A Short Account of the History of Mathematics · W. W. Rouse Ball
The nineteenth century saw the creation of numerous new departments of pure mathematics--notably of a theory of numbers, or higher arithmetic; of theories of forms and groups, or a higher algebra; of theories of functions of multiple periodicity, or a higher trigonometry; and of a general theory of functions, embracing extensive regions of higher analysis. Further, the developments of synthetic and analytical geometry created what practically were new subjects. The foundations of the subject and underlying assumptions (notably in arithmetic, geometry, and the calculus) were also subjected to a rigorous scrutiny. Lastly, the application of mathematics to physical problems revolutionized the foundations and treatment of that subject. Numerous Schools, Journals, and Teaching Posts were established, and the facilities for the study of mathematics were greatly extended.
Developments, such as these, may be taken as opening a new period in the history of the subject, and I recognize that in the future a writer who divides the history of mathematics as I have done would probably treat the mathematics of the seventeenth and eighteenth centuries as forming one period, and would treat the mathematics of the nineteenth century as commencing a new period. This, however, would imply a tolerably complete and systematic account of the development of the subject in the nineteenth century. But evidently it is impossible for me to discuss adequately the mathematics of a time so near to us, and the works of mathematicians some of whom are living and some of whom I have met and known. Hence I make no attempt to give a complete account of the mathematics of the nineteenth century, but as a sort of appendix to the preceding chapters I mention the more striking features in the history of recent pure mathematics, in which I include theoretical dynamics and astronomy; I do not, however, propose to discuss in general the recent application of mathematics to physics.
In only a few cases do I give an account of the life and works of the mathematicians mentioned; but I have added brief notes about some of those to whom the development of any branch of the subject is chiefly due, and an indication of that part of it to which they have directed most attention. Even with these limitations it has been very difficult to put together a connected account of the mathematics of recent times; and I wish to repeat explicitly that I do not suggest, nor do I wish my readers to suppose, that my notes on a subject give the names of all the chief writers who have studied it. In fact the quantity of matter produced has been so enormous that no one can expect to do more than make himself acquainted with the works produced in some special branch or branches. As an illustration of this remark I may add that the committee appointed by the Royal Society to report on a catalogue of periodical literature estimated, in 1900, that more than 1500 memoirs on pure mathematics were then issued annually, and more than 40,000 a year on scientific subjects.
Most histories of mathematics do not treat of the work produced during this century. The chief exceptions with which I am acquainted are R. d'Adhémar's L'uvre mathématique du xixe siècle; K. Fink's Geschichte der Mathematik, Tübingen, 1890; E. J. Gerhardt's Geschichte der Mathematik in Deutschland, Munich, 1877; S. Günther's Verm. Unt. zur Geschichte der mathematischen Wissenschaften, Leipzig, 1876, and Ziele und Resultate der neueren mathematisch-historischen Forschung, Erlangen, 1876; J. G. Hagen, Synopsis der höheren Mathematik, 3 volumes, Berlin, 1891, 1893, 1906; a short dissertation by H. Hankel, entitled Die Entwickelung der Mathematik in den letzten Jahrhunderten, Tübingen, 1885; a Discours on the professors at the Sorbonne by C. Hermite in the Bulletin des sciences mathématiques, 1890; F. C. Klein's Lectures on Mathematics, Evanston Colloquium, New York and London, 1894; E. Lampe's Die reine Mathematik in den Jahren 1884--1899, Berlin, 1899; the eleventh and twelfth volumes of Marie's Histoire des sciences, in which are some notes on mathematicians who were born in the last century; P. Painlevé's Les Sciences mathématiques au xix ^e siècle; a chapter by D. E. Smith in Higher Mathematics, by M. Merriman and R. S. Woodward, New York, 1900; and V. Volterra's lecture at the Rome Congress, 1908, "On the history of mathematics in Italy during the latter half of the nineteenth century."
A few histories of the development of particular subjects have been written--such as those by Isaac Todhunter on the theories of attraction and on the calculus of probabilities; those by T. Muir on determinants, that by A. von Braunmühl on trigonometry, that by R. Reiff on infinite series, that by G. Loria, Il passato ed il presente delle principali teorie geometriche, and that by F. Engel and P. Stäckel on the theory of parallels. The transactions of some of the scientific societies and academies also contain reports on the progress in different branches of the subject, while information on the memoirs by particular mathematicians is given in the invaluable volumes of J. C. Poggendorff's Biographisch-literarisches Handwörterbuch zur Geschichte der exacten Wissenschaften, Leipzig. The Encyklopädie der mathematischen Wissenschaften, which is now in course of issue, aims at representing the present state of knowledge in pure and applied mathematics, and doubtless in some branches of mathematics it will supersede these reports. The French translation of this encyclopaedia contains numerous and valuable additions. I have found these authorities and these reports useful, and I have derived further assistance in writing this chapter from the obituary notices in the proceedings of various learned Societies. I am also indebted to information kindly furnished me by various friends, and if I do not further dwell on this, it is only that I would not seem to make them responsible for my errors and omissions.
A period of exceptional intellectual activity in any subject is usually followed by one of comparative stagnation; and after the deaths of Lagrange, Laplace, Legendre, and Poisson, the French school, which had occupied so prominent a position at the beginning of this century, ceased for some years to produce much new work. Some of the mathematicians whom I intend to mention first, Gauss, Abel, and Jacobi, were contemporaries of the later years of the French mathematicians just named, but their writings appear to me to belong to a different school, and thus are properly placed at the beginning of a fresh chapter.
