. 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. PROP. XIV. PROB. TO describe a square that shall be equal to a given See figure. Let A be the given rectilineal figure ; it is required to describea square that shall be equal to A. Describe * the rectangular parallelogram BCDE equal tt) the a figure A. If, then, the sides o
. 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. PROP. XIV. PROB. TO describe a square that shall be equal to a given See figure. Let A be the given rectilineal figure ; it is required to describea square that shall be equal to A. Describe * the rectangular parallelogram BCDE equal tt) the a figure A. If, then, the sides of it BE, ED are equal to one another, it a ^ ^.^^^ jr is a square, andwhat was requir-ed is now done :but if they are notequal, produce one \ / •B of them BE to F,and make EF e-qual to ED, andbisect BF in G;and from the centre G, at the distance GB, or GF, describe thesemicircle BHF, and produce DE to H, and join GH ; therefore,because the straight line BF is divided into two equal parts inthe point G, and into two unequal at E, the rectangle BE,EF, together with the square of EG, is equal ^ to the square ofb 5, : but GF is equal to GH; therefore the rectangle BE, EF, I. 66 THE ELEMENTS, Sec. Book II. together with the square of EG, is equal to the square of GH;*--v-—^ but the squares of HE, EG are equal « to the square of GH:c therefore the rectangle BE, EF, together with the square ofEG, is equal to the squares of HE, EG: take away the squareof EG, which is common to both; and the i^emaining rect-angle BE, EF is equal to the square of EH: but the rectanglecontained by BE, EF is the parallelogram BD, because EF isequal to ED ; therefore BD is equal to the square of EH ; butBD is equal to the rectilineal figure A; therefore the rectilinealfigure A is equal to the square of EH: wherefore a square hasbeen made equal to the given rectilineal figure A, viz. the squaredescribed upon EH. Which was to be done. THE ELEMENTS OF EUCLID- BOOK III. DEFINITIONS. I. JljQUAL ci
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Keywords: ., bookauthoreuclid, bookcentury1800, booksubje, booksubjectgeometry
