Cases in which there is only one real root which is It appears therefore that the expression which the preceding process has given for x, furnishes the general solution of an equation of 9 dimensions, of which the proposed cubic equation is a factor and we are not authorized to conclude, from this investigation, that there is any symbolical expression which can be formed, which is capable of expressing simultaneously the three roots of the cubic equation arithmeti- be both positive, and if cal. 4 be greater than then and T 27 (Art. 968) are arithmetical and real, and the three roots are of which the first alone is arithmetical, the other two being imaginary. 4 Where one root only is real, but not arithmetical. If r be negative, q positive, and greater than σ and T are negative and real: in this case there is only one real root, which is not arithmetical: the other roots are imaginary. Where one If q be negative, then σ and are real, but with different root only is real which signs: there is, therefore, only one real root, which is arithmetical is arithmeor not, according as r is positive or negative: the other roots tical or not, according are imaginary. as r is posi tive or negative. Case in which all the roots are real, and one only, or at most two, are arith metical. are imaginary: they have therefore no real and arithmetical cube roots. These three roots of the equation are real, and one at least, but not more than two, of them, are arithmetical: for their sum is equal to zero*. the cible case of cubic 974. The term irreducible has been applied to the case The irreduof cubic equations considered in the last Article, where three roots, though all of them are real and one of them, at least, equations: why so arithmetical, are not capable of being determined, as in all called. other cases, by the extraction of roots, or the other processes of Arithmetic and Arithmetical Algebra: the difficulty which Geome trical representa was thus presented to the earlier writers on Algebra*, to whom the principles of Symbolical Algebra were almost entirely unknown, was insurmountable, and involved the solution, as the preceding investigation shews, of the problem of trisecting an angle, which was also beyond the province of plane Geometry it is not the only case in which the separation of Arithmetical and Symbolical Algebra, and of plane and the higher Geometry in which the different conic sections and other curves appear, will be found to be marked by common limits. 975. The principles of interpretation, which we have given in a former Chapter (xxx1.), will enable us to represent the two tion of the portions of which the several roots in the preceding solution cubic equa- are composed as well as the roots themselves. roots of a tion in the irreducible case. Let the angle BAC =0, and let the angle BAP= COS √(cos + sin and (cos-√1 sin), 3 will be expressed in magnitude and in position by AP and Ap, and their symbolical sum, or 2 COS by 2AQ: in a similar 3 3 manner, the values of the other two roots will be the symbolical sums of AP' and Ap', and AP" and Ap", which are respectively equivalent to 2AQ' and 2AQ”. An excellent account of the progress of the researches and discoveries of Tartalea, Cardan, Ferrari, Bombelli, Vieta, Des Cartes, and other early algebraists, in the solution of cubic and biquadratic equations, is given by Montucla in his Histoire des Mathématiques, Tom. I. Pars III. Livre 111. If the angle be between 0 and 90°, or if its cosine be posi- Case in which one or two of are both greater than 90°, and their the three roots are tical. cosines are negative: in other words, one root of the correspond- arithmeing equation is positive, and two are negative: but if be between 90° and 180°, or if its cosine be negative, then and 2 п — 0 are less than 90°, and 3 2π+0 0 3 greater than 90o; or in other words, two of the three real roots are positive, and one of them is negative. 976. Before we proceed to the consideration of numerical Problem, examples of cubic equations, we shall notice the following pro- the origin illustrating blem as well calculated to illustrate the origin of their multiple of the ambisolutions, and to exhibit the composition of their coefficients. "To find three numbers, whose sum shall be equal to a, the sum of whose products shall be equal to b and their continued product equal to c." Let x, y, z be the three numbers required; then the conditions of the problem give us x + y + z = a, xy+xz+yz= b, xyz= c. From the first equation we get guous solutions of cubic equations. Inasmuch as x may represent indifferently any one of the three unknown numbers or quantities which were assumed to be represented by x, y, z, which are all similarly involved in the original equations assumed, it must equally represent them Composition of the coefficients of a cubic equation. all; or in other words, x will admit of three values, which are those of the several unknown symbols*. If the sum of these three values or a=0, then one of them at least is positive, and two negative, or two of them positive, and one negative, when all the roots are real: or otherwise one of the three values is real, whether positive or negative, and the symbolical sum of the two others, which are imaginary, is equal to it with a different sign. 977. In considering the relations which the coefficients of a cubic equation bear to its roots, it is evident likewise, from the result of the preceding elimination, (and the same may be easily proved from other considerations†), that the coefficient (a) of the second • The same remark applies to the values of the unknown symbol in an equation, which results by elimination from any system of equations which are symmetrical with respect to the several unknown symbols which they involve: thus, if we eliminate y from the system of symmetrical equations we get the quadratic equation where has two values, which are those of x and y in the proposed system of equations; for x and y are obviously interchangeable with each other. But if the system of equations had been x3+ y3 = 35, which are not symmetrical with respect to a and y, the equation resulting from the elimination of y would have been a cubic equation, in which x has one real value only, the other two being imaginary. |