A History of Mathematics/Recent Times/Theory of Numbers

by Florian Cajori

 

THEORY OF NUMBERS.

 

"Mathematics, the queen of the sciences, and arithmetic, the queen of mathematics." Such was the dictum of Gauss, who was destined to revolutionise the theory of numbers. When asked who was the greatest mathematician in Germany, Laplace answered, Pfaff. When the questioner said he should have thought Gauss was, Laplace replied, "Pfaff is by far the greatest mathematician in Germany; but Gauss is the greatest in all Europe."^[83] Gauss is one of the three greatest masters of modern analysis,—Lagrange, Laplace, Gauss. Of these three contemporaries he was the youngest. While the first two belong to the period in mathematical history preceding the one now under consideration, Gauss is the one whose writings may truly be said to mark the beginning of our own epoch. In him that abundant fertility of invention, displayed by mathematicians of the preceding period, is combined with an absolute rigorousness in demonstration which is too often wanting in their writings, and which the ancient Greeks might have envied. Unlike Laplace, Gauss strove in his writings after perfection of form. He rivals Lagrange in elegance, and surpasses this great Frenchman in rigour. Wonderful was his richness of ideas; one thought followed another so quickly that he had hardly time to write down even the most meagre outline. At the age of twenty Gauss had overturned old theories and old methods in all branches of higher mathematics; but little pains did he take to publish his results, and thereby to establish his priority. He was the first to observe rigour in the treatment of infinite series, the first to fully recognise and emphasise the importance, and to make systematic use of determinants and of imaginaries, the first to arrive at the method of least squares, the first to observe the double periodicity of elliptic functions. He invented the heliotrope and, together with Weber, the bifilar magnetometer and the declination instrument. He reconstructed the whole of magnetic science.

Carl Friedrich Gauss^[47] (1777–1855), the son of a bricklayer, was born at Brunswick. He used to say, jokingly, that he could reckon before he could talk. The marvellous aptitude for calculation of the young boy attracted the attention of Bartels, afterwards professor of mathematics at Dorpat, who brought him under the notice of Charles William, Duke of Brunswick. The duke undertook to educate the boy, and sent him to the Collegium Carolinum. His progress in languages there was quite equal to that in mathematics. In 1795 he went to Göttingen, as yet undecided whether to pursue philology or mathematics. Abraham Gotthelf Kästner, then professor of mathematics there, and now chiefly remembered for his Geschichte der Mathematik (1796), was not an inspiring teacher. At the age of nineteen Gauss discovered a method of inscribing in a circle a regular polygon of seventeen sides, and this success encouraged him to pursue mathematics. He worked quite independently of his teachers, and while a student at Göttingen made several of his greatest discoveries. Higher arithmetic was his favourite study. Among his small circle of intimate friends was Wolfgang Bolyai. After completing his course he returned to Brunswick. In 1798 and 1799 he repaired to the university at Helmstadt to consult the library, and there made the acquaintance of Pfaff, a mathematician of much power. In 1807 the Emperor of Russia offered Gauss a chair in the Academy at St. Petersburg, but by the advice of the astronomer Olbers, who desired to secure him as director of a proposed new observatory at Göttingen, he declined the offer, and accepted the place at Göttingen. Gauss had a marked objection to a mathematical chair, and preferred the post of astronomer, that he might give all his time to science. He spent his life in Göttingen in the midst of continuous work. In 1828 he went to Berlin to attend a meeting of scientists, but after this he never again left Göttingen, except in 1854, when a railroad was opened between Göttingen and Hanover. He had a strong will, and his character showed a curious mixture of self-conscious dignity and child-like simplicity. He was little communicative, and at times morose.

