8

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

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

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

becaufe GQ, KT are at right angles to the fame plane, they are i. 6. 11. 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, 15. 11. in the fame plane with them, are parallel k to one another.

and not 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 the fo- Book XI. lid EQ is equal to the folid AE; and the folid FY to the folid CF; because they are upon the fame bafes and of the fame altitude. 1. 29. or therefore the folid AE is equal to the folid CF. Wherefore folid parallelepipeds, &c. Q. E. D.

PROP. XXXII. THEOR.

30. 11.

SOLID parallelepipeds which have the fame altitude, see X.

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.

2

To the straight line FG apply the parallelogram FH equal to a. Cor.4.8. AE, fo that the angle FGH be equal to the angle LCG; and complete the folid parallelepiped GK upon the bafe FH, one of whofe infifting lines is FD, whereby the folids CD, GK must be of the fame altitude. therefore the folid AB is equal to the folid GK, because b. 31. 11. they are upon equal bafes AE,

B

D

K

C

is parallel to its oppofite planes, the bafe HF is to the bafe FC, as c. 25. the folid 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 bafe 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 prifms upon triangular bafes, 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 paralle lepipeds AB, CD, in the firft of which let MO, and in the other let GQ be one of the infifting lines. and becaufe the folid parallelepipeds AB, CD have the fame altitude, they are to one another as the bafe AE is to the bafe CF; wherefore the prifms, which are thei?

Book XI. halves, are to one another as the base AE to the base CF; that is, as the triangle AEM to the triangle CFG.

d. 28. 11.

PROP. XXXIII. THEOR.

SIMILAR 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 EM 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 reason, the parallelogram MK is fimilar and equal to

CR, and alfo OE to

B X

D

FD. therefore three

K

C

F

A

L

M

b. C. 11. therefore the folid KO is equal and fimilar to the folid CD. complete the parallelogram GK, and complete the folids EX, LP upon the bafes GK, KL, fo that EH be an infifling 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 c. 1. 6. 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 as GE to EL, fo is GK to KL; and as HE to EM, fo is PE to KM. there

fore

fore as the parallelogram AG to the parallelogram GK, fo is GK Book X1. to KL, and PE to KM. but as AG to GK, fo is the folid AB to

the folid EX; and as GK to KL, fodis the folid EX to the folid PL; d. 25. 11. 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, so is the parallelogram AG to the parallelogram GK, and the straight line AE to the straight 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 straight 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 described from the first to the fimilar folid fimilarly defcribed from the second; because the first straight line has to the fourth, the triplicate ratio of that which it has to the fecond,

PRO P. D. THEOR.

SOLID parallelepipeds contained by parallelograms see N.

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 AR 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 DX, and AO to DH.

Book XI.

a

Produce MA, NA, OA to P, Q, R, fo that AP be equal to DL, AQ to DK, and AR to DII; 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 a. C. 11. equal to the folid CD. complete likewife the folid AY the base of which is AS, and of which AO is one of its infifting straight lines. Take any ftraight line a, and as MA to AP, fo make a to b; and as NA to AQ, so 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 flraight 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 foþ. 32. 11. lid AY, as b the bafe AE to the bafe AS; that is, as the straight line

c. 25.11. a is to c. and the folid AY is to the folid AX, as the base OQ is to the bafe QR; that is, as the ftraight line OA to AR; that is, as the ftraight line c to the ftraight line d. and because 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 d. Def.A. 5. 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.

PROP.