3. Suppose that the positive value [delta] is an inferior limit to the difference between two real roots of the equation; or rather (since the foregoing expression would imply the existence of real roots) suppose that there are not two real roots such that their difference taken positively is = or < [delta]; then, [gamma] being any value whatever, there is clearly at most one real root between the limits [gamma] and [gamma] + [delta]; and by what precedes there is such real root or there is not such real root, according as [f]([gamma]), [f]([gamma] + [delta]) have opposite signs or have the same sign. And by dividing in this manner the interval [beta] to [alpha] into intervals each of which is = or < [delta], we should not only ascertain the number of the real roots (if any), but we should also separate the real roots, that is, find for each of them limits [gamma], [gamma] + [delta] between which there lies this one, and only this one, real root. In particular cases it is frequently possible to ascertain the number of the real roots, and to effect their separation by trial or otherwise, without much difficulty; but the foregoing was the general process as employed by Joseph Louis Lagrange even in the second edition (1808) of the _Traite de la resolution des equations numeriques_;[2] the determination of the limit [delta] had to be effected by means of the "equation of differences" or equation of the order 1/2 n(n - 1), the roots of which are the squares of the differences of the roots of the given equation, and the process is a cumbrous and unsatisfactory one. 4. The great step was effected by the theorem of J.C.F. Sturm (1835)--viz. here starting from the function [f](x), and its first derived function [f]'(x), we have (by a process which is a slight modification of that for obtaining the greatest common measure of these two functions) to form a series of functions [f](x), [f]'(x), [f]2(x), ... [f]_n(x) of the degrees n, n - 1, n - 2 ... 0 respectively,--the last term [f]_n(x) being thus an absolute constant. These lead to the immediate determination of the number of real roots (if any) between any two given limits [beta], [alpha]; viz. supposing [alpha] > [beta] (that is, [alpha] nearer to +[oo]), then substituting successively these two values in the series of functions, and attending only to the signs of the resulting values, the number of the changes of sign lost in passing from [beta] to [alpha] is the required number of real roots between the two limits. In particular, taking [beta], [alpha] = -[oo], +[oo] respectively, the signs of the several functions depend merely on the signs of the terms which contain the highest powers of x, and are seen by inspection, and the theorem thus gives at once the whole number of real roots.