A History of Mathematics/Recent Times/Algebra
ALGEBRA.
The progress of algebra in recent times may be considered under three principal heads: the study of fundamental laws and the birth of new algebras, the growth of the theory of equations, and the development of what is called modern higher algebra.
We have already spoken of George Peacock and D. F. Gregory in connection with the fundamental laws of algebra. Much was done in this line by De Morgan.
Augustus De Morgan (1806–1871) was born at Madura (Madras), and educated at Trinity College, Cambridge. His scruples about the doctrines of the established church prevented him from proceeding to the M.A. degree, and from sitting for a fellowship. In 1828 he became professor at the newly established University of London, and taught there until 1867, except for five years, from 1831–1835. De Morgan was a unique, manly character, and pre-eminent as a teacher. The value of his original work lies not so much in increasing our stock of mathematical knowledge as in putting it all upon a thoroughly logical basis. He felt keenly the lack of close reasoning in mathematics as he received it. He said once: "We know that mathematicians care no more for logic than logicians for mathematics. The two eyes of exact science are mathematics and logic: the mathematical sect puts out the logical eye, the logical sect puts out the mathematical eye; each believing that it can see better with one eye than with two." De Morgan saw with both eyes. He analysed logic mathematically, and studied the logical analysis of the laws, symbols, and operations of mathematics; he wrote a Formal Logic as well as a Double Algebra, and corresponded both with Sir William Hamilton, the metaphysician, and Sir William Rowan Hamilton, the mathematician. Few contemporaries were as profoundly read in the history of mathematics as was De Morgan. No subject was too insignificant to receive his attention. The authorship of "Cocker's Arithmetic" and the work of circle-squarers was investigated as minutely as was the history of the invention of the calculus. Numerous articles of his lie scattered in the volumes of the Penny and English Cyclopædias. His Differential Calculus, 1842, is still a standard work, and contains much that is original with the author. For the Encyclopædia Metropolitana he wrote on the calculus of functions (giving principles of symbolic reasoning) and on the theory of probability. Celebrated is his Budget of Paradoxes, 1872. He published memoirs "On the Foundation of Algebra" (Trans. of Cam. Phil. Soc., 1841, 1842, 1844, and 1847).
In Germany symbolical algebra was studied by Martin Ohm, who wrote a System der Mathematik in 1822. The ideas of Peacock and De Morgan recognise the possibility of algebras which differ from ordinary algebra. Such algebras were indeed not slow in forthcoming, but, like non-Euclidean geometry, some of them were slow in finding recognition. This is true of Grassmann's, Bellavitis's, and Peirce's discoveries, but Hamilton's quaternions met with immediate appreciation in England. These algebras offer a geometrical interpretation of imaginaries. During the times of Descartes, Newton, and Euler, we have seen the negative and the imaginary, , accepted as numbers, but the latter was still regarded as an algebraic fiction. The first to give it a geometric picture, analogous to the geometric interpretation of the negative, was H. Kühn, a teacher in Danzig, in a publication of 1750–1751. He represented by a line perpendicular to the line , and equal to in length, and construed as the mean proportional between and . This same idea was developed further, so as to give a geometric interpretation of , by Jean-Robert Argand (1768–?) of Geneva, in a remarkable Essai (1806).[70] The writings of Kühn and Argand were little noticed, and it remained for Gauss to break down the last opposition to the imaginary. He introduced as an independent unit co-ordinate to 1, and as a "complex number." The connection between complex numbers and points on a plane, though artificial, constituted a powerful aid in the further study of symbolic algebra. The mind required a visual representation to aid it. The notion of what we now call vectors was growing upon mathematicians, and the geometric addition of vectors in space was discovered independently by Hamilton, Grassmann, and others, about the same time.
