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the straight lines LK, KA, AE, are all equal, and also the straight lines KO, AY, EF, which make equal angles with LK, KA, AE, the parallelograms LO, KY, AF, are equal and similar (Pl. Ge. VI. 20); and likewise the parallelograms KX, KB, AG; as also (II. 4) the parallelograms LZ, KP, ÁR, because they are opposite planes. For the same reason, the parallelograms EC, HQ, MS, are equal; and the parallelograms HG, HI, IN, as also HD, MU, NT; therefore three planes of the solid LP are equal and similar to three planes of the solid KR, as also to three planes of the solid AV; but the three planes opposite to these three are equal and similar to them in the several solids; therefore the solids LP, KR, AV, are contained by equal and similar planes. And because the planes LZ, KP, AR, are parallel, 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 (1. 15); and the same is true of the other contiguous planes; therefore the solids LP, KR, and AV, are equal to one another (II. 3). For the same reason, the three solids ED, HU, MT, are equal to one another; therefore what multiple soever the base LF is of the base AF, the same multiple is the solid LV of the solid AV; for the same reason, whatever multiple the base NF is of the base HF, the same multiple is the solid NV of the solid ED; and if the base LF be equal to the base NF, the solid LV is equal to the solid NV; and if the base LF be greater than the base NF, the solid LV is greater than the solid NV; and if less, less. Since then there are four magnitudes, namely, the two bases AF, FH, and the two solids AV, ED, and of the base AF and solid AV, the base LF and solid LV are any equimultiples whatever; and of the base FH and solid ED, the base FN and solid NV are any equimultiples whatever; and it has been proved, that if the base LF is greater than the base FN, the solid LV is greater than the solid NV; and if equal, equal; and if less, less. Therefore (Pl. Ge. V. Def. 10), as the base AF is to the base FH, so is the solid AV to the solid ED.

Scholium.-This proposition may be demonstrated by the principle in the twenty-seventh proposition of the additional Fifth Book, thus :

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Let the parallelopiped AF be cut by a plane GH parallel to either of its sides AE or BF, then AG: GB=AK:KB, For, let the bases AK, KB, be commensurable, and hence the sides AH, HB, E are so too. Let AH and HB contain a common measure Z 4 and 3 times respectively. Divide AH into 4 and HB into 3 equal parts in L, M, N, O, and P, and through these points let planes LS, MT, &c. pass, parallel to AE, then AG will be divided into 4, and HF into 3, equal parallelopipeds, AS, LT, &c. The figures AS, LT, &c. are parallelopipeds, for their opposite sides are parallel (II. Def. 5), and hence the opposite sides are equal and similar parallelograms (II. 4). Also the parallelograms AQ, LR, are equal, for ALLM, and AM is parallel to CR. For a similar reason EQSR; and also AELS. The parallelograms opposite to these are also equal (II. 4); therefore the two parallelopipeds AS, LT, are contained by the same number of equal and similar parallelograms, similarly situated. The parallel planes AE, LS, are cut by the plane EK; therefore the inclination of AE and EQ is equal to that of LS and SR; and the same may be proved of the inclinations of the other sides of AS and LT. Hence (II. 3) AS = LT. It may be similarly proved that the parallelopipeds LT, MU, NG, HV, OW, and PF, are equal. Hence AG: GB = 4:3; but AK:KB 4:3; therefore AG: GB=AK: KB. The same proportion is similarly proved, whatever be the number of times that AH: HB contain their common measure, when commensurable; hence the proportion exists when they are incommensurable (Pl. Ge. Aď. V. 27).

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COR. Because the parallelogram AF (former figure) is to the parallelogram FH as YF to FC (Pl. Ge. VI. 1), therefore the solid AV is to the solid ED as YF to FC.

PROPOSITION VI. THEOREM.

If a solid parallelopiped be cut by a plane passing through the diagonals of two of the opposite planes, it shall be cut in two equal prisms.

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Let AB be a solid parallelopiped, and DE, CF, the diagonals of the opposite parallelograms AH, GB, namely, 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 same plane with it, CD, FE, are parallel (I. 8); wherefore the diagonals CF, DE, are in the plane in which the parallels are, and are themselves parallels (1. 14); and the plane CDEF shall cut A the solid AB into two equal parts.

