obtained by Means of imaginary Quantities. 93 
Convinced in ray own mind, that there can be neither para- 
doxes nor mysteries inherent and inexplicable in a system of 
characters of . our own invention, and combined according to 
rules, the origin and extent of which we can precisely ascertain, 
I have endeavoured, in the present memoir, to shew why certain 
conclusions obtained through the means of imaginary quantities 
are necessarily true: to effect this is my prime object; a sub- 
ordinate one is, to shew that the method founded on imaginary 
symbols is commodious, and proper to he adopted, because of 
easy and extensive application. 
It has been already observed, that demonstration ultimately 
depends on observations made on individual objects, and that 
a conclusion expressed by certain characters and signs, if gene- 
ral, must be true in each particular case that presents itself, on 
assigning specific values to the signs. After affixing a signifi- 
cation to the symbols x, +> &c. the product of (a -j- b) and 
{c -{- d ) can be proved equal to ( ac ) -j- ( ad ) -{- {be) -{- {bd); 
\ina—b, a can be proved equal , a, b, c, See. being the signs- 
of real quantities ; but nothing can be affirmed concerning the 
product of {a -f b s/' ' — 1), and {c dv' — 1 ), nor concern- 
ing the form na — b v/ — 1 ; and all that can be meant by the. 
form {a 4- b */ — - 1 ) x {c -}- d s/ — 1 ) is, that the characters 
are to be combined after the same manner that the signs of 
real quantities are; so that [a -- f bV — 1) x {c + d >/ — 1), 
and ac + ad V — 1 ebs/ — 1 — bd, are two forms equi- 
valent to each other, not proved equivalent, but put so, by ex- 
tending the rule demonstrated for the signs of real quantities to 
characters that are insignificant. 
