And although theoretically, in order to complete by a finite number of operations the separation of the real roots, we still need to know the value of the before-mentioned limit [delta]; yet in any given case the separation may be effected by a limited number of repetitions of the process. The practical difficulty is when two or more roots are very near to each other. Suppose, for instance, that the theorem shows that there are two roots between 0 and 10; by giving to x the values 1, 2, 3, ... successively, it might appear that the two roots were between 5 and 6; then again that they were between 5.3 and 5.4, then between 5.34 and 5.35, and so on until we arrive at a separation; say it appears that between 5.346 and 5.347 there is one root, and between 5.348 and 5.349 the other root. But in the case in question [delta] would have a very small value, such as .002, and even supposing this value known, the direct application of the first-mentioned process would be still more laborious. 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 [beta], [alpha] there lies one, and only one, real root of the equation; [f]([beta]) 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 [beta], [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]([beta]), then the root is between [beta], [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 [beta], at which [f]([theta]) has the sign of [f]([beta]), 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.