* 28. 11.

:

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 they shall be to one another as their bases. Complete the parallelograms AE, CF, 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 CF: wherefore the prisms, which are their halves, are to one another, as the base AE to the base CF; that is, as the triangle AEM to the triangle CFG.

PROPOSITION XXXIII.

THEOR.-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 shall have 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, each to each, 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
solids AB, CD are similar; therefore
the parallelogram KL is similar and

* 24. 11.

* C. 11.

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 parallelograms of the solid KO are equal and similar to three parallelograms of the solid CD: and the three opposite ones in each solid are equal * and similar to these: therefore the solid KO is equal * and similar to the solid CD. Complete the parallelogram GK; and upon the bases GK, KL, complete the solids EX, LP, 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,

• 1.6.

25.11. 25. 11.

25.11.

are similar, and by permutation, as AE is to CF, so is EG to FN, and so is EH to FR: but 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, so * is the parallelo- * 1. 6. gram AG to the parallelogram GK; and as GE to EL, so is GK to KL; and as HE to EM, so is PE to KM: therefore 1.6. as the parallelogram AG to the parallelogram GK, so is GK to KL, and PE to KM: but as AG to GK, so is the solid AB to the solid EX; and as GK to KL, so is the solid EX to the solid PL; and as PE to KM, so 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; there- † 11 Def. 5. fore 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: but 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. Therefore, similar solid parallelopipeds, &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 ratio of that which it has to the second.

PROPOSITION D.

THEOR.-Solid parallelopipeds which are contained by paral- See N. lelograms 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, which are equiangular, each to each, to the parallelograms CH, CK, CL, which

*C. 11.

† 12. 6.

32. 11.

* 25. 11.

contain the solid CD: the ratio which the solid AB has to the solid CD, shall be the same with that which is compounded of the ratios of the sides AM to DL, AN to DK, and AO to DH.

*

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 AO 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 VI. 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, as *the base AE to

D

B

K
H

Y

P

M

R

T

And the

the base AS; that is, as the straight line a is to c. 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 æquali, the solid AB is to the solid AX, or CD which is equal to it, as the straight line a is to d. * Def. A. 5. But the ratio of a to d is said to be compounded* of the ratios of a to b, b to c, 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 cach: and 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 AO to DH. Q. E. D.

See N.

PROPOSITION XXXIV.

THEOR.—The bases and altitudes of equal solid parallelopipeds, are reciprocally proportional: and if the bases and altitudes be reciprocally proportional, the solid parallelopipeds are equal.

K

BR

D

X

Let AB, CD be two solid parallelopipeds: and first, let the insisting straight lines AG, EF, LB, HK; CM, NX, OD, PR be at right angles to the bases. If the solid AB be equal to the solid CD, their bases shall be reciprocally proportional to their altitudes; that is, as the base EH is to the base NP, so shall CM be to AG. If the base EH be equal to the base NP, then because the solid AB is likewise equal to the solid CD, CM shall be equal to AG: because if the bases EH, NP be equal, but the altitudes AG, CM be not equal, neither shall the solid AB be equal to the solid CD: but the solids are equal, by the hypothesis; therefore the altitude CM is not unequal to the altitude AG; that is, they are equal. Wherefore, as the base EH to the base NP, so is CM to AG.

H

P

Α E C

RD

K B

M

X

F

G

T

P

A

E

CN

Next, let the bases EH, NP not be equal, but EH greater than the other: then since the solid AB is equal to the solid CD, CM is therefore greater than AG: for if it be not, neither also in this case would the solids AB, CD be equal, which, by the hypothesis, are equal. Make then CT equal to AG, and complete the solid parallelopiped CV, of which the base is NP, and altitude CT. Because the solid AB is equal to the solid CD, therefore the solid AB is to the solid CV, as *the solid CD to the solid CV: but as the solid AB to the solid CV, so* is the base EH to the base NP; for the solids AB, CV are of the same altitude: and as the solid CD to CV, so is the base MP to the base PT, and so is the straight line MC to CT and CT is equal to AG; therefore as the base EH to the base NP, so is MC to AG. Wherefore the bases of the solid parallelopipeds AB, CD are reciprocally proportional to their altitudes.

Let now the bases of the solid parallelopipeds AB, CD be reciprocally proportional to their altitudes, viz. as the base EH is to the base NP, so let CM be to AG: the solid AB shall be equal to the solid CD.

If the base EH be equal to the base NP, then, since EH is to NP as the al

titude of the solid CD is to the altitude of the solid AB,there

*7. 5.

32.11.

25. 11.

* 1.6.

fore the altitude of CD is equal to the altitude of AB: but A. 5.

* 31. 11.

* A. 5.

* 32. 11.

† 1. 6.

* 25. 11.

† 9.5.

solid parallelopipeds upon equal bases, and of the same altitude, are equal to one another; therefore the solid AB is equal to the solid CD.

K B

And be

R D

But let the bases EH, NP be unequal, and let EH be the greater of the two: therefore, since, as the base EH to the base NP, so is CM the altitude of the solid CD to AG the altitude of AB, CM is greater * than AG. Therefore, as before, take CT equal to AG, and complete the solid CV. cause the base EH is to the base NP, as CM to AG, and that AG is equal to CT, therefore the base EH is to the base NP, as MC to CT. But as the base EH is to NP, so is the solid AB to the solid CV; for the solids AB, CV are of the same altitude: and as MC to CT, so is the base MP to the base PT, and the solid CD to the solid * CV therefore as the solid AB to the solid CV, so is the solid CD to the solid CV; that is, each of the solids AB, CD has the same ratio to the solid CV; and therefore the solid AB is equal to the solid CD.

H

F

P

A E C

Second general case. Let the insisting straight lines FE, BL, GA, KH; XN, DO, MC, RP not be at right angles to the bases of the solids.

In this case, likewise, if the solids AB, CD be equal, their bases shall be reciprocally proportional to their altitudes, viz. the base EH shall be to the base NP, as the altitude of the solid CD to the altitude of the solid AB.

From the points F, B, K, G ; X, D, R, M, draw perpendicu lars to the planes in which are the bases EH, NP, meeting those planes in the points S, Y, V, T ; Q, I, U, Z ; and complete the solids FV, XU, which are parallelopipeds, as was proved in the last part of Prop.

* 29. or 30. equal

11.

H

LSP

A. E

to the solid DZ, being upon the same base XR, and of the same altitude; therefore the solid BT is equal to the solid DZ: but the bases are reciprocally proportional to the