1 14 Mr. Woodhouse on the Truth of Conclusions 
Euler, confounding the common meaning of logarithms with 
their scientific definition, granted that the log. (i) i was equal 
log. ( — 1 }*, and endeavoured to reconcile the contradictions 
that immediately followed from such a concession. 
The arguments intended to prove that the logarithms of 
negative quantities were real, may easily be shewn to be nuga- 
tory. Euler, certainly too much attached to mere calculation, 
instead of directly opposing them, sought to divert their force. 
D’Alembert asserted, that the two progressions i, 2, 3, &c. 
— 1, — 2, — 3, &c. might have the same series of logarithms, 
0, p, q, r, See. This is true, if — 2 means 2 x ( — 1), — 3, 
3 x ( — 1), &c. or the progression — 1, — 2, — 3, &c. is the 
same as 1 x ( — 1), 2 ( — 1), 3 ( — 1), &c. wherein ( — 1) 
is considered as an unit, or as (x) a sign of a real quan- 
tity. But the question is thus evaded; since — 1, — 2, — 3, 
— , &c. is brought precisely under the same predicament as 
1, 2, 3, 4, &c. The only real point of inquiry could he, whe- 
ther, consistently with the system of logarithms established for 
positive quantities, the logarithms of negative quantities were 
real. 
A second argument brought by Bernouilli and D’Alembert 
was, that since a : — a — a' a == ( — af log. (a) 2 
=^= log. (— af ••• 2 log. (+ a) = 2 log. {—a) log. (a) = 
log (#). This proposition, affirmed by D’Alembert to be strictly 
true, viz. (a : ■ — a : : — -a : a), was granted to be so by Euler, 
although it ought to have been denied ; since, thus abstractedly 
proposed it is absurd and unintelligible, and impossible to be 
proved. If, however,* ( — af be assumed = [af, ( — i) z = 
* 1 have explained, in a former note, for what reasons, and in what circumstances, 
— a x — « is the same as a x a. 
