2^g THIRD REPORT— 1835. 



tifies the assertion, which we have made above, that quantities 

 or their symbols affected by the signs +, — , or cos 9 + \^ — \. 

 sin 6, are only distinguished from each other by the greater or 



less facility of their interpretation. 



The geometrical interpretation of the sign V — 1, when 

 applied to symbols denoting lines, though more than once 

 suggested by other authors, was first formally maintained by 

 M. Buee in a paper in the Philosojihical Transactions for 1806*, 

 which contains many original, though very imperfectly deve- 

 loped views upon the meaning and application of algebraical 

 signs. In the course of the same year a small pamphlet was pub- 

 lished at Paris by M. Argand, entitled Essai sur une Maniere 

 de repr^senter les Quanfites Imaginaires, dans les Construc- 

 tions G^om^triques, written apparently without any knowledge 

 of M. Buee's paper. In this memoir M. Argand arrives at this 

 proposition. That the algebraical sum f of two lines |, estimated 

 both according to magnitude and dii'ection, would be the dia- 

 gonal of the parallelogram which might be constructed upon 

 them, considei'ed both with respect to direction and magnitude, 

 which is, in fact, the capital conclusion of this theory. This 

 memoir of M. Argand seems, however, to have excited very 

 little attention ; and his views, which were chiefly founded upon 

 analogy, were too little connected with, or rather dependent 

 upon, the great fundamental principles of algebra, to entitle 

 his conclusions to be received at once into the great class 

 of admitted or demonstrated truths. It would appear that 

 M. Argand had consulted Legendre upon the subject of his me- 

 moir, and that a favourable mention of its contents was made 

 by that great analyst in a letter which he wrote to the brother 

 of M. J. F. Fran9ais, a mathematician of no inconsiderable 

 eminence. It was the inspection of this letter, upon the death 

 of his brother, which induced M. Franpais to consider this 

 subject, and he published, in the fourth volume of Gergonne's 

 Annales des Mathematiqties for 1813, a very curious memoir 

 upon it, containing views more extensive, and more completely 

 developed than those of M. Argand, though generally agreeing 

 with them in their character, and in the conclusions deduced 

 from them. This publication led to a second memoir upon the 

 same theory from M. Argand, and to several observations 

 upon it, in the same Journal, from MM. Servois, Franpais, 

 and Gergonne, in which some of the most prominent objections 

 to it were proposed, and partly, though very imperfectly, an- 



* This paper was read in 1805. f -ta somme dirigee. 



I Lignes dirigies. 



