2 Lamb, A New Version of ArgancTs Proof. 



for all finite values o{ x and j^ which do not make R — o. 

 Hence there must be some finite value of s which makes 

 R = o,f{z) = o. 



The leading idea of this argument is identical with 

 that of Argand's original proof,* but the theorem that a 

 function satisfying (4) and the other conditions indi- 

 cated cannot have a minimum (or maximum) value allows 

 the reasoning to be put very succinctly. 



The method involves, of course, the assumption that 

 a continuous function (in this case a function of two 

 variables) does actually attain its lower limit. Scruples 

 on points such as this did not as a rule gain currency 

 until a much later period ; but it is of interest to note, 

 as a matter of mathematical history, that the stringency 

 of Argand's argument was questioned at the outset, on 

 precisely this ground, by Servois.f The reply made by 

 ArgandJ is hardly successful from a modern standpoint, 

 and indeed the general recognition of the fact that there 

 is an assumption in the matter, and the formulation of a 

 regular demonstration by Weierstrass, belong to quite 

 recent times. 



2. The proof above given forms, in a way, the counter- 

 part of a well-known investigation by Cauchy.§ If ds be 

 an element of arc of a curve in the plane xy, and dn an 



* Ann. de Math., t. 4 , p. 133 ; t. 5., p. 197 {1815). A modern version 

 is given in Chrystal's Algebra, t. i, c. 12, § 22, and in many Continental 

 works. 



t Ann. de Math., t. 4, p. 222. " Ce n'est point assez, ce me 



semble, de trouver des valeurs de x qui donnent au polynome des valeurs 

 sans cesse decroissantes ; il faut, de plus, que la loi du decroissement amene 

 necessairement le polynome a zero, ou qu'elle soit telle que zero ne soit pas, 

 si Ton peut s'exprimer ainsi, f asyrnptote du polynome." 



Xlbid., p. 197, 



§ Reproduced by Todhunter and Burnside in their treatises on the 

 71ieory of Equations. 



