OF NEGATIVE AND OF IMAGINARY QUANTITIES. 
93 
continuously above and below the line of the apsides are reciprocal to each 
other ; and their product equal to the square of either ordinate at the zero 
point, which, in the position of the curves corresponding to the natural system, 
is when the fluxions of x and y are equal. 
It is obvious from inspecting this figure, that if m be a natural number and 
M its logarithm, both in the scale of (a), and M be taken in the scale of (&), or 
— (a), it will become the logarithm of — - ; but M in the scale of ( b ) is also 
the logarithm of m in the same scale ; and M taken as at first in the scale of 
(a), or — ( b ), is the logarithm of 
If the two curves are moved on each other, so that the two ordinates mea- 
suring 2.302585 in reference to the former unit be made continuous, and that 
length be taken as a new unit, they will then represent the tabular logarithms 
in both scales. 
I flatter myself with having now clearly established the principle, that all 
properties belonging to the scales of (a) and (b) are mutually interchange- 
able ; and that consequently imaginary quantities will be found in the even 
roots of either scale, as the universal antecedent is taken in the other. And 
this leads to the question, 
What are imaginary quantities ? 
I answer. Creations merely of arbitrary definition, endowed with properties 
at the pleasure of him that defines them, the whole dispute respecting imagi- 
nary quantities turning on the point contested from the earliest times between 
the hostile sects of Realists and Nominalists, descending through Plato, Ari- 
stotle and the Stoics, to the Philosophers of Alexandria, and from them to the 
Schoolmen, who imbittered their discussions with theological controversy and 
persecution. 
If the conceptions of the mind, all abstractions and generalizations, were 
considered as substantial forms, possessing existences distinct from the intelli- 
gences contemplating them ; or, as some writers have expressed themselves 
autousia gaudentes, the ^<pou immutable and eternal, which united to matter 
constitute the universe, nothing could be equally absurd with the supposition 
of impossible or imaginary quantities; but according to the theory now so uni- 
