3.ax.

f byp. and taken together lefs than K. Therefore I ff the circle FENK) the polyg. ELFMNHO (the g30.3.& circle FEN- the fegm. EL LF, &c.) In the 1. poft. 1. circle ABTg conceive a like polygonon AKBSh I. 12. CTDV infcribed. therefore fince AKBSCTDV. ELFMGNHO ACq, EGq k the circle 19. ax. 1. ABT. I. and the polyg. AKBSCTDV the m 14. 5. circle ABT. the polyg. ELFMGNHO m fhall be I. but before, I was ELFMGNHO. which is repugnant.

k hyp.

n hyp.

Again, if it be poffible, let I be the circle EFN. Therefore because ACq.EGq n:: the circle ABT. I; and inverfely I. the circle ABT. EGq, ACq. fuppofe I. the circle ABT the circle EFN. K. otherefore the circle ABT-K. p and PII.5. EGq. ACq:: the circle EFN. K. which is fhewn to be repugnant.

0 14.5.

Therefore it must be concluded, that I is = to the circle EFN. Which was to be dem.

Coroll

Hence it follows, that as a circle is to a cirdcle, fo is a polygonon described in one to a like polygonon defcribed in the other.

E

G

PRO P. III.

Every Pyramide 4BDC having a triangular bafe,may be divided into two Pyramides AEGH, HIKC, equal, and like one to the other, having bafes triangular, and like to the whole ABDC; and into D two equal Prifms, BFGEIH, EGDHIK; which two Prifms are greater than the half of the whole Pyramide ABDC.

Divide the fides of the pyramide into two parts at the pointsE,F,G,H,I,K,and join the right lines EF,FG,GE,EI,IF,FK,KG,GH,HE.Becaufe the fides of the pyramide are proportionally

cut, a thence HI,AB; and GF, AB; and IF,DC; a 2. 6. and HG, DC,&c. are parallels, and confequently • HI FG; and GH,FI are alfo parallels, therefore

it is apparent that the triangles ABD, AEG,EBF, FDG, HIK, b are equiangular, and that the four b 29. 1. laft are e equal : in like manner the triangles c 26. I. ACB,AHE,EIB,HIC,FGK are equiangular; and the four laft are equal one to the other. Alfo the triangles BFI, FDK, IKC, EGH; and lastly, the triangles AHG, GDK, HKC, EFI are like and equal. Moreover the triangles HIK to ADB,and EGH to BDC, and EFI to ADC, and FGK to ABC, dare parallel. From whence it evidently d 15. 11. follows, firft that the pyramides AEGH, HIKC are equal, and e like to the whole ABDC, and to e 1o. def. one another. Next, that the folids BFGEIH, II. FGDIHK, are prisms, and that of equal heighth, as being placed between the parallel planes ABD, HIK, but the bale BFGE is f double of the base fz. FDG, wherefore the faid prifms are equal; g 40. II, whereof the one BFGEIH is greater than the pyramide BFEI, that is, then AEGH, the whole than its part; and confequently the two prifms are greater than the two pyramides, and fo exceed the half of the whole pyramide ABDC Which was to be dem.

PROP. IV.

ax.

Le

If there be two pyramides ABCD, EFGH, of the fame altitude, having triangular bases ABC, LFG;

and

a 15.5. b 22.6.

and either of them be divided into two pyramides (AILM, MNOD; and EPRS, STVH) equal one to the other, and like to the whole, and into two equal prifms (IBKLMN, KLCNMO; and PFQRST, QRGTSV) and if in like manner either of thofe pyramides made by the former divifion be divided, and this be done continually; then as the bafe of one pyramide is to the base of the other pyramide, fo are all the prifins which are in one pyramide, to all the prifms which are in the other pyramide, being equal in multitude.

For (applying the conftruction of the precedent Prop.) BC. KC a :: FG.QG. b therefore the c2.6.&c.triangle ABC is to the like triangle LKC as d16. 5. EFG is to c the like RQG. therefore by permuefch. 34. tation ABCEFG d: LKC, RQG e: the prifm KLCNMO. QRGTSV (for thefe are of equal altitude) f: IBKLMN. PFQRST. g wherefore the triangle ABC. EFG: the prifm KLCMNO +IBKLMN. the prifm QRGTSV +PFQRST. Which was to be dem.

II.

£ 7.5. 12.5

But if the pyramides MNOD, AILM; and EPRS, STVH, be further divided, in like manner the four new prifms made hereby fhall be to the four produced before, as the bafes MNO and AIL are to the bases STV, and EPR; that is, as LKC to RQG, or as ABC to EFG. ↳ wherefore all the prifms of the pyramide ABCD. are to all the priims of the pyramide EFGH as the base ABC is to the bafe LFG,

PROP

Pyramides ABCD, EFGH, being under the fame altitude, having triangular bafes ABC, EFG, are one to another as their bafes ABC, EFG, are.

Let the triangles ABC. EFG :: ABCD. X. I fay X is equal to the pyramide EFGH. For if it be poffible, let X be EFGH. and let the excefs be Y, divide the pyramide EFGH into prifms and pyramides, and the other pyramides in like manner, a till the pyramides left EPRS, SVTH, be less than the folid Y. Therefore fince a 1. 10. the pyramide EFGH X Y, it is manifeft that the remaining prifms PFQRST, QRGTSV are greater than the folid X. Conceive the pyramide ABCD divided after the fame manner; bb4. II. then will be the prifm IBKLMN-KLCNMO. PFORST QRGTSV :: ABC. EFG the e hyp. pyr. ABCD. X. d therefore X the prifm PFQ- d 14. 5. RST QRGTSV; which is contrary to that which was affirmed before.

Again, conceive Xthe pyr. EFGH. and

make the pyr. EFGH. Y:: X the pýr, ABCD ee hyp. and :: EFG. ABC. Becaufe EFGHf-X,g thence Y cor. 4. 5. the pyr. ABCD. which is fhewn before to be f fuppof. impoffible. Therefore I conclude, that X is equal g 14. 5. to the pyr. EFGH. Which was to be dem.

PROP.

2

PRO P. VI.

M

a 5. 12. b 18. 5.

c zz. s.

Pyramides ABCDEF, GHIKLM, confifting under the fame altitude, and having polygonous bafes ABCDE, GHIKL, are to one another as their bafes ABCDE, FGHIKL are.

Draw the right lines AC,AD,GI,GK, then is the base ABC.ACD a :: the pyr. ABCF.ACDF. b therefore by compofition ABCD. ACD: the pyr. ABCDF. ACDF. a but also ACD. ADE:: the pyr. ACDF. ADEF. c therefore of equality ABCD. ADE :: ABCDF. ADEF. and b thence by compofition ABCDE. ADE :; the pyr. ABC5. 12. DEF. ADEF. moreover ADE. GKL :: the pyr. ADEF. GKLM; and as before, and inverfely. GKL. GHIKL:: the pyr. GKLM. GHIKLM. ctherefore again of equality ABCDE.GHIKL::the pyr. ABCDEF. GHIKLM. Which was to be dem. F KA

2.5.12. B

4. 5.

If the bafes have not fides of equal multitude, the demonftra-" tion will proceed thus. The base ABC. GHI e :: the pyr.ABCF. GHIK. e and ACD. GHI DH I :: the pyr. ACDF.GHIK. ƒ therefore the base ABCD. GHI :: the pyr. ABCDF.GHIK. e Moreover the bafe ADE.GHI the pyr. ADEF. GHIK. f therefore the bafe ABCDE, GHI :: the pyr. ABCDEFGHIK.

PROP.