EQUATION 501 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. Lagrauge s method (1767) gives the root as a continued fraction a 4- r . . . i where a is a positive or negative o ~r c T* 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 -f 1 ; the value of x is therefore = a + , where y is positive and greater than 1 ; from the given equation for x, writing therein x a+ - , we form an equation of / the same order for i/, and this equation will have one, and only one, positive root greater than 1 ; hence finding for it the limits 6, 6+1 (where b is = or >1), we have y = lj + -, where z is positive and greater than 1; and so on that is, we thus obtain the successive denominators 6, c, d . . of the continued fraction. The method is theo retically 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 commensur able root (that is, a root equal to a rational fraction) is found accurately, since, when such root exists, the continued frac tion terminates. 14. Newton s method (1711), as perfected by Fourier (1831), may be roughly stated as follows. If x = y be an approximate value of any root, and y + h the correct value, then f(y + h) = , that is, ~ and then, if h be so small that the terms after the second may be neglected, /(y) + hf (y) = , that is, h = - ---, or fn the new approximate value is x = y- ~~ > ail( i so on, as often as we please. It will be observed that so far nothing has been assumed as to the separation of the roots, or even as to the existence of a real root ; y has been taken as the approximate value of a root, but no precise meaning has been attached to this expression. The question arises, what are the conditions to be satisfied by y in order that the process may by successive repetitions actually lead to a certain real root of the equation ; or say that, y being an approximate value of a certain real root, the new value y - TTT-. may be a more approximate value. Referring to the figure, it is easy to see that that if OC represent the assumed value y, then, drawing the ordinate CP to meet the curve in P, and the tangent PC to meet the axis in C , ws shall have OC as the new approximate value of the root. But observe that there is here a real root OX, and that the curve beyond X is convex to the axis ; under these conditions the point C is nearer to X than was C ; and, starting with C instead of C, and pro ceeding in like manner to draw a new ordinate and tangent, and so on as often as we please, we approximate continually, and that with great rapidity, to the true value OX. But if C had been taken on the other side of X, where the curve is concave to the axis, the new point C might or might not be nearer to X than was the point C ; and in this case the method, if it succeeds at all, does so by acci dent only, i.e., it may happen that C or some subsequent point comes to be a point C, such that OC is a proper approximate value of the root, and then the subsequent approximations proceed in the same manner as if this value had been assumed in the first instance, all the pre ceding work being wasted. It thus appears that for the proper application of the method we require more than the mere separation of the roots. In order to be able to approximate to a certain root a, = OX, we require to know that, between OX and some value ON, the curve is always convex to the axis (analytically, between the two values, f(x) and /"(.*) must have always the same sign). When this is so, the point C may be taken anywhere on the proper side of X, and within the portion XN of the axis ; and the process is then the one already explained. The approxima tion is in general a very rapid one. If we know for the required root OX the two limits OM, ON such that from M to X the curve is always concave to the axis, while from X to N it is always convex to the axis, then, taking D anywhere in the portion MX and (as before) C in the portion XN, drawing the ordinates DQ, CP, and join ing the points P, Q by a line which meets the axis in D , also constructing the point C by means of the tangent at P as before, we have for the required root the new limits OD , OC ; and proceeding in like manner with the points D , C , and so on as often as we please, we obtain at each step two limits approximating more and more nearly to the required root OX. The process as to the point D , trans lated into analysis, is the ordinate process of interpola tion. Suppose OD = (3, OC = a, we have approximately /( ft + h) =f(/3) + -j, whence if the root is ft + h then A.-fc=M. /() ~f(ft) Returning for a moment to Homer s method, it may be remarked that the correction h, to an approximate value a, is therein found as a quotient the same or such as the quotient /(a) -=-/(a) which presents itself in Newton s method. The difference is that with Homer the integer part of this quotient is taken as the presumptive value of h, and the figure is verified at each step. With Newton the quotient itself, developed to the proper number of decimal places, is taken as the value of h ; if too many decimals are taken, there would be a waste of work ; but the error would correct itself at the next step. Of course the calculation should be conducted without any such waste of work. Next as to the theory of imaginaries. 15. It will be recollected that the expression number and the correlative epithet numerical were at the outset used in a wide sense, as extending to imaginaries. This extension arises out of the theory of equations by a process analogous to that by which number, in its original most restricted sense of positive integer number, was extended to have the meaning of a real positive or negative magni tude susceptible of continuous variation. If for a moment number is understood in its most
restricted sense as meaning positive integer number, tko