(703 Art. XII. A Treatise on the Resolution of the higher Equations in Algebra By W. Lea. 4to. pp. viii. 40. Price 5s. Johnson and Co. 1811. MR. LEA is an author of whom the mathematical world knows but little at present, but who, as far as we can judge from the specimen before us, possesses the capacity of making himself known to excellent purpose. His principal design in this treatise, is to reduce to one, several of the apparently independent methods, which have been proposed at different times, by some of the most able mathematicians, to solve equations. The principle he has adopted for this purpose, though not entirely new, is extremely simple and satisfactory; and in its developement the author has evinced a considerable acquaintance with the works of other algebraists, as well as great ingenuity in deducing and comparing the chief rules of resolution. The principle of operation consists merely in comparing different resolutions of general problems. Thus, having given the two equations y+py+go, and y-x-m = o, the author finds r and y, and thence deduces the solution of a general quadratic. From the equations y3 + qu+ ro, and 2 — xy + m = o, he finds x and y, and thence deduces the solution of a general cubic. Then, from the equations yao, and x2 xy by+co, he forms and resolves a general biquadratic. These operations being effected, he proceeds thus: § 9. If now we examine the equations assumed in the three examples we have given, we shall find that the first in each example is comprised in the general one The second equation in the first example is y· −x+m= 0, all these equations are comprised then in the more general one y-P = o, P representing any function whatever of x. But since only the first power of y enters in this equation, it is evident we may make it much more general by introducing the higher powers; let us then for our second equation assume 3m+Pymi + QyTM + RyTM-3 + U=0, P, Q, &c. being any functions whatever of x; and we will now proceed to shew, that it is easy from these equations to deduce the different methods of resolution, which have at different times been proposed. 10. Assuming y" + py11 + qy12 + ry13...... + 1 = 1 ...... and yTM + Py—' + Qym→→2 + Rym-3........ + U = •, the coefficients P, QR, &c. being any functions whatever of: It is required to deduce any number of resolutions of general equations of the third and fourth degree, and of particular forms of the higher equations; also the different principles by which Cardan resolved a cubic equation, and Ferrari, Descartes, and Bezout, a biquadratic; Demoivre's resolu tion of a recipro al equation; and the general theories proposed by Tschirnhaus, Waring, Euler, &c. Let us suppose the different values of y in the second equation to be y = = a, y = ß, y = 7, &c. substituting them successively, in our first we have +4=0 +u=0 +u=0 Now each of these equations answers only to one value of y; but if we multiply them continually, we form an equation, which evidently contains all its different values; and it is plain the result will be the same, whatever change we make in the order of the quantities a, ß, y, &c. this result can then only involve similar functions of these quantities, and may thus be rationally expressed by means of the coefficients P, Q, R, &c. of our second equation: and since P, Q, R, &c. represent functions of x, substituting in place of them their values, we obtain an equation in which only is contained with known quantities. § 11. If now (as in Article I, and II,) we can by any means determine the roots of this equation, conversely y, that is the root of the ge neral equation y + pya¬1 +qy"—2 + ry —3.... + u = 0, will thus be known. -2 Or supposing (as in art. 5, and 6,) we are able to resolve our two assumed equations, conversely x, the root of the equation formed in the manner described in the last article will also be known; and the succeeding problems will serve as examples of the almost infinite variety of solutions which may be thus obtained. § 12. Now, in order to obtain the different required principles of solution, it is only necessary to assume the two comparatively particular equations + ty2+vy + u = 0, and + Fy - U =0: where U only, the coefficient of the last term of the second equation is a function of x, and that the very particular function U = Gx2+Hx + K Lx2 + Mx + N 13. Let us first make n = 3, p = 0, m = 1, A = 1, and U1 = x2 + K, -q, or what is the same thing, make at once U *q our equations then become x which are those assumed in Art. II. and the resolution depends on dividing the root y of the proposed cubic into two such parts x, and 9 - that their product beg; which is the principle of Cardan, or rather of Tartalea, and is probably that by which Scipio Ferreus obtained his resolution. 14. Next make n = 2, m = 1, and A= 1; then our two equations become ty2+vy + u = 0, and y = V1 = Gx+Hx+K Lx2 + Mx + N If we further make t : 1, vo, G = 1, H = 0, L = 0, and M = 1, our last become y2 + u = o, and y = " the two equations x + N assumed in Art. VI, and which were first assumed by Bezout, in the Paris Acts for 1764.* 15. Now in the last article, if we substitute in the first equation for 3 its value in the second, we form the biquadratic (Gx2 + Hx + K)2 + v(Gx2 + Hx + K) x (Lx2 + Mx + N) u (Lx2 + Mx + No; from which, by assigning particular values to five of the nine coefficients t, v, G, H, &c. we may obtain almost any number of different resolutions of a general biquadratic. 