CH. XVI] THREE GEOMETRICAL PROBLEMS 347 



triangle whose sides are equal respectively to the perimeter of 

 the circle and the radius of the circle. Half the ratio of these 

 lines is a numher, usually denoted by nr. 



That this number is incommensurable had been long sus- 

 pected, and has been now demonstrated. The earliest analytical 

 proof of it was given by Lambert* in 1761 ; in 1803 Legendref 

 extended the proof to show that ir' was also incommensurable ; 

 and recently LindemannJ has shown that ir cannot be the root 

 of a rational algebraical equation. 



An earlier attempt by James Gregory to give a geometrical 

 demonstration of this is worthy of notice. Gregory proved § 

 that the ratio of the area of any arbitrary sector to that of 

 the inscribed or circumscribed polygons is not expressible by 

 a finite number of algebraical terms. Hence he inferred that 

 the quadrature was impossible. This was accepted by Montucla, 

 but it is not conclusive, for it is conceivable that some particular 

 sector might be squared, and this particular sector might be the 

 whole circle. 



In connection with Gregory's proposition above cited, I may 

 add that Newton || proved that in any closed oval an arbitrary 

 sector bounded by the curve and two radii cannot be expressed 

 in terms of the co-ordinates of the extremities of the arc by a 

 finite number of algebraical terms. The argument is condensed 

 and difficult to follow: the same reasoning would show that 

 a closed oval curve cannot be represented by an algebraical 

 equation in polar co-ordinates. From this proposition no 



* Mtmoires de VAcadimit de Berlin for 1761, Berlin, 1768, pp. 265— 



322. 



t Legendre's Geometry, Brewster's translation, Edinburgh, 1824, pp. 239— 



245. 



% TJeber die Zahl ir, Mathcmatische Annalen, Leipzig, 1882, vol. xx, pp, 21S — 

 225. The proof leads to the conclusion that, if x is a root of a rational 

 integral algebraical equation, then e x cannot be rational : hence, if 7r» was the 

 root of such an equation, e** could not be rational ; but e" is equal to - 1, and 

 therefore is rational; hence iri cannot be the root of suoh an algebraical 

 equation, and therefore neither can t. 



§ Vera Circuit et Hyperbolae Quadratura, Padua, 1668 : this is reprinted in 

 Hujgens's Opera Varia, Leyden, 1724, pp. 405—462. 



U Principia, bk. i, seotion vi, lemma xxviii. 



