Arthur Cayley

“ Léonhard Euler, in his paper on curvature in the Berlin Memoirs for 1760, had considered, not the normals of the surface, but the normals of the plane sections through a particular normal, so that the question of the intersection of successive normals of the surface had never presented itself to him. Monge’s memoir just referred to gives the ordinary differential equation of the curves of curvature, and establishes the general theory in a very satisfactory manner ”

“ A quartic equation is solvable by radicals, and it is to be remarked that the existence of such a solution depends on the existence of 3-valued functions such as ab + cd of the four roots (a, b, c, d) : by what precedes ab + cd is the root of a cubic equation, which equation is solvable by radicals: hence ab + cd can be found by radicals; and since abcd is a given function, ab and cd can then be found by radicals. But by what precedes, if ab be known then any similar function, say a + b, is obtainable rationally ”

“ But an equation with real coefficients may have as well imaginary as real roots, and we have further the theorem that for any such equation the imaginary roots enter in pairs, viz. α + βi being a root, then α − βi will be also a root. It follows that if the order be odd, there is always an odd number of real roots, and therefore at least one real root. ”

“ The general theory of the solvability of an equation by radicals depends fundamentally on A. T. Vandermonde’s remark (1770) that, supposing an equation is solvable by radicals, and that we have therefore an algebraical expression of x in terms of the coefficients, then substituting for the coefficients their values in terms of the roots, the resulting expression must reduce itself to any one at pleasure of the roots a, b, c ... ”

“ By what precedes there is nothing in this that might not have been expected; the algebraical solution makes the solution depend on the extraction of the cube root of a number, and there was no reason for expecting this to be a real number. It is well known that the case in question is that wherein the three roots of the numerical cubic equation are all real; if the roots are two imaginary, one real, then contrariwise the quantity under the cube root is real; and the algebraical solution gives the numerical one. ”

“ The method gives all the partitions of a number, but we may consider different classes of partitions: the partitions into a given number of parts, or into not more than a given number of parts; or the partitions into given parts, either with repetitions or without repetitions, &c. It is possible, for any particular class of partitions, to obtain methods more or less easy for the formation of the partitions either of a given number or of the successive numbers 1, 2, 3, &c. And of course in any case, having obtained the partitions, we can count them and so obtain the number of partitions. ”

“ Real Theory. 2. Postponing all consideration of imaginaries, we take in the first instance the coefficients to be real, and attend only to the real roots (if any) ; that is, p1, p2, ... pn are real positive or negative quantities, and a root a, if it exists, is a positive or negative quantity such that an − p1an−1 ... ± pn = 0, or say, ƒ (a) = 0.
It is very useful to consider the curve y = ƒ (x) ,—or, what would come to the same, the curve Ay = ƒ (x) ,—but it is better to retain the first-mentioned form of equation, drawing, if need be, the ordinate y on a reduced scale. ”

“ In vol. iii. we have memoirs relating to the proof of the theorem that every numerical equation has a real or imaginary root, the memoir on the Hypergeometric Series, that on Interpolation, and the memoir Determinatio attractionis—in which a planetary mass is considered as distributed over its orbit according to the time in which each portion of the orbit is described, and the question (having an implied reference to the theory of secular perturbations) is to find the attraction of such a ring. ”

“ We know that between the limits β, α there lies one, and only one, real root of the equation; ƒ (β) and ƒ (α) have therefore opposite signs. Suppose any intermediate value is θ; in order to determine by Sturm’s theorem whether the root lies between β, θ, or between θ, α, it would be quite unnecessary to calculate the signs of ƒ (θ) ,ƒ′ (θ) , ƒ2 (θ) ...; only the sign of ƒ (θ) is required ”

