. 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. f LM to P the same with it; and be-cause the ratio of the figure BD to BG is given, find this ratio bythe 53d dat. and make the ratio of P to Q the same. Also, be-cause the ratio of CB to BA is given, make the ratio of Q to Rthe same ; and take LN equal to R ; through the pomi M drawOM parallel to LN, and
. 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. f LM to P the same with it; and be-cause the ratio of the figure BD to BG is given, find this ratio bythe 53d dat. and make the ratio of P to Q the same. Also, be-cause the ratio of CB to BA is given, make the ratio of Q to Rthe same ; and take LN equal to R ; through the pomi M drawOM parallel to LN, and complete the parallelogram NLOS; then/ this is similar to the parallelogram BF. Because the angle ABK is equal to NLM, and the angle ABEto NLO, the angle KBE is equal to MLO ; and the anglesEKE, LMO are equal, because the angle ABK is equal toNLlNI; therefore the triangles BKE, LMO are equiangular toone another ; wherefore as BE to BK, so is LO to LM ; andbecause as the figure BF to BD, so is the straight line LMto P ; and as BD to BG, so is P to Q ; ex xquali, as BF,that is Miirl to BG, so is LM to Q: but BB is to^ BG, asKB to BC ; as therefore KB to BC, so is LM to Q ; and be-cause BE hi to BK, as LO to LM; and as BK to BC, so isLM to Q : and as BC to BA, so Q was made to R; there-. DATA. 43» fore, ex aquali, as BE to BA, so is LO to R, that is to LN ;and the angles ABE, NLO arc equal ; therefore the parallelo-gram BF is similar to LS. PROP. LXX. 62, ra IF two straight lines have a given ratio to one ano-SeeNote,thcr, and upon one of them be described a rectihiiealfigure given in species, and upon the other a paraiklo-gram having a given angle; if the figure have a givenratio to the paralielogram, tlie paralleiogrLm is given inspecies. Let the two straight lines AB, CD have a given ratio to oneanother, and upon AB let the figure AEB given in species bedescribed, and upon CD the parallelogram DF having the givenangle FCD ; if the ratio of AEB to DF be given, the parallelo-gr
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Keywords: ., bookauthoreuclid, bookcentury1800, booksubje, booksubjectgeometry
