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 joining 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', translated into analysis, is the ordinate process of interpolation. Suppose OD = [beta], OC = [alpha], we have approximately [f]([beta] + h) = [f]([beta]) + h{[f]([alpha]) - [f]([beta])} / ([alpha] - [beta]), whence if the root is [beta] + h then h = - ([alpha] - [beta])[f]([beta]) / {[f]([alpha]) - [f]([beta])}. Returning for a moment to Horner's method, it may be remarked that the correction h, to an approximate value [alpha], is therein found as a quotient the same or such as the quotient [f]([alpha]) / [f]'([alpha]) which presents itself in Newton's method. The difference is that with Horner 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. _Imaginary Theory_. 7. 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 magnitude susceptible of continuous variation.