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a 41. I.

Book XII GM, MH, HN, NE, and if tangents are drawn from the points K, L, M, N, and parallelograms compleated upon EF, FG, GH, and HE; then each of the triangles EKF, FLG, GMH, HNE, will be equal to half the parallelogram, and therefore greater than half the fegment of the circle it ftands in; if the remaining fegments are bifected, and triangles drawn, as before; and this be continued till the fegments are lefs than the excefs by which the circle EFGH exceeds the figure S6. Let thefe be the fegments cut off by the right lines EK, KF, FL, LG, GM, MH, HN, NE; then the remaining polygon EKFLGMHN will be greater than the figure S.

b Lem.

C I.

d hyp.

e 11. S.

f 14. S.

Defcribe the polygon AXBOCPDR, in the circle ABCD, fimilar to the polygon EKFLGMHN ; then, as the fquare of BD is to the fquare of FH, fo is the circle ABCD to the figure S4; and, as the polygon AXBOCPDR to the polygon EKFLGMHN, fo is the circle ABCD to the figure Se; but the circle ABCD is greater than the polygon in it; therefore the figure S is greater than the polygon EKFLGMHN f; but it is lefs; which is abfurd; therefore the fquare of BD to the fquare of FH is not as the circle ABCD to fome figure lefs than the circle EFGH. In like manner, it is proved that the fquare of FH to the fquare of BD is not to the circle EFGH as fome figure lefs than the circle ABCD.

Laftly, the fquare of BD to the fquare of FH, is not as the circle ABCD to fome figure greater than the circle EFGH.

For, if poffible, let it be to the figure T, greater than the circle EFGH; then, inversely, the fquare of FH is to the fquare of BD as the figure T is to the circle ABCD; but, because T is greater than the circle EFGH, T will be to the circle ABCD as the circle EFGH is to fome figure lefs than the circle ABCD, which is proved impoffible; therefore the fquare of BD to the fquare of FH is not as the circle ABCD to fome figure less than the circle EFGH, nor to one greater; therefore, as the fquare of BD is to the fquare of FH, fo is the circle ABCD to the circle EFGH. Wherefore, &c.

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E

VERY pyramid having a triangular bafe, may be divided into two pyramids, equal and fimilar to one another, having triangular bafes, and fimilar to the whole pyramid; and into two equal prifms; which two prifms are greater than the half of the whole pyramid.

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

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34. I

C 29. I.

d

4. 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, and KH to AB b; but AE 2. 6. is equal to KH; for each is equal to EB; therefore the two fides AE, AH, are equal to the two fides KH, HM, the angle MHK equal to HAE; therefore the bafes EH, MK, are equal, and the triangle AEH equal and fimilar to KHM. For the fame reafon, 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 whose base is the triangle HKL, and vertex the point Mf; and, because HK is parallel to f def. Io. AB, the fide of the triangle AMB, the triangles AMB, ** MHK, are fimilar. 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; therefore the pyramid ABCM is fimilar to HKLM; but the pyramid AEHG is proved fimilar to HKLM, therefore fimilar to one a- h 21. 6, nother, and fimilar to ABCM.

Again, because BN is equal to NC, the parallelogram BG is double the triangle GNC; therefore the prifm contained by i 41. 1ệ 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 thefe 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.

2 4 6.

b 22. 6.

c 32. and

28. 11.

d 3.

PROP. IV. TH E, O R.

IF F 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 bafe of the other, fo are all the prifms 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 bafes 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 conftructed fimilar to the pyramid ABCM; then, all the triangles defcribed in the base ABC being fimilar to the whole, and to one another; and also thofe 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 another d, as alfo the two prifms in the pyramid DEFH; wherefore the prifm, whose base is the parallelogram EGNB, and oppofite base the right line KH, is to the prifm, whose bafe is the triangle NGC, and oppofite base the triangle HKL, fo is the prism whose base is the parallegram EPRQ, and oppofite bafe the right line ST to the prifm whofe bafe is the triangle RQF, and oppofite to it the triangle STY, compound. as the prifms EBNGKH, GNCLHK, together, are to the prifm GNCLHK, fo the prifms PEQRST, ROFSTY, together, are to ROFSTY; altern. as the prifms EBNGKH, GNCLHK, together, are to the prisms PEORST, RQFSTY, together, fo the prifm GNCLHK to RQFSTY; but the prifm GNCLHK is to the prifm RQFSTY

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 base DEF, fo are the two prisms in the pyramid ABCM to the two prisms in the pyramid DEFH. For the fame reason, the prifms 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 prifms in the pyramid ABCM are to all the prifms in the pyramid DEFH as the bafe ABC to the DEF. Wherefore, &c.

P

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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.

a 4.

If not, let the bafe 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 Frifms 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 prisms in the pyramid ABCM to the prifms in the pyramid DEFH, 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 prifms in it; therefore the folid Z is greater than the prisms in DEFH; and likewife lefs ; which c 14. 5. is abfurd; therefore the base ABC is not to the base DEF as the pyramid ABCM to fome folid less than DEFH. For the fame reason, the base DEF is not to the base ABC as the pyramid DEFH to fome folid less than ABCM; but ABC is not to DEF as ABCM is to fome folid I, greater than DEFH. For, if poffible, DEF is to ABC as I to ABCM 4; but the folid I is d inverf. greater than DEFH; then, as I is to ABCM, fo is DEFH to fome folid lefs than ABCM; which is proved abfurd; therefore ABC to DEF is not as ABCM to fome folid greater than DEFH; but it was also proved not to be to fome folid less than DEFH; therefore ABC is to DEF as ABCM is to DEFH.

For

12. 5.

Book XII For the fame reason, in the pyramids ABCDEM, FGHKLN, fig. 6. the pyramid ABCM is to the pyramid ACDM as the bafe ABC is to the base 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 def. 5.

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a 34. I.

b 5. and 6.

E

VERY prifm, having a triangular bafe, may be divided into three pyramids, equal to one another, and having trian gular bafes.

Let there be a prifm, 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 BD, the pyramid whofe bafe is the triangle ABD, and vertex the point C, is equal to the pyramid whofe bafe is the triangle EBD, and vertex the point Cb; but the pyramid whofe bafe 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 whose base is the triangle EBC, and vertex the point D.

Again, becaufe FCBE is a parallelogram whofe diameter is CE, the triangle ECF is equal to the triangle CBE ; therefore the pyramid whofe bafe is the triangle BEC, and vertex the point D, is equal to the pyramid whofe bafe is the triangle CEF, and vertex the point Db: But the pyramid whofe base is the triangle BEC, and vertex the point D, has been proved equal to the pyramid whose base is the triangle ABD, and vertex the point C; therefore, alfo the pyramid whofe bafe 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 bafes. And, because the pyramid whofe bafe is the triangle ABD, and vertex the point C, is the fame with the pyramid whofe bafe is the triangle ABC,

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