. 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. a SI. 1- draw BG parallel » to CA, and through F draw FH parallelto ED: then each of p . p. tt the figures GBCA, ^ A u ti DEFH i5 a parallel-ogram, and they b are equal ^ to oneanother,because theyare upon equal basesBC, EF, and be-tween the same pa- rallels BF, GH ; and the triangle ABC is t
. 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. a SI. 1- draw BG parallel » to CA, and through F draw FH parallelto ED: then each of p . p. tt the figures GBCA, ^ A u ti DEFH i5 a parallel-ogram, and they b are equal ^ to oneanother,because theyare upon equal basesBC, EF, and be-tween the same pa- rallels BF, GH ; and the triangle ABC is the half <= of the pa-rallelogram GBCA, because the diameter AB bisects it; and thetriangle DEF is the half ^ of the parallelogram DEFH, becausethe diameter DF bisects it: but the halves of equal things are d 7. Ax. equal ^; therefore the triangle ABC is equal to the triangleDEF. Whprefore, triangles, 8cc. Q, E. D. PROP. XXXIX. THEOR. EQUAL triangles upon the same base, and uponthe same side of it, are between the same parallels. Let the equal triangles ABC, DBC be upon the same baseBC, and upon the same side of it; they are between the sameparallels. Join AD ; AD is parallel to BC ; for, if it is not, through the» SI. 1- point A draw ^ AE parallel to BC, and join EC: the triangle. OF EUCLID. 43 ABC is equal •» to the triangle EBC, because it is upon the same Book BC, and between the same paral-lels BC, AE: but the triangle ABC isequal to the triangle BDC ; thereforealso the triangle BDC is equal to thetriangle EBC, the greater to the less,which is impossible: therefore AE isnot parallel to BC. In the same man-ner, it can be demonstrated that no otherline but AD is parallel to BC ; AD istherefore parallel to E. D.
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Keywords: ., bookauthoreuclid, bookcentury1800, booksubje, booksubjectgeometry