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Because the triangle CGF is equal (Pl. Ge. I. 34) to the triangle CBF, and the triangle DAE to DHE; and that the parallelogram CA is equal (II. 4) and similar to the opposite one BE; and the parallelogram GE to CH; therefore the planes which contain the prisms CAE, CBE, are equal and similar, each to each; and they are also equally inclined to one another, because the planes AC, EB, are parallel, as also AF and BD, and they are cut by the plane CE; therefore the prism CAE is equal to the prism CBE (II. 3), and the solid AB is cut into two equal prisms by the plane CDEF.

Def. The insisting straight lines of a parallelopiped, mentioned in the following propositions, are the sides of the parallelograms betwixt the base and the plane parallel to it.

PROPOSITION VII. THEOREM.

Solid parallelopipeds upon the same base, and of the same altitude, the insisting straight lines of which are terminated in the same straight lines in the plane opposite to the base, are equal to one another.

Let the solid parallelopipeds AH, AK, be upon the same base AB, and of the same altitude, and let their insisting straight lines AF, AG, LM, LN, be terminated in the same straight line FN; and CD, CE, BH, BK, be terminated in the same straight line DK; the solid AH is equal to the solid AK.

Because CH, CK, are parallelograms, CB is equal (Pl. Ge. I. 34) to each of the opposite sides 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 (Pl. Ge. I. 38) to the triangle BHK ;

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and the parallelogram DG is equal (Pl. Ge. I. 36) to the parallelogram HN. For the same reason, the triangle AFG is equal to the triangle LMN, and the parallelogram CF is equal (II. 4) to the parallelogram BM, and CG to BN; for they are opposite. Therefore, the planes which contain the prism DAG are similar and equal to those which contain the prism HLN, each to each; and the contiguous planes are also equally inclined to one another (I. 15), because that the parallel planes AD and LH, as also AÉ and LK, are cut by the same plane DN; therefore the prisms DAG, HLN, are equal (II. 3). If therefore the prism LNH be taken from the solid, of which the base is the parallelogram AB, and FDKN the plane opposite to the base; and if from this same solid there be taken the prism AGD, the remaining solid, namely, the parallelopiped AH, is equal to the remaining parallelopiped AK.

PROPOSITION VIII. THEOREM.

Solid parallelopipeds upon the same base, and of the same altitude, the insisting straight lines of which are not terminated in the same straight lines in the plane opposite to the base, are equal to one another.

Let the parallelopipeds CM, CN, be upon the same base AB, and of the same altitude, but their insisting straight lines AF, AG, LM, LN, CD, CE, BH, BK, not terminated in the same straight lines; the solids CM, CN, are equal to one another.

Produce FD, MH, and NG, KE, and let them meet one another in the points O, P, Q, R; and join AO, LP, BQ, CR. Because the planes (II. Def. 5) LBHM and ACDF

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are parallel, and because the plane LBHM is that in which are the parallels LB, MHPQ, and in which also is the figure BLPQ; and because the plane ACDF is that in which are the parallels AC, FDOR, and in which also is the figure CAOR ; therefore the figures M BLPQ, CAOR, are in parallel planes. In like manner, because the planes ALNG and CBKE are parallel, and the plane ALNG

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is that in which are the parallels AL, OPGN, and in which also is the figure ALPO; and the plane CBKE is that in which are the parallels CB, RQEK, and in which also is the figure CBQR; therefore the figures ALPO, CBQR, are in parallel planes. But the planes ACBL, ORQP, are also parallel; therefore the solid CP is a parallelopiped. Now the solid parallelopiped CM is equal to the solid parallelopiped CP; because they are upon the same base, and their insisting straight lines AF, AO, CD, CR, LM, LP, BH, BQ, are in the same straight lines FR, MQ; and the solid CP is equal to the solid CN; for they are upon the same base ACBL, and their insisting straight lines AO, AG, LP, LN, CR, CE, BQ, BK, are in the same straight lines ON, RK; therefore the solid CM is equal to the solid CN.

PROPOSITION IX. THEOREM.

Solid parallelopipeds which are upon equal bases, and of the same altitude, are equal to one another.

Let the solid parallelopipeds AE, CF, be upon equal bases AB, CD, and be of the same altitude; the solid AE is equal to the solid CF.

Case 1. Let the insisting straight lines be at right angles to the bases AB, CD, and let the bases be placed in the same plane, and so as that the sides CL, LB, be in a straight line; therefore the straight line LM, which is at right angles

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