There is no mathematician of this century whose writings have had a greater effect than those of Gauss; nor is it on only one branch of the science that his influence has left a permanent mark. I cannot, therefore, commence my account of the mathematics of recent times better than by describing very briefly his more important researches.
Gauss. Karl Friedrich Gauss was born at Brunswick on April 23, 1777, and died at Göttingen on February 23, 1855. His father was a bricklayer, and Gauss was indebted for a liberal education (much against the will of his parents, who wished to profit by his wages as a labourer) to the notice which his talents procured from the reigning duke. In 1792 he was sent to the Caroline College, and by 1795 professors and pupils alike admitted that he knew all that the former could teach him: it was while there that he investigated the method of least squares, and proved by induction the law of quadratic reciprocity. Thence he went to Göttingen, where he studied under Kästner: many of his discoveries in the theory of numbers were made while a student here. In 1798 he returned to Brunswick, where he earned a somewhat precarious livelihood by private tuition.
In 1799 Gauss published a demonstration that every integral algebraical function of one variable can be expressed as a product of real linear or quadratic factors. Hence every algebraical equation has a root of the form a + bi , a theorem of which he gave later two other distinct proofs. His Disquisitiones Arithmeticae appeared in 1801. A large part of this had been submitted as a memoir to the French Academy in the preceding year, and had been rejected in a most regrettable manner; Gauss was deeply hurt, and his reluctance to publish his investigations may be partly attributable to this unfortunate incident.
The next discovery of Gauss was in a totally different department of mathematics. The absence of any planet in the space between Mars and Jupiter, where Bode's law would have led observers to expect one, had been long remarked, but it was not till 1801 that any one of the numerous group of minor planets which occupy that space was observed. The discovery was made by G. Piazzi of Palermo; and was the more interesting as its announcement occurred simultaneously with a publication by Hegel in which he severely criticised astronomers for not paying more attention to philosophy,--a science, said he, which would at once have shewn them that there could not possibly be more than seven planets, and a study of which would therefore have prevented an absurd waste of time in looking for what in the nature of things could never be found. The new planet was named Ceres, but it was seen under conditions which appeared to render it impracticable to forecast its orbit. The observations were fortunately communicated to Gauss; he calculated its elements, and his analysis put him in the first rank of theoretical astronomers.
The attention excited by these investigations procured for him in 1807 the offer of a chair at Petrograd, which he declined. In the same year he was appointed director of the Göttingen Observatory and professor of Astronomy there. These offices he retained to his death; and after his appointment he never slept away from his Observatory except on one occasion when he attended a scientific congress at Berlin. His lectures were singularly lucid and perfect in form, and it is said that he used here to give the analysis by which he had arrived at his various results, and which is so conspicuously absent from his published demonstrations; but for fear his auditors should lose the thread of his discourse, he never willingly permitted them to take notes.
I have already mentioned Gauss's publications in 1799, 1801, and 1802. For some years after 1807 his time was mainly occupied by work connected with his Observatory. In 1809 he published at Hamburg his Theoria Motus Corporum Coelestium, a treatise which contributed largely to the improvement of practical astronomy, and introduced the principle of curvilinear triangulation; and on the same subject, but connected with observations in general, we have his memoir Theoria Combinationis Observationum Erroribus Minimis Obnoxia, with a second part and a supplement.
Somewhat later he took up the subject of geodesy, acting from 1821 to 1848 as scientific adviser to the Danish and Hanoverian Governments for the survey then in progress; his papers of 1843 and 1866, Ueber Gegenstände der höhern Geodäsie, contain his researches on the subject.
Gauss's researches on electricity and magnetism date from about the year 1830. His first paper on the theory of magnetism, entitled Intensitas Vis Magneticae Terrestris ad Mensuram Absolutam Revocata, was published in 1833. A few months afterwards he, together with W. E. Weber, invented the declination instrument and the bifilar magnetometer; and in the same year they erected at Göttingen a magnetic observatory free from iron (as Humboldt and Arago had previously done on a smaller scale) where they made magnetic observations, and in particular showed that it was practicable to send telegraphic signals. In connection with this Observatory Gauss founded an association with the object of securing continuous observations at fixed times. The volumes of their publications, Resultate aus der Beobachtungen des magnetischen Vereins for 1838 and 1839, contain two important memoirs by Gauss: one on the general theory of earth-magnetism, and the other on the theory of forces attracting according to the inverse square of the distance.
Gauss, like Poisson, treated the phenomena in electrostatics as due to attractions and repulsions between imponderable particles. Lord Kelvin, then William Thomson (1824--1907), of Glasgow, shewed in 1846 that the effects might also be supposed analogous to a flow of heat from various sources of electricity properly distributed.
In electrodynamics Gauss arrived (in 1835) at a result equivalent to that given by W. E. Weber of Göttingen in 1846, namely, that the attraction between two electrified particles e and e' , whose distance apart is r , depends on their relative motion and position according to the formula e e' r^-2 1 + (rr - 12r^2)^2 c^-2 .
Gauss, however, held that no hypothesis was satisfactory which rested on a formula and was not a consequence of a physical conjecture, and as he could not frame a plausible physical conjecture he abandoned the subject.
