Elements of geometry and trigonometry . PKOBLKM IV. To divide a f^iven line into two parts, such that thr rryenter partshall be a proportional between the whole line and theother part. fOD Let AB be the given line. At the extremity B of the lineAB, erect the perpendicular BCequal to the half of AB ; from thepoint C, as a centre, with the ra-dius CB, describe a semicircle ;draw AC cutting the circumfe-rence in D ; and take AF=AD : Athe line AB will be divided at the point F in the manner required ; that is, we shall have AB : AF : : AF : FB. For, AB being perpendicular to the r
Elements of geometry and trigonometry . PKOBLKM IV. To divide a f^iven line into two parts, such that thr rryenter partshall be a proportional between the whole line and theother part. fOD Let AB be the given line. At the extremity B of the lineAB, erect the perpendicular BCequal to the half of AB ; from thepoint C, as a centre, with the ra-dius CB, describe a semicircle ;draw AC cutting the circumfe-rence in D ; and take AF=AD : Athe line AB will be divided at the point F in the manner required ; that is, we shall have AB : AF : : AF : FB. For, AB being perpendicular to the radius at its extremity,is a tangent ; and if AC be produced till it again meets thecircumference in E, we shall have AE : AB : : AB : AD(Prop. XXX.) ; hence, by division, AE—AB : AB : : AB—AD : AD, But since the radius is the half of AB, the diame-ter DE is equal to AB, and consequently AE—AB-AD=AF ;also, because AF=AD, we have AB^—AD—FB ; henceAF : AB : : FB : AD or AF ; whence, by exchanging theextremes for the means, AB : AF : : AF ; FB. Schdium, This sort of division of the line AB is called di-vision in extreme and mean ratio : the use of it will be per-ceived in a future part of the
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Keywords: ., bo, bookcentury1800, booksubjectgeometry, booksubjecttrigonometry