16. Let us make t 1, G= 1 and L = 1, v = o, u =— o, that is, let us then we have (x2 + Hx + k)' — (Mx + N)2 consider a biquadratic as formed by the difference of the squares of x2 + H≈ + K, and Mx + N, which is the principle of Ferrari. § 17. Next make t = 0, V = 1, uo, G: equation in Art. XV, then becomes =0; =1, and L = 1, our (x + Hx + K) × (x2 + Mx + N)o, that is, let us consider a biquadratic as formed by the multiplication of two quadratics, which is the principle of Descartes. 18. Now reverting to the two equations in Art. XII, let us make deduce Demoivre's solution of a reciprocal equation. 19. If next we make m=n → 1, and Ux, they become y2 + py11 + qy" +ry"-3 -3...... + u = 0, and $20. Now the second assumed equation remaining the same as in the * For this deduction I am indebted to Mr. Woodhouse, = first make the coefficients of all the terms be0, we then have -3 last article, let us in the -- re = x, the equations of Bezout; and if in second deduced from the first, it becomes + C2 √ u2¬3 + &c. = x, Аo √ uo1 + B √ u Thus then we perceive that this boasted formula, and the way Bezout proposes to resolve equations, are as to principle an exact conversion of the method of Tschirnhaus; his consisting in assuming x == Ay11 + By+Cy3+ &c. to transform a general equation of the nth degres into another of the same dimensions which shall want all its terms except the first and last; theirs in assuming x = Ay" + By + Cy"—3 + &c. for we have shewn Waring and Euler's formula to be the same as to principle as Bezout's equations) to transform an equation of the 1 degree, which wants all its terms except the first and last, into a general one. § 21. Lastly, let us make m = 1, A = 1, U1 x + K = 1, x + N and all the coefficients of our first assumed equation between the first and last = o, we then have x + K x + N which are the equations proposed by Bezout in 1762. 22. Not only may these different methods, as we have shewn, be deduced from our assumed equations; but the resolutions obtained from them may also, as will be seen in the succeeding problems, be obtained from the method laid down in art. 10 and 1.' pp. 5-9. - = From this quotation, our readers will be able to form a tolerable conception of Mr. Lea's method. He pursues it through a variety of problems, of which we regret that we can only speak very concisely. Thus, in his second problem, assuming ty2 + vy + u = 0, and y — P: = 0, he deduces different solutions of a general biquadratic. He draws, for example, from the same principle, the separate methods of Ferrari, Descartes, Bezout, and Euler; as well as explains the necessary limitation in the method of Ferrari, first shewn, we believe, by Mr. Wood. In his third problem, Mr. Lea assumes y+py + qy®—± + ry"¬3 + u = 0, and y- Po, in order to deduce the solution of particular forms of the higher equations. Under this problem he treats five different examples, among which is the well known reciprocal equation of Demoivre. The fourth and fifth problems exhibit a variety of solutions of general cubics, and biquadratics, and particular forms of the higher equations. In the fifth problem too, Euler's new me. thod for biquadratics (given in his Algebra) is shewn to coinci le with that obtained from the general theory of Waring and Bezout. Mr. Lea, in his sixth problem, assumes y+uo, and y — Py + 2 = 0, in order to shew how the solution of particular forms of the higher equations may be deduced; and to form one, of which Waring's equations 3.1, 3. 2, 3. 3, 5. 1, and 5. 2, p. 169 to 172 of his Meditationes Algebraicæ, may be only particular cases. This he effects, so as to give Waring's equations 5. I, and 5. 2, under a more simple and convenient form. In his seventh problem, assuming y+u =0, and y-Py+ 2y R = 0, it is required to shew how the solutions of particular forms of equations may be deduced; and to form one of which Waring's equations 4. 1, 4.2, 6. 1, and 6. 2. p. 170 to 173, of his Meditationes Algebraicæ, may be only particular cases. Here again his processes are marked with his usual ingenuity, and his results with his usual success. In the eighth and ninth problems, our author proceeds by still different assumptions, to deduce general solutions of cubics and biquadratics, and particular solutions of some of the higher equations; and his examples are, as usual, extremely well chosen. In the course of his investigation, he points out the excellences of preceding authors in the same department, as well as in certain cases shews their defects. Thus, he remarks, very properly, that Simpson, at p. 151 of his Algebra, and Maclaurin, at p. 229, should have noticed the case, in which the rules given by them, at those places, fails; namely, the case, in which the four roots of an equation of the form + ex3 + } e2 x2 + rf e3 x + 256 €1 o, whose four roots are equal to each other, and to He shews also, and it is a matter of no small importance to know, that the method of surd divisors is, with respect to biquadratics, completely useless. e. Our mathematical readers may judge from this analysis, that we think very favourably of Mr. Lea's treatise. In truth, we have been dissatisfied with nothing respecting it, but its magnitude. An author of so much talent ought not to confine himself to such narrow limits, nor to leave untouched many other subjects in this department of analysis, which, we are persuaded, lie quite within the compass of his powers. We hope soon to meet him again in the fairy land of these speculations; and in the mean while beg to recommend his treatise to those, who wish for a clue to lead them through some of the mazes and intricacies in which younger travellers in these regions are now and then apt to lose their way. |