A new epoch in the theory of numbers dates from the publication of his Disquisitiones Arithmeticœ, Leipzig, 1801. The beginning of this work dates back as far as 1795. Some of its results had been previously given by Lagrange and Euler, but were reached independently by Gauss, who had gone deeply into the subject before he became acquainted with the writings of his great predecessors. The Disquisitiones Arithmeticœ was already in print when Legendre's Théorie des Nombres appeared. The great law of quadratic reciprocity, given in the fourth section of Gauss' work, a law which involves the whole theory of quadratic residues, was discovered by him by induction before he was eighteen, and was proved by him one year later. Afterwards he learned that Euler had imperfectly enunciated that theorem, and that Legendre had attempted to prove it, but met with apparently insuperable difficulties. In the fifth section Gauss gave a second proof of this "gem" of higher arithmetic. In 1808 followed a third and fourth demonstration; in 1817, a fifth and sixth. No wonder that he felt a personal attachment to this theorem. Proofs were given also by Jacobi, Eisenstein, Liouville, Lebesgue, A. Genocchi, Kummer, M. A. Stern, Chr. Zeller, Kronecker, Bouniakowsky, E. Schering, J. Petersen, Voigt, E. Busche, and Th. Pepin.^[48] The solution of the problem of the representation of numbers by binary quadratic forms is one of the great achievements of Gauss. He created a new algorithm by introducing the theory of congruences. The fourth section of the Disquisitiones Arithmeticœ, treating of congruences of the second degree, and the fifth section, treating of quadratic forms, were, until the time of Jacobi, passed over with universal neglect, but they have since been the starting-point of a long series of important researches. The seventh or last section, developing the theory of the division of the circle, was received from the start with deserved enthusiasm, and has since been repeatedly elaborated for students. A standard work on Kreistheilung was published in 1872 by Paul Bachmann, then of Breslau. Gauss had planned an eighth section, which was omitted to lessen the expense of publication. His papers on the theory of numbers were not all included in his great treatise. Some of them were published for the first time after his death in his collected works (1863–1871). He wrote two memoirs on the theory of biquadratic residues (1825 and 1831), the second of which contains a theorem of biquadratic reciprocity.

Gauss was led to astronomy by the discovery of the planet Ceres at Palermo in 1801. His determination of the elements of its orbit with sufficient accuracy to enable Olbers to rediscover it, made the name of Gauss generally known. In 1809 he published the Theoria motus corporum coelestium, which contains a discussion of the problems arising in the determination of the movements of planets and comets from observations made on them under any circumstances. In it are found four formulæ in spherical trigonometry, now usually called "Gauss' Analogies," but which were published somewhat earlier by Karl Brandon Mollweide of Leipzig (1774–1825), and earlier still by Jean Baptiste Joseph Delambre (1749–1822).^[44] Many years of hard work were spent in the astronomical and magnetic observatory. He founded the German Magnetic Union, with the object of securing continuous observations at fixed times. He took part in geodetic observations, and in 1843 and 1846 wrote two memoirs, Ueber Gegenstände der höheren Geodesie. He wrote on the attraction of homogeneous ellipsoids, 1813. In a memoir on capillary attraction, 1833, he solves a problem in the calculus of variations involving the variation of a certain double integral, the limits of integration being also variable; it is the earliest example of the solution of such a problem. He discussed the problem of rays of light passing through a system of lenses.

Among Gauss' pupils were Christian Heinrich Schumacher, Christian Gerling, Friedrich Nicolai, August Ferdinand Möbius, Georg Wilhelm Struve, Johann Frantz Encke.

Gauss' researches on the theory of numbers were the starting-point for a school of writers, among the earliest of whom was Jacobi. The latter contributed to Crelle's Journal an article on cubic residues, giving theorems without proofs. After the publication of Gauss' paper on biquadratic residues, giving the law of biquadratic reciprocity, and his treatment of complex numbers, Jacobi found a similar law for cubic residues. By the theory of elliptical functions, he was led to beautiful theorems on the representation of numbers by 2, 4, 6, and 8 squares. Next come the researches of Dirichlet, the expounder of Gauss, and a contributor of rich results of his own.

