io8 Mr. Woodhouse on the Truth of Conclusions 
e xV —i anc j e y V _ i eX p an ded, and then their terms com- 
bined according to the rules for the multiplication of quantities, 
or whether e( x + J>) ^ ~ 1 be immediately expanded, by writing 
[x -f- y) s/ — i for at, in the series for e x . 
The other demonstrations examined will appear conducted on 
the same principle, which is simple, and of easy and immediate 
application : hence, although the symbol s/ — 1 be beyond the 
power of arithmetical computation, the operations in which it is 
introduced are intelligible, and deserve, if any operations do, 
the name of reasoning. 
It is almost superfluous to observe, that if the operations by 
means of imaginary symbols have appeared to be necessarily 
true, the arguments founded on the analogy subsisting be- 
tween the circle and hyperbola must be abandoned, as unsatis- 
factory. What has been proved concerning the properties of 
lines appertaining to a circle, has been so without any mention 
of the hyperbola ; and I may say, without danger of refutation, 
that the demonstrations would be strictly true, if such a curve 
as the hyperbola had never been invented. Add to this, that 
imaginary expressions are useful in leading to just conclusions, 
in investigations purely algebraical. 
The chief purpose of this Paper is fulfilled, if it has appeared 
that the operations with imaginary symbols possess the evidence 
and rigour of mathematical demonstration : whether it is conve- 
nient to use imaginary quantities in analytical investigation, must 
be determined on the grounds of abridgment and extensive ap- 
plication. In the cases that I have considered, imaginary expres- 
sions are not, I know, indispensably necessary : they are excluded 
from each of three different methods for the solution of propo- 