Such conjectures were proposed by Riemann in 1858, and by C. Neumann, now of Leipzig, and E. Betti (1823--1892) of Pisa in 1868, but Helmholtz in 1870, 1873, and 1874 showed that they were untenable. A simpler view which regards all electric and magnetic phenomena as stresses and motions of a material elastic medium had been outlined by Michael Faraday (1791--1867), and was elaborated by James Clerk Maxwell (1831--1879) of Cambridge in 1873; the latter, by the use of generalised co-ordinates, was able to deduce the consequences, and the agreement with experiment is close. Maxwell concluded by showing that if the medium were the same as the so-called luminiferous ether, the velocity of light would be equal to the ratio of the electromagnetic and electrostatic units, and subsequent experiments have tended to confirm this conclusion. The theories previously current had assumed the existence of a simple elastic solid or an action between matter and ether.
The above and other electric theories were classified by J. J. Thomson of Cambridge, in a report to the British Association in 1885, into those not founded on the principle of the conservation of energy (such as those of Ampère, Grassmann, Stefan, and Korteweg); those which rest on assumptions concerning the velocities and positions of electrified particles (such as those of Gauss, W. E. Weber, Riemann, and R. J. E. Clausius); those which require the existence of a kind of energy of which we have no other knowledge (such as the theory of C. Neumann); those which rest on dynamical considerations, but in which no account is taken of the action of the dielectric (such as the theory of F. E. Neumann); and, finally, those which rest on dynamical considerations and in which the action of the dielectric is considered (such as Maxwell's theory). In the report these theories are described, criticised, and compared with the results of experiments.
Gauss's researches on optics, and especially on systems of lenses, were published in 1840 in his Dioptrische Untersuchungen.
From this sketch it will be seen that the ground covered by Gauss's researches was extraordinarily wide, and it may be added that in many cases his investigations served to initiate new lines of work. He was, however, the last of the great mathematicians whose interests were nearly universal: since his time the literature of most branches of mathematics has grown so fast that mathematicians have been forced to specialise in some particular department or departments. I will now mention very briefly some of the most important of his discoveries in pure mathematics.
His most celebrated work in pure mathematics is the Disquisitiones Arithmeticae, which has proved a starting-point for several valuable investigations on the theory of numbers. This treatise and Legendre's Théorie des nombres remain standard works on the theory of numbers; but, just as in his discussion of elliptic functions Legendre failed to rise to the conception of a new subject, and confined himself to regarding their theory as a chapter in the integral calculus, so he treated the theory of numbers as a chapter in algebra. Gauss, however, realised that the theory of discrete magnitudes or higher arithmetic was of a different kind from that of continuous magnitudes or algebra, and he introduced a new notation and new methods of analysis, of which subsequent writers have generally availed themselves. The theory of numbers may be divided into two main divisions, namely, the theory of congruences and the theory of forms. Both divisions were discussed by Gauss. In particular the Disquisitiones Arithmeticae introduced the modern theory of congruences of the first and second orders, and to this Gauss reduced indeterminate analysis. In it also he discussed the solution of binomial equations of the form x^n = 1 : this involves the celebrated theorem that it is possible to construct, by elementary geometry, regular polygons of 2^m (2^n + 1) sides, where m and n are integers and 2^n + 1 is a prime--a discovery he had made in 1796. He developed the theory of ternary quadratic forms involving two indeterminates. He also investigated the theory of determinants, and it was on Gauss's results that Jacobi based his researches on that subject.
The theory of functions of double periodicity had its origin in the discoveries of Abel and Jacobi, which I describe later. Both these mathematicians arrived at the theta functions, which play so large a part in the theory of the subject. Gauss, however, had independently, and indeed at a far earlier date, discovered these functions and some of their properties, having been led to them by certain integrals which occurred in the Determinatio Attractionis, to evaluate which he invented the transformation now associated with the name of Jacobi. Though Gauss at a later time communicated the fact to Jacobi, he did not publish his researches; they occur in a series of note-books of a date not later than 1808, and are included in his collected works.
Of the remaining memoirs in pure mathematics the most remarkable are those on the theory of biquadratic residues (wherein the notion of complex numbers of the form a + bi was first introduced into the theory of numbers), in which are included several tables, and notably one of the number of the classes of binary quadratic forms; that relating to the proof of the theorem that every algebraical equation has a real or imaginary root; that on the summation of series; and, lastly, one on interpolation. His introduction of rigorous tests for the convergency of infinite series is worthy of attention. Specially noticeable also are his investigations on hypergeometric series; these contain a discussion of the gamma function. This subject has since become one of considerable importance, and has been written on by (among others) Kummer and Riemann; later the original conceptions were greatly extended, and numerous memoirs on it and its extensions have appeared. I should also mention Gauss's theorems on the curvature of surfaces, wherein he devised a new and general method of treatment which has led to many new results. Finally, we have his important memoir on the conformal representation of one surface upon another, in which the results given by Lagrange for surfaces of revolution are generalised for all surfaces. It would seem also that Gauss had discovered some of the properties of quaternions, though these investigations were not published until a few years ago.
