upon BC, CX are defcribed 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 tri22. 6.angle 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 e, the perpendiculars drawn from the points G, H to the bafes ABC, DEF, which, by the hypothefis, are equal to one another, fhail be cut each into two equal 1711d 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, RVFSTY are of the fame altitude; and therefore, as the base LXC to

the

11.

the bafe 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 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 base is the pa rallelogram 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 base is the parallelogramb 4. §i PEVR, and oppofite fide TS, to the prifm of which the bafe 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 bafe ABC to the bafe DEF, fo 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 py ramid OMNG be to the two prifms in the pyramid STYH; But the hafe OMN is to the bafe STY, as the bafe ABC to the bafe DEF; therefore, as the bafe ABC to the bafe 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 prifins made by dividing the pyramids AKLO and DPRS, and of all made by the fame number of divifions. Q. E. D.

See N.

PROP. V. THEOR.

PYRAMIDS of the fame altitude which have triangular bafes, are to one another as their bafes.

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 base DEF, fo is the py ramid ABCG to the pyramid DEFH.

For, if it be not fo, the bafe 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 a 3. 12 prifms: Therefore thefe 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 excefs 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 bate b 4. 12. ABC to the bafe DEF, fo b are the prifms in the pyramid ABCG to the pritms in the pyramid DEFH: But as the bale 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 e alfo the folid Q is greater than the prifms in the pyramid DEFH. But it is allo lefs, which is impoffible. Therefore the bafe ABC is not to

C 14. 5.

the

This may be explained the fame way as at the note in propofition 1. in the like cafe.