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and so as that the sides CL, LB be in a straight line; there- Book XI. fore the straight line LM, which is at right angles to the plane v in which the bases are, in the point L, is common a to the two a 13.11. folids AE, CF; let the other insisting lines of the solids be AG, HK, BE ; DF, OP, CN : And first, let the angle ALB be equal to the angle CLD; then AL, LD are in a straight line b. b 14. I. Produce OD, ÅB, and let them meet in Q, and complete the solid parallelepiped LR, the base of which is the parallelogram LQ, and of which LM is one of its insisting straight lines : Therefore, because the parallelogram AB is equal to CD, as the base AB is to the base LQ, lo is the base CD to the same c 7. Š• LQ: And because the solid parallelepiped AR is cut by the plane LMEB, which is parallel to the opposite planes AK, DR; as the base AB is to the base LQ, fo is d the solid AE to the d 25. 11. solid LR: For the same reason, because the folid parallelepiped CR is cut by the plane LMFD, which is parallel to the opposite planes CP, BR; as P
R the base CD to the base LQ, so is the
E folid CF to the so
G lid LR : But as the
K base AB to the base
Q LQ, so the base CD to the base LQ, as before was proved : С
AS H T folid AE to the solid LR, fo is the folid CF to the folid LR; and therefore the folid AE is equal to the folid CF.
. But let the solid parallelepipeds SE, CF be upon equal bases SB, CD, and be of the fame altitude, and let their insisting straight lines be at right angles to the bases; and place the bases SB, CD in the same plane, so that CL, LB be in a straight line; and let the angles SLB, CLD be unequal ; the folid SE is also in this case equal to the solid CF: Produce DL, TS until they meet in A, and from B draw BH parallel to DA; and
let HB, OD produced meet in Q, and complete the solids AE, | LR: Therefore the solid AE, of which the base is the parallelo
gram LE, and AK the one opposite to it, is equalf to the fo- f 29. 11. lid SE, of which the base is LE, and to which SX is opposite; for they are upon the same base LE, and of the same altitude, and their infifting straight lines, viz. LA, LS, BH, BT; MG, MV, EK, EX are in the same straight lines AT, GX : And be
8 35. I.
Book XI. cause the parallelogram AB is equal a to SB, for they are uș. m on the fame base LB, and between the same parallels LB, AT ; and that the base
E the base AB is equal
X to the base CD, and
L the solid AE is e.
AS HT qual to the solid ČF; but the solid AE is equal to the solid SE, as was demon. strated; therefore the folid 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 bafes AB, CD; in this case likewise the folid AE is equal to the folid CF: From
the points G, K, E, M; N, S, F, P, draw the straight lines h 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, ÚY: Then, because GQ, KT, are at right
i 6. II.
angles to the same plane, they are parallel i to one another : And MG, EK are parallels ; therefore the planes MO, 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 k 15.11. plane with them, are parallel to one another : For the same
reason, the planes MV, GT are parallel to one another: There fore the solid QE is a parallelepiped : In like manner, it may be proved, that the solid YF is a parallelepiped : But, from what has been demonstrated, the folid EQ is equal to the folid FY, because they are upon equal bases MK, PS, and of the same altitude, and have their infifting straight lines at right angles
to the bases : And the solid EQ is equal to the solid AE ; and Book XI. the folid FY to the solid CF; because they are upon the fame bases and of the same altitude: Therefore the solid AE is equal [ 29. or 30. to the solid CF. Wherefore solid parallelepipeds, &c. Q. E. D.' 11.
PRO P. XXXII. THE O R.
OLID parallelepipeds which have the same altitude, See N.
are to one another as their bases.
Let AB, CD be folid parallelepipeds 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 folid CD.
To the straight line FG apply the parallelogram FH equal" a Cor. 45.5 10 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 insisting lines is FD, whereby the folids CD, GK must be of the fame altitude: Therefore the solid AB is equal to 31. . to the solid B
D K GK, because they are upon
QI equal bases
OP AE, FH, and
F are of the same altitude : And because the solid parallelepi
A M C G H ped CK is cut by the plane DG which is parallel to its opposite planes, the base HF is to the base FC, as the solid HD to the folid DC: But c 25. Ik the base HF is equal to the base AE, and the folid GK to the folid AB: Therefore, as the base AE to the base CF, so is the solid AB to the solid CD. Wherefore folid parallelepipeds, &c. Q. E. D.
Cor. From this it is manifest that prisms upon triangular ba, ses, of the fame altitude, are to one another as their bafes.
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 solid 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 solid parallelepipeds 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 d, are to one manother, as the base AE to the base CF; that is, as the triangle 28. II. AEM to the triangle CFG.
IMILAR folid parallelepipeds are one to another in the triplicate ratio of their homologous fides.
Let AB, CD be similar folid parallelepipeds, and the fide 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 these produced take EK c. qual to CF, EL equal to FN, and EM equal to FR ; and com. plete the parallelogram KL, and the folid KO: Because KE, EL are equal to CF, FN, and the angleKEL 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 parallelogram KL is fimilar and equal to the parallelogram CN: For the same reafon, the parallelogram MK is fimilar and equal to CR, and also OE to FD: There
X fore three paralle
H lograms of the so
P lid KO are equal
IK the three opposite
E ones in each folid
a 24. Ir.
b C. 11.
and as HE to EM, fo is PE to KM: Therefore as the parallelo- Book XI, gram AG to the parallelogram GK, so is GK to KL, and PE to KM: But as AG to GK, so d is the solid AB to the solidc 1.6. EX; and as GK to KL, fod is the solid EX to the folid PL ;d 25. II, and as PE to KM, fod is the folid PL to the folid KO: And therefore as the folid 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 folid 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 tri. plicate 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 itraight line CF. Therefore the folid AB has to the solid CD, the triplicate ratio of that which the fide AE has to the homologous fide 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, fo is the folid parallelepiped described from the first to the similar solid fimilarly described from the second ; because the first straight line has to the fourth the triplicate ratio of that which it has to the second.
OLID 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 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 solid CD is the same with that which is compounded of the ratios of the fides AM to DL, AN to DK, and AO to DH.