. Memoirs and proceedings of the Manchester Literary & Philosophical Society. the 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,su
. Memoirs and proceedings of the Manchester Literary & Philosophical Society. the 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 parabola and the circle y = - x4 I Z x~ +y* —^ x + 4:i-<z = sin 3 o,. Fig. II. Use of Parabola. Manchester Memoirs, Vol. lix. (1915), No. 13. 17 Fig. 11 shows the method of construction. A OB isthe angle to be trisected. 0 is the origin and OB theaxis of x. OC\s made unit length, 0F=0D = 2\0C) = 2. 0S=~ , <9£ = -^ 2*^= #2 - OFsin ADB=2 sin 3a = 8 ^ The parabola has its vertex at 0 and focus at 5. Thecircle has its centre at H and passes through O. PQ isdrawn parallel to OB through the intersection P, and OQis made equal to OC; this is unit length. Then if theordinate of P is ;, sin 006 = —-= = y = sin a = sin . OQ 3 Hence the angle QOC is one-third of the angle A OB. (4) Oilier Methods derived fi om those already given. In any of the preceding cases (IV. I, 2 or 3) the equations of the circle and conic may be combined to form an equation of the second degree, which will be the equation of another conic intersecting the circle in the four points common to the circle and the original conic. For example : in the method of Clairaut (IV, 2, Fig.
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Keywords: ., bookcentury1800, bookdecade1880, booksubjectscience, bookyear1888
