Elements of geometry and trigonometry . neAG ; and taking AC of any magnitude, applyit five times upon AG ; join the last pointof division G, and the extremity B, by thestraight line GB ; then draw CI parallel to _GB : AI will be the fifth part of the line pAB ; and thus, by applying AI five times ^-upon AB, the line AB will be divided intofive equal parts. For, since CI is parallel to GB, the sides AG, AB, are cutproportionally in C and I (Prop, XV.). But AC is the fifthpart of AG, hence AI is the fifth part of AB, Secondly. Let it be pro-posed to divide the line ABinto parts proportional tot
Elements of geometry and trigonometry . neAG ; and taking AC of any magnitude, applyit five times upon AG ; join the last pointof division G, and the extremity B, by thestraight line GB ; then draw CI parallel to _GB : AI will be the fifth part of the line pAB ; and thus, by applying AI five times ^-upon AB, the line AB will be divided intofive equal parts. For, since CI is parallel to GB, the sides AG, AB, are cutproportionally in C and I (Prop, XV.). But AC is the fifthpart of AG, hence AI is the fifth part of AB, Secondly. Let it be pro-posed to divide the line ABinto parts proportional tothe given lines P, Q, A. draw the indefi-nite line AG ; make AC =P, CD=Q, DErrR; jointhe extremities E and B ;and through the points C, D, draw CI, DF, parallel to EB ; the line AB will be dividedinto parts AI, IF, FB, proportional to the given lines P, Q,R. For, by reason of the CI, DF, EB, the parts AI,IF, FB, are proportional to the parts AC, CD, DE ; and byconstruction, these are equal to the given lines P, Q, BOOK IV. 99 PROBLEM II.
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