OF NEGATIVE QUANTITIES. «* 251 



matics, we may observe that great use is made of impossible roots ; and we 

 may fairly conclude that if these quantities are of so great service to mathe- 

 maticians, even while they are ignorant of their real nature, tliey will be of 

 much greater service when the true theory respecting them is known ; we may 

 reasonably expect, that our knowledge of algebra will be increased when the 

 nature of impossible roots is understood in the same manner as that of possible 

 roots ; these are the general advantages which we shall derive from the geo- 

 metric representation of the square roots of negative quantities : but there is 

 one particular advantage, and that, one of the greatest importance, which 

 arises from the definition of addition ; addition is performed in the same manner 

 as composition of motion in dynamics, therefore any question in dynamics 

 where the motion of the bodies is confined to one plane, becomes a mere 

 question of algebra, the laws of motion being contained in the definitions of 

 algebra. .' 



Before I conclude this paper, it will be proper to take notice of two works 

 which have appeared on this subject ; the first a paper in the Philosophical 

 Transactions, for the year 1806, p. 23 : intitled "Memoire sur les Quantit^s 

 Imaginaires, par M. Bue'e ;" the second, a work intitled " La vraie Th^orie 

 des Quantites Negatives et des Quantit^s pr6tendues Imaginaires, par C. V. 

 MouREY, Paris, 1828." I was not aware of the existence of M. Bue'e's paper 

 till November 1827, when my treatise was in the press: at that time I read 

 his paper, and also the article upon it in the Edinburgh Review of July 1808. 

 M. Bue'e begins with stating that the negative sign has two different signifi- 

 cations in algebra ; viz. that if algebra be considered as a universal arithmetic, 

 the negative sign is a sign of subtraction, but that if algebra be considered as 

 a mathematical language, the negative sign is a sign of a quality ; on this 

 point he makes the following observations : 



''Consider^s comme signes d'op^rations arithm^tiques, -|- et — sont les 

 signes, I'un de I'addition, I'autre de la soustraction." 



" Consid^res comme signes d' operations geometriques, ils indiquent des di- 

 rections oppos6es. Si I'un, par exemple, signifie qu'une ligne doit etre tir^e de 

 gauche a droite, I'autre signifie qu'elle doit etre tiree de droite a gauche." 



" Mis devant une quantite, q, ils peuvent designer, comme je I'ai dit, deux 

 operations arithmetiques oppos6es dont cette quantite est le sujet. Devant 



2 k2 



