95 
obtained by Means of imaginary Quantities. 
which it was necessary either to combine or separate ; and, to 
obtain general rules for their combination and separation, the 
first algebraists feigned forms similar to what really presented 
themselves in specific cases : * thus, in questions producing 
• It Ms been already observed, that, the determination of general rules for algebraic 
operations was posterior to the actual solutions of problems. To obtain a rule for the 
multiplication of algebraic quantities, a form such as a — b -f c — m, was proposed 
to be multiplied by d — e+ f—n; since it was necessary to have a law for the mul- 
tiplication of the signs , a general one was established, that like signs multiplied 
produce -f , unlike — , either from proving such law when (a -f c ) was > b -f m, 
and (d +./) > (« + «)> or from remarking that, in the solution of problems, the ob- 
servance of such law always produced true conclusions. It is very certain that the 
tnind can form no idea of an abstract negative quantity ; and therefore nothing can be 
affirmed concerning the multiplication of — a, and — b, nor of ,a — b, and ,c — d, 
if a is < b, or e < d. Let us attend, however, to the real meaning of negative quan- 
tities, and to the cause of their appearance in the solution of problems. The rule for 
transposition is, that quantities may be transferred from one side of the equation to 
the other, changing their signs. By virtue of this rule, an equation may appear under 
the form — x r= a — b, \a < b), or x — y ~ a — b, x < y, a < b) ; which equa- 
tions, abstractedly considered, may appear absurd, but become intelligible by means 
of the equations x zzb — a, y — x — b — a, to which they are significant, and to 
which' they may be immediately reduced. Suppose now, — x — a — b is to be mul- 
tiplied by — z xx nr — ?r(7H < «) ; if the forms be reduced to their equivalent ones, 
x r: b — a, and z ~n — m, and then multiplied, the product may be proved 
xz ~bn — bm — an + a m. Now, let (a — b) be multiplied by (m — n), in the 
same manner as it ought to be if a were > b, m > n, and the product is, 
(am — an — bm + bn), or (bn — bm — an -f am), the same as arose from mul- 
tiplying b — a by n — m, and which is equal xz; hence _ x x — x must be put 
xz; hence, in multiplication, we are sure to have right results by always observing the 
law that the product of like signs is +, of unlike — . In a similar manner it may be 
proved, that the product of x — y — a — b, by m — n — d — c, will be truly ex- 
pressed by combining the quantities accprding to the same law for the signs. It is evi- 
dent how much the establishment of this law must facilitate calculation; since, without 
considering whether „r is greater or less than y, the product of (x —y) and of (m — n) 
may immediately be put down. Such equations a sx — yzza — b, ( < y) must fre- 
quently occur in calculation, unless every step of the process be rendered extremely 