William Rowan Hamilton (1805–1866) was born of Scotch parents in Dublin. His early education; carried on at home, was mainly in languages. At the age of thirteen he is said to have been familiar with as many languages as he had lived years. About this time he came across a copy of Newton's Universal Arithmetic. After reading that, he took up successively analytical geometry, the calculus, Newton's Principia, Laplace's Mécanique Céleste. At the age of eighteen he published a paper correcting a mistake in Laplace's work. In 1824 he entered Trinity College, Dublin, and in 1827, while he was still an undergraduate, he was appointed to the chair of astronomy. His early papers were on optics. In 1832 he predicted conical refraction, a discovery by aid of mathematics which ranks with the discovery of Neptune by Le Verrier and Adams. Then followed papers on the Principle of Varying Action (1827) and a general method of dynamics (1834–1835). He wrote also on the solution of equations of the fifth degree, the hodograph, fluctuating functions, the numerical solution of differential equations.
The capital discovery of Hamilton is his quaternions, in which his study of algebra culminated. In 1835 he published in the Transactions of the Royal Irish Academy his Theory of Algebraic Couples. He regarded algebra "as being no mere art, nor language, nor primarily a science of quantity, but rather as the science of order of progression." Time appeared to him as the picture of such a progression. Hence his definition of algebra as "the science of pure time." It was the subject of years' meditation for him to determine what he should regard as the product of each pair of a system of perpendicular directed lines. At last, on the 16th of October, 1843, while walking with his wife one evening, along the Royal Canal in Dublin, the discovery of quaternions flashed upon him, and he then engraved with his knife on a stone in Brougham Bridge the fundamental formula . At the general meeting of the Irish Academy, a month later, he made the first communication on quaternions. An account of the discovery was given the following year in the Philosophical Magazine. Hamilton displayed wonderful fertility in their development. His Lectures on Quaternions, delivered in Dublin, were printed in 1852. His Elements of Quaternions appeared in 1866. Quaternions were greatly admired in England from the start, but on the Continent they received less attenttion. P. G. Tait's Elementary Treatise helped powerfully to spread a knowledge of them in England. Cayley, Clifford, and Tait advanced the subject somewhat by original contributions. But there has been little progress in recent years, except that made by Sylvester in the solution of quaternion equations, nor has the application of quaternions to physics been as extended as was predicted. The change in notation made in France by Hoüel and by Laisant has been considered in England as a wrong step, but the true cause for the lack of progress is perhaps more deep-seated. There is indeed great doubt as to whether the quaternionic product can claim a necessary and fundamental place in a system of vector analysis. Physicists claim that there is a loss of naturalness in taking the square of a vector to be negative. In order to meet more adequately their wants, J. W. Gibbs of Yale University and A. Macfarlane of the University of Texas, have each suggested an algebra of vectors with a new notation. Each gives a definition of his own for the product of two vectors, but in such a way that the square of a vector is positive. A third system of vector analysis has been used by Oliver Heaviside in his electrical researches.
Hermann Grassmann (1809–1877) was born at Stettin, attended a gymnasium at his native place (where his father was teacher of mathematics and physics), and studied theology in Berlin for three years. In 1834 he succeeded Steiner as teacher of mathematics in an industrial school in Berlin, but returned to Stettin in 1836 to assume the duties of teacher of mathematics, the sciences, and of religion in a school there.[71] Up to this time his knowledge of mathematics was pretty much confined to what he had learned from his father, who had written two books on "Raumlehre" and "Grössenlehre." But now he made his acquaintance with the works of Lacroix, Lagrange, and Laplace. He noticed that Laplace's results could be reached in a shorter way by some new ideas advanced in his father's books, and he proceeded to elaborate this abridged method, and to apply it in the study of tides. He was thus led to a new geometric analysis. In 1840 he had made considerable progress in its development, but a new book of Schleiermacher drew him again to theology. In 1842 he resumed mathematical research, and becoming thoroughly convinced of the importance of his new analysis, decided to devote himself to it. It now became his ambition to secure a mathematical chair at a university, but in this he never succeeded. In 1844 appeared his great classical work, the Lineale Ausdehnungslehre, which was full of new and strange matter, and so general, abstract, and out of fashion in its mode of exposition, that it could hardly have had less influence on European mathematics during its first twenty years, had it been published in China. Gauss, Grunert, and Möbius glanced over it, praised it, but complained of the strange terminology and its "philosophische Allgemeinheit." Eight years afterwards, Bretschneider of Gotha was said to be the only man who had read it through. An article in Crelle's Journal, in which Grassmann eclipsed the geometers of that time by constructing, with aid of his method; geometrically any algebraic curve, remained again unnoticed. Need we marvel if Grassmann turned his attention to other subjects,—to Schleiermacher's philosophy, to politics, to philology? Still, articles by him continued to appear in Crelle's Journal, and in 1862 came out the second part of his Ausdehnungslehre. It was intended to show better than the first part the broad scope of the Ausdehnungslehre, by considering not only geometric applications, but by treating also of algebraic functions, infinite series, and the differential and integral calculus. But the second part was no more appreciated than the first. At the age of fifty-three, this wonderful man, with heavy heart, gave up mathematics, and directed his energies to the study of Sanskrit, achieving in philology results which were better appreciated, and which vie in splendour with those in mathematics.
