a calculus proposed by the inventor, Mr Landen, as a substitute for the method of fluxions. The object of this substitution was to avoid introducing the idea of motion, and of quantities infinitely or indefinitely small, into mathematical investigation. The residual analysis accordingly proceeds, by taking the difference of the same function of a variable quantity in two different states of that quantity, and expressing the relation of this difference to the difference between the two states of the said variable quantity itself. This relation being first expressed generally, is then considered in the case when the difference of the two states of the variable quantity is $= 0$; and by that means it is evident, that the same thing is done as when the fluxion of a function of a variable quantity is assigned by the ordinary methods.
The evolution of the functions, considered in this very general view, requires the assistance of a new theorem, discovered by Mr Landen, and remarkable for its simplicity, as well as its great extent. It is, that if $x$ and $y$ are any two variable quantities,
$$\frac{x^n - y^n}{x - y} = x^{n-1} + x^{n-2}y + \cdots + y^{n-1}$$
where $m$ and $n$ are any integer numbers.
This theorem is the basis of the calculus; and from the expressions $x^n - y^n$, and $x - y$ having the form of
$$\frac{x^n - y^n}{x - y} = x^{n-1} + x^{n-2}y + \cdots + y^{n-1}$$
we have
$$x^n - y^n = (x - y)(x^{n-1} + x^{n-2}y + \cdots + y^{n-1})$$
which algebraists call residuals, the ingenious inventor gave to his whole method the name of the residual analysis.
The first account of this method was published by Mr Landen in 1758, under the title of a *Discourse concerning the Residual Analysis*. The first book of the Residual Analysis itself was published in 1764; and contained an explanation of the principles of the new calculus, with its application to several of the most considerable problems belonging to the direct method of fluxions. The second book was intended to give the solution of many of the most difficult problems that belong to the inverse method of fluxions, or to the integral calculus; but it has never been published: a circumstance which every one, who has taken the trouble to study the first part of the work, will very much regret.
If we estimate the value of the residual analysis from the genius, profound knowledge, and extensive views required to the discovery of it, it will rank high among works of invention; but if, on the other hand, we estimate its value by its real practical utility, as an instrument of investigation, we must rate it much lower. When compared with the fluxionary calculus, which it was intended to supersede, its principles, though in appearance more rigorous, are much less easily apprehended, much less luminous, and less direct in their application; and therefore, as a means of extending the bounds of mathematical science, it must ever be regarded as vastly inferior to the latter (a).