. 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. Xn. PROB. TO find a fourth proportional to three given straightlines. Let A, B, C be the three given straight lines; it is required tofind a fourth j)roportional to A, B, C OF EUCLID. 169 Take two straight lines DE, DF, containing any angle EDF ; Book VIand upon these make DG equalto A, GE equal to
. 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. Xn. PROB. TO find a fourth proportional to three given straightlines. Let A, B, C be the three given straight lines; it is required tofind a fourth j)roportional to A, B, C OF EUCLID. 169 Take two straight lines DE, DF, containing any angle EDF ; Book VIand upon these make DG equalto A, GE equal to B, and DHequal to C ; and havint^ joinedGH, draw EF parallel» to itthrough the point E : and be-cause GH is parallel to EF, oneof the sides of the triangleDEF, DG is to GE, as DH toHFb; but DG is equal to A,GE to B, and DH to C ; there-fore, as A is to B, so is C to HF. Wherefore to the three given straight lines A, B, C a fourthproportional HF is found. Which was to be b%G, PROP. XIII. PROB. TO find a mean proportional between two givenstraight lines. Let AB, BC be the two given straight lines ; it is required tofind a mean proportional between them. Place AB, BC in a straight line, and upon AC descrih^ thesemicircle ADC, and from thepoint B draw * BD at right an-gles to AC, and join AD, DC. Because the angle ADC in asemicircle is a right angle *>, andbecause in the right angled tri-angle ADC, DB is drawn fromthe right angle perpendicular tothe base, DB is a mean propor-tional between AB, BC, the segments of the base <=: therefore be- c Cor. 8,tween the two given straight lines AB, BC a mean proportional ^DB is found. Which was to be done.
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Keywords: ., bookauthoreuclid, bookcentury1800, booksubje, booksubjectgeometry
