94 
MR. DAVIES GILBERT ON THE NATURE 
versally prevalent as scarcely to admit of a single exception among all those 
who make the powers of the human mind the subject of their peculiar research ; 
Classifications, Abstractions, Generalizations are allowed to be mere creatures 
of the reasoning faculty, existing nowhere but in the mind by which they are 
contemplated. To such unsubstantial existences any qualities may be imputed; 
but the only one known or useful in algebra, is the supposed even root of a 
real quantity taken in the scale opposite to that which has given the universal 
antecedent. The sign or mark indicating the extraction impossible to be per- 
formed, veils the real quantity, and renders it of no actual value until the sign 
is taken away by an involution, or the reverse of the supposed operation which 
that mark or sign represents, although by its arbitrary essence the quantity so 
veiled is in the mean time made applicable to all the purposes for which real 
quantities are used in all kinds of formulae. 
While therefore the sign of the supposed extraction of a root remains, the 
quantity to which it is prefixed has no more than a potential existence ; but 
it stands ready to exist in energy whenever that sign is removed. 
Consequently, without experience, it is impossible to know whether an im- 
plicit function of such an ideal quantity, will or will not be cleared by deve- 
lopment of the symbol indicating the supposed extraction of a root, that is, 
whether any actual value does or does not belong to such a function. 
Subject to the above conditions, namely, that the quantity veiled by the sign 
of a supposed extraction shall be treated in expansions and formulae according 
to the laws applicable to real quantities, and that it shall exist in energy when- 
ever an involution has reversed the supposed extraction of an even root, — 
Let (A) be supposed equal to */ — 1 ^ ~ 1 to find (A) ; 
Then, according to the established laws of real quantities arbitrarily extended 
to these that are imaginary, the log. of A = -/ - 1 X log. of v' — 1 ; but 
by a well known theorem, 
the log. of T = (n/~ - ,/=|) - i (./ ^T 2 - ^=f) 
+ + ( n /-1 
And this series X — 1 becomes - 2 X §-f + 7- ! &c. each alternate term 
