€)4t Mr. Woodhouse on the Truth of Conclusions 
X I 
In like manner (a -{- 6) can never be proved equal to 
j V — i _ x V' — i — i . A . . , . . . , 
a -f xv — 1 a b-\- &c. it is only an abridged 
symbol for the series ; there can be no ambiguity in the mean- 
X — — I 
ing of [a -f- b ) , since it is intended to represent the series 
X ✓ — I 
which arises from developing (a b) , after the same 
manner that [a + b) x is developed. 
The symbol v/ — 1 might arise from translating questions of 
which the statement involved a contradiction of ideas into alge- 
braic language, and reasoning on them, as if they really admitted 
a solution. For instance, if it were required to divide the number 
12 into two such parts, that their product should equal 37, 
this question in algebraic language would be 12 x — = 37 ; 
an absurd statement, since no real number can be assigned to x 
that verifies it ; but, according to the rules for transposition, the 
equation is# — x z = 37, is equivalent to x 1 — 1 2 x 36 = — 1 . 
If x were the sign of a real quantity, x — 6, or 6 — x, would be 
the square root of x * — 12 x 36; if therefore =f= (a: — 6) be 
put for the square root, it is put so by extending the rule proved 
for real quantities to this case ; and the radical placed over the 
symbol — 1, shews that such extension has been assumed ; hence 
x — 6 = - j- \/ — 1 is an expression of which the origin is 
known, being derived from x 1 — 12 x 36 == — 1. 
In the present inquiry, it is immaterial how the symbol \/ — 1 
originated : I think its origin most probably accounted for thus. 
The determination of general rules for the combination of alge- 
braic quantities, was probably posterior to the actual solution of 
many problems, effected by particular artifices. During the solu- 
tions, certain similar parcels of characters presented themselves. 
