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a 41, 1.

CI.

d hyp.

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f 14. s.

Book XII GM, MH, HN, NE, and if tangents are drawn from the points W 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 segment of the circle it stands in ; if the remaining segments are bifected, and triangles drawn, as before ; and this be continued till the segments are less than the excess by

which the circle EFGH exceeds the figure. Sb. Let these be b Lem.

the segments 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.

Describe the polygon AXBOCPDR, in the circle ABCD, similar to the polygon EKFLGMHN '; then, as the square of BD is to the square of FH, so is the circle ABCD to the figure Sd, and, as the polygon AXBOCPDR to the polygon EKFLGMHN, fo is the circle ABCD to the figure S'; but the circle ABCD is greater than the polygon in it; therefore the figure S is greater than the polygon EKFLGMHNf; but it is less; which is absurd; therefore the square of BD to the square of FH is not as the circle ABCD to some figure less than the circle EFGH. In like manner, it is proved that the square of FH to the square of BD is not to the circle EFGH as some figure less than the circle ABCD.

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

For, if possible, let it be to the figure T, greater than the circle EFGH; then, inversely, the square of FH is to the square 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 some figure less than the circle ABCD, which is proved impossible; therefore the square 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 square of BD is to the square of FH, fo is the circle ABCD to the circle EFGH. Wherefore, &c.

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E

VERY Dyramid having a triangular base, may be divided

into two pyramids, equal and similar to one another, having triangular bases, and similar to the whole pyramid ; and into two equal prisms; which two prisms are greater than the half of the whole pyramid

C29. I,

4. I.

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 similar to one another, having triangular bases, and similar to the whole; and into two equal prisms; which two prisms are greater than the half of the whole pyramid.

For, bisect 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 D; but AE a 2. 6. is equal to KH; for each is equal to EB; therefore the two sides AE, AH, are equal to the two fides KH, HM, the angle MHK equal to HAEC, therefore the bases EH, MK, are equal, and the triangle AEH equal and fimilar to KHM. For the same reason, AHG is equal and similar 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 c 10. IL triangles KML, EHG, are equal and similar; therefore the pyramid, whose base is the triangle AEG, and vertex the point H, is equal and similar to the pyramid whose base is the triangle HKL, and vertex the point Mf; and, because HK is parallel to f def. Ie. AB, the fide of the triangle AMB, the triangles AMB, 11 MHK, are fimilar & For the same reason, the triangle MBC is 6 2. 6. similar 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 similar to HKLM; but the pyramid AEHG is proved similar to HKLM, therefore fimilar to one a-21. 6, nother", and fimil r to ABCM.

Again, because BN is equal to NC; the parallelogram BG is double the triangle GNC i ; therefore the prism contained byi 43. I. the two triangles BKN, EHG, and the three parallelograms BG,BH, KG, is equal to the prism contained by the two triangles NGC, KHL, and the three parallelograms KG, GL, NL, the one of which is constitute upon the parallelogram BG, and opposite to it the right line KH; the other upon the triangle GNC, and opposite to it the triangle KHL; and the parallelogram BG is double the triangle GNC, and have the same altitude; therefore they are equal; but either of thefe prisms is greater than the k 40. II. pyramid AEGH, or HKLM; for the prism EBNGHK is greater than the pyramid EBNK, which is equal to the pyramid AEGHf, or HKLM; wherefore the prifm EBNGHK is greater than the pyramid AEGH, or HKLM; therefore the prism GNCHKL, is likewise greater than the pyramid HKLM; but the prisms are equal; therefore, together, are greater than the two pyramids ; therefore the whole pyramid is divid into two equal pyramids fimilar to the whole, and to one another, and

into

Book XII into two equal prisms, which two prisms together are greater

than half the pyramid.

PRO P. IV. THE, O R.

IF F there are two Pyramids of the same altitude, having trian

gular bases, and each of them divided into two pyramids equal to one another, and similar to the whole, and into two equal prisms; and, if each pyramid be divided in the same manner, and this be done continually; then, as the base of the one pyramid is to the base of the other, so are all the prisms in the one pyramid to all the prisms in the other, being equal in number.

a 4, 6.

b 22. 6.

