Arthur Cayley

“ In general, an indeterminate equation results when the number of unknowns exceeds by unity the number of equations. If, on the other hand, we have two equations connecting two unknowns, it is possible to solve the equations separately for one unknown, and then if we equate these values we obtain an equation in one unknown, which is soluble if its degree does not exceed the fourth. ”

“ It is convenient to mention here the theorem that, x being determined as above by an equation of the order n, any rational and integral function whatever of x, or more generally any rational function which does not become infinite in virtue of the equation itself, can be expressed as a rational and integral function of x, of the order n − 1, the coefficients being rational functions of the coefficients of the equation. ”

“ The Greeks could not solve this equation, which also arose in the problems of duplicating a cube and trisecting an angle, by the ruler and compasses, but only by mechanical curves such as the cissoid, conchoid and quadratrix. Such solutions were much improved by the Arabs, who also solved both cubics and biquadratics by means of intersecting conics; at the same time, they developed methods, originated by Diophantus and improved by the Hindus, for finding approximate roots of numerical equations by algebraic processes. ”

“ 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 ”

“ Any determinant
formed out of the elements of the original determinant, by selecting the lines and columns at pleasure, is termed a minor of the original determinant; and when the number of lines and columns, or order of the determinant, is n−1, then such determinant is called a first minor; the number of the first minors is=n 2, the first minors, in fact, corresponding to the several elements of the determinant—that is, the coefficient therein of any term whatever is the corresponding first minor. ”

“ Such equations are called simultaneous; and a simultaneous system is a series of equations equal in number to the number of unknowns. Such a system is not always soluble, for it may happen that one equation is implied by the others; when this occurs the system is called porismatic or poristic. An identity differs from an equation inasmuch as it cannot be solved, the terms mutually cancelling ”

“ 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 ”

“ 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. ”

“ An equation states an equality existing between two classes of quantities, distinguished as known and unknown; these correspond to the data of a problem and the thing sought. It is the purpose of the mathematician to state the unknowns separately in terms of the knowns; this is called solving the equation, and the values of the unknowns so obtained are called the roots or solutions. The unknowns are usually denoted by the terminal letters, . ”

“ The far-reaching discoveries of Sylvester and Cayley rank as one of the most important developments of pure mathematics. Numerous new fields were opened up, and have been diligently explored by many mathematicians. ”

“ 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 ... ”

“ The first minors, each divided by the determinant itself, form a system of elements inverse to the elements of the determinant.
A determinant is symmetrical when every two elements symmetrically situated in regard to the dexter diagonal are equal to each other ”

“ 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. ”

“ Quadratic equations arose in the Greek investigations in the doctrine of proportion, and although they were presented and solved in a geometrical form, the methods employed have no relation to the generalized conception of algebraic geometry which represents a curve by an equation and vice versa. The simplest quadratic arose in the construction of a mean proportional (x) between two lines (a, b) , or in the construction of a square equal to a given rectangle; for we have the proportion a:x = x:b; i.e. x2 = ab. A more general equation, viz. ”

“ 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. ”

“ 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 ”

“ Determinants composed of binomial coefficients have been studied by V. von Zeipel; the expression of definite integrals as determinants by A. Tissot and A. Enneper, and the expression of continued fractions as determinants by Jacobi, V. Nachreiner, S. Günther and E. Fürstenau. (See T. Muir, Theory of Determinants, 1906) .
↑ The expression, a linear function, is here used in its narrowest sense, a linear function without constant term ”

“ 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. ”

“ The latter can only yield approximate values: when theoretically exact constructions are available there is a source of error in the draughtsmanship, and when the constructions are only approximate, the accuracy of the results is more problematical. ”

“ 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. ”

“ 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 ”

“ 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. ”

“ 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. ”

“ 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 ”

“ 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 ”

“ 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 ”

“ 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.) ”

“ 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. ”