Elements of geometry and trigonometry . BOOK IV. 1U7 PROBLEM XVII. To construct a rectangle that shall he equivalent to a givensquarey and the difference of whose adjacent sides shall heequal to a given line. Suppose C equal to the given square, and AB the differenceof the sides. Upon the given line AB as a diame-ter, describe a semicircle : at the ex-tremity of the diameter draw the tan-gent AD, equal to the side of the squareC ; through the point D and the centreO draw the secant DF ; then will DEand DF be the adjacent sides of therectangle required. For, first, the difference of these sides
Elements of geometry and trigonometry . BOOK IV. 1U7 PROBLEM XVII. To construct a rectangle that shall he equivalent to a givensquarey and the difference of whose adjacent sides shall heequal to a given line. Suppose C equal to the given square, and AB the differenceof the sides. Upon the given line AB as a diame-ter, describe a semicircle : at the ex-tremity of the diameter draw the tan-gent AD, equal to the side of the squareC ; through the point D and the centreO draw the secant DF ; then will DEand DF be the adjacent sides of therectangle required. For, first, the difference of these sidesis equal to the diameter EF or AB ;secondly, the rectangle DE, DF, isequal to AD^ (Prop. XXX.) ; hence that rectangle is equivalentto the given square PROBLEM XVIII. To find the common measure^ if there is one, between the diagonaland the side of a square. Let ABCG be ?ny square what-ever, and AC its diagonal. We must first ap[)ly CB uponCA, as often as it may be containedthere. For this purpose, let thesemicircle DBE be described, fromthe centre C, with the radius is evident that CB is containedonce in AC, with the remainderAD ; tl»e result of the first operationis therefore the (juoticnt 1, with the remainder AD, which lat-ter must now !>e com[)ared with BC, or its e(iual AB. We might here take AF--AD, and actually apply it uponAB ; we should find it to be contained twice witli a remain-der ; but as that remainder, and those which succeed it, con-
Size: 1437px × 1739px
Photo credit: © The Reading Room / Alamy / Afripics
License: Licensed
Model Released: No
Keywords: ., bo, bookcentury1800, booksubjectgeometry, booksubjecttrigonometry
