Let there be a pyramid, whose base is the triangle ABC, and vertex the point M, then the pyramid ABCM may be divided into two pyramids, equal and fimilar to one another, having triangular bases, and fimilar to the whole; and into two equal prifms; which two prifms are greater than the half of the whole pyramid.

Book XII

C 29. I.

For, bifect AB, BC, AC, MA, MB, MC, in the points E, N, G, H, K, L; join EH, EG, EK, EN, HG, HK, HL, KL, KN, NG; then, because AE is equal to EB, and AH to HM, EH is parallel to MB 2, and KH to AB b; but Al a 2. 6. b 34. I. is equal to KH; for each is equal to EB; therefore the two fide: AE, AH, are equal to the two fides KH, HM, the angle MHK equal to HAE; therefore the bafes EH, MK, are equal, an à 4. 1. the triangle AEH equal and fimilar to KHM. For the fame reason, AHG is equal and fimilar to MHL. And, because AE, AG, are equal and parallel to HK, HL, each to each, the angle KHL is equal to EAG, and the base KL to EG ; therefore the 10. 11. triangles KML, EHG, are equal and fimilar; therefore the pyramid, whose base is the triangle AEG, and vertex the point H, is equal and fimilar to the pyramid whofe bafe is the triangle HKL, and vertex the point Mf; and, because HK is parallel to def. 10. AB, the fide of the triangle AMB, the triangles AMB, **. MHK, are fimilars. For the fame reafon, the triangle MBC is 8 2. 6. fimilar to MKL, and the triangle AMC to MHL; but the angles KHL, BAC, are equal, and the triangles fimilar e; therefore the pyramid ABCM is fimilar to HKLM; but the pyramid AEHG is proved fimilar to HKLM, therefore fimilar to one ħ 21. 6 nother, and similar to ABCM.

i

Again, because BN is equal to NC, the parallelogram BG is double the triangle GNC ; therefore the prifm contained by 41. të the two triangles BKN, EHG, and the three parallelograms BG,BH, KG, is equal to the prifm contained by the two triangles NGC, KHL, and the three parallelograms KG, GL, NL, the one of which is conftitute upon the parallelogram BG, and oppofite to it the right line KH; the other upon the triangle GNC, and oppofite to it the triangle KHL; and the parallelogram BG is double the triangle GNC, and have the fame altitude; therefore they are equal; but either of these prifms is greater than the k 40. II. pyramid AEGH, or HKLM; for the prifm EBNGHK is greater than the pyramid EBNK, which is equal to the pyramid AEGH, or HKLM; wherefore the prifm EBNGHK is greater than the pyramid AEGH, or HKLM; therefore the prifm GNCHKL, is likewife greater than the pyramid HKLM; but the prifms are equal; therefore, together, are greater than the two pyramids; therefore the whole pyramid is divided into two equal pyramids fimilar to the whole, and to one another, and

into

Book XII into two equal prifms, which two prifms together are greater than half the pyramid.

a 4. 6.

b 22. 6.

€ 32. and

28. 11.

d 3.

PROP. IV. THE OR.

IF there are two pyramids of the fame altitude, having triangular bafes, and each of them divided into two pyramids equal to one another, and fimilar to the whole, and into two equal prifms; and, if each pyramid be divided in the fame manner, and this be done continually; then, as the base of the one pyramid is to the base of the other, fo are all the prisms in the one pyramid to all the prifms in the other, being equal in number.

Let there be two pyramids of the fame altitude, having the triangular bases ABC, DEF, and vertices the points M, H, and each of them divided into two pyramids, equal to one another, and fimilar to the whole, and into two equal prifms; and if, in like manner, each of the pyramids made by the former divifion be fuppofed divided, and this be done continually; then, as the bafe ABC is to the bafe DEF, fo are all the prifms in the pyramid ABCM to all the prifms in the pyramid DEFH, being equal in number.

