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 accordingly 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 difference 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 difference of the two states of the variable quantity is = 0; and by that means it is obvious, that the same thing is done as when the function of a variable quantity 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 simplicity and great extent. It is, that
\[ \frac{m}{n} \times \frac{n}{m} = \frac{m}{n} \]
if \( x \) and \( v \) are any two variable quantities
\[ \frac{m}{n} \times \frac{n}{m} = \frac{m}{n} \]
\[ 1 + \frac{v}{x} + \frac{v^2}{x^2} + \frac{v^3}{x^3} + \cdots (m) \]
\[ 1 + \left( \frac{v}{x} \right)^n + \left( \frac{v}{x} \right)^n + \left( \frac{v}{x} \right)^n + \cdots (n) \]
where \( m \) and \( n \) are any integer numbers.
This theorem is the basis of the calculus, and from the expressions \( \frac{m}{n} - \frac{n}{m} \), 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 The first book of the analysis appeared in 1764, which contained an explanation of the principles of the new calculus, with its application to problems of the direct method of fluxions, and the second book solved several problems of the inverse method, but it was never published.
If we estimate the value of this analysis by its practical utility, it may be said to possess no great merit. Its principles are much less easily apprehended than the fluxionary calculus; they are not so luminous, and less direct in their application, as well as inferior to it for enlarging the boundaries of mathematical science.
RESIDUAL Figure, in Geometry, the figure remaining after the subtraction of the less from the greater.
RESIDUAL Root, is a root composed of two members only connected by the sign — or minus. Thus, \(a - b\), or \(5 - 3\), is a residual root; and is so called, because its true value is no more than the residue, or difference between the parts \(a\) and \(b\) or \(5\) and \(3\), which in this case is \(2\).