. 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. to the triangle ABC. Because,as was shown, GF is to FK, as AB, BC to the parallelogramAC ; and FK is to its half, as AC is to its half, which is the tri-angle ABC ; therefore, ex equali^ GF is to the half of FK, asAB, BC rectangle is to the triangle ABC. 5^. IF two parallelograms be equiangular
. 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. to the triangle ABC. Because,as was shown, GF is to FK, as AB, BC to the parallelogramAC ; and FK is to its half, as AC is to its half, which is the tri-angle ABC ; therefore, ex equali^ GF is to the half of FK, asAB, BC rectangle is to the triangle ABC. 5^. IF two parallelograms be equiangular, as a side ofthe first to a side of the second, so is the other sideof the second to the straight line to which tlie otherside of the first has the same ratio ^vhich the first pa-rallelogram has to the second. And consequently,if tlc ratio of the first parallelogram to the second begiven, the ratio of the other side of the first to thatstraight line is given; and if the ratio of tlie otherside of tlie first to that straight line be given, the ratioof t!-ic fii-st parallelogram to the second is given. Let iVC, DF be two equiangular parallelograms, as BC, aside of the first, is to EF, a side of the second, so is DE, theo-tlier side of the second, to the straiglit line to which AB, the. 416 EUCLIDS other side of the first has the same ratio wliich AC has to DF. Produce the straight line AB, and make as BC to EF, soDE to BG, and complete the parallelo-gram BGHC ; therefore, because BC orGH, is to EF, as DE to BG, the sidesabout the equal angles BGH, DEF are& 14. 6. reciprocally proportional ; wherefore athe parallelogram BH is equal to DF ;and AB is to BG, as the parallelogramAC is to BH , that is, to DF ; as there-fore BC is to EF, so is DE to BG, Avhichis the straight line to which AB has thesame ratio that AC has to DF. And if the ratio of the parallelogram AC to DF be given, thenthe Iatio of the straight line AB to BGis given ; and if the ratioof AB to the straight line BG be
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Keywords: ., bookauthoreuclid, bookcentury1800, booksubje, booksubjectgeometry
