. Memoirs and proceedings of the Manchester Literary & Philosophical Society. ows that r cos /3 = d,which is the condition that the hyperbola shall passthrough the point where OB cuts the tangent to thecircle at A (compare Fig. J). The method of Pappus is deduced by simply taking(Jj = 7r/2. (2) Use of a hyperbola of eccentricity curve may be used in a very simple manner totrisect a circular arc, and hence to trisect an angle. Thefollowing constructions, which only differ slightly, aregiven by Newton (<?), Pappus (/;) and Clairaut (<;), and arerestated in the Mathematical Recreat
. Memoirs and proceedings of the Manchester Literary & Philosophical Society. ows that r cos /3 = d,which is the condition that the hyperbola shall passthrough the point where OB cuts the tangent to thecircle at A (compare Fig. J). The method of Pappus is deduced by simply taking(Jj = 7r/2. (2) Use of a hyperbola of eccentricity curve may be used in a very simple manner totrisect a circular arc, and hence to trisect an angle. Thefollowing constructions, which only differ slightly, aregiven by Newton (<?), Pappus (/;) and Clairaut (<;), and arerestated in the Mathematical Recreations of W. Ball. 14 GEE AND Adamson, Trisecting an Angle. Method of Newton.—Let ABC {Fig. 9) be the angleto be trisected. On BC take any point E, and throughE draw EM perpendicular to BC Construct a hyper-bola of eccentricity =2, with EM as directrix and B ascorresponding focus. Let F and H be the vertices, sothat BF= 2FE and BH= 2HE. Draw BO perpendicularto AB, meeting ME at 0. With 0 as centre and OB orOH as radius, describe a circle cutting at P the branch of. Fig. 9. Method of Sir I. Newton. the hyperbola which passes through F. Join PB. Thenthe angle ABP will be one-third of the angle ABC Toprove this, draw PQ parallel to BH, intersecting ME atN. join QH, PH. Then PQ=2PN=PB= QH. There-fore the angle ABP, which is equal to the angle BHP, isequal to half the angle PBH and to one-third of theangle ABC. Method of Pappus.—Let ABC {Fig. 10) be the angleto be trisected. As before {Fig. 9) construct the hyper-bola and determine the point 0. With 0 as centre andOB or OH as radius describe a segment of a circle cuttingME produced at 5. Join SB, SH. With 5 as centreand SB or SH as radius describe a circle cutting at P Manchester Memoirs, Vol. //It. (1915), No. 13. 15the branch of the hyperbola which passes through F. A \ M
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