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Book XI. cause the parallelogram AB is equal 8 to SB, for they are upon

the same base LB, and between the same parallels LB, AT ; & 35. I.

and that the base
SB is equal to the

R
base CD; therefore N

M

E
the base AB is equal
to the base CD, and

G

X

K the angle ALB is

D

Q equal to the angle

0
CLD: Therefore,

B
by the first case, C L
the solid AE is e.

AS HT
qual to the solid
CF; but the solid AE is equal to the solid SE, as was demon-
strated ; therefore the solid SE is equal to the solid CF.

But, if the inGfting straight lines AG, HK, BE, LM; CN, RS, DF, OP, be not at right angles to the bases AB, CD; in this case likewise the solid AE is equal to the solid CF: From

the points G, K, E, M, N, S, F, P, draw the straight lines b 11. 11. GO, KT, EV, MX; NY, SZ, FI, PU, perpendicular to the

plane in which are the bases AB, CD; and let them meet it in
the points Q, T, V, X; Y, Z, I, U, and join QT, TV, VX,
XQ; YZ, ZI, IU, UY: Then, because GQ, KT, are at right
M E

P
G

s K

N
B
I X

D

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R А н от i 6. II. angles to the same plane, they are parallel i to one another :

And MG, EK are parallels; therefore the planes MQ, ET, of which one paffes through MG, GQ, and the other through

EK, KT which are parallel to MG, GQ, and not in the same k 15. II. plane with them, are parallel to one another: For the same

reason, the planes MV, GT are parallel to one another : Therefore the solid QE is a parallelepiped : In like manner, it may be proved, that the folid YF is a parallelepiped : But, from what has been demonstrated, the folid EQ is equal to the solid FY, because they are upon equal bases MK, PS, and of the same altitude, and have their inlifting straight lines at right angles

to

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II.

to the bases: And the folid EQ is equal to the folid AE; and Book XI.
the solid FY to the solid CF, because they are upon the same,
bases and of the same altitude : Therefore the folid AE is equal : 29. or 36.
to the folid CF. Wherefore folid parallelepipeds, &c. Q. E. D.

THEOR.

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PRO P. XXXII.

SOLID parallelepipeds which have the same altitude, are see N.

to one another as their bases.

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F

E

Let AB, CD be solid parallelepipeds of the same altitude : They are to one another as their bafes ; that is, as the base AE to the base CF, so is the solid AB to the solid CD.

To the straight line FG apply the parallelogram FH equal a Cor.45.I.
to AE, so that the angle FGH be equal to the angle LCG;
and complete the folid parallelepiped GK upon the base FH,
one of whose inlifting lines is FD, whereby the folids CD, GK
muft be of the fame altitude : Therefore the folid AB is equal ob 31Ir.
the solid

D K
B
GK,

because
they are upon

Q
equal

bases
N

OP
AE, FH, and

L
are of the same
altitude : And
because the fo.
lid parallelepi-
A M M

G H
ped CK is cut
by :he plane DG which is parallel to its opposite planes, the base
HF is to the base FC, as the solid HD to the solid DC: But c 25. II.
the base HF is equal to the base AE, and the solid GK to the
solid AB : Therefore, as the base AE to the base CF, so is the
solid AB to the solid CD. Wherefore solid parallelepipeds, &c.'
Q. E. D.

Cor. From this it is manifest that prisms upon triangular bases, of the same altitude, are to one another as their bases.

Let the prisms, the bases of which are the triangles AEM, CFG, and NBO, PDQ the triangles opposite to them, have the same altitude; and complete the parallelograms AE, CF, and the folid parallelepipeds AB, CD, in the first of which let MO, and in the other let GQ be one of the insisting lines. And because the folid parallelepipeds AB, CD have the same altitude, they are to one another as the base AE is to the base

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CF ;

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Book Xi CF; wherefore the prisms, which are their halves d, are to one Manother, as the base A E to the base CF ; that is, as the triangle à 28. 11. AEM to the triangle CFG.

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Similar solid parallelepipeds are one to another in the

triplicate ratio of their homologous fides.

G

Let AB, CD be similar solid parallelepipeds, and the side AE homologous to the fide CF: The solid AB has to the folid CD, the triplicate ratio of that which AE has to CF.

Produce AE, GE, HE, and in these produced take EK equal to CF, EL equal to FN, and EM equal to FR, and complete the parallelogram KL, and the folid KO : Because KE, EL are equal to CF, FŇ, and the angle KEL equal to the angle CFN, because it is equal to the angle AEG which is equal to CFN, by reason that the folids AB, CD are similar; therefore the paral. lelogram KL is similar and equal to the parallelogram CN: For the same reason, the parallelogram MK is similar and equal to CR, and also OE to FD : There

B X fore three paralle

D

H Н lograms of the fo

P lid KO are equal

R
and similar to three N
parallelograms of
the solid CD: And

KI
C
the three opposite

F

A ones in each folid

L a 24. II. are equal a and

M
similar to these :
Therefore the fo.

0 b C. II. lid KO is equal

and fimilar to the solid CD: Complete the parallelogram GK, and complete the folids EX, LP upon the bases GK, KL, so that EH be an infitting Itraight ine in each of them, whereby they must be of the fame altitude with the folid AB: And because the folids AB, CD are fimilar, and, by permutation, as AE is to CF, so is EG to FN, and so is EH to FR ; and FC is equal to EK, and FN to EL, and FR to EM ; therefore as AE to EK, fo is EG to EL, and so is HE to EM: But, as AE to EK, fo is the parallelogram AG to the parallelogram GK; and as GE to EL, so is c GK to KL;

and

1.6.

and ás HE to EM, so is PE to KM: Therefore as the parallelo- Book XI. gram AG to the parallelogram GK, fo is GK to KL, and PE to KM: But as AG to GK, so d is the solid AB to the folid

CI. 6.

d 25. II. EX; and as GK to KL, so is the folid EX to the folid PL ; and as PE to KM, so d is the solid PL to the solid KO: And therefore as the solid AB to the solid EX, so is EX to Pl., and PL to KO: But if four magnitudes be continual propor. tionals, the first is said to have to the fourth the triplicate ratio of that which it has to the second : Therefore the solid AB has to the folid KO, the triplicate ratio of that which AB has to EX : But as AB is to EX, so is the parallelogram AG to the parallelogram GK, and the straight line AE to the straight line EK. Wherefore the solid AB has to the solid KU, the tri. plicate ratio of that which A E has to EK. And the solid KO is equal to the solid CD, and the straight line EK is equal to the straight line CF. Therefore the folid AB has to the folid CD, the triplicate ratio of that which the Gde A E has to the homologous Gde CF, &c. Q. E, D.

Cor. From this it is manifeft, that, if four straight lines be continual proportionals, as the first is to the fourth, so is the solid parallelepiped described from the first to the fimilar solid fimilarly described from the second; because the first ftraight line has to the fourth the triplicate ratio of that which it has to the second,

1

PRO P. D. THE O R.

OLID
SOLID parallelepipeds contained by parallelograms Sce N,

equiangular to one another, each to cach, that is, of which the solid angles are equal, each to each, have to one another the ratio which is the same with the ratio compounded of the ratios of their fides.

Let AB, CD be solid parallelepipeds, of which AB is contained by the parallelograms AE, AF, AG equiangular, each to each, to the parallelograms CH, CK, CL which contain the solid CD. The ratio which the folid AB has to the folid CD is the same with that which is compounded of the ratios of the des AM to DL, AN to DK, and AO to DH.

Produce

Book XI. Produce MA, NA, OA to P, Q, R, so that AP be equal W to DL, AQ to DK, and AR to DĦ; and complete the solid

parallelepiped AX contained by the parallelograms AS, AT,

AV fimilar and equal to CH, CK, CL, each to each. Therea C. II. fore the solid AX is equal to the solid CD. Complete likewise

the folid AY, the base of which is AS, and of which AO is one of its ingifting straight lines. Take any straight line a, and as MA to AP, so make a to b; and as NA to AQ, so make b to c; and as AO to AR, fo c to d: Then, because the parallelogram AE is equiangular to AS, AE is to AS, as the Itraight line a to c, as is demonstrated in the 23. Prop. Book 6. and the folids AB, AY, being betwixt the parallel planes

B., EAS, are of the same altitude. Therefore the solid AB b 32. II.

ist : solid AY, as the base AE to the base AS ; that is, as the itraight line a is to c. And the solid AY is to the solid

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< 25.11. AX, as the base OQ is to the base QR ; that is, as the straight

line OA to AR; that is, as the straight line c to the straight line d. And because the folid AB is to the solid AY, as a is to c, and the solid AY to the solid AX, as c is to d; ex aequali, the solid AB is to the solid AX, or CD which is equal to it,

as the straight line a is to d. But the ratio of a to d'is said to d def. A. 5. be compounded d of the ratios of a to b, b to c, and c to d,

which are the fame with the ratios of the sides MA to AP, NA to AQ, and OA to AR, each to each. And the Edes AP, AQ, AR are equal to the sides DL, DK, DH, each to each. Therea fore the solid AB has to the solid CD the ratio which is the same with that which is compounded of the ratios of the sides AM to DL, AN to DK, and ÁO to DH. Q. E. D.

PROP

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