g6 Mr. Woodhouse on the Truth of Conclusions 
quadratic equations, forms such as — yx-\- 10, x'-\- 3 X — 10, 
appeared ; and therefore, to obtain a general rule for the solution, 
of all like forms, x z ± ax t±=b was invented ; and the solution, 
being made general, was necessarily extended to those cases 
which admitted no real answer. When such an extension is 
assumed, it is always indicated by the symbol %/ — 1; and 
hence, to know what operations are to be performed with the 
symbol \/ — 1 , it is necessary to recur to the quadratic forms 
from which it is arbitrarily derived. 
I now proceed to shew how sines, cosines, See. may be ex- 
pressed by means of exponential expressions ; and, for the sake 
of perspicuity, I avoid all fluxionary operations, and adhere to 
a purely algebraical calculus. 
To find the form for the developement of e x , let y = e x , 
or y = 1 -j- e — 1] = 1 + * — i| , n being any quantity 
which disappears of itself in the value of y. 
Now 1 e — i\ = 1 -f » (« - 1) + ( e — 1 )* -f 
See. — (arranging the terms according to the powers of rc) 
1 -f Arc -{- Brc 2 -f Crc 3 +, See. 
A = [e — 1 ) — \ [e — 1 y -{- i (e — 1 ) : — See. the values 
of B, C, &c. it is unnecessary to investigate, since they disap- 
pear in the calculation. 
tedious by considerations on the relative value of quantities, and unless the rule for 
transposition t>e clogged with needless limitations : an abstract negative quantity is 
indeed unintelligible ; but -x~ — a, or x — y — a — b (x < y), are perfectly in- 
telligible by means of their equivalent equations, x — a, y — x — b — a , to which 
they can be immediately reduced. The tendency of the reform proposed to be intro- 
duced into algebra is, it appears to me, to destroy the chief advantages of that art; 
its compendious and expeditious methods of calculation. 
