346 THREE GEOMETRICAL PROBLEMS [CH. XVI 



this hyperbola which passes through H cut the circle in P. 

 Draw PM perpendicular to OG and produce it to cut the circle 

 in Q. Then by the focus and directrix property we have 

 AP : PM = AH : HL = 2 : 1, .-. AP = 2 . PM = PQ. Hence, by 

 symmetry, AP = PQ = QR. :. AOP = POQ = QOR. 



I may conclude by giving the solution which Chasles* 

 regards as the most fundamental. It is equivalent to the 

 following proposition. If OA and OB are the bounding radii 

 of a circular arc AB, then a rectangular hyperbola having OA 

 for a diameter and passing through the point of intersection of 

 OB with the tangent to the circle at A will pass through one 

 of the two points of trisection of the arc. 



Several instruments have been constructed by which 

 mechanical solutions of the problem can be obtained. 



The Quadrature of the Circle^. 



The object of the third of the classical problems was the 

 determination of a side of a square whose area should be equal 

 to that of a given circle. 



The investigation, previous to the last two hundred years, 

 of this question was fruitful in discoveries of allied theorems, 

 but in more recent times it has been abandoned by those who 

 are able to realize what is required. The history of this subject 

 has been treated by competent writers in such detail that I shall 

 content myself with a very brief allusion to it. 



Archimedes showedj (what possibly was known before) that 

 the problem is equivalent to finding the area of a right-angled 



* Traits des sections coniquet, Paris, 1865, art. 37, p. 36. 



t See Montuela's Histoire des Recherches sur la Quadrature du Gercle, 

 edited by P. L. Lacroix, Paris, 1831 ; also various articles by A. De Morgan, 

 and especially his Budget of Paradoxes, London, 1872. A popular sketoh of 

 the subject has been compiled by H. Schubert, Die Quadratw des Zirkels, 

 Hamburg, 1889 ; and since the publication of the earlier editions of these 

 Recreations Prof. F. Radio of Zurich has given an analysis of the arguments of 

 Archimedes, Huygens, Lambert, and Legendre on the subject, with an intro- 

 duction on the history of the 'problem, Leipzig, 1892. 



X Archimedis Opera, KAkXov /Urpri<ns, prop, i, ed. Torelli, pp. 203 — 205 ; ed. 

 Heiberg, vol. i, pp. 258—261, vol. in, pp. 269—277. 



