344 THREE GEOMETRICAL PROBLEMS [CH. XVI 



BG : OD = OD : GE = GE : OA Hence OD and GE are two 

 mean proportionals between any two lines BG and OA. 



The Trisection of an Angle* - 



The trisection of an angle is the second of these classical 

 problems, but tradition has not enshrined its origin in romance. 

 The following two constructions are among the oldest and best 

 known of those which have been suggested; they are quoted 

 by Pappus f, but I do not know to whom they were due 

 originally. 



The first of them is as follows. Let A OB be the given 

 angle. From any point P in OB draw PM perpendicular to 

 OA. Through P draw PR parallel to OA. On MP take a point 

 Q so that if OQ is produced to cut PR in R then QR = 2 . OP. 

 If this construction can be made, then AOR = ^AOB. The 

 solution depends on determining the position of R. This was 

 effected by a construction which may be expressed analytically 

 thus. Let the given angle be tan -1 (bja). Construct the hyper- 

 bola xy = ab, and the circle (x - a) 2 + (y - by = 4 (a 8 + ¥). Of 

 the points where they cut, let x be the abscissa which is greatest, 

 then PR — x— a, and tan -1 (b/x) = £ tan -1 (b/a). 



The second construction is as follows. Let A OB be the 

 given angle. Take OB = OA, and with centre and radius OA 

 describe a circle. Produce AO indefinitely and take a point 

 on it external to the circle so that if GB cuts the circumference 

 in D then CD shall be equal to OA. Draw OE parallel to GDB. 

 Then, if this construction can be made, A0E=^AOB. The 

 ancients determined the position of the point G by the aid of 

 the conchoid : it could be also found by the use of the conic 

 sections. 



I proceed to give a few other solutions, confining myself to 

 those effected by the aid of conies. 



* On the bibliography of the subject see the supplements to L'Intermgdiaire 

 des Mathgmaticiens, Paris, May and June, 1904. 



+ Pappus, Mathematieae Collectiones, bk. iv, props. 32, 33 (ed. Commandino, 

 Bonn, 1670, pp. 97 — 99). On the application to this problem of the traditional 

 Greek methods of analysis see Geometrical Analysis, by J. Leslie, Edinburgh, 

 second edition, 1811, pp. 245—247. 



