P

Book XI. lines AT, GX. and because the parallelogram AB is equal to SB, for they are upon the fame bafe LB, and between the fame parallels LB, AT; and

8. 35. 1.

that the bafe SB is
equal to the bafe CD;
therefore the bafe AB
is equal to the bafe
CD. and the angle

F

R

to the folid CF; but the folid AE is equal to the folid SE, as was demonftrated; therefore the folid SE is equal to the folid CF.

But if the infifting ftraight lines AG, HK, BE, LM; CN, RS, DF, OP, be not at right angles to the bafes AB, CD; in this cafe likewise the folid AE is equal to the folid CF. from the points G, K, E, M, N, S, F, P, draw the ftraight lines GQ, KT, EV, MX; h. 11. 11. NY, SZ, FI, PU, perpendicular to the plane in which are the bafes AB, CD; and let them meet it in the points Q, T, V, X;

i. 6. 11.

A HQ T
AH

Y, Z, I, U, and join QT, TV, VX, XQ; YZ, ZI, IU, UY. then because GQ, KT, are at right angles to the fame plane, they are parallel 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, k. 15. 11. and not in the fame plane with them, are parallel k to one another. for the fame reafon, the planes MV, GT are parallel to one another. therefore the folid QE is a parallelepiped. in like manner, it may be proved, that the folid YF is a parallelepiped. but, from what has been demonftrated, the folid EQ is equal to the folid FY, because they are upon equal bafes MK, PS, and of the fame altitude, and

have their infifting straight lines at right angles to the bafes. and Book XI.. the folid EQ is equal to the folid AE; and the folid FY ton the folid CF; because they are upon the fame bafes and of the 1.29. or 30. fame altitude. therefore the folid AE is equal to the folid CF. Wherefore folid parallelepipeds, &c. Q. E. D.

SOLID

PROP. XXXII. THEOR.

II.

OLID parallelepipeds which have the fame altitude, Sce N. are to one another as their bafes.

Let AB, CD be folid parallelepipeds of the fame altitude. they are to one another as their bafes; that is, as the bafe AE to the bafe CF, fo the folid AB to the folid CD.

To the ftraight line FG apply the parallelogram FH equal to a.Cor.45.1. AE, fo that the angle FGH be equal to the angle LCG; and complete the folid parallelepiped GK upon the bafe FII, one of whofe infifting lines is FD, whereby the folids CD, GK muft be of the

fame altitude. therefore the folid AB is equal to the folid GK, b. 31. 17. because they are upon equal bafes AE, FH, and are

D

K

B

c

to its oppofite planes, the base HF is to the bafe FC, as the folid c. 25. 14. HD to the folid DC. but the bafe HF is equal to the bafe AE, and the folid GK to the folid AB. therefore as the base AE to the bafe CF, fo is the folid AB to the folid CD. Wherefore folid parallelepipeds, &c. Q. E. D.

Cor. From this it is manifeft that prifins upon triangular bases, of the fame altitude, are to one another as their bafes.

Let the prifms the bafes of which are the triangles AEM, CFG, and NBO, PDQ the triangles oppofite to them, have the fame altitude; and complete the parallelograms AE, CF, and the folid parallelepipeds AB, CD, in the firft of which let MO, and in the other let GQ be one of the infifting lines, and because the folid parallelepipeds AB, CD have the fame altitude, they are to one

Book XI. another as the bafe AE is to the bafe CF; wherefore the prifms, which are their halves 4, are to one another as the bafe AE to the d. 28. 11. base CF; that is, as the triangle AEM to the triangle CFG.

SIMI

IMILAR folid parallelepipeds are one to another in the triplicate ratio of their homologous fides.