Common to the Ausdehnungslehre and to quaternions are geometric addition, the function of two vectors represented in quaternions by and , and the linear vector functions. The quaternion is peculiar to Hamilton, while with Grassmann we find in addition to the algebra of vectors a geometrical algebra of wide application, and resembling Möbius's Barycentrische Calcul, in which the point is the fundamental element. Grassmann developed the idea of the "external product," the "internal product," and the "open product." The last we now call a matrix. His Ausdehnungslehre has very great extension, having no limitation to any particular number of dimensions. Only in recent years has the wonderful richness of his discoveries begun to be appreciated. A second edition of the Ausdehnungslehre of 1844 was printed in 1877. C. S. Peirce gave a representation of Grassmann's system in the logical notation, and E. W. Hyde of the University of Cincinnati wrote the first text-book on Grassmann's calculus in the English language.
Discoveries of less value, which in part covered those of Grassmann and Hamilton, were made by Saint-Venant (1797–1886), who described the multiplication of vectors, and the addition of vectors and oriented areas; by Cauchy, whose "clefs algébriques" were units subject to combinatorial multiplication, and were applied by the author to the theory of elimination in the same way as had been done earlier by Grassmann; by Justus Bellavitis (1803–1880), who published in 1835 and 1837 in the Annali delle Scienze his calculus of æquipollences. Bellavitis, for many years professor at Padua, was a self-taught mathematician of much power, who in his thirty-eighth year laid down a city office in his native place, Bassano, that he might give his time to science.[65]
The first impression of Grassmann's ideas is marked in the writings of Hermann Hankel (1839–1873), who published in 1867 his Vorlesungen über die Complexen Zahlen. Hankel, then docent in Leipzig, had been in correspondence with Grassmann. The "alternate numbers" of Hankel are subject to his law of combinatorial multiplication. In considering the foundations of algebra Hankel affirms the principle of the permanence of formal laws previously enunciated incompletely by Peacock. Hankel was a close student of mathematical history, and left behind an unfinished work thereon. Before his death he was professor at Tübingen. His Complexe Zahlen was at first little read, and we must turn to Victor Schlegel of Hagen as the successful interpreter of Grassmann. Schlegel was at one time a young colleague of Grassmann at the Marienstifts-Gymnasium in Stettin. Encouraged by Clebsch, Schlegel wrote a System der Raumlehre which explained the essential conceptions and operations of the Ausdehnungslehre.
