obtained by Means of imaginary Qii antities. 115 
( 1 ) 4 , then the equality log. ( — af — log. (a) 2 becomes intel- 
ligible ; since it means that the measure of the ratio between 
( — a) 2 and ( — 1 ) 2 is equal the measure of the ratio between (a) 2 
and ( — 1 ) 2 ; but then this argument becomes the same as the 
former, and is equally illusory ; for — a and — 1 are in fact 
made a and 1. If logarithms be defined, the measures of ratios 
existing between real quantities, then it is absurd to attempt 
deducing the logarithms of negative quantities from any reason- 
ing on the relation that 1 has to — 1 ; since there is no neces- 
sary connection between 1 and — 1 ; and, independently of 
certain assumptions, the ratio of 1 : — 1 is perfectly unintel- 
ligible. Indeed the question admits no other meaning than, that 
I originally assigned it : if a form demonstrated for positive 
quantities be extended, then certain symbols may be exhibited, 
which, agreeably to such extension, are called the logarithms of 
negative quantities. 
Other arguments than those I have mentioned, were drawn 
from the theories of curves and fluxions, not only foreign to 
the question, which was purely algebraical, but of small weight ; 
had they been of greater, the inquiry would necessarily have 
been diverted on the nature of the connection existing between 
these theories and algebra. 
In this controversy, the predominancy of the “ Esprit Geo- 
metrique v is remarkable ; if, in an inquiry purely mathematical, 
any ambiguity or paradox presents itself, the most simple and 
natural method is, to recur to the original notions on which 
calculation has been founded. Instead of pursuing this method, 
the controvertists sought to derive illustration from obscure 
doctrines, or to discover the latent truth amidst the complex 
forms and involutions of analysis. 
