Book III.

F a folid be contained by fix planes, two and two of which are parallel, the oppofite planes are fimilar and equal parallelograms.

Let the folid CDGH be contained by the parallel planes AC, GF, BG, CE; FB, AE: its oppofite planes are similar and equal parallelograms.

Ᏼ

H

G

Because the two parallel planes BG, CE, are cut by the plane AC, their common fections AB, CD are parallel a. A-a14. 2,Sup. gain, because the two parallel planes BF, AE, are cut by the plane AC, their common fections AD, BC are parallel: and AB is parallel to CD; therefore AC is a parallelogram. In like manner, it may be proved that each of the figures CE, FG, GB, BF, AE is a parallelogram: join AH, DF; and because AB s parallel to DC, and BH to CF; the two ftraight lines AB, BH, A which meet one another, are parallel to DC and CF, which meet one another; wherefore, though the first two are not in the fame plane with the other two, they contain equal angles b; the angle ABH is b 9. 2. Sup. therefore equal to the angle DCF. And because AB, BH, are equal to DC, CF, and the angle ABH equal to the angle DCF; therefore the base AH is equal to the base DF, and © 4. I. the triangle ABH to the triangle DCF: For the fame reafon, the triangle AGH is equal to the triangle DEF; and therefore the parallelogram BG is equal and fimilar to the parallelogram CE. In the fame manner, it may be proved, that the parallelogram AC is equal and fimilar to the parallelogram GF, and the parallelogram AE to BF. Therefore, if a folid, &c. Q. E. D.

D

Supplement

IF

PROP. III. THEOR.

F a folid parallelepiped be cut by a plane parallel to two of its oppofite planes, it will be divided into two folids, which will be to one another as their bafes.

Let the folid parallelepiped ABCD be cut by the plane EV, which is parallel to the oppofite planes AR, HD, and divides the whole into the folids ABFV, EGCD; as the bafe AEFY to the bafe EHCF, fo is the folid ABFV to the folid EGCD.

Produce AH both ways, and take any number of straight lines HM, MN, each equal to EH, and any number AK, KL each equal to EA, and complete the parallelograms LO, KY, HQ, MS, and the folids LP, KR, HU, MT then, because

:

R V D U

T

a 20. 6.

the straight lines LK, KA, AE are all equal, and also the ftraight lines KO, AY, EF, which make equal angles with LK, KA, AE, the parallelograms LO, KY, AF are equal and fimilar a: and likewife the parallelograms KX, b 2. 3. Sup. KB, AG; as alfo b the parallelograms LZ, KP, AR, becaufe they are "oppofite planes. For the fame reafon, the parallelograms EC, HQ, MS are equal a; and the parallelograms HG, HI, IN, as alfo b HD, MU, NT; there fore three planes of the folid LP, are equal and fimilar to three planes of the folid KR, as alfo to three planes of the folid AV: but the three planes oppofite to these three are equal and fimilar to them b in the feveral folids; therefore the folids LP, KR, AV are contained by equal and fi

milar

milar planes. And because the planes LZ, KP, AR are pa- Book III. railel, and are cut by the plane XV, the inclination of LZ

to XP is equal to that of KP to PB; or of AR to BV c: and 15.2.Sup the fame is true of the other contiguous planes, therefore the folids LP, KR, and AV, are equal to one another d. For thed 1. 3. Sup. fame reason, the three folids ED, HU, MT are equal to one another; therefore what multiple foever the base LF is of the base AF, the fame multiple is the folid LV of the solid AV.; for the fame reason, whatever multiple the bafe NF is of the base HF, the fame multiple is the folid NV of the folid ED: And if the base LF be equal to the base NF, the folid LV is equal d to the folid NV; and if the bafe LF be greater than the base NF, the folid LV is greater than the folid NV; and if lefs, less. Since then there are four magnitudes, viz. the two bases AF, FH, and the two folids AV, ED, and of the base AF and folid AV, the base LF and folid LV are any equimultiples whatever; and of the bafe FH and folid ED, the base FN and folid NV are any equimultiples whatever; and it has been proved, that if the bafe LF is greater than the bafe FN, the folid LV is greater than the folid NV; and if equal, equal; and if less, less: Therefore e, as the base AF is to the bafe FH, fo is e 5. def. 5. the folid AV to the folid ED. Wherefore, if a folid, &c. Q. E. D.

COR. Because the parallelogram AF is to the parallelogram FH as YF to FC, therefore the folid AV to the folid ED f 1.6. as YF to FC.

F a folid parallelepiped be cut by a plane paffing through the diagonals of two of the oppofite planes, it shall be cut in two equal prifms.

Let AB be a folid parallelepiped, and DE, CF the diagonals of the oppofite parallelograms AH, GB, viz. those which are drawn betwixt the equal angles in each; and because CD, FE are each of them parallel to GA, though not in the a 8. 2. Sup: fame plane with it, CD, FE are parallel a; wherefore the diagonals CF, DE are in the plane

in which the parallels are, and are

b 14.2.Sup. themselves parallels b; and the plane CDEF fhall cut the folid AB into two equal parts.

C 34. I.

Because the triangle CGF is equal c to the triangle CBF, and the triangle DAE to DHE; and that the d 2.3 Sup. parallelogram CA is equald and fimilar to the oppofite one BE; and the parallelogram GE to CH: therefore the planes which contain

G

C.

B

the prisms CAE, CBE, are equal and fimilar, each to each; and they are alfo equally inclined to one another, because the planes AC, EB are parallel, as alfo AF and BD, and they are et. 3. Sup. cut by the plane CE e. Therefore the prism CAE is equal to the prifm CBE e, and the folid AB is cut into two equal prifms by the plane CDEF. Q. E. D.

N. B. The infifting straight lines of a parallelepiped, mentioned in the following propofitions, are the fides of the parallelograms betwixt the base and the plane parallel to it.

PROP

Book III.

SOLI

PROP. V. THEOR.

OLID parallelepipeds upon the fame bafe, and of the fame altitude, the infifting ftraight lines of which are terminated in the fame ftraight lines in the plane oppofite to the bafe, are equal to one

another.

Let the folid parallelepipeds AH, AK be upon the fame bafe AB, and of the fame altitude, and let their infifting ftraight lines AF, AG, LM, LN, be terminated in the fame ftraight line FN, and CD, CE, BH, BK be terminated in the fame straight line DK; the folid AH is equal to the folid AK.

Because CH, CK are parallelograms, CB is equal a to each a 34. 1. of the oppofite fides DH, EK; wherefore DH is equal to EK: add, or take away the common part HE; then DE is equal to HK: Wherefore also the triangle CDE is equal b to b 38. 1. the triangle BHK and the parallelogram DG is equal to the parallelogram HN. For the fame reason, the triangle AFG is equal to the triangle LMN, and the parallelogram CF is equal to the parallelogram BM, and CG to BN; for d 2. 3. Sup:

d

36. I.

they are oppofite, Therefore the planes which contain the prifm DAG are fimilar and equal to those which contain the prifm HLN, each to each; and the contiguous planes are alfo equally inclined to one another e, because that the parallel e15.2. Sup. planes AD and LH, as alfo AE and LK, are cut by the fame

plane DN therefore the prifms DAG, HLN are equal f. If f 1. 3 Sup.

S3

therefore