obtained by Means of imaginary Quantities. 99 
To remove all doubt and occasion of cavil, it is to be under- 
stood, that [e x * * — 1 -f- e- x v - l ) means, that the terms of the 
series which e x v — 1 represents, are to be connected with the 
terms of the series that e~~ x v - 1 represents, according to the 
rules obtaining for the addition of real quantities : again, that 
x s/ — 1 — x s/ — 1 is put equal o, not by bringing x V — 1 
under the predicament of quantity, and making It the subject 
of arithmetical computation, but by giving to 4" anc J “ their 
proper signification when used with real quantities, and then 
they designate reverse operations : again, that is equal 
to x, not because it is true that a quantity multiplied and di- 
vided by the same number remains the same, but because 
X - - ZA means, that x is to be combined with v/ — 1 after the 
v — 1 
manner that real quantities are in multiplication, and then di- 
vided after the manner that real quantities are in division ; and 
therefore, since the two operations are the reverse of each other, 
. 1 and x must be equivalent expressions. * 
To facilitate the solution of the propositions demonstrated by 
means of imaginary quantities, I previously observe, that, A be- 
ing any symbol whatever, A x [e xV ~ l -f e~ xx/ ~ — e y ' / ~-~ l ) i 
and A e xx/ — 1 -\- A e— x ^ — 1 — A e$ ^ ~~ *, are equivalent ex- 
* After this manner ought to be interpreted what Maclaurin and Bernouilli 
have indistinctly expressed, concerning the compensation that ought to take place when 
real quantities are represented by means of imaginary symbols. — Bern. Vol. I. No. 70. 
Macl. Fluxions, Art. 699, 763. 
O 2 