In the theory of attractions we have a paper on the attraction of homogeneous ellipsoids: the already-mentioned memoir of 1839, on the theory of forces attracting according to the inverse square of the distance; and the memoir, Determinatio Attractionis, in which it is shown that the secular variations, which the elements of the orbit of a planet experience from the attraction of another planet which disturbs it, are the same as if the mass of the disturbing planet were distributed over its orbit into an elliptic ring in such a manner that equal masses of the ring would correspond to arcs of the orbit described in equal times.
The great masters of modern analysis are Lagrange, Laplace, and Gauss, who were contemporaries. It is interesting to note the marked contrast in their styles. Lagrange is perfect both in form and matter, he is careful to explain his procedure, and though his arguments are general they are easy to follow. Laplace, on the other hand, explains nothing, is indifferent to style, and, if satisfied that his results are correct, is content to leave them either with no proof or with a faulty one. Gauss is as exact and elegant as Lagrange, but even more difficult to follow than Laplace, for he removes every trace of the analysis by which he reached his results, and studies to give a proof which, while rigorous, shall be as concise and synthetical as possible.
Dirichlet. One of Gauss's pupils to whom I may here allude is Lejeune Dirichlet, whose masterly exposition of the discoveries of Jacobi (who was his father-in-law) and of Gauss has unduly overshadowed his own original investigations on similar subjects. Peter Gustav Lejeune Dirichlet was born at Düren on February 13, 1805, and died at Göttingen on May 5, 1859. He held successively professorships at Breslau and Berlin, and on Gauss's death in 1855 was appointed to succeed him as professor of the higher mathematics at Göttingen. He intended to finish Gauss's incomplete works, for which he was admirably fitted, but his early death prevented this. He produced, however, several memoirs which have considerably facilitated the comprehension of some of Gauss's more abstruse methods. Of Dirichlet's original researches the most celebrated are those dealing with the establishment of Fourier's theorem, those in the theory of numbers on asymptotic laws (that is, laws which approximate more closely to accuracy as the numbers concerned become larger), and those on primes.
It is convenient to take Gauss's researches as the starting-point for the discussion of various subjects. Hence the length with which I have alluded to them.
The Theory of Numbers, or Higher Arithmetic. The researches of Gauss on the theory of numbers were continued or supplemented by Jacobi, who first proved the law of cubic reciprocity; discussed the theory of residues; and, in his Canon Arithmeticus, gave a table of residues of prime roots. Dirichlet also paid some attention to this subject.
Eisenstein. The subject was next taken up by Ferdinand Gotthold Eisenstein, a professor at the University of Berlin, who was born at Berlin on April 16, 1823, and died there on October 11, 1852. The solution of the problem of the representation of numbers by binary quadratic forms is one of the great achievements of Gauss, and the fundamental principles upon which the treatment of such questions rest were given by him in the Disquisitiones Arithmeticae. Gauss there added some results relating to ternary quadratic forms, but the general extension from two to three indeterminates was the work of Eisenstein, who, in his memoir Neue Theoreme der höheren Arithmetik, defined the ordinal and generic characters of ternary quadratic forms of an uneven determinant; and, in the case of definite forms, assigned the weight of any order or genus; but he did not consider forms of an even determinant, nor give any demonstrations of his work.
Eisenstein also considered the theorems relating to the possibility of representing a number as a sum of squares, and showed that the general theorem was limited to eight squares. The solutions in the cases of two, four, and six squares may be obtained by means of elliptic functions, but the cases in which the number of squares is uneven involve special processes peculiar to the theory of numbers. Eisenstein gave the solution in the case of three squares. He also left a statement of the solution he had obtained in the case of five squares; but his results were published without proofs, and apply only to numbers which are not divisible by a square.
Henry Smith. One of the most original mathematicians of the school founded by Gauss was Henry Smith. Henry John Stephen Smith was born in London on November 2, 1826, and died at Oxford on February 9, 1883. He was educated at Rugby, and at Balliol College, Oxford, of which latter society he was a fellow; and in 1861 he was elected Savilian professor of Geometry at Oxford, where he resided till his death.
The subject in connection with which Smith's name is specially associated is the theory of numbers, and to this he devoted the years from 1854 to 1864. The results of his historical researches were given in his report published in parts in the Transactions of the British Association from 1859 to 1865. This report contains an account of what had been done on the subject to that time together with some additional matter. The chief outcome of his own original work on the subject is included in two memoirs printed in the Philosophical Transactions for 1861 and 1867; the first being on linear indeterminate equations and congruences, and the second on the orders and genera of ternary quadratic forms. In the latter memoir demonstrations of Eisenstein's results and their extension to ternary quadratic forms of an even determinant were supplied, and a complete classification of ternary quadratic forms was given.
Smith, however, did not confine himself to the case of three indeterminates, but succeeded in establishing the principles on which the extension to the general case of n indeterminates depends, and obtained the general formulae--thus effecting the greatest advance made in the subject since the publication of Gauss's work. In the account of his methods and results which appeared in the Proceedings of the Royal Society, Smith remarked that the theorems relating to the representation of numbers by four squares and other simple quadratic forms, are deducible by a uniform method from the principles there indicated, as also are the theorems relating to the representation of numbers by six and eight squares. He then proceeded to say that as the series of theorems relating to the representation of numbers by sums of squares ceases, for the reason assigned by Eisenstein, when the number of squares surpasses eight, it was desirable to complete it. The results for even squares were known. The principal theorems relating to the case of five squares had been given by Eisenstein, but he had considered only those numbers which are not divisible by a square, and he had not considered the case of seven squares. Smith here completed the enunciation of the theorems for the case of five squares, and added the corresponding theorems for the case of seven squares.