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

Produce AE, GE, HE, and in thefe produced take EK equal to CF, EL equal to FN, and LM equal to FR; and complete the parallelogram KL, and the folid KO. because KE, EL are equal to CF, FN, and the angle KEL equal to the angle CFN, because the angle AEG is equal to CFN, by reafon that the folids AB, CD are fimilar; therefore the parallelogram KL is fimilar and equal to the parallelogram CN. for the fame reafon, the parallelogram MK is fimilar and equal to CR, and alfo OE to FD. therefore three parallelograms

of the folid KO are
equal and fimilar to
three parallelograms

в х

G

ΑΣ

L

M

C. 3. 6.

folid CD. complete

the parallelogram GK, and complete the folids EX, LP upon the bafes GK, KL, fo that EH be an infifting ftraight line 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, fo is EG to FN, and fo 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 fo is HE to EM. but as AE to EK, fo is the parallelogram AG to the parallelogram GK; and

C

3

c

as GE to EL, fo is GK to KL; and as HE to EM, fo is PE Book XI. to KM. therefore as the parallelogram AG to the parallelogram GK, fo is GK to KL, and PE to KM. but as AG to GK, fod is c. 1. 6. the folid AB to the folid EX; and as GK to KL, fod is the folid d. 25. 15. EX to the folid PL; and as PE to KM, fod is the folid PL to the folid KO. and therefore as the folid AB to the folid EX, fo is EX to PL, and PL to KO. but if four magnitudes be continual proportionals, the first is faid to have to the fourth the triplicate ratio of that which it has to the fecond. therefore the folid AB has to the folid KO, the triplicate ratio of that which AB has to EX. but as AB is to EX, fo is the parallelogram AG to the parallelogram GK, and the ftraight line AE to the ftraight line EK. wherefore the folid AB has to the folid KO, the triplicate ratio of that which AE has to EK. and the folid KO is equal to the folid CD, and the straight line EK is equal to the ftraight line CF. Therefore the folid AB has to the folid CD, the triplicate ratio of that which the fide AE has to the homologous fide CF. Q. E. D.

COR. From this it is manifeft, that if four ftraight lines be continual proportionals, as the first is to the fourth, fo is the folid parallelepiped defcribed from the firft to the fimilar folid fimilarly defcribed from the fecond; because the firft ftraight line has to the fourth, the triplicate ratio of that which it has to the fecond.

PROP. D. THEOR.

SOLID parallelepipeds contained by parallelograms See 7,

equiangular to one another, each to each, that is, of which the folid angles are equal, each to each; have to one another the ratio which is the fame with the ratio compounded of the ratios of their fides.

Let AB, CD be folid 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 folid CD. the ratio which the folid AB has to the folid CD is the fame with that which is compounded of the ratios of the fides AM to DL, AN to DK, and AO to DH.

230

Book XI.

a. C. 11.

b. 32. II

Produce MA, NA, OA to P, Q, R, fo that AP be equal to DL, AQ___ to DK, and AR to DH; and complete the folid parallelepiped AX contained by the parallelograms AS, AT, AV fimilar and equal to CH, CK, CL, each to each. therefore the folid AX is equal to the folid CD. complete likewife the folid AY the bafe of which is AS, and of which AO is one of its infifting ftraight lines. Take any ftraight line a, and as MA to AP, fo make a to b; and as NA to AQ, fo make b to c; and as OA to AR, fo c to d. then because the parallelogram AE is equiangular to AS, AE is to AS, as the ftraight line a to c, as is demonftrated in the 23. Prop. Book 6. and the folids AB, AY, being betwixt the parallel planes BOY, EAS, are of the fame altitude. therefore the folid AB is to the folid AY, as the bafe AE to the bafe AS; that is, as the ftraight line a is to c. and the folid AY

C. 25. II.

d. Def. A.5.

is to the folid AX, as the bafe OQ is to the bafe QR; that is,

as the straight line OA to AR; that is, as the straight line c to the ftraight line d. and becaufe the folid AB is to the folid AY, as a is to c, and the folid AY to the folid AX, as c is to d; ex aequali, the folid AB is to the folid AX, or CD which is equal to it, as the ftraight line a is to d. but the ratio of a to d is faid to be compounded of the ratios of a to b, b to c, and c to d, which are the fame with the ratios of the fides MA to AP, NA to AQ, and OA to AR, each to each. and the fides AP, AQ, AR are equal to the fides DL, DK, DH, each to each. Therefore the folid AB has to the folid CD the ratio which is the fame with that which is compounded of the ratios of the fides AM to DL, AN to DK, and AO to DH. Q. E. D.