r ༧° and if represent the greatest negative coefficient found in this It is only necessary to consider here the positive limit, as this alone relates to the real roots of the proposed equation. less than the square of the smallest difference between the roots of the proposed equation, we may find its square root, or at least, take the rational number next below this root; this number, which I shall designate by k, will represent the difference which must exist between the several numbers to be substituted. We thus form the two series, from which we are to take only the terms, comprehended between the limits to the smallest and the greatest positive roots, and those to the smallest and the greatest negative roots of the proposed equation. Substituting these different numbers, we shall arrive at a series of results, which will show by the changes of the sign that take place, the several real roots, whether positive or negative. 218. Let there be, for example, the equation x3-7x+7=0, from which, in art. 208., was derived the equation making z = 1 23 42 22 + 441 z 49 = 0; , and, after substituting this value, arranging the re we must, therefore, take k = or <This condition will be ✓10 = fulfilled, if we make k; but it is only necessary to suppose k; for by putting 9 in the place of v in the preceding equation, we obtain a positive result, which must become greater, when a greater value is assigned to v, since the terms v3 and 9 v2 already destroy each other, and 4 v exceeds 5. The highest limit to the positive roots of the proposed equation x37x+7= 0, is 8, and that to the negative roots-8; we must, therefore, substitute for x the numbers Ꮖ We may avoid fractions by making x = for in this case the differences between the several values of a will be triple of those between the values of x, and, consequently, will exceed unity; we shall then have only to substitute, successively, The signs of the results will be changed between + 4 and + 5, between 5 and 6, and between shall have for the positive values, x4 and 5 5 and 6 9 and 10, so that we and the negative value of a will be found between Knowing now the several roots of the proposed equation within , we may approach nearer to the true value by the method explained in art. 215. 219. The methods employed in the example given in art. 215., and in the preceding article, may be applied to an equation of any degree whatever, and will lead to values approaching the several real roots of this equation. It must be admitted, however, that the operation becomes very tedious, when the degree of the proposed equation is very elevated; but in most cases it will be unnecessary to resort to the equation (D), or rather its place may be supplied by methods, with which the study of the higher branches of analysis will make us acquainted.* I shall observe, however, that by substituting successively the numbers 0, 1, 2, 3, &c. in the place of x, we shall often be led to suspect the existence of roots, that differ from each other by a quantity less than unity. In the example upon which we have been employed, the results are + 7, + 1, + 1, + 13, which begin to increase after having decreased from + 7 to + 1. From this order being reversed it may be supposed, that between the numbers + 1 and + 2 there are two roots either equal or nearly equal. To verify this supposition, the unknown quantity should be multiplied. Making x = = 1/0 we find y y3 — 700 y + 7000 = 0, an equation which has two positive roots, one between 13 and 14, and the other between 16 and 17. The number of trials necessary for discovering these roots is not great; for it is only between 10 and 20, that we are to search for y; and the values of this unknown quantity being determined in whole numbers, we may find those of r within one tenth of unity. 220. When the coefficients in the equation proposed for resolution are very large, it will be found convenient to transform this equation into another, in which the coefficients shall be reduced to smaller numbers. If we have, for example 80 x3 1998 x2 14937x+5000 = 0, we may make x = 10z; the equation then becomes 2482319,98 22- 14,937 z+0,5 = 0. If we take the entire numbers, which approach nearest to the coefficients in this result, we shall have 248 23 20 22 15 z+ 0,5 = 0. It may be readily discovered, that z has two real values, one between 0 and 1, the other between 1 and 2, whence it follows, that those of the proposed equation are between 0 and 10, and 10 and 20. * A very elegant method given by Lagrange for avoiding the use of the equation (D) may be found in the Traité de la Résolution des Equations numériques. I shall not here enter into the investigation of imaginary roots, as it depends on principles we cannot at present stop to illustrate; I shall pursue the subject in the Supplement. 221. Lagrange has given to the successive substitutions a form which has this advantage, that it shows immediately what approaches we make to the true root by each of the several operations, and which does not presuppose the value to be known within one tenth. Let a represent the entire number immediately below the root sought; to obtain this root, it will be only necessary to augment a by a fraction; we have, therefore, x = a + The equation involving y, with which we are furnished by substituting this value in the proposed equation, will necessarily have one root greater than unity; taking b to represent the entire number immediately below this root, we have for the second approximation x = a + But b having the same relation to y, which a has to x, we may, in the equation involving y, make y = b+ and y' will necessarily be 1 y 1 greater than unity; representing by b' the entire number immediately below the root of the equation in y', we have substituting this value in the expression for x, we have for the third approximation to x. We may find a fourth by making 1 y' = '+; for if b" designate the entire number immediately y" 222. I shall apply this method to the equation x3 — 7 x + 7 = 0. We have already seen (218), that the smallest of the positive roots of this equation is found between 1 and, that is, between 1 and 2; we make, therefore, x 1+ ; we shall then have y The limit to the positive roots of this last equation is 5, and by substituting, successively, 0, 1, 2, 3, 4, in the place of y, we immediately discover, that this equation has two roots greater than unity, one between 1 and 2, and the other between 2 and 3. Hence x = 1 + 1, and x = 1 + 1/1, These two values correspond to those, which were found above between and, and between and, and which differ from each other by a quantity less than unity. In order to obtain the first, which answers to the supposition of y= 1, to a greater degree of exactness, we make we then have 1 y = 1 + y' y13 — 2 y12 — y' + 1 = 0. We find in this equation only one root greater than unity, and that is between 2 and 3, which gives whence y=1+1=3, x = 1 + 3 号。 Again, if we suppose y' = 2; we shall be furnished with the equation + we find the value of y' to be between 4 and 5; taking the smallest of these numbers, 4, we have 13 y' = 2+, y=1+ ¦ = 1, x = 1 + 13 It would be easy to pursue this process, by making y" = 4 + |