This paper was the occasion of a dramatic incident in the history of mathematics. Fourteen years later, in ignorance of Smith's work, the demonstration and completion of Eisenstein's theorems for five squares were set by the French Academy as the subject of their "Grand prix des sciences mathématiques." Smith wrote out the demonstration of his general theorems so far as was required to prove the results in the special case of five squares, and only a month after his death, in March 1883, the prize was awarded to him, another prize being also awarded to H. Minkowski of Bonn. No episode could bring out in a more striking light the extent of Smith's researches than that a question, of which he had given the solution in 1867, as a corollary from general formulae which governed the whole class of investigations to which it belonged, should have been regarded by the French Academy as one whose solution was of such difficulty and importance as to be worthy of their great prize. It has been also a matter of comment that they should have known so little of contemporary English and German researches on the subject as to be unaware that the result of the problem they were proposing was then lying in their own library.
J. W. L. Glaisher of Cambridge has recently extended these results, and investigated, by the aid of elliptic functions, the number of representations of a number as the sum of 2n squares where n is not greater than 9.
Among Smith's other investigations I may specially mention his geometrical memoir, Sur quelques problèmes cubiques et biquadratiques, for which in 1868 he was awarded the Steiner prize of the Berlin Academy. In a paper which he contributed to the Atti of the Accademia dei Lincei for 1877 he established a very remarkable analytical relation connecting the modular equation of order n , and the theory of binary quadratic forms belonging to the positive determinant n . In this paper the modular curve is represented analytically by a curve in such a manner as to present an actual geometrical image of the complete systems of the reduced quadratic forms belonging to the determinant, and a geometrical interpretation is given to the ideas of "class," "equivalence," and "reduced form." He was also the author of important papers in which he succeeded in extending to complex quadratic forms many of Gauss's investigations relating to real quadratic forms. He was led by his researches on the theory of numbers to the theory of elliptic functions, and the results he arrived at, especially on the theories of the theta and omega functions, are of importance.
Kummer. The theory of primes received a somewhat unexpected development by E. E. Kummer of Berlin, who was born in 1810 and died in 1893. In particular he treated higher complex members of the form a + sum bj , where j is a complex root of j^p - 1 = 0 , p being a prime. His theory brought out the unexpected result that the proposition that a number can be resolved into the product of powers of primes in one and only one way is not necessarily true of every complex number. This led to the theory of ideal primes, a theory which was developed later by J. W. R. Dedekind. Kummer also extended Gauss's theorems on quadratic residues to residues of a higher order, and wrote on the transformations of hypergeometric functions.
The theory of numbers, as treated to-day, may be said to originate with Gauss. I have already mentioned very briefly the investigations of Jacobi, Dirichlet, Eisenstein, Henry Smith, and Kummer. I content myself with adding some notes on the subsequent development of certain branches of the theory.
The distribution of primes has been discussed in particular by P. L. Tchebycheff (1821--1894) of Petrograd, G. F. B. Riemann, and J. J. Sylvester. Riemann's short tract on the number of primes which lie between two given numbers affords a striking instance of his analytical powers. Legendre had previously shown that the number of primes less than n is approximately n/(loge n - 1.08366) ; but Riemann went farther, and this tract and a memoir by Tchebycheff contain nearly all that has been done yet in connection with a problem of so obvious a character, that it has suggested itself to all who have considered the theory of numbers, and yet which overtaxed the powers even of Lagrange and Gauss. In this paper also Riemann stated that all the roots of Gamma(1/2s + 1)(s - 1)pi^-delta/2zeta(s) are of the form 1/2 + it where t is real. It is believed that the theorem is true, but as yet it has defied all attempts to prove it. Riemann's work in this connection has proved the starting-point for researches by J. S. Hadamard, H. C. F. von Mangoldt, and other recent writers.
The partition of numbers, a problem to which Euler had paid considerable attention, has been treated by A. Cayley, J. J. Sylvester, and P. A. MacMahon. The representation of numbers in special forms, the possible divisors of numbers of specified forms, and general theorems concerned with the divisors of numbers, have been discussed by J. Liouville (1809--1882), the editor from 1836 to 1874 of the well-known mathematical journal, and by J. W. L. Glaisher of Cambridge. The subject of quadratic binomials has been studied by A. L. Cauchy; of ternary and quadratic forms by L. Kronecker (1823--1891) of Berlin; and of ternary forms by C. Hermite of Paris.
The most common text-books are, perhaps, that by O. Stolz of Innspruck, Leipzig, 1885--6; that by G. B. Mathews, Cambridge, 1892; that by E. Lucas, Paris, 1891; and those by P. Bachmann, Leipzig, 1892--1905. Possibly it may be found hereafter that the subject is approached better on other lines than those now usual.
The conception of Number has also been discussed at considerable length during the last quarter of the nineteenth century. Transcendent numbers had formed the subject of two memoirs by Liouville, but were subsequently treated as a distinct branch of mathematics, notably by L. Kronecker and G. Cantor. Irrational numbers and the nature of numbers have also been treated from first principles, in particular by K. Weierstrass, J. W. R. Dedekind, H. C. R. Méray, G. Cantor, G. Peano, and B. A. W. Russell. This subject has attracted much attention of late years, and is now one of the most flourishing branches of modern mathematics. Transfinite, cardinal, and ordinal arithmetic, and the theory of sets of points, may be mentioned as prominent divisions. The theory of aggregates is related to this subject, and has been treated by G. Cantor, P. du Bois-Raymond, A. Schönflies, E. Zermelo, and B. A. W. Russell.
