CH. XVI] THREE GEOMETRICAL PROBLEMS 343 



but I will enunciate only those proposed respectively by Vieta, 

 Descartes, Gregory of St Vincent, and Newton. 



Vieta's construction is as follows*. Describe a circle, centre 

 0, whose radius is equal to half the length of the larger of the 

 two given lines. In it draw a chord AB equal to the smaller 

 of the two given lines. Produce AB to E so that BE = AB. 

 Through A draw a line AF parallel to OE. Through draw a 

 line DOCFG, cutting the circumference in D and C, cutting AF 

 in F, and cutting BA produced in G, so that GF=0A. If this 

 line can be drawn then AB : GG=GG : GA = GA : CD. 



Descartes pointed outf that the curves 



a? = ay and a? + y 1 = ay + bx 

 cut in a point (x, y) such that a : x = x : y = y :b. Of course 

 this is equivalent to the first solution given by Menaechmus, 

 but Descartes preferred to use a circle rather than a second 

 conic. 



Gregory's construction was given in the form of the following 

 theorem J. The hyperbola drawn through the point of inter- 

 section of two sides of a rectangle so as to have the two other 

 sides for its asymptotes meets the circle circumscribing the 

 rectangle in a point whose distances from the asymptotes are 

 the mean proportionals between two adjacent sides of the rect- 

 angle. This is the geometrical expression of the proposition 

 that the curves xy = ab and a? + y* = ay+bx cut in a point 

 (x, y) such that a : x = x : y = y :b. 



One of the constructions proposed by Newton is as follows§. 

 Let OA be the greater of two given lines. Bisect OA in B. 

 With centre and radius OB describe a circle. Take a point 

 C on the circumference so that BG is equal to the other of the 

 two given lines. From draw ODE cutting AG produced in 

 D, and BG produced in E, so that the intercept DE— OB. Then 



* Opera Mathematica, ed. Sohooten, Leyden, 1646, prop, v, pp. 242—243. 



+ Geometria, bk. m, ed. Sehooten, Amsterdam, 1659, p. 91. 



J Gregory of St Vincent, Opus Geometrician Quadraturae Circuit, Antwerp, 

 1647, bk. vi, prop. 138, p. 602. 



§ Arithmetica Univertalis, Ralphson's (second) edition, 1728, p. 242 ; see 

 also pp. 243, 245. 



