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PRO P. XXVII.
lepiped similar, and similarly situated to one given. Let AB be the given straight line, and CD the given folid pa. rallelepiped. It is required from AB to describe a solid paralle
lepiped limilar, and fimilarly situated to CD. in 26. 11. At the point A of the given straight line AB, make a solid
angle equal to the solid angle at C, and let BAK, KAH, HAB be the three plane angles which contain it, so that BAK be e
qual to the angle ECG, and KAH to GCF, and HAB to b 12. 6. FCE: And as EC to CG, so make b BA to AK ; and as GC to
CF, so make b KA to AH; wherefore, ex aequali“, as EC to
parallelograms of the folid AL are similar to three of the folid d 24. 11.
CD; and the three opposite ones in each folid are equal and similar to these, each to each. Also, because the plane angles which contain the solid angles of the figures are equal, each to
each, and fituated in the same order, the folid angles are e e B. II. qual, each to each. Therefore the solid AL is similar f to the $11. def. II. folid CD. Wherefore from a given straight line AB a solid pa
rallelepiped AL has been described similar, and similarly situated to the given one CD. Which was to be done.
PRO P. XXVIII.
a solid parallelepiped be cut by a plane passing thro’ See N.
the diagonals of two of the opposite planes ; it shall be cut in two equal parts.
Let AB be a solid parallelepiped, and DE, CF, the diagonals of the opposite 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, and not in the same plane with it, CD, FF are parallel ; wherefore the diagonals CF, a 9, 11. DE are in the plane in which the parallels are, and are themselves paral
B lelsb: And the plane CDEF shall cut
b 16. 11 the folid AB into two equal parts.
G Because the triangle CGF is equal to the triangle CBF, and the triangle
C 34. I. DAE to DHE; and that the paral
D lelogram CA is equal d and similar to
H d 24.11 the opposite one BE ; and the parallelogram GE to CH : Therefore the A
E prism contained by the two triangles CGF, DAE, and 'the three parallelograms CA, GE, EC, is equal to the prism contained by the two triangles CBF, C. sI. DHE, and the three parallelograms BE, CH, EC; because they are contained by the same number of equal and similar planes, alike fituated, and none of their solid angles are contained by more than three plane angles. Therefore the folid AB is cut into two equal parts by the plane CDEF. Q. E. D.
· N. B. The insisting straight lines of a parallelepiped, mentioned in the next and some following propositions, are the * Gides of the parallelograms betwixt the base and the opposite
plane parallel to it.'
SOLID parallelepipeds upon the fame base, and of the Sec N.
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.
Book XI. Let the foli! parallelepipeds AH, AK be upon the same ta fe WAB, and of the fame altitude, and let their infifting straight See the fi- lines AF, AG, LM, LN, be terminated in the same straight line gures below. FN, and CD, CE, BH, BK be terminated in the same straight
line DK; the solid AH is equal to the solid AK.
First, Let the parallelograms DG, HN, which are opposite to the base AB, have a common side HG: Then, because the folid AH is cut by the plane AGHC passing through the diagonals AG, CH of the opposite planes ALGF, CBHD, AH is cut into two equal parts * by the plane AGHC: Therefore the folid AH is double of the
B the diagonals LG, BH of the opposite planes ALNG, A L CBKH, the solid AK is double of the same prism which is contained betwixt the triangles ALG, CBH. Therefore the folid AH is equal to the folid AK.
But, let the parallelograms DM, EN oppofite to the base, have no common side : Then, because CH, CK are parallelograms, CB is equal to each of the opposite 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 tric 38. 1. angle CDE is equal to the triangle BHK : And the parallelod 36. S.
gram DG is equal d to the parallelogram HN: For the same
reason, the triangle AFG is equal to the triangle LMN, and C24.11. the parallelogram CF is equal to the parallelogram BM, and D H E
E H K
b 34. S.
taken from the folid of which the base is the parallelogram Book XI. AB, and in which FDKN is the one opposite to it; and if from this same folid there be taken the prism AFGCDE; the remaining folid, viz. the parallelepiped AH, is equal to the remaining parallelepiped AK. Therefore solid parallelepipeds, &c. Q. E. D.
OLID parallelepipeds upon the same base, and of the
See Ni fame 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 parallelepipeds CM, CN be upon the fame base AB, and of the fame 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: And because the plane LBHM is parallel to the opposite
С plane ACDF, and that the plane LBHM is that in which are the parallels LB, MHPQ, in which also is the figure BLPQ; and the plane ACDF is that in which are the parallels AC, FDOR, in which also is the figure CAOR ; therefore the figures BLPO, CAOR are in parallel planes : In like manner, because the plane ALNG is parallel to the opposite plane CBKE, and that the plane ALNG is that in which are the parallels
Book XI. AL, OPGN, in which also is the figure ALPO; and the plane
CBKE is that in which are the parallels CB, RQEK, in which also is the figure CBQR ; therefore the figures ALPO, CBQR are in parallel planes : And the planes ACBL, OROP are parallel ; therefore the solid CP is a parallelepiped : But the so
lid CM, of which the base is ACBL, to which FDHM is the a 29. 11. opposite parallelogram, is equal to the solid CP of which, the
base is the parallelogram ACBL, to which ORQP is the one opposite; 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 folid CP is equal to the fold CN; for they are upon the same base ACBL, and their insisting straight lines AO, AG, LP, LN; CR, CE, BO, BK are in the same straight lines ON, RK: Therefore the folia CM is equal to the folid CN. Wherefore solid parallelepipeds, &c. Q. E. D.
PRO P. XXXI. THE O R. OLID parallelepipeds which are upon equal bases, , and of the same altitude, are equal to one another.
Let the solid parallelepipeds AE, CF, be upon equal bases AB, CD, and be of the same altitude; the solid AE is equal to the solid CF.
First, Let the insisting straight lines be at right angles to the bases AB, CD, and let the bases be placed in the same plane,