Peter Gustav Lejeune Dirichlet^[88] (1805–1859) was born in Düren, attended the gymnasium in Bonn, and then the Jesuit gymnasium in Cologne. In 1822 he was attracted to Paris by the names of Laplace, Legendre, Fourier, Poisson, Cauchy. The facilities for a mathematical education there were far better than in Germany, where Gauss was the only great figure. He read in Paris Gauss' Disquisitiones Arithmeticœ, a work which he never ceased to admire and study. Much in it was simplified by Dirichlet, and thereby placed within easier reach of mathematicians. His first memoir on the impossibility of certain indeterminate equations of the fifth degree was presented to the French Academy in 1825. He showed that Fermat's equation, $\scriptstyle {x^{n}+y^{n}=z^{n}}$, cannot exist when $\scriptstyle {n=5}$. Some parts of the analysis are, however, Legendre's. Euler and Lagrange had proved this when $\scriptstyle {n}$ is 3 and 4, and Lamé proved it when $\scriptstyle {n=7}$. Dirichlet's acquaintance with Fourier led him to investigate Fourier's series. He became docent in Breslau in 1827. In 1828 he accepted a position in Berlin, and finally succeeded Gauss at Göttingen in 1855. The general principles on which depends the average number of classes of binary quadratic forms of positive and negative determinant (a subject first investigated by Gauss) were given by Dirichlet in a memoir, Ueber die Bestimmung der mittleren Werthe in der Zahlentheorie, 1849. More recently F. Mertens of Graz has determined the asymptotic values of several numerical functions. Dirichlet gave some attention to prime numbers. Gauss and Legendre had given expressions denoting approximately the asymptotic value of the number of primes inferior to a given limit, but it remained for Riemann in his memoir, Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse, 1859, to give an investigation of the asymptotic frequency of primes which is rigorous. Approaching the problem from a different direction, Patnutij Tchebycheff, formerly professor in the University of St. Petersburg (born 1821), established, in a celebrated memoir, Sur les Nombres Premiers, 1850, the existence of limits within which the sum of the logarithms of the primes $\scriptstyle {P}$, inferior to a given number $\scriptstyle {x}$, must be comprised.^[89] This paper depends on very elementary considerations, and, in that respect, contrasts strongly with Riemann's, which involves abstruse theorems of the integral calculus. Poincaré's papers, Sylvester's contraction of Tchebycheff's limits, with reference to the distribution of primes, and researches of J. Hadamard (awarded the Grand prix of 1892), are among the latest researches in this line. The enumeration of prime numbers has been undertaken at different times by various mathematicians. In 1877 the British Association began the preparation of factor-tables, under the direction of J. W. L. Glaisher. The printing, by the Association, of tables for the sixth million marked the completion of tables, to the preparation of which Germany, France, and England contributed, and which enable us to resolve into prime factors every composite number less than 9,000,000.

Miscellaneous contributions to the theory of numbers were made by Cauchy. He showed, for instance, how to find all the infinite solutions of a homogeneous indeterminate equation of the second degree in three variables when one solution is given. He established the theorem that if two congruences, which have the same modulus, admit of a common solution, the modulus is a divisor of their resultant. Joseph Liouville (1809–1882), professor at the Collége de France, investigated mainly questions on the theory of quadratic forms of two, and of a greater number of variables. Profound researches were instituted by Ferdinand Gotthold Eisenstein (1823–1852), of Berlin. Ternary quadratic forms had been studied somewhat by Gauss, but the 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 uneven determinant; and, in case of definite forms, assigned the weight of any order or genus. But he did not publish demonstrations of his results. In inspecting the theory of binary cubic forms, he was led to the discovery of the first covariant ever considered in analysis. He showed that the series of theorems, relating to the presentation of numbers by sums of squares, ceases when the number of squares surpasses eight. Many of the proofs omitted by Eisenstein were supplied by Henry Smith, who was one of the few Englishmen who devoted themselves to the study of higher arithmetic.

Henry John Stephen Smith^[90] (1826–1883) was born in London, and educated at Rugby and at Balliol College, Oxford. Before 1847 he travelled much in Europe for his health, and at one time attended lectures of Arago in Paris, but after that year he was never absent from Oxford for a single term. In 1861 he was elected Savilian professor of geometry. His first paper on the theory of numbers appeared in 1855. The results of ten years' study of everything published on the theory of numbers are contained in his Reports which appeared in the British Association volumes from 1859 to 1865. These reports are a model of clear and precise exposition and perfection of form. They contain much original matter, but the chief results of his own discoveries were printed in the Philosophical Transactions for 1861 and 1867. They treat of linear indeterminate equations and congruences, and of the orders and genera of ternary quadratic forms. He established the principles on which the extension to the general case of $\scriptstyle {n}$ indeterminates of quadratic forms depends. He contributed also two memoirs to the Proceedings of the Royal Society of 1864 and 1868, in the second of which he remarks that the theorems of Jacobi, Eisenstein, and Liouville, relating to the representation of numbers by 4, 6, 8 squares, and other simple quadratic forms are deducible by a uniform method from the principles indicated in his paper. Theorems relating to the case of 5 squares were given by Eisenstein, but Smith completed the enunciation of them, and added the corresponding theorems for 7 squares. The solution of the cases of 2, 4, 6 squares may be obtained by elliptic functions, but when the number of squares is odd, it involves processes peculiar to the theory of numbers. This class of theorems is limited to 8 squares, and Smith completed the group. In ignorance of Smith's investigations, the French Academy offered a prize for the demonstration and completion of Eisenstein's theorems for 5 squares. This Smith had accomplished fifteen years earlier. He sent in a dissertation in 1882, and next year, a month after his death, the prize was awarded to him, another prize being also awarded to H. Minkowsky of Bonn. The theory of numbers led Smith to the study of elliptic functions. He wrote also on modern geometry. His successor at Oxford was J. J. Sylvester.