Let there be two pyramids of the same 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 similar to the whole, and into two equal prisms; and if, in like manner, each of the pyramids made by the former division be supposed divided, and this be done continually; then, as the base ABC is to the base DEF, so are all the prisms in the pyramid ABCM to all the prisms 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 similar 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 ROF; and, as BC is to NC, fo is EF to FQ; therefore ABC is to NGC as, DEF is to ROFb. And, altern. as ABC is to DEF, so is NGC to RQF; but, as NGC is to RQF, so is the prism GNCLHK to the prism RQFYST° : But the two prisms in the pyramid ABCM are equal to one another d, as also the two prisms in the pyramid DEFH; wherefore the prism, whose base is the parallelogram EGNB, and opposite baise the right line KH, is to the prism, whose base is the triangle NGC, and opposite base the triangle HKL, so is the prism whose base is the parallegram EP'RQ, and opposite bale the right line ST to the prism whose base is the triangle ROF, and opposite to it the triangle STY', compound. as the prisms EBNGKH, GNCLHK, together, are to the prism GNCLHK, so the prisms PEQRST, ROFSTY, together, are to ROESTY; altern. as the prisms EBNGKH, GNCLHK, together, are to the prisms PEORST, RQFSTY, together, so the prism GNCLHK to RQFSTY ; but the prism GNCLHK is to the prism RQFSTY

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as the base GNC to the base ROF and fo is the base ABC to Book XII the base DEF; therefore, as the base ABC is to the base DEF, 1o are the two prisms in the pyramid ABCM to the two prisms in the pyramid DEFH. For the same reason, the prisms in the pyramids HKLM, STYH, or any other pyramids made by any of the former divisions, are to each other as their bases; wherefore, all the prisms in the pyramid ABCM are to all the prisms in the pyramid DEFH as the bafe ABC to the DEF. Wherefore, &c.

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P

YR AMIDS of the same altitude, having triangular or

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First, let ABCM, DEFH, be pyramids of the same altitude, having the triangular bases ABC, DEF; then the pyramid ABCM is to the pyramid DEFH as the base ABC is to DEF; and so 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 some folid Z, less than the pyramid DEFH, which divide into two pyramids equal to each other, and into two F risms which are greater than half of the whole pyramid ; and, if the pyramids made by the former division be divided in the fame manner, till some pyramids in the pyramid DEFH is found less than the excess by which the pyramid DEFH exceeds Z. Let these pyramids be DPRS, STYH. Let the pyramid ABCM be divided into the same number of similar parts, as the pyramid DEFH; then are the prisms in the pyramid ABCM to the prisms in the pyramid DEFH", as the base ABC is to DEF; but the base ABC is to the base DEF as the pyramid ABCM to the solid Zb; therefore the pyramid ABCM is to the b hyp. solid Z as the prisms in ABCM to the prisms in DEFH; but ABCM is greater than the prisms in it; therefore the solid Z is greater than the prisms in DEFH°; and likewise less b; which c 14. s. is absurd ; therefore the base ABC is not to the base DEF as the pyramid ABCM to some folid less than DEFH. For the same reason, the base DEF is not to the base ABC as the pyramid DEFH to some solid less than ABCM ; but ABC is not to DEF as ABCM is to some solid I, greater than DEFH. For, if possible, DEF is to ABC as I to ABCM 4; but the solid Iis d inverí. greater than DEFH; then, as I is to ABCM, fo is DEFH to some folid less than ABCM ; which is proved abfurd; therefore ABC to DEF is not as ABCM to some solid greater than DEFH; but it was also proved not to be to some folid less than DEFH; therefore ABC is to DEF as ABCM is to DEFH.

For

Book XII

For the same 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 same 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 less, lefs ; ' therefore, as the base ABCDE is to the base FGHKL, so is the pyramid ABCDEM to the pyramid FGHKLN"; Wherefore, &c.

c def. s.

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VERY prism, having a triangular base, may be divided

into three pyramids, equal to one another, and having triangular bases.

E

a 34. I.

Let there be a prism, whose base is the triangle ABC, and the oppofite base to that the triangle DEF; then the prism ABCDEF

may

be divided into three equal pyramids, having triangular bases.

Por, join BD, EC, CD; then, because ABCD is a paralle.

logram, whose diameter is BD, the pyramid whose base is the bs. and 6. triangle ABD, and vertex the point C, is equal to the pyramid

whose base is the triangle EBD, and vertex the point C b; but
the pyramid whose base is the triangle EBD, and vertex the
point C, is equal to the pyramid whose base is the triangle
EBC, and vertex the point D ; for they are contained by the
fame plains ; therefore the pyramid whose base 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, because FCBE is a parallelogram whose diameter is
CE, the triangle ECF is equal to the triangle CBE ; therefore
the pyramid whose base is the triangle BEC, and vertex the
point D, is equal to the pyramid whole base is the triangle CEF,
and vertex the point D 6But the pyramid whose base is the tri-
angle 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, also the pyramid whose base is the triangle
CEF, and vertex the point D, is equal to the pyramid whose
base 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 pyra-
mid whose base is the triangle ABD, and vertex the point C,
is the fame with the pyramid whose base is the triangle ABC,

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