Elliptic and Abelian Functions, or Higher Trigonometry. The theory of functions of double and multiple periodicity is another subject to which much attention has been paid during this century. I have already mentioned that as early as 1808 Gauss had discovered the theta functions and some of their properties, but his investigations remained for many years concealed in his note-books; and it was to the researches made between 1820 and 1830 by Abel and Jacobi that the modern development of the subject is due. Their treatment of it has completely superseded that used by Legendre, and they are justly reckoned as the creators of this branch of mathematics.
Abel. Niels Henrick Abel was born at Findoe, in Norway, on August 5, 1802, and died at Arendal on April 6, 1829, at the age of twenty-six. His memoirs on elliptic functions, originally published in Crelle's Journal (of which he was one of the founders), treat the subject from the point of view of the theory of equations and algebraic forms, a treatment to which his researches naturally led him.
The important and very general result known as Abel's theorem, which was subsequently applied by Riemann to the theory of transcendental functions, was sent to the French Academy in 1826, but was not printed until 1841: its publication then was due to inquiries made by Jacobi, in consequence of a statement on the subject by B. Holmboe in his edition of Abel's works issued in 1839. It is far from easy to state Abel's theorem intelligently and yet concisely, but, broadly speaking, it may be described as a theorem for evaluating the sum of a number of integrals which have the same integrand, but different limits--these limits being the roots of an algebraic equation. The theorem gives the sum of the integrals in terms of the constants occurring in this equation and in the integrand. We may regard the inverse of the integral of this integrand as a new transcendental function, and if so the theorem furnishes a property of this function. For instance, if Abel's theorem be applied to the integrand (1 - x^2)^-1/2 it gives the addition theorem for the circular (or trigonometrical) functions.
The name of Abelian function has been given to the higher transcendents of multiple periodicity which were first discussed by Abel. The Abelian functions connected with a curve f(x, y) are of the form int u dx where u is a rational function of x and y . The theory of Abelian functions has been studied by a very large number of modern writers.
Abel criticised the use of infinite series, and discovered the well-known theorem which furnishes a test for the validity of the result obtained by multiplying one infinite series by another. He also proved the binomial theorem for the expansion of (1 + x)^n when x and n are complex. As illustrating his fertility of ideas I may, in passing, notice his celebrated demonstration that it is impossible to express a root of the general quintic equation in terms of its coefficients by means of a finite number of radicals and rational functions; this theorem was the more important since it definitely limited a field of mathematics which had previously attracted numerous writers. I should add that this theorem had been enunciated as early as 1798 by Paolo Ruffini, an Italian physician practising at Modena; but I believe that the proof he gave was deficient in generality.
Jacobi. Carl Gustav Jacob Jacobi, born of Jewish parents at Potsdam on Dec. 10, 1804, and died at Berlin on Feb. 18, 1851, was educated at the University of Berlin, where he obtained the degree of Doctor of Philosophy in 1825. In 1827 he became extraordinary professor of Mathematics at Königsberg, and in 1829 was promoted to be an ordinary professor. This chair he occupied till 1842, when the Prussian Government gave him a pension, and he moved to Berlin, where he continued to live till his death in 1851. He was the greatest mathematical teacher of his generation, and his lectures, though somewhat unsystematic in arrangement, stimulated and influenced the more able of his pupils to an extent almost unprecedented at the time.
Jacobi's most celebrated investigations are those on elliptic functions, the modern notation in which is substantially due to him, and the theory of which he established simultaneously with Abel, but independently of him. Jacobi's results are given in his treatise on elliptic functions, published in 1829, and in some later papers in Crelle's Journal; they are earlier than Weierstrass's researches which are mentioned below. The correspondence between Legendre and Jacobi on elliptic functions has been reprinted in the first volume of Jacobi's collected works. Jacobi, like Abel, recognised that elliptic functions were not merely a group of theorems on integration, but that they were types of a new kind of function, namely, one of double periodicity; hence he paid particular attention to the theory of the theta function. The following passage, in which he explains this view, is sufficiently interesting to deserve textual reproduction:--
E quo, cum universam, quae fingi potest, amplectatur periodicitatem analyticam elucet, functiones ellipticas non aliis adnumerari debere transcendentibus, quae quibusdam gaudent elegantiis, fortasse pluribus illas aut maioribus, sed speciem quandam iis inesse perfecti et absoluti.
Among Jacobi's other investigations I may specially single out his papers on Determinants, which did a great deal to bring them into general use; and particularly his introduction of the Jacobian, that is, of the functional determinant formed by the n^2 partial differential coefficients of the first order of n given functions of n independent variables. I ought also to mention his papers on Abelian transcendents; his investigations on the theory of numbers, to which I have already alluded; his important memoirs on the theory of differential equations, both ordinary and partial; his development of the calculus of variations; and his contributions to the problem of three bodies, and other particular dynamical problems. Most of the results of the researches last named are included in his Vorlesungen über Dynamik.
