CH. XVI] THREE GEOMETRICAL PROBLEMS 345 



Among other constructions given by Pappus* I may quote 

 the following. Describe a hyperbola whose eccentricity is two. 

 Let its centre be G and its vertices A and A'. Produce CA' 

 to S so that A'S = CA'. On AS describe a segment of a 

 circle to contain the given angle. Let the orthogonal bisector 

 of AS cut this segment in 0. With centre and radius OA or 

 OS describe a circle. Let this circle cut the branch of the 

 hyperbola through A' in P. Then SOP = $S0A. 



In modern times one of the earliest of the solutions by a 

 direct use of conies was suggested by Descartes, who effected 

 it by the intersection of a circle and a parabola. His con- 

 structionf is equivalent to finding the points of intersection, 

 other than the origin, of the parabola y*=\ as and the circle. 

 a? + y 2 — if-oc + 4cm/ = 0. The ordinates of these points are given 

 by the equation 4y" = 3y — a. The smaller positive root is the 

 sine of one-third of the angle whose sine is a. The demonstra- 

 tion is ingenious. 



One of the solutions proposed by Newton is practically 

 equivalent to the third one which is quoted above from Pappus. 

 It is as followsj. Let A be the vertex of one branch of a 

 hyperbola whose eccentricity is two, and let S be the focus of 

 the other branch. On AS describe the segment of a circle con- 

 taining an angle equal to the supplement of the given angle. 

 Let this circle cut the S branch of the hyperbola in P. Then 

 PAS will be equal to one-third of the given angle. 



The following elegant solution is due to Clairaut§. Let 

 AOB be the given angle, Take OA = OB, and with centre 

 and radius OA describe a circle. Join AB, and trisect it in 

 H, K, so that AH = HK = KB. Bisect the angle A OB by OG 

 cutting AB in L. Then AH= 2.HL. With focus A, vertex 

 H, and directrix OG, describe a hyperbola. Let the branch of 



* Pappus, Mathematicae Collectiones, bk. iv, prop. 34, pp. 99 — 104. 



t Geomctria, bk. in, ed. Sohooten, Amsterdam, 1659, p. 91. 



I Arithmctica Universalis, problem xhi, Ralphson's (second) edition, London, 

 1728, p. 148; see also pp. 243—245. 



§ I believe that this was first given by Clairaut, but I have mislaid my 

 reference. The construction occurs as an example in the Geometry of Conies, 

 by 0. Taylor, Cambridge, 1881, No. 308, p. 126. 



