. 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. and that the plane LBHM is that in which are the parallels LB,MHPQ, in which also is the figure BLPQ ; and the plane ACDFis that in which are the parallels AC, FDOR, in which also isthe figure C AOR; therefore the figures BLPQ, C AOR are inparallel planes: in like manner, because the plane ALNG is pa-rall
. 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. and that the plane LBHM is that in which are the parallels LB,MHPQ, in which also is the figure BLPQ ; and the plane ACDFis that in which are the parallels AC, FDOR, in which also isthe figure C AOR; therefore the figures BLPQ, C AOR are inparallel planes: in like manner, because the plane ALNG is pa-rallel CO the opposite plane CBKE, and that the plane ALNG isthat in which are the parallels AL, OPGN, in which also is the 9,32 THE ELEMENTS BookXI. figure ALPO ; and the plane CBKE is that in which are the pa-^—V— rallels CB, RQEK, in which also is the figure CBQR; thereforethe figures ALPO, CBQR are in parallel planes: and the planesACBL, ORQP are parallel; therefore the solid CP is a paralle-lepiped : but the solid CM, of which the base is ACBL, to whicha FDHM is the opposite parallelogram, is equal ^ to the solid CP,of which the base is the parallelogram ACBL, to which ORQP. A C is the one opposite, because they are apon the same base, andtheir insisting straight lines AF, AO, CD, CR; LM, LP, BH,BQ are in the same straight lines FR, MQ: and the solid CP isequal » 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 OL, RK; therefore thesolid CM is equal to the solid CN. Wherefore, solid parallelepi-peds, &c. Q. E. D. PROP. XXXL THEOR. SeeN. SOLID 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 so-lid CF. First, Let the insisting straight lines be at right angles to thebases AB, CD, and let the bases be
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Keywords: ., bookauthoreuclid, bookcentury1800, booksubje, booksubjectgeometry
