. Memoirs and proceedings of the Manchester Literary & Philosophical Society. 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
. Memoirs and proceedings of the Manchester Literary & Philosophical Society. 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. Fig. 10. Methods of Pappus and Clairaut. Draw PQ parallel to BH cutting ME at N. Join BP,QH and SP. Then, as before, the arcs BP, PQ and QH are L PBH= IL PSH= L BSP= ~ L BSH~ It ABC. 2 3 3 Method of Clair ant. This method is the most direct ofthe three. In Fig. 10 let BSHbe the angle to be S as centre describe a circle cutting off SB and BH. Trisect BH at F and K. Bisect the angle BSHby SM, cutting BH at E. Construct a hyperbola ofeccentricity =2, with B as focus and 5M as directrix, andlet the branch which passes through .Fcut the circle at PS. Then the angle BSP is one-third of the angleBSHy as previously proved. It can easily be shown that the circle with centre 5cuts the hyperbola at H, P, and at two other points Pu Ps,such that PPyP, is an equilateral triangle. 16 Gee AND A DAMSON, Trisecting an Angle. (3) Use of a parabola. This method for trisecting an angle is given byDescartes and is re-stated in Balls MathematicalRecreations. The
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Keywords: ., bookcentury1800, bookdecade1880, booksubjectscience, bookyear1888
