RESIDUAL ANALYSIS, 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 infi-
nitely or indefinitely small, into mathematical investiga-
tion. The residual analysis accordingly proceeds, by tak-
ing the difference of the same function of a vari-
able 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 gen-
erally, 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 theo-
rem, discovered by Mr Landen, and remarkable for its
simplicity, as well as its great extent. It is, that if

\begin{aligned} x \text{ and } v \text{ are any two variable quantities, } \frac{x^m}{x} = \frac{v^m}{v} \\ = x^{m-1} \times \frac{1 + \frac{v}{x} + \frac{v^2}{x^2} + \frac{v^3}{x^3} + \dots (m)}{1 + \left(\frac{v}{x}\right)^{\frac{m}{n}} + \left(\frac{v}{x}\right)^{\frac{2m}{n}} + \left(\frac{v}{x}\right)^{\frac{3m}{n}} + \dots (n)} \end{aligned}

where m and n are any integer numbers.

This theorem is the basis of the calculus; and from
the expressions x^{\frac{m}{n}} = v^{\frac{m}{n}}, and x = v having the form of
SUPPL. VOL. II. Part II.

what 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 con-
cerning 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 con-
siderable 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 be-
long to the inverse method of fluxions, or to the inte-
gral calculus; but it has never been published: a cir-
cumstance 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 ap-
pearance more rigorous, are much less easily appre-
hended, much less luminous, and less direct in their ap-
plication; and therefore, as a means of extending the
bounds of mathematical science, it must ever be regard-
ed as vastly inferior to the latter (A).