For, let the pyramid DEFH be constructed fimilar to the pyramid ABCM; then, all the triangles described in the base ABC being fimilar to the whole, and to one another; and also those in DEF, being equal in number to the triangles in ABC; then ABC will be fimilar to NGC, and DEF to RQF; and, as BC is to NC, fo is EF to FQ; therefore ABC is to NGC as DEF is to RQFb. And, altern. as ABC is to DEF, fo is NGC to RQF; but, as NGC is to RQF, fo is the prism GNCLHK to the prifm RQFYST: But the two prifms in the pyramid ABCM are equal to one anotherd, as alfo the two prifms in the pyramid DEFH; wherefore the prifm, whofe bafe is the parallelogram EGNB, and oppofite bafe the right line KH, is to the prifm, whose base is the triangle NGC, and oppofite base the triangle HKL, fo is the prifm whofe bafe is the parallegram EPRQ, and oppofite bafe the right line ST to the prism whose base is the triangle RQF, and oppofite to it the triangle STY, compound. as the prifms EBNGKH, GNCLHK, together, are to the prism GNCLHK, so the prisms PEQRST, ROFSTY, together, are to ROFSTY; altern. as the prisms EBNGKH, GNCLHK, together, are to the prifms PEORST, RQFSTY, together, fo the prifm GNCLĤK to RQFSTY; but the prifm GNCLHK is to the prism RQFSTY

as

as the bafe GNC to the bafe RQF and fo is the bafe ABC to Book XII the base DEF; therefore, as the bafe ABC is to the bafe DEF, n fo are the two prifms in the pyramid ABCM to the two prisms in the pyramid DEFH. For the fame reason, the prisms in the pyramids HKLM, STYH, or any other pyramids made by any of the former divifions, are to each other as their bafes; wherefore, all the prisms in the pyramid ABCM are to all the prisms in the pyramid DEFH as the base ABC to the DEF. Wherefore, &c.

P

PROP. V. and VI. THEOR.

YRAMIDS of the fame altitude, having triangular or
polygonous bafes, are to one another as their bafes.

First, let ABCM, DEFH, be pyramids of the fame altitude, having the triangular bafes ABC, DEF; then the pyramid ABCM is to the pyramid DEFH as the bafe ABC is to DEF; and fo are any number of pyramids to their triangular bases.

If not, let the base ABC be to the base DEF as the pyramid ABCM is to fome folid Z, lefs than the pyramid DEFH, which divide into two pyramids equal to each other, and into two Frisms which are greater than half of the whole pyramid; and, if the pyramids made by the former divifion be divided in the fame manner, till fome pyramids in the pyramid DEFH is found lefs than the excess by which the pyramid DEFH exceeds Z. Let thefe pyramids be DPRS, STYH. Let the pyramid ABCM be divided into the fame number of fimilar parts, as the pyramid DEFH; then are the prifms in the pyramid ABCM to the prifms in the pyramid DEFH 2, as the bafe ABC is to DEF; but the bafe ABC is to the bafe DEF as the pyramid ABCM to the folid Z; therefore the pyramid ABCM is to the hyp. folid Z as the prifms in ABCM to the prifms in DEFH; but ABCM is greater than the priims in it; therefore the folid Z is greater than the prisms in DEFH; and likewife lefs ; which c 14. 5. is abfurd; therefore the bafe ABC is not to the bafe DEF as the pyramid ABCM to fome folid lefs than DEFH. For the fame reason, the base DEF is not to the bafe ABC as the pyramid DEFH to fome folid lefs than ABCM; but ABC is not to DEF as ABCM is to fome folid I, greater than DEFH. For, if poffibie, DEF is to ABC as I to ABCM d; but the folid 1 is d inverf. greater than DEFH; then, as I is to ABCM, fo is DEFH to fome folid less than ABCM; which is proved abfurd; therefore ABC to DEF is not as ABCM to fome folid greater than DEFH; but it was alfo proved not to be to fome folid lefs than DEFH; therefore ABC is to DEF as ABCM is to DEFH.

b.

For

a 34. 1.

b 5, and 6.

