obtained by Means of imaginary Quantities. 1 1 1 
-~ 4 =; hence (s/ — l )^ - " 1 == e~*, or (\/ ■ — i)^ - *, is an 
abridged symbol for the series 1 + ■ — - — ; — - — - -f- & c, 
I do not pretend to say, that such expressions as the above 
are likely to occur in investigation, and to be practically use- 
ful ; my sole concern is to shew, that they are perfectly intelli- 
gible, and the necessary consequences of certain assumptions. 
The paradoxes and contradictions mutually alleged against 
each other, by mathematicians engaged in the controversy* 
concerning the application of logarithms to negative and im- 
possible quantities, may be employed as arguments against the 
use of those quantities in investigation. The paradoxes and 
contradictions will quickly disappear, by adopting the same 
mode of explanation that has been already employed in this 
paper. The memoir of Euler is in some parts erroneous, and 
frequently unsatisfactory. 
The use of a mathematical definition is, to deduce from it the 
properties of the thing defined ; and, whatever definition of lo- 
garithms be taken, we either have immediately, or may deduce 
for the purpose of computation, an expression such as y = e x , in 
which x is the logarithm of e to the base e ; the develope- 
ment of e x has been proved to be 1 -{- A x -fi —— + T~ 7 ~J 4 
A being = (e - 1) — ± (e — 1 )‘ + -§■(* — 1 ) 3 — &c. 
* This controversy exercised along time the abilities of Leibnitz, Bernouilli, 
Euler, D’Alembert, and Foncenex. The Commercium Philosophicum et Ma- 
thematicum, published at Lausanne, in 1745. and containing the letters of the two 
first controvertists, I have never seen; but I presume, that all the essential arguments 
of the controversy are to be found in Foncenex’s Memoir, (Vol. I. Mem. de Turin,.) 
in Euler’s, (Mem. de Berlin, 1749.) and in D’Alembert’s Opuscules, Vol. I, 
