2.6.

+10. 11.

angle A EG is alfo equal and fimilar to the Triangle HKL; wherefore the Pyramid whofe Bafe is the TriDef.10.11. angle A EG, and Vertex the Point H, ‡ is equal and fimilar to the Pyramid whofe Bafe is the Triangle HKL, and Vertex the Point D. And because H K is drawn parallel to the Side A B of the Triangle AD B, the Triangle A D B** shall be equiangular to the Triangle D KH, and they have their Sides proportional; therefore the Triangle ADB is fimilar to the Triangle DHK. And, for the fame Reason, the Triangle DBC is fimilar to the Triangle D KL; and the Triangle ADC to the Triangle DHL. And fince the two Right Lines BA, AC, touching each other, are parallel to the two Lines KH, HL, touching each other, not being in the fame Plane with them, these fhall contain equal Angles +: therefore the Angle BAC is equal to the Angle K HL: And BA is to A C, as KH is to HL; wherefore the Triangle ABC is fimilar to the Triangle HKL; and fo the Pyramid, whofe Bafe is the Triangle A B C, and Vertex the Point D, is fimilar to the Pyramid, whose Base is the Triangle HKL, and Vertex the Point D. But the Pyramid, whose Base is the Triangle H K L, and Vertex the Point D, has been proved fimilar to the Pyramid whose Base is the Triangle A EG, and Vertex the Point H; therefore the Pyramid whole Bafe is the Triangle A B C, and Vertext the Point D is fimilar to the Pyramid whose Base is the Triangle AEG, and Vertex the Point H: Wherefore both the Pyramids AEGH, HKL D, are fimilar to the whole Pyramid ABCD. And becaufe B F is equal to F C, the Parallelogram E B F G will be double to the Triangle GFC; and fince there are two Prifms of equal Altitude, one of which has that Parallelogram for a Base, and the other the Triangle, and the Parallelogram is double to the Triangle; whofe Prisms will be equal to one another: Therefore the Prism contained under the two Triangles BKF, EHG, and the three Parallelograms E B F G, E B KH, KHG F, is equal to the Pritms contained under the two Triangles GFC, HKL, and the three Parallelograms K FCL, LCGH, HKFG. And it is manifeft, that each of thofe Prifms, the Base of one of which is the Parallellogram EBGF; and the oppofite Bafe to that the

40. II.

Right Line K H, and the Bafe of the other, the Triangle GFC, and the oppofite Bafe to this the Triangle KLH, are greater than either of the Pyramids, whofe Bafes are the Triangles A EG, HKL, and Vertices the Points H and D. For fince, if the Right Lines EF, EH, be joined, the Prifm, whofe Bafe is the Parallelogram EBF G, and the oppofite Bafe to that the Right Line K H, is greater than the Pyramid, whofe Bafe is the Triangle E B F, and Vertex the Point K. But the Pyramid whofe Bafe is the Triangle EBF, and Vertex the Point K, is equal to the Pyramid whofe Bafe is the Triangle A E G, and Vertex the Point H; for they are contained under equal and fimilar Planes. Wherefore the Prifm, whofe Bafe is the Parallelogram E BF G, and the oppofite Bafe to it the Right Line HK, is greater than the Pyramid whofe Bafe is the Triangle A EG, and Vertex the Point H. But the Prifm whofe Bafe is the Parallelogram EBFG, and the oppofite Bafe to it the Right Line HK, is equal to the Prifm whofe Bafe is the Triangle GFC, and the oppofite Bafe to this the Triangle HKL; and the Pyramid whofe Ba'e is the Triangle A EG, and Vertex the Point H, is equal to the Pyramid whofe Bafe is the Triangle H K L, and Vertex the Point D: Therefore the two Prisms aforefaid are greater than the faid two Pyramids, whose Bafes are the Triangles A E G, HK L, and Vertices the Points H, D: And fo the whole Pyramid, whose Base is the Triangle A B C, and Vertex the Point D, is divided into two equal Pyramids, fimilar to each other, and to the Whole, and into two equal Prifms, which two Prifms together, are greater than half of the whole Pyramid. Therefore, every 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; which was to be demonftrated.

23.6.

PROPOSITION IV.
THEOREM.

If there are two Pyramids of the fame Altitude,
baving triangular Bafes, and each of them be di-
vided into two Pyramids, equal to one another,
and fimilar to the Whole, as also into two equal
Prifms; and if, in like manner, each of the two
Pyramids, made by the former Divifion, be di-
vided, and this be done continually; then, as the
Bafe of one Pyramid is to the Bafe 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 Multitude.

L

ET there be two Pyramids of the fame Altitude, having the triangular Bafes A B C, DEF, whofe Vertices are the Points G, H; and let each of them be 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 conceived to be divided, and this be done continually; I fay, as the Bafe A B C is to the Bafe D E F, fo are all the Prisms that are in the Pyramid ABC G, to all the Prifms that are in the Pyramid D E F H, being equal in Multitude.

