XXI. A new method of solving numerical equations of all orders, by continuous approximation

Abstract

Abstract 1. The process which it is the object of this Essay to establish, being nothing else than the leading theorem in the Calculus of Derivations, presented under a new aspect, may be regarded as a universal instrument of calculation, extending to the composition as well as analysis of functions of every kind. But it comes into most useful application in the numerical solution of equations. 2. Arbogast’s developement of ϕ ( α + βx + γx2 + δx3 + ∈x4 + .... ) (See Calc, des Der. § 33 ) supposes all the coefficients within the parenthesis to be known previously to the operation of ϕ. To the important cases in which the discovery of γ, δ, &c. depends on the previous developement of the partial functions ϕ ( α + βx ), ϕ ( α + βx + γx2), &c.