CHAPTER IV 
THE REPRESENTATION OF POINTS OF A PLANE 
§ 1. Representation by fheans of point-pairs. 
We had frequent occasion to remark in the course of the 
last chapter that the methods for representing the points of 
a curve, or at least some of them, were perfectly adequate to 
represent all the points of a complex plane. The reason for 
explaining them in that chapter, instead of waiting until the 
present one, was that the writers who first discovered them 
were more interested in the more restricted problem. We now 
return to these methods and consider them from the broader 
point of view, and in comparison with other methods which 
have been devised for representing all the points of a com 
plex plane. 
We showed on p. 75 that the usual Gauss representation of 
the complex points of a single line could be described in such 
geometrical terms as to suggest an immediate extension to the 
representation of all finite points of the plane, and mentioned 
in that connexion the name of Laguerre. This admirable 
geometer seems to have been the first writer to really appre 
hend the scope and meaning of the problem.* His ideas were 
greatly developed by two others. Gaston Tarry studied the 
elementary properties of the representation with great patience 
and a wealth of detail.f Eduard Study reworked the whole 
subject in its wider aspects, bringing to the discussion that 
profundity of vision which is characteristic of all of his mathe- 
* ‘ Sur l’emploi des imaginaires en géométrie’, Collected Works, Paris, 1906, 
vol. ii, pp. 88 fF. 
t Tarry’s papers are found under a variety of titles in the Proceedings of the 
Association française pour V Avancement des Sciences, Toulouse, 1887, Oran, 1888, 
Paris, 1889, and Marseilles, 1891.