For, fince BX is equal to XC, and AL to LC, XL fhall be parallel to AB, and the Triangle ABC fimilar to the Triangle LXC. For the fame Reafon, the Triangle DEF fhall be alfo fimilar to the Triangle RQF: And becaufe BC is double to C X, and EF to FQ; it fhall be, as BC is to C X, fo is EF to FQ: And fince there are defcribed upon BC, CX, Rightlined Figures A B C, L XC, fimilar and alike fituate and upon EF, FQ, Right-lined Figures DEF, RQF, fimilar and alike fituate; therefore, as the Triangle BAC is to the Triangle LX C, fo is + the Triangle DEF to the Triangle RQF; and (by Alternation) as the Triangle A B C is to the Triangle D E F, fo is the Triangle LXC to the Triangle ROF. But as the Tri128. and angle LXC is to the Triangle RQF, fo is the Prifm, whofe Bafe is the Triangle LXC, and the oppofite Bafe

† 22.6.

32. 11.

*

to that the Triangle OM N, to the Prifm, whofe Bafe
is the Triangle RQF, and the oppofite Bafe to that the
Triangle STY; therefore, as the Triangle ABC is to
the Triangle D E F, fo is the Prifm whofe Bafe is the
Triangle LXC, and the oppofite Bafe to that the Tri-
angle OMN, to the Prifm whofe Base is the Triangle
RQF, and the oppofite Bafe to that the Triangle
STY: And because the two Prifms that are in the Py-
ramid A BCG are equal to one another, as also those
two that are in the Pyramid DEFH; it fhall be, as
the Prifm whofe Bafe is the Parallelogram K L X B,
and the oppofite Bafe to that the Right Line M O, is
to the Prism whofe Base is the Triangle LX C, and
the oppofite Base to that the Triangle O M N, fo is
the Priim whofe Bafe is the Parallelogram E PR Q,
and the oppofite Bafe to that the Right Line S T, to
the Prism whofe Bafe is the Triangle RQF, and the
opposite Base to that the Triangle STY: Therefore,
(by compounding), as the Prifms KB XLMO,
LXCMNO, together, are to the Prifin LXCMNO,
fo the Prifms PEQRST, RQFSTY, together,
are to the Prifm RQFSTY: And (by Alternation),
as the Priims KBXLMO, LXCM NO, together,
are to the Prilms PEQRST, ROFSTY, toge-
ther, fo is the Prifm LX CMNO to the Prum
RQFSTY: But as the Prifm LXCM NO is to
the P ifm RQFSTY, fo has the Bafe LXC heen
proved to be to the Bafe RFQ; and fo the Bafe
ABC to the Bafe DEF: Therefore, alto, as the Tri-
angle A B C is to the Triangle DEF, fo are the two
Prisms that are in the Pyramid ABCG, to the two
Pritms that are in the Pyramid DEFH. If, in the fame
manner each of the Pyramids OMNG, ST YH,
made by the former Divifion, be divided, it fhall be, as
the Bale OMN is to the Bafe S T Y, fo the two
Pifms that are in the Pyramid OMNG, to the two
Priims that are in the Pyramid S TY H. But as the
Bafe OMN is to the Bafe STY, fo is the Base ABC
to the Base DEF: Therefore, as the Bale ABC is to
the Bafe DE F, fo are the two Prifins that are in the
Pyramid A B C G, to the two Prifms that are in the
Pyramid DEFH; and fo the two Prifms that are in
the Pyramid OMNG, to the two Prisms that are in
the Pyramid STYH; and fo the four to the four. We

R 4

demon

4 of this.

By Hyp.

demonftrate the fame of Prisms made by the Divifion of the Pyramids AKLO, DPRS; and, of all other Prifms, being equal in Multitude; which was to be demonftrated.

PROPOSITION V.

THEOREM.

Pyramids of the fame Altitude, and having triangular Bafes, are to one another as their Bafes.

LET there be two Pyramids of the fame Altitude, baving the triangular Bases A B C, DEF, whose Vertices are the Points G, H. I fay, as the Base ABC is to the Bafe DE F, fo is the Pyramid A B C G to the Pyramid D E F H.

For, if it be not fo, then it fhall be as the Bafe ABC is to the Bafe DEF, fo is the Pyramid ABCG 1o fome Solid, greater or lefs than the Pyramid DEFH. Firft, let it be to a Solid lefs, which let be Z; and divide the Pyramid DEFH into two Pyramids equal to each other, and fimilar to the Whole, and into two equal Prifms; then these two Prifms are greater than the Half of the whole Pyramid: And, again, let the Pyramid, made by the former Divifion, be divided after the fame manner; and let this be done continually, until the Pyramids in the Pyramid DEFH are lefs than the Excefs by which the Pyramid D E F H, exceeds the Solid Z. Let thefe, for Example, be the Pyramids D'PRS, STYH; then the Prifms remaining in the Pyramids D E F H, are greater than the Solid Z: Alfo, let the Pyramid ABCG be divided into the fame Number of fimilar Parts as the Pyramid DEFHis; and then, as the Bafe A B C is to the Bafe DEF, fo the Prisms that are in the Pyramid ABCG, to the Prisms that are in the Pyramid DEFH. But as the Bafe A B C is to the Bafe DEF, fo is the Pyramid ABCG to the Solid Z; and therefore, as the Pyramid A B CG is to the Solid Z, fo are the Prisms that are in the Pyramid A BCG, to the Prisms that are in the Pyramid DEFH. But the Pyramid ABCG is greater than the Prifms that are in it; wherefore, alfo, the folid Z is greater than

the