. The Elements of Euclid : viz. the first six books, together with the eleventh and twelfth : the errors, by which Theon, or others, have long ago vitiated these books, are corrected, and some of Euclid's demonstrations are restored : also, the book of Euclid's Data, in like manner corrected. it may be proved, that the parallelogram AC ^^--^ c G ^ ^?^ E GF EUCLID. 225 is equal and similar to the parallelogram GF, and the AE to BF. Therefore, if a solid, 8cc. Q. E. D. *—v— PROP. XXV. THEOR. IF a solid parallelepiped be cut by a plane parallel See two of its opposite p
. The Elements of Euclid : viz. the first six books, together with the eleventh and twelfth : the errors, by which Theon, or others, have long ago vitiated these books, are corrected, and some of Euclid's demonstrations are restored : also, the book of Euclid's Data, in like manner corrected. it may be proved, that the parallelogram AC ^^--^ c G ^ ^?^ E GF EUCLID. 225 is equal and similar to the parallelogram GF, and the AE to BF. Therefore, if a solid, 8cc. Q. E. D. *—v— PROP. XXV. THEOR. IF a solid parallelepiped be cut by a plane parallel See two of its opposite planes, it divides the whole intotwo solids, the base of one of which shall be to the ^base of the other as the one solid is to the other. Let the solid parallelepiped ABCD be cut by the plane EV,which is parallel to the opposite planes AR, HD, and divides thewhole into the two solids ABFV, EGCD; as the base AEFY ofthe first is to the base EHCF of the other, so is the solid ABFVto the solid EGCD. Produce AH both ways, and take any number of straight linesHM, MN, each equal to EH, and any number AK, KL, eachequal to EA, and complete the parallelograms LO, KY, HQ, MS,and the solids LP, KR, HU, MT; then, because the straightlines LK, KA, AE are all equal, the parallelograms LO, KY, AF. are equal a: and likewise the parallelog»^inis KX, KB, AG »; a 36. also ^ the parallelograms LZ, KP, A^i because they are op- b planes: for the same reason, parallelograms EC, HQ,MS are equal =; and the parallelr&iams HG, HI, IN, as also^HD, MU, NT: therefore three planes of the solid LP are equaland similar to three planes of «:he solid KR, as also to three planesof the solid A V: but the -hree planes opposite to these threeare equal and similar *> tJ them in the several solids, and noneof their solid angles a--e contained by more than three plane an-gles : therefore the three solids LP, KR, are equal c to one c C. 11,another: iov the same reason, the three solid
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Keywords: ., bookauthoreuclid, bookcentury1800, booksubje, booksubjectgeometry
