Therefore we muft conclude that I Circ. EFN. QE. D.

COROL.

Hence as one Circle is to another, fo is a Polygon defcrib'd in the one, to a fimilar Polygon defcrib'd in the other.

one ABDC; and

into two equal Prifms BFGEHI, FGDIHK ; which two Prifms together are greater than the half of the whole Pyramid ABDC.

Bifect the Sides of the Pyramid in the Points E, F, G, H, I, K, and join the Points by the right Lines, EF, FG, GE, EI, IF, FK, KG, GH, HE.

a

Now because the Sides of the Pyramid are cut proportionally, therefore HI, AB; and GF, AB, and IF, DC; and HG, DC, &c. are parallel; confequently alfo HI, FG, and GH, FI, are parallel. Whence it is manifeft, that the Triangles ABD, AEG, EBF, FDG, HIK are equiangular; and the four laft are equal. In like manner the Triangles ACB, AHE, ÉIB, HIC, FGK, are equiangular, and the four last are equal; likewife the Triangles BFI, FDK, IKC, EGH; and at length the Triangles

.

C

AHG,

b

AHG, GDK, HKC, EFI, are equal and fimilar. But the Triang. HIK, is a parallel to a 15. 11. ADB, and EGH to BDC, and EFI to ADC, and FGK to ABC. From whence it plainly follows, (1.) that the Pyramids AEGH, HIKC, are equal; and fimilar to one another, and to 10 def. the whole one ABDC. Again, the Solids BFG II. EIH, FGDIHK, are Prifms of equal Altitudes, viz. fituate between parallel Planes ABD, HIK. but the Bafe BFGE, is the 2 ax. 1. Double of the Bafe FDG. And fo the faid Prifms are equal. One whereof BFGEIH is a greater than the Pyramid BEFI, that is AEGH, the Whole than the Part; and confequently the two Prifms are greater than the two Pyramids, and fo greater than the half of the whole Pyramid ABDC. Q. E. D.

23

d

PROP. IV.

C

If there be two Pyramids ABCD, EFGH, of the fame Altitude, having Triangular Bafes ABC,

E

40. II.

R

H

B

K

EFG, and each of them be divided into two Pyra mids (AILM, MNOD; and EPRS, STVH) equal to one another, and fimilar to the whole, and into two equal Prisms, IBKLM, KLCNMO; and PFQRST, QRGTSV. And if in like manner, each of the two Pyramids made by the former Divifion, be divided, and this be done continually; then as the Bafe of one Pyramid is to the Bafe

O 4

of

b22.6.

of the other Pyramid, fo are all the Prisms that are in one Pyramid, to all the Prifms that are in the other Pyramid being equal in Number.

C 26.. fimilar c C

:

For (ufing here the Conftruction of the last 15.1. Problem) BC: KC:: FG QG. Therefore b the Triang. ABC is to the Triang. LKC fimilar to it, as EFG is to the Triang. RQG to it. Whence, by Permutation fchol. 34 ABC EFG LKC RQG: Prism : a :: KLCNMO QRGTSV (for these have the fame Altitude) :: IBKLMN: PFQRST. Whence the Triang. ABC; EFG :: Prim KLCNMO+IBKLMN: Prifm QRGTSV +PFQRST. Q. E.D.

11.

e

7.5.

12. 5.

h.12.5.

f

And if the Pyramids MNOD, AILM; and EPRS, STVH be in like manner further divided; the four new Prifms here made to the four there made, will be as the Bafes MNO and AIL to the Bases STV, and EPR, that is as LKC to RQG; or as " ABC to EFG. Wherefore all the Prifms of the Pyramid ABCD to all the Prisms of EFGH, fhall be as the Base ABC to the Base EFG. 2. E. D.

PROP. V.

Pyramids ABCD, EFGH of the fame Altitude having Triangular Bafes ABC, EFG, are to one another as thofe Bafes ABC, EFG.

=

6

Let the Triang. ABC: EFG:: ABCD : X. I fay X Pyr. EFGH; for, if poffible, let X Pyr. EFGH; and let Y be the Excels. Divide the Pyramid EFGH into Prifms and Pyramids, and the remaining Pyramids in like manner until the remaining Pyramids EPRS, 1. 10. STVH, become less than the Solid Y. Therefore fince the Pyr. EFGH = X + Y; it is manifeft that the remaining Prifms PFQRST, QRGTSV, are greater than the Solid X. Now conceive the Pyramid ABCD to be divided in like manner; then will the Prifm IBKLMN 4. 12. b + KLCNMO: PFQRST + QRGTSV :: ABC EFG :: Pyr. ABCD X. Therefore X- Prism PFQRST + QRGTSV

is contrary to what was firft affirmed. Again, let X

:

d

which

byp. 14. 5.

e

hyp. &•

Suppof.

Pyr. EFGH, and fuppofe the Pyr. EFGH Y X Pyr. ABCD :: EFG ABC. Becaufe EFGHX, there- cor. 4. 5. fore fhall Y Pyr. ABCD. which is impof- 8 14. 5. fible from what has been faid. Therefore we are to conclude that X= Pyr. EFGH. Q. E.D.

PROP VI

Pyramids ABCDEF, GHIKLM, having polygoneous Bafes ABCDE, GHIKL, are to one another as the Bafes ABCDE, GHIKL.

5. 12.

b18.5.

€ 22.5.

d 5. 12.

5. 12.

24. 5.

с

Draw the right Lines AC, AD, GI, GK. then the Bafe ABC ACD:: Pyr. ABCF: ACDF. And fo by Compofition, ABCD: ACD:: Pyr. ABCDF: ACDF. But likewife ACD: ADE:: Pyr. ACDF: ADEF. Therefore by Equality ABCD: ADE :: ABCDF: ADEF. Therefore by Compofition, ABCDE; ADE Pyr. ABCDEF: ADEF. Again ADE: GKL :: Pyr. ADEF: GKLM; and, as at firft, and inverfely, GKL: GHIKL :: Pyr. GKLM: GHIKLM. Therefore again by Equality, ABCDE: GHIKL :: Pyr. AB CDEFGHIKLM. QE. D.

f

e

d

C

If the Bafes have unequal Numbers of Sides, the Demonftration is thus. The Bafe ABC: GHI :: Pyr. ABCF : GHIK. But ACD: GHI :: Pyr. ACDF : GHIK. Therefore the Bafe ABCD GHI Pyr. ABCDF: GHIK; but likewife the Bafe ADE: GHI:: Pyr. ADEF: GHIK. Therefore the Bafe ABCDE: GHI :: Pyr. ABCDEFGHIK. Q. E. D.

PROP. VII.

Every Prifm ABCDEF having a Triangular Bafe, may be divided into three Pyramids, ACBF, ACDF, CDFE, equal to one another, and having Triangular Bafes.

Draw