OF NEGATIVE AND OF IMAGINARY QUANTITIES. 
95 
vanishing. But 2 — f + f + f — | &c. = the quadrantial arc of a circle to 
radius unity. 
Therefore the log. of A = — quad. arc. 
And A = e ~ quad ' arc = 0.2078796. 
Consequently, */ — 1 ^ 1 is an abbreviated mark or symbol, according to the 
above arbitrary conditions, for the radix of the natural system of logarithms 
raised to a negative power, indicated by the quadrantial arc of a circle to radius 
unity. And in the event of — 1 ^ 1 ever occurring in the solution of a 
problem, e -quadl arc or 2.71828 - 1 " 5 ' 08 or 0.2078796 may be substituted for it. 
And this is what practically happens in regard to all expressions apparently 
imaginary, which are found to represent real quantities, as is well known in 
cases of circular arcs and logarithms. These mental abstractions have more- 
over extended the bounds of analysis far beyond the utmost limits it could 
otherwise have attained ; they have bestowed harmony and simplicity of form 
on its most recondite investigations, and eminently has their use been im- 
portant in equations, by resolving them into a number of simple factors equal 
to the dimensions of the equation in its highest term. 
It appears from these considerations, that several ingenious mathematicians 
have taken an incorrect view of ideal quantities, by mistaking incidental pro- 
perties for those which constitute their essence ; as, for example, when they 
are supposed to be principles of perpendicularity, because they may in some 
cases indicate extension at right angles to the direction here designated by (a) 
and (b), but with an equal degree of propriety might the actually existing 
square root of a quantity be considered as the principle of obliquity ; because, 
in certain cases, it indicates the hypothenuse of a right-angled triangle. 
I would here notice an error in reasoning (as it appears to me), fallen into 
by all authors who have endeavoured to explain the mode of arriving at a true 
conclusion respecting the sines and cosines of multiple arcs ; which reasoning 
imputes actual properties to ideal quantities, instead of deriving them all from 
mere arbitrary convention. 
Given the sine, and consequently the cosine of an arc, to find the sine and 
cosine of n times that arc: 