Ernst Eduard Kummer (1810–1893), professor in the University of Berlin, is closely identified with the theory of numbers. Dirichlet's work on complex numbers of the form $\scriptstyle {a+ib}$, introduced by Gauss, was extended by him, by Eisenstein, and Dedekind. Instead of the equation $\scriptstyle {x^{4}-1=0}$, the roots of which yield Gauss' units, Eisenstein used the equation $\scriptstyle {x^{2}-1=0}$ and complex numbers $\scriptstyle {a+b\rho }$ ($\scriptstyle {\rho }$ being a cube root of unity), the theory of which resembles that of Gauss' numbers. Kummer passed to the general case $\scriptstyle {x^{n}-1=0}$ and got complex numbers of the form $\scriptstyle {\alpha =\alpha _{1}A_{1}+\alpha _{2}A_{2}+\alpha _{3}A_{3}+\cdots }$, where $\scriptstyle {\alpha _{i}}$ are whole real numbers, and $\scriptstyle {A_{i}}$ roots of the above equation.^[59] Euclid's theory of the greatest common divisor is not applicable to such complex numbers, and their prime factors cannot be defined in the same way as prime factors of common integers are defined. In the effort to overcome this difficulty, Kummer was led to introduce the conception of "ideal numbers." These ideal numbers have been applied by G. Zolotareff of St Petersburg to the solution of a problem of the integral calculus, left unfinished by Abel (Liouville's Journal, Second Series, 1864, Vol. IX.). Julius Wilhelm Richard Dedekind of Braunschweig (born 1831) has given in the second edition of Dirichlet's Vorlesungen über Zahlentheorie a new theory of complex numbers, in which he to some extent deviates from the course of Kummer, and avoids the use of ideal numbers. Dedekind has taken the roots of any irreducible equation with integral coefficients as the units for his complex numbers. Attracted by Kummer's investigations, his pupil, Leopold Kronecker (1823–1891) made researches which he applied to algebraic equations.

On the other hand, efforts have been made to utilise in the theory of numbers the results of the modern higher algebra. Following up researches of Hermite, Paul Bachmann of Münster investigated the arithmetical formula which gives the automorphics of a ternary quadratic form.^[89] The problem of the equivalence of two positive or definite ternary quadratic forms was solved by L. Seeber; and that of the arithmetical automorphics of such forms, by Eisenstein. The more difficult problem of the equivalence for indefinite ternary forms has been investigated by Edward Selling of Würzburg. On quadratic forms of four or more indeterminates little has yet been done. Hermite showed that the number of non-equivalent classes of quadratic forms having integral coefficients and a given discriminant is finite, while Zolotareff and A. N. Korkine, both of St. Petersburg, investigated the minima of positive quadratic forms. In connection with binary quadratic forms. Smith established the theorem that if the joint invariant of two properly primitive forms vanishes, the determinant of either of them is represented primitively by the duplicate of the other.

The interchange of theorems between arithmetic and algebra is displayed in the recent researches of J. W. L. Glaisher of Trinity College (born 1848) and Sylvester. Sylvester gave a Constructive Theory of Partitions, which received additions from his pupils, F. Franklin and G. S. Ely.

The conception of "number" has been much extended in our time. With the Greeks it included only the ordinary positive whole numbers; Diophantus added rational fractions to the domain of numbers. Later negative numbers and imaginaries came gradually to be recognised. Descartes fully grasped the notion of the negative; Gauss, that of the imaginary. With Euclid, a ratio, whether rational or irrational, was not a number. The recognition of ratios and irrationals as numbers took place in the sixteenth century, and found expression with Newton. By the ratio method, the continuity of the real number system has been based on the continuity of space, but in recent time three theories of irrationals have been advanced by Weierstrass, J. W. R. Dedekind, G. Cantor, and Heine, which prove the continuity of numbers without borrowing it from space. They are based on the definition of numbers by regular sequences, the use of series and limits, and some new mathematical conceptions.