5. Supposing the separation once effected, the determination of the single real root which lies between the two given limits may be effected to any required degree of approximation either by the processes of W.G. Horner and Lagrange (which are in principle a carrying out of the method of Sturm's theorem), or by the process of Sir Isaac Newton, as perfected by Joseph Fourier (which requires to be separately considered).
First as to Horner and Lagrange. We know that between the limits ß, [alpha] there lies one, and only one, real root of the equation; [f](ß) and [f]([alpha]) have therefore opposite signs. Suppose any intermediate value is [theta]; in order to determine by Sturm's theorem whether the root lies between ß, [theta], or between [theta], [alpha], it would be quite unnecessary to calculate the signs of [f]([theta]),[f]'([theta]), [f]2([theta]) ...; only the sign of [f]([theta]) is required; for, if this has the same sign as [f](ß), then the root is between ß, [theta]; if the same sign as [f]([alpha]), then the root is between [theta], [alpha]. We want to make [theta] increase from the inferior limit ß, at which [f]([theta]) has the sign of [f](ß), so long as [f]([theta]) retains this sign, and then to a value for which it assumes the opposite sign; we have thus two nearer limits of the required root, and the process may be repeated indefinitely.
Horner's method (1819) gives the root as a decimal, figure by figure; thus if the equation be known to have one real root between 0 and 10, it is in effect shown say that 5 is too small (that is, the root is between 5 and 6); next that 5.4 is too small (that is, the root is between 5.4 and 5.5); and so on to any number of decimals. Each figure is obtained, _not_ by the successive trial of all the figures which precede it, but (as in the ordinary process of the extraction of a square root, which is in fact Horner's process applied to this particular case) it is given presumptively as the first figure of a quotient; such value may be too large, and then the next inferior integer must be tried instead of it, or it may require to be further diminished. And it is to be remarked that the process not only gives the approximate value [alpha] of the root, but (as in the extraction of a square root) it includes the calculation of the function [f]([alpha]), which should be, and approximately is, = 0. The arrangement of the calculations is very elegant, and forms an integral part of the actual method. It is to be observed that after a certain number of decimal places have been obtained, a good many more can be found by a mere division. It is in the progress tacitly assumed that the roots have been first separated.