Multiple algebra was powerfully advanced by Peirce, whose theory is not geometrical, as are those of Hamilton and Grassmann. Benjamin Peirce (1809–1880) was born at Salem; Mass., and graduated at Harvard College, having as undergraduate carried the study of mathematics far beyond the limits of the college course.[2] When Bowditch was preparing his translation and commentary of the Mécanique Céleste, young Peirce helped in reading the proof-sheets. He was made professor at Harvard in 1833, a position which he retained until his death. For some years he was in charge of the Nautical Almanac and superintendent of the United States Coast Survey. He published a series of college text-books on mathematics, an Analytical Mechanics, 1855, and calculated, together with Sears C. Walker of Washington, the orbit of Neptune. Profound are his researches on Linear Associative Algebra. The first of several papers thereon was read at the first meeting of the American Association for the Advancement of Science in 1864. Lithographed copies of a memoir were distributed among friends in 1870, but so small seemed to be the interest taken in this subject that the memoir was not printed until 1881 (Am. Jour. Math., Vol. IV., No. 2). Peirce works out the multiplication tables, first of single algebras, then of double algebras, and so on up to sextuple, making in all 162 algebras, which he shows to be possible on the consideration of symbols , , etc., which are linear functions of a determinate number of letters or units , , , , etc., with coefficients which are ordinary analytical magnitudes, real or imaginary,—the letters , , etc., being such that every binary combination , , , etc., is equal to a linear function of the letters, but under the restriction of satisfying the associative law.[56] Charles S. Peirce, a son of Benjamin Peirce, and one of the foremost writers on mathematical logic, showed that these algebras were all defective forms of quadrate algebras which he had previously discovered by logical analysis, and for which he had devised a simple notation. Of these quadrate algebras quaternions is a simple example; nonions is another. C. S. Peirce showed that of all linear associative algebras there are only three in which division is unambiguous. These are ordinary single algebra, ordinary double algebra, and quaternions, from which the imaginary scalar is excluded. He showed that his father's algebras are operational and matricular. Lectures on multiple algebra were delivered by J. J. Sylvester at the Johns Hopkins University, and published in various journals. They treat largely of the algebra of matrices. The theory of matrices was developed as early as 1858 by Cayley in an important memoir which, in the opinion of Sylvester, ushered in the reign of Algebra the Second. Clifford, Sylvester, H. Taber, C. H. Chapman, carried the investigations much further. The originator of matrices is really Hamilton, but his theory, published in his Lectures on Quaternions, is less general than that of Cayley. The latter makes no reference to Hamilton.
The theory of determinants[73] was studied by Hoëné Wronski in Italy and J. Binet in France; but they were forestalled by the great master of this subject, Cauchy. In a paper (Jour. de l'ecole Polyt., IX., 16) Cauchy developed several general theorems. He introduced the name determinant, a term previously used by Gauss in the functions considered by him. In 1826 Jacobi began using this calculus, and he gave brilliant proof of its power. In 1841 he wrote extended memoirs on determinants in Crelle's Journal, which rendered the theory easily accessible. In England the study of linear transformations of quantics gave a powerful impulse. Cayley developed skew-determinants and Pfaffians, and introduced the use of determinant brackets, or the familiar pair of upright lines. More recent researches on determinants appertain to special forms. "Continuants" are due to Sylvester; "alternants," originated by Cauchy, have been developed by Jacobi, N. Trudi, H. Nägelbach, and G. Garbieri; "axisymmetric determinants," first used by Jacobi, have been studied by V. A. Lebesgue, Sylvester, and Hesse; "circulants" are due to E. Catalan of Liège, W. Spottiswoode (1825–1883), J. W. L. Glaisher, and R. F. Scott; for "centro-symmetric determinants" we are indebted to G. Zehfuss. E. B. Christoffel of Strassburg and G. Frobenius discovered the properties of "Wronskians," first used by Wronski. V. Nachreiner and S. Günther, both of Munich, pointed out relations between determinants and continued fractions; Scott uses Hankel's alternate numbers in his treatise. Text-books on determinants were written by Spottiswoode (1851), Brioschi (1854), Baltzer (1857), Günther (1875), Dostor (1877), Scott (1880), Muir (1882), Hanus (1886).
Modern higher algebra is especially occupied with the theory of linear transformations. Its development is mainly the work of Cayley and Sylvester.
Arthur Cayley, born at Richmond, in Surrey, in 1821, was educated at Trinity College, Cambridge.[74] He came out Senior Wrangler in 1842. He then devoted some years to the study and practice of law. On the foundation of the Sadlerian professorship at Cambridge, he accepted the offer of that chair, thus giving up a profession promising wealth for a very modest provision, but which would enable him to give all his time to mathematics. Cayley began his mathematical publications in the Cambridge Mathematical Journal while he was still an undergraduate. Some of his most brilliant discoveries were made during the time of his legal practice. There is hardly any subject in pure mathematics which the genius of Cayley has not enriched, but most important is his creation of a new branch of analysis by his theory of invariants. Germs of the principle of invariants are found in the writings of Lagrange, Gauss, and particularly of Boole, who showed, in 1841, that invariance is a property of discriminants generally, and who applied it to the theory of orthogonal substitution. Cayley set himself the problem to determine a priori what functions of the coefficients of a given equation possess this property of invariance, and found, to begin with, in 1845; that the so-called "hyper-determinants" possessed it. Boole made a number of additional discoveries. Then Sylvester began his papers in the Cambridge and Dublin Mathematical Journal on the Calculus of Forms. After this, discoveries followed in rapid succession. At that time Cayley and Sylvester were both residents of London, and they stimulated each other by frequent oral communications. It has often been difficult to determine how much really belongs to each.
