# A History of Mathematics/Recent Times/Analysis

ANALYSIS.
Under this head we find it convenient to consider the subjects of the differential and integral calculus, the calculus of variations, infinite series, probability, and differential equations. Prominent in the development of these subjects was Cauchy.
Augustin-Louis Cauchy (1789-1857) was born in Paris, and received his early education from his father. Lagrange and Laplace, with whom the father came in frequent contact, foretold the future greatness of the young boy. At the École Centrale du Panthéon he excelled in ancient classical studies. In 1805 he entered the Polytechnic School, and two years later the École des Ponts et Chaussées. Cauchy left for Cherbourg in 1810, in the capacity of engineer. Laplace's Mécanique Céleste and Lagrange's Fonctions Analytiques were among his book companions there. Considerations of health induced him to return to Paris after three years. Yielding to the persuasions of Lagrange and Laplace, he renounced engineering in favour of pure science. We find him next holding a professorship at the Polytechnic School. On the expulsion of Charles X., and the accession to the throne of Louis Philippe in 1830, Cauchy, being exceedingly conscientious, found himself unