upon

the same b base LB, and between the same parallels LB, AT: and that the base SB is equal to the base CD; therefore the

Book XI.

for they are un

base AB is equal

P

to the base CD,

and the angle ALB

is equal to the an-
gle CLD: there- O
fore, by the first
case, the solid AE

F

R

lid AE is equal to

h 11. 11.

the solid SE, as was demonstrated; therefore the solid SE is equal to the solid CF.

But if the insisting 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 GQ, KT, EV, MX; NY, SZ, FI, PU, perpendicularh 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

C

R

Y

i 6. 11.

k 15. 11.

A

Ꮓ

H Q T angles to the same plane, they are paralleli to one another: and MG, EK are parallels; therefore the plane MQ, ET, of which one passes through MG, GQ, and the other through EK, KT, which are parallel to MG, GQ, and not in the same plane with them, are parallel to one another: for the same reason the planes MV, CT are parallel to one another: therefore the solid QE is a parallelopiped: in like manner, it may be proved, that the solid YF is a parallelopiped: but, from what has been demonstrated, the solid EQ is equal to the solid FY, because they are upon equal bases MK, PS, and of the same altitude, and have their insisting straight lines at right 29. or 30. angles to the bases: and the solid EQ is equal to the solid

11.

AE; and the solid FY to the solid CF; because they are upon Book XI. the same bases and of the same altitude: therefore the solid

AE is equal to the solid CF.

Wherefore solid parallelo

pipeds, &c. Q. E. D.

PROP. XXXII. THEOR.

SOLID parallelopipeds which have the same alti- See Note, tude, are to one another as their bases.

Let AB, CD be solid parallelopipeds of the same altitude: they are to one another as their bases; 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 equala a Cor. 45. to AE, so that the angle FGH be equal to the angle LCG, and complete the solid parallelopiped GK upon the base FH, one of whose insisting lines is FD, whereby the solids CD, GK must be of the same titude: therefore the solid AB

is equal to the

solid GK, be

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, 11. the base HF is equal to the base AF, 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 parallelopipeds, &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, CET and the solid parallelopipeds AB, CD, in the first of which let MO, and in the other let GQ be one of the insisting lines. And because the solid parallelopipeds AB, CD have the same altitude, they are to one another as the base AE is to the base

Book XI. CF; wherefore the prisms, which are their halves", aré to oneanother as the base AE to the base CF; that is, as the triangle AEM to the triangle CFG.

d 28. 11.

PROP. XXXIII. THEOR.

a 24.11.

b C. 11,

c1.6

SIMILAR solid parallelopipeds are one to another in the triplicate ratio of their homologous sides.

Let AB, CD be similar solid parallelopipeds, and the side AE homologous to the side CF: the solid AB has to the solid 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 solid KO: because KE, EL are equal to CF, FN, and the angle KEL equal to the angle CFN, because it is equel to the angle AEG, which is equal to CFN, by reason that the solids AB, CD are similar; therefore the parallelogram 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. Therefore three paral

:

lelograms of the
solid KO are e-
qual and similar
to three paral-
lelograms of the
solid CD and
the three oppo-
site ones in each
solid are equala
and similar to
these: therefore
the solid KO is
equal and simi-

B

X

lar to the solid CD: complete the parallelogram GK, and complete the solids EX, LP upon the bases GK, KL, so that EH be an insisting straight line in each of them, whereby they must be of the same altitude with the solid AB and because the solids AB, CD are similar, 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, so is EG to EL, and so is HE to EM: but, as AE to EK, soc is the parallelogram AG to the parallelogram GK; and as GE to EL, so ist

GK to KL, and as HE to EM, soc is PE to KM: therefore as Book XI. the parallelogram AG to the parallelogram GK, so is GK to KL, and PE to KM: but as AG to GK, sod is the solid AB c 1.6. to the solid EX; and as GK to KI., sod is the solid EX to the b 25.11. ́solid PL; and as PE to KM, sod 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 proportionals, 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 solid 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 KO the triplicate ratio of that which AE 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 solid AB has to the solid CD the triplicate ratio of that which the side AE has to the homologous side CF, &c. Q. E. D.

COR. From this it is manifest, that, if four straight lines be continual proportionals, as the first is to the fourth, so is the solid parallelopiped described from the first to the similar solid similarly described from the second; because the first straight line has to the fourth the triplicate ratie of that which it has to the second,

PROP. D. THEOR

SOLID parallelopipeds contained by parallelograms See Note equiangular to one another, each to each, 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 sides.

Let AB, CD be solid parallelopipeds, 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 solid AB has to the solid CD, is the same with that which is compounded of the ratios of the sides AM to DL, AN to DK, and AO to DH,

Book XI.

a C. 11.

b 32.11.

© 25. 11,

Produce MA, NA, OA to P, Q, R, so that AP be equal to DL, AQ to DK, and AR to DH; and complete the solid parallelopiped AX contained by the parallelograms AS, AT, AV similar and equal to CH, CK, CL, each to each. Therefore the solid AX is equal to the solid CD. Complete likewise the solid AY, the base of which is AS, and of which AO is one of its insisting 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, so c to d: then, because the parallelogram AE is equiangular to AS, AE is to AS, as the straight line a to c, as is demonstrated in the 23d prop. book' 6, and the solids AB, AY, being betwixt the parallel planes BOY, EAS, are of the same altitude. Therefore the solid AB is to the solid AY, asb the base AE to the base AS; that is, as the straight line a is to c. And the solid AY is to the solid 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 solid 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 aquali, 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 d def. A. 5. to d is said to be compounded of the ratios of a to b, b to c,

And

and c to d, which are the same with the ratios of the sides
MA to AP, NA to AQ, and OA to AR, each to each.
the sides AP, AQ, AR are equal to the sides DL, DK, DH,
each to each. Therefore 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 AQ to
DH. Q. É. D...