RESIDUAL analysis, a calculus invented by Mr Landen, and proposed as a substitute for the method of fluxions. The design of it was to avoid introducing the idea of motion, and of quantities infinitely small, into mathematical investigation. The residual analysis ac- cordingly proceeds, by taking the difference of the same function of a variable quantity in two different states of that quantity, and denoting the relation of this differ- ence to the difference between the two states of the said variable quantity. This relation being first generally expressed, is next considered in the case when the dif- ference of the two states of the variable quantity is zero; and by that means it is obvious, that the same thing is done as when the function of a variable quan- tity is assigned by the ordinary methods.

The evolutions of the functions, considered in this very general view, requires the aid of a new theorem, discovered by Mr Landen, and remarkable for its sim- plicity and great extent. It is, that

if \(x\) and \(v\) are any two variable quantities

\[ \frac{x^n - v^n}{x - v} = \frac{1 + \frac{v^3}{x^3} + \frac{v^4}{x^4} + \cdots}{(m)} \]

where \(m\) and \(n\) are any integer numbers.

This theorem is the basis of the calculus, and from the expressions \(x^n - v^n\), and \(x - v\) having the form of what algebraists denominate residuals, the inventor gave to his method the name of the residual analysis.

Mr Landen published the first account of this method in 1758, which he denominated A Discourse concerning RESISTANCE OF FLUIDS.