For the fame reason, in the pyramids ABCDEM, FGHKLN, fig. . the pyramid ABCM is to the pyramid ACDM as the bate ABC is to the bafe ACD; and ACDM is to ADEM as ACD is to AED; therefore the whole ABCDEM is to ABCM as ABCDE to ABC d. For the fame reason, as FGHKLN is to FGHKL as FGHN is to FGH; if ABCDE is equal to FGHKL, the pyramid ABCDEM is equal to FGHKLN; if greater, greater, and, if lefs, lefs; therefore, as the bafe ABCDE is to the base FGHKL, fo is the pyramid ABCDEM to the pyramid FGHKLN; Wherefore, &c.

E

PRO P. VIL THE OR.

VERY prism, having a triangular base, may be divided into three pyramids, equal to one another, and having triangular bafes.

Let there be a prism, whose base is the triangle ABC, and the oppofite bafe to that the triangle DEF; then the prifm ABCDEF may be divided into three equal pyramids, having triangular bafes.

For, join BD, EC, CD; then, because ABCD is a paralle. logram, whofe diameter is BD2, the pyramid whose base is the triangle ABD, and vertex the point C, is equal to the pyramid whofe bafe is the triangle EBD, and vertex the point CD; but the pyramid whose base is the triangle EBD, and vertex the point C, is equal to the pyramid whofe bafe is the triangle EBC, and vertex the point D; for they are contained by the fame plains; therefore the pyramid whofe bafe is the triangle ABD, and vertex the point C, is equal to the pyramid whofe base is the triangle EBC, and vertex the point D.

Again, because FCBE is a parallelogram whofe diameter is CE, the triangle ECF is equal to the triangle CBE 2; therefore the pyramid whose base is the triangle BEC, and vertex the point D, is equal to the pyramid whofe base is the triangle CEF, and vertex the point Db: But the pyramid whose base is the triangle BEC, and vertex the point D, has been proved equal to the pyramid whofe bafe is the triangle ABD, and vertex the point C; therefore, also the pyramid whose base is the triangle CEF, and vertex the point D, is equal to the pyramid whofe bafe is the triangle ABD, and vertex the point C; therefore the prism ABCDEF is divided into three pyramids, equal to one another, and having triangular bases. And, because the pyramid whose base is the triangle ABD, and vertex the point C, is the fame with the pyramid whofe bafe is the triangle ABC, and

1

and vertex the point D, for they are contained by the fame Book XII plains; and the pyramid whose base is the triangle ABD, and vertex the point C, has been proved to be a third part of the prifm, having the fame bafe, viz. the triangle ABC, and the oppofite base the triangle DEF: Which was required.

COR. I. Hence every pyramid is a third part of a prism, having the fame base, and an equal altitude; for, if the base of a prifm be of any other figure, it can be divided into prifms, having triangular bafes.

II. Prisms of the fame altitude are to one another as their bafes.

S

P R O P. VIII. THE O R.

IMILAR pyramids, having triangular bafes, are in the
triplicate ratio of their homologous fides.

Let the two pyramids whofe bafes are ABC, DEF, and vertices the points G, H, be fimilar and alike fituate, then the pyramids ABCG, DEFH, are to one another in the triplicate ratio of BC to EF.

a 29. II.

For, compleat the parallelopipedons BGML, EHPO, then each contain two equal prifms, having triangular bafes ; and a pyramid is one third of a prifm, having the fame bafe and altitude ; but fimilar folid parallalelopipedons are to one another b7. in the triplicate ratio of their homologous fides, and parts have © 33. 11. the fame proportion as their like multiples 4; therefore the pyra- d is. s. mids ABCG, DEFH, are to one another in the triplicate ratio

d

of their homologous fides. Wherefore, &c.

COR. Hence fimilar pyramids, having polygonous bafes, are to one another in the triplicate ratio of their homologous ñdes.

T

PRO P. IX. THE O R.

HE bafes and altitudes of equal pyramids, having triangu lar bafes, are reciprocally proportional; and those pyramids, having triangular bafes, whofe bafes and altitudes are reciprocally proportional, are equal.