“ Returning for a moment to Horner’s method, it may be remarked that the correction h, to an approximate value α, is therein found as a quotient the same or such as the quotient ƒ (α) ÷ ƒ′ (α) which presents itself in Newton’s method. The difference is that with Horner the integer part of this quotient is taken as the presumptive value of h, and the figure is verified at each step. With Newton the quotient itself, developed to the proper number of decimal places, is taken as the value of h; if too many decimals are taken, there would be a waste of work ”

“ The mode of division is more clearly seen for a larger value of n; thus, for n= 7 a prime root is = 3, and the arrangement of the roots is r, r3, r2, r6, r4, r5. We may form either 3 periods each of 2 terms, P1, P2, P3 = r + r6, r3 + r4, r2 + r5 respectively; or else 2 periods each of 3 terms, P1, P2 = r + r2 + r4, r3 + r6 + r5 respectively; in each case the symmetrical functions of the periods are rationally determinable: thus in the case of the two periods P1 + P2 = −1, P1P2 = 3 + r + r2 + r3 + r4 + r5 + r6, = 2 ”

“ C in the portion XN, drawing the ordinates DQ, CP, and joining the points P, Q by a line which meets the axis in D′, also constructing the point C′ by means of the tangent at P as before, we have for the required root the new limits OD′, OC′; and proceeding in like manner with the points D′, C′, and so on as often as we please, we obtain at each step two limits approximating more and more nearly to the required root OX. The process as to the point D′, translated into analysis, is the ordinate process of interpolation. ”

“ MONGE, GASPARD (1746–1818) , French mathematician, the inventor of descriptive geometry, was born at Beaune on the 10th of May 1746. He was educated first at the college of the Oratorians at Beaune, and then in their college at Lyons—where, at sixteen, the year after he had been learning physics, he was made a teacher of it. Returning to Beaune for a vacation, he made, on a large scale, a plan of the town, inventing the methods of observation and constructing the necessary instruments ”

“ It has already been shown how the several roots of the equation xn − 1 = 0 can be expressed in the form cos 2sπ/n + i sin 2sπ/n, but the question is now that of the algebraical solution (or solution by radicals) of this equation. There is always a root = 1 ”

“ A glance through the index will show the vast extent which the theory has assumed, and the form of general conclusions arrived at; thus, in book iii., the algebraical applications comprise Abelian equations, equations of Galois; the geometrical ones comprise Q. Hesse’s equation, R. F. A. Clebsch’s equations, lines on a quartic surface having a nodal line, singular points of E. E. Kummer’s surface, lines on a cubic surface, problems of contact ”

“ In vol. iv. we have the memoir Allgemeine Auflösung, on the graphical representation of one surface upon another, and the Disquisitiones generales circa superficies curvas. (An account of the treatment of surfaces which he originated in this paper will be found in the article Surface.) And in vol. v. we have a memoir On the Attraction of Homogeneous Ellipsoids, and the already mentioned memoir Allgemeine Lehrsätze, on the theory of forces attracting according to the inverse square of the distance. (A. Ca.) ”

“ Suppose for a moment that t1, t2, t3 are all known; then the equations being linear in y1, y2, y3 these can be expressed rationally in terms of the coefficients and of t1, t2, t3; that is, y1, y2, y3 will be known. ”

“ In 1807 he was appointed director of the Göttingen observatory, an office which he retained to his death: it is said that he never slept away from under the roof of his observatory, except on one occasion, when he accepted an invitation from Baron von Humboldt to attend a meeting of natural philosophers at Berlin. ”

“ On the formation of the Senate he was appointed a member of that body, with an ample provision and the title of count of Pelusium; but on the fall of Napoleon he was deprived of all his honours, and even excluded from the list of members of the reconstituted Institute. He died at Paris on the 28th of July 1818. ”

“ It was shown by Sir James Cockle and Robert Harley (1858–1859) in connexion with the Jerrardian form, and by Arthur Cayley (1861) , that Lagrange’s resolvent equation of the sixth order can be replaced by a more simple sextic equation occupying a like place in the theory. ”