Riemann. Georg Friedrich Bernhard Riemann was born at Breselenz on Sept. 17, 1826, and died at Selasca on July 20, 1866. He studied at Göttingen under Gauss, and subsequently at Berlin under Jacobi, Dirichlet, Steiner, and Eisenstein, all of whom were professors there at the same time. In spite of poverty and sickness he struggled to pursue his researches. In 1857 he was made professor at Göttingen, general recognition of his powers soon followed, but in 1862 his health began to give way, and four years later he died, working, to the end, cheerfully and courageously.
Riemann must be esteemed one of the most profound and brilliant mathematicians of his time; he was a creative genius. The amount of matter he produced is small, but its originality and power are manifest--his investigations on functions and on geometry, in particular, initiating developments of great importance.
His earliest paper, written in 1850, was on the general theory of functions of a complex variable. This gave rise to a new method of treating the theory of functions. The development of this method is specially due to the Göttingen school. In 1854 Riemann wrote his celebrated memoir on the hypotheses on which geometry is founded: to this subject I allude below. This was succeeded by memoirs on elliptic functions and on the distribution of primes: these have been already mentioned. He also investigated the conformal representation of areas, one on the other: a problem subsequently treated by H. A. Schwarz and F. H. Schottky, both of Berlin. Lastly, in multiple periodic functions, it is hardly too much to say that in his memoir in Borchardt's Journal for 1857, he did for the Abelian functions what Abel had done for the elliptic functions. A posthumous fragment on linear differential equations with algebraic coefficients has served as the foundation of important work by L. Schlesinger.
I have already alluded to the researches of Legendre, Gauss, Abel, Jacobi, and Riemann on elliptic and Abelian functions. The subject has been also discussed by (among other writers) J. G. Rosenhain (1816--1887) of Königsberg, who wrote (in 1844) on the hyperelliptic, and double theta functions; A. Göpel (1812--1847) of Berlin, who discussed hyperelliptic functions; L. Kronecker of Berlin, who wrote on elliptic functions; L. Königsberger of Heidelberg and F. Brioschi (1824--1897) of Milan, both of whom wrote on elliptic and hyperelliptic functions; Henry Smith of Oxford, who discussed the transformation theory, the theta and omega functions, and certain functions of the modulus; A. Cayley of Cambridge, who was the first to work out (in 1845) the theory of doubly infinite products and determine their periodicity, and who has written at length on the connection between the researches of Legendre and Jacobi; and C. Hermite of Paris, whose researches are mostly concerned with the transformation theory and the higher development of the theta functions.
Weierstrass. The subject of higher trigonometry was put on a somewhat different footing by the researches of Weierstrass. Karl Weierstrass, born in Westphalia on October 31, 1815, and died at Berlin on February 19, 1897, was one of the greatest mathematicians of the nineteenth century. He took no part in public affairs; his life was uneventful; and he spent the last forty years of it at Berlin, where he was professor.
With two branches of pure mathematics--elliptic and Abelian functions, and the theory of functions--his name is inseparably connected. His earlier researches on elliptic functions related to the theta functions, which he treated under a modified form in which they are expressible in powers of the modulus. At a later period he developed a method for treating all elliptic functions in a symmetrical manner. Jacobi had shown that a function of n variables might have 2n periods. Accordingly Weierstrass sought the most general expressions for such functions, and showed that they enjoyed properties analogous to those of the hyperelliptic functions. Hence the properties of the latter functions could be reduced as particular cases of general results.
He was naturally led to this method of treating hyperelliptic functions by his researches on the general theory of functions; these co-ordinated and comprised various lines of investigation previously treated independently. In particular he constructed a theory of uniform analytic functions. The representation of functions by infinite products and series also claimed his especial attention. Besides functions he also wrote or lectured on the nature of the assumptions made in analysis, on the calculus of variations, and on the theory of minima surfaces. His methods are noticeable for their wide-reaching and general character. Recent investigations on elliptic functions have been largely based on Weierstrass's method.
Among other prominent mathematicians who have recently written on elliptic and hyperelliptic functions, I may mention the names of G. H. Halphen (1844--1889), an officer in the French army, whose investigations were largely founded on Weierstrass's work; F. C. Klein of Göttingen, who has written on Abelian functions, elliptic modular functions, and hyperelliptic functions; H. A. Schwarz of Berlin; H. Weber of Strassburg; M. Nöther of Erlangen; H. B. L. Stahl of Tübingen; F. G. Frobenius of Berlin; J. W. L. Glaisher of Cambridge, who has in particular developed the theory of the zeta function; and H. F. Baker of Cambridge.
The usual text-books of to-day on elliptic functions are those by J. Tannery and J. Molk, 4 volumes, Paris, 1893--1901; by P. E. Appell and E. Lacour, Paris, 1896; by H. Weber, Brunswick, 1891; and by G. H. Halphen, 3 volumes, Paris, 1886--1891. To these I may add one by A. G. Greenhill on the Applications of Elliptic Functions, London, 1892.
The Theory of Functions. I have already mentioned that the modern theory of functions is largely due to Weierstrass and H. C. R. Méray. It is a singularly attractive subject, and has proved an important and far-reaching branch of mathematics. Historically its modern presentation may be said to have been initiated by A. Cauchy, who laid the foundations of the theory of synectic functions of a complex variable. Work on these lines was continued by J. Liouville, who wrote chiefly on doubly periodic functions. These investigations were extended and connected in the work by A. Briot (1817--1882), and J. C. Bouquet (1819--1885), and subsequently were further developed by C. Hermite.
