FROM NUMBER TO QUATERNION. 67 



those ideas are shared by many, and it is not rare to see even 

 professors of mathematics working under that delusion that com- 

 plex quantities are properly without existence. 



Complex or so called imaginary quantities are, truly speaking, 

 no more imaginary than the real quantities. A negative quantity 

 may be an imaginary solution for a certain problem whilst on 

 the other hand a complex quantity may be a real solution for some 

 kinds of questions treated in an appropriate manner.* 



I trust that the present paper will help a few persons to under- 

 stand clearly the real nature of complex quantities in general, and 

 will clearly show that the calculus of quaternions is a new algebra, 

 a generalization of the old one. 



Historical. — A short historical account of the question we are 

 .about to treat may prove interesting, and at any rate may serve 

 to rectify certain historical errors in Tait's quaternions.^ 



It is in a remarkable pamphlet printed at Paris in 1806: " Essai 

 sur une maniere de representer les quantites imaginaires dans 

 les constructions geometriques," that an otherwise unknown 

 Genevese mathematician Argand, gave the representation of the 

 so called imaginary quantities. £ 



The geometrical constructions, now in general use, which have 

 thrown such a light on the operations of complex quantities, are 

 all found in that pamphlet for the first time ; so that it is inexact 

 to say, as Tait does in his treatise on quaternions: — "In the 

 present century Argand, Warren and others extended the results 

 of Wallis and de Moire. They attempted to express as a line the 

 product of two lines each represented by a symbol such as a + b v^— i. 

 To a certain extent they succeeded, but simplicity was not gained 



* I shall always remember the exclamation of a bachelor of science — a 

 clever physicist — to whom I was explaining the theory of elliptic functions 

 when I came to speak of functions of a complex argument, ** You do not 

 mean it has any physical application ! " He holds a very different idea 

 just now. 



f Being unable to get the third edition, I can only speak about the 

 second edition. 



% However according to a recent work : "A History of Mathematics " 

 by F. Cajori, New York, 1894, the first one to represent quantities of form 

 b \/—i was H. Kuhn, a teacher of Dantzig, iD a publication of 1750-1751. 

 Unfortunately no exact reference is given. 



