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, they 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 Quantites 
Imaginaires, par M. Bue'e the second, a work intitled “ La vraie Theorie 
des Quantites Negatives et des Quantites pretendues 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 : 
“Consideres comme signes d’operations arithmetiques, + et — sont les 
signes, l’un de l’addition, l’autre de la soustraction.” 
“ Consideres comme signes d’ operations geometriques, ils indiquent des di- 
rections opposes. Si l’un, par exemple, signifie qu’une ligne doit etre tiree de 
gauche a droite, l’autre signifie qu’elle doit etre tiree de droite a gauche.” 
“ Mis devant une quantite, q, ils peuvent designer, comme je 1’ai dit, deux 
operations arithmetiques opposees dont cette quantite est le sujet. Devant 
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