Next I may refer to the researches on the theory of algebraic functions which have their origin in V. A. Puiseux's memoir of 1851, and G. F. B. Riemann's papers of 1850 and 1857; in continuation of which H. A. Schwarz of Berlin established accurately certain theorems of which the proofs given by Riemann were open to objection. To Riemann also we are indebted for valuable work on modular functions which has been recently published in his Nachträge. Subsequently F. C. Klein of Göttingen connected Riemann's theory of functions with the theory of groups, and wrote on automorphic and modular functions; H. Poincaré of Paris also wrote on automorphic functions, and on the general theory with special applications to differential equations. Quite recently K. Hensel of Marburg has written on algebraic functions; and W. Wirtinger of Vienna on Abelian functions.
I have already said that the work of Weierstrass shed a new light on the whole subject. His theory of analytical functions has been developed by G. Mittag-Leffler of Stockholm; and C. Hermite, P. E. Appell, C. E. Picard, E. Goursat, E. N. Laguerre, and J. S. Hadamard, all of Paris, have also written on special branches of the general theory; while E. Borel, R. L. Baire, H. L. Lebesgue, and E. L. Lindelöf have produced a series of tracts on uniform functions which have had a wide circulation and influence.
As text-books I may mention the Theory of Functions of a Complex Variable, by A. R. Forsyth, second edition, Cambridge, 1900; Abel's Theorem by H. F. Baker, Cambridge, 1897, and Multiple Periodic Functions by the same writer, Cambridge, 1907; the Théorie des fonctions algébriques by P. E. Appell and E. Goursat, Paris, 1895; parts of C. E. Picard's Traité d'Analyse, in 3 volumes, Paris, 1891 to 1896; the Theory of Functions by J. Harkness and F. Morley, London, 1893; the Theory of Functions of a Real Variable and of Fourier's Series by E. W. Hobson, Cambridge, 1907; and Die Theorie des Abel'schen Functionen by H. B. L. Stahl, Leipzig, 1896.
Higher Algebra. The theory of numbers may be considered as a higher arithmetic, and the theory of elliptic and Abelian functions as a higher trigonometry. The theory of higher algebra (including the theory of equations) has also attracted considerable attention, and was a favourite subject of study of the mathematicians whom I propose to mention next, though the interests of these writers were by no means limited to this subject.
Cauchy. Augustin Louis Cauchy, the leading representative of the French school of analysis in the nineteenth century, was born at Paris on Aug. 21, 1789, and died at Sceaux on May 25, 1857. He was educated at the Polytechnic school, the nursery of so many French mathematicians of that time, and adopted the profession of a civil engineer. His earliest mathematical paper was one on polyhedra in 1811. Legendre thought so highly of it that he asked Cauchy to attempt the solution of an analogous problem which had baffled previous investigators, and his choice was justified by the success of Cauchy in 1812. Memoirs on analysis and the theory of numbers, presented in 1813, 1814, and 1815, showed that his ability was not confined to geometry alone. In one of these papers he generalised some results which had been established by Gauss and Legendre; in another of them he gave a theorem on the number of values which an algebraical function can assume when the literal constants it contains are interchanged. It was the latter theorem that enabled Abel to show that in general an algebraic equation of a degree higher than the fourth cannot be solved by the use of a finite number of purely algebraical expressions.
To Abel, Cauchy, and Gauss we owe the scientific treatment of series which have an infinite number of terms. In particular, Cauchy established general rules for investigating the convergency and divergency of such series, rules which were extended by J. L. F. Bertrand (1822--1900) of Paris, Secretary of the French Académie des Sciences, A. Pringsheim of Munich, and considerably amplified later by E. Borel, by M. G. Servant, both of Paris, and by other writers of the modern French school. In only a few works of an earlier date is there any discussion as to the limitations of the series employed. It is said that Laplace, who was present when Cauchy read his first paper on the subject, was so impressed by the illustrations of the danger of employing such series without a rigorous investigation of their convergency, that he put on one side the work on which he was then engaged and denied himself to all visitors, in order to see if any of the demonstrations given in the earlier volumes of the Mécanique céleste were invalid; and he was fortunate enough to find that no material errors had been thus introduced. The treatment of series and of the fundamental conceptions of the calculus in most of the text-books then current was based on Euler's works, and was not free from objection. It is one of the chief merits of Cauchy that he placed these subjects on a stricter foundation.
On the restoration in 1816 the French Academy was purged, and, incredible though it may seem, Cauchy accepted a seat procured for him by the expulsion of Monge. He was also at the same time made professor at the Polytechnic; and his lectures there on algebraic analysis, the calculus, and the theory of curves, were published as text-books. On the revolution in 1830 he went into exile, and was first appointed professor at Turin, whence he soon moved to Prague to undertake the education of the Comte de Chambord. He returned to France in 1837; and in 1848, and again in 1851, by special dispensation of the Emperor was allowed to occupy a chair of mathematics without taking the oath of allegiance.
His activity was prodigious, and from 1830 to 1859 he published in the Transactions of the Academy, or the Comptes Rendus, over 600 original memoirs and about 150 reports. They cover an extraordinarily wide range of subjects, but are of very unequal merit.
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