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. Lagrange's method (1767) gives the root as a continued fraction a + 1/b + 1/c + ..., where a is a positive or negative integer (which may be = 0), but b, c, ... are positive integers. Suppose the roots have been separated; then (by trial if need be of consecutive integer values) the limits may be made to be consecutive integer numbers: say they are a, a + 1; the value of x is therefore = a + 1/y, where y is positive and greater than 1; from the given equation for x, writing therein x = a + 1/y, we form an equation of the same order for y, and this equation will have one, and only one, positive root greater than 1; hence finding for it the limits b, b + 1 (where b is = or > 1), we have y = b + 1/z, where z is positive and greater than 1; and so on--that is, we thus obtain the successive denominators b, c, d ... of the continued fraction. The method is theoretically very elegant, but the disadvantage is that it gives the result in the form of a continued fraction, which for the most part must ultimately be converted into a decimal. There is one advantage in the method, that a commensurable root (that is, a root equal to a rational fraction) is found accurately, since, when such root exists, the continued fraction terminates. 6. Newton's method (1711), as perfected by Fourier(1831), may be roughly stated as follows. If x = [gamma] be an approximate value of any root, and [gamma] + h the correct value, then [f]([gamma] + h) = 0, that is,