An elementary treatise on the differential calculus, in which the method of limits is exclusively made use of, by the Rev. M. O'Brien.

48

THE DIFFERENTIAL NOTATION.

Origin of 83. Again it may be asked, what is the reason of using
the ndf() the letter d before f(x) and x, to represent the terms of the
tion ~dx 'ratio by which we denote /f( )? This question may be
and of the
term differ- answered by the following very brief account of the origin of
ential. this notation and of the term differential.
f (M') -f(x), and x'-x are the differences between corresponding values of f(x) and x, and they are often represented by a S prefixed to f (G) and x, in this manner, viz.
Sf(x), S3; ~ being simply an abbreviation of the words
"the difference between two values of."
Now, Sf(x) and Sx become zero when v'= ix, and then
their ratio X ceases to be a definite quantity; but so long
as ' is not actually equal to x the ratio is a definite
quantity. We may therefore conceive Sf(x) and Bx as small
as we please though not actually zero, without rendering our
conception of the ratio at all vague or difficult: indeed
it is just as easy to conceive that a definite ratio subsists between-, /f(x) and 3x when they are in a state of extreme
smallness as when they are of ordinary magnitude for our
idea of a ratio is quite independent of the actual magnitude
of the quantities composing it.
THe differences Sf(x) and 4x when in a state of extreme smallness were called diferentials by Leibnitz (i.e.
minute differences), and the symbols df(x) and dx were
made use of by him to represent them; d, like S, being
simply an abbreviation of the words "differential of."
In all calculations into which these differentials entered
he supposed them to be what are called infinitesimals, i.e.
quantities so small, that they may without error be neglected compared with ordinary quantities, and on this supimpossible to represent what are called total differentials without this notation. It is
peculiarly adapted to the case of definite and multiple integrals in the Integral Calculus. And it is a very expressive notation, which makes it peculiarly convenient in
mixed mathematics; e.g. in the case of the principle of virtual velocities applied to
an example.