James Joseph Sylvester was born in London in 1814, and educated at St. Johns College, Cambridge. He came out Second Wrangler in 1837. His Jewish origin incapacitated him from taking a degree. In 1846 he became a student at the Inner Temple, and was called to the bar in 1850. He became professor of natural philosophy at University College, London; then, successively, professor of mathematics at the University of Virginia, at the Royal Military Academy in Woolwich, at the Johns Hopkins University in Baltimore, and is, since 1883, professor of geometry at Oxford. His first printed paper was on Fresnel's optic theory, 1837. Then followed his researches on invariants, the theory of equations, theory of partitions, multiple algebra, the theory of numbers, and other subjects mentioned elsewhere. About 1874 he took part in the development of the geometrical theory of link-work movements, originated by the beautiful discovery of A. Peaucellier, Capitaine du Génie à Nice (published in Nouvelles Annales, 1864 and 1873), and made the subject of close study by A. B. Kempe. To Sylvester is ascribed the general statement of the theory of contravariants, the discovery of the partial differential equations satisfied by the invariants and covariants of binary quantics, and the subject of mixed concomitants. In the American Journal of Mathematics are memoirs on binary and ternary quantics, elaborated partly with aid of F. Franklin, now professor at the Johns Hopkins University. At Oxford, Sylvester has opened up a new subject, the theory of reciprocants, treating of the functions of a dependent variable and the functions of its differential coefficients in regard to , which remain unaltered by the interchange of and . This theory is more general than one on differential invariants by Halphen (1878), and has been developed further by J. Hammond of Oxford, McMahon of Woolwich, A. R. Forsyth of Cambridge, and others. Sylvester playfully lays claim to the appellation of the Mathematical Adam, for the many names he has introduced into mathematics. Thus the terms invariant, discriminant, Hessian, Jacobian, are his.
The great theory of invariants, developed in England mainly by Cayley and Sylvester, came to be studied earnestly in Germany, France, and Italy. One of the earliest in the field was Siegfried Heinrich Aronhold (1819–1884), who demonstrated the existence of invariants, and , of the ternary cubic. Hermite discovered evectants and the theorem of reciprocity named after him. Paul Gordan showed, with the aid of symbolic methods, that the number of distinct forms for a binary quantic is finite. Clebsch proved this to be true for quantics with any number of variables. A very much simpler proof of this was given in 1891, by David Hilbert of Königsberg. In Italy, F. Brioschi of Milan and Faà de Bruno (1825–1888) contributed to the theory of invariants, the latter writing a text-book on binary forms, which ranks by the side of Salmon's treatise and those of Clebsch and Gordan. Among other writers on invariants are E. B. Christoffel, Wilhelm Fiedler, P. A. McMahon, J. W. L. Glaisher of Cambridge, Emory McClintock of New York. McMahon discovered that the theory of semi-invariants is a part of that of symmetric functions. The modern higher algebra has reached out and indissolubly connected itself with several other branches of mathematics—geometry, calculus of variations, mechanics. Clebsch extended the theory of binary forms to ternary, and applied the results to geometry. Clebsch, Klein, Weierstrass, Burckhardt, and Bianchi have used the theory of invariants in hyperelliptic and Abelian functions.
In the theory of equations Lagrange, Argand, and Gauss furnished proof to the important theorem that every algebraic equation has a real or a complex root. Abel proved rigorously that the general algebraic equation of the fifth or of higher degrees cannot be solved by radicals (Crelle, I., 1826). A modification of Abel's proof was given by Wantzel. Before Abel, an Italian physician, Paolo Ruffini (1765–1822), had printed proofs of the insolvability, which were criticised by his countryman Malfatti. Though inconclusive, Ruffini's papers are remarkable as containing anticipations of Cauchy's theory of groups.[76] A transcendental solution of the quintic involving elliptic integrals was given by Hermite (Compt. Rend., 1858, 1865, 1866). After Hermite's first publication, Kronecker, in 1858, in a letter to Hermite, gave a second solution in which was obtained a simple resolvent of the sixth degree. Jerrard, in his Mathematical Researches (1832–1835), reduced the quintic to the trinomial form by an extension of the method of Tschirnhausen. This important reduction had been effected as early as 1786 by E. S. Bring, a Swede, and brought out in a publication of the University of Lund. Jerrard, like Tschirnhausen, believed that his method furnished a general algebraic solution of equations of any degree. In 1836 William R. Hamilton made a report on the validity of Jerrard's method, and showed that by his process the quintic could be transformed to any one of the four trinomial forms. Hamilton defined the limits of its applicability to higher equations. Sylvester investigated this question, What is the lowest degree an equation can have in order that it may admit of being deprived of consecutive terms by aid of equations not higher than th degree. He carried the investigation as far as , and was led to a series of numbers which he named "Hamilton's numbers." A transformation of equal importance to Jerrard's is that of Sylvester, who expressed the quintic as the sum of three fifth-powers. The covariants and invariants of higher equations have been studied much in recent years.
Abel's proof that higher equations cannot always be solved algebraically led to the inquiry as to what equations of a given degree can be solved by radicals. Such equations are the ones discussed by Gauss in considering the division of the circle. Abel advanced one step further by proving that an irreducible equation can always be solved in radicals, if, of two of its roots, the one can be expressed rationally in terms of the other, provided that the degree of the equation is prime; if it is not prime, then the solution depends upon that of equations of lower degree. Through geometrical considerations, Hesse came upon algebraically solvable equations of the ninth degree, not included in the previous groups. The subject was powerfully advanced in Paris by the youthful Evariste Galois (born, 1811; killed in a duel, 1832), who introduced the notion of a group of substitutions. To him are due also some valuable results in relation to another set of equations, presenting themselves in the theory of elliptic functions, viz. the modular equations. Galois's labours gave birth to the important theory of substitutions, which has been greatly advanced by C. Jordan of Paris, J, A, Serret (1819–1885) of the Sorbonne in Paris, L. Kronecker (1823–1891) of Berlin, Klein of Göttingen, M. Nöther of Erlangen, C. Hermite of Paris, A. Capelli of Naples, L. Sylow of Friedrichshald, E. Netto of Giessen. Netto's book, the Substitutionstheorie, has been translated into English by F. N. Cole of the University of Michigan, who contributed to the theory. A simple group of 504 substitutions of nine letters, discovered by Cole, has been shown by E. H. Moor of the University of Chicago to belong to a doubly-infinite system of simple groups. The theory of substitutions has important applications in the theory of differential equations. Kronecker published, in 1882, his Grundzüge einer Arithmetischen Theorie der Algebraischen Grössen.
Since Fourier and Budan, the solution of numerical equations has been advanced by W. G. Horner of Bath, who gave an improved method of approximation (Philosophical Transactions, 1819). Jacques Charles François Sturm (1803–1855), a native of Geneva, Switzerland, and the successor of Poisson in the chair of mechanics at the Sorbonne, published in 1829 his celebrated theorem determining the number and situation of roots of an equation comprised between given limits. Sturm tells us that his theorem stared him in the face in the midst of some mechanical investigations connected with the motion of a compound pendulum.[77] This theorem, and Horner's method, offer together sure and ready means of finding the real roots of a numerical equation.
The symmetric functions of the sums of powers of the roots of an equation, studied by Newton and Waring, was considered more recently by Gauss, Cayley, Sylvester, Brioschi. Cayley gives rules for the "weight" and "order" of symmetric functions.
The theory of elimination was greatly advanced by Sylvester, Cayley, Salmon, Jacobi, Hesse, Cauchy, Brioschi, and Gordan. Sylvester gave the dialytic method (Philosophical Magazine, 1840), and in 1852 established a theorem relating to the expression of an eliminant as a determinant. Cayley made a new statement of Bézout's method of elimination and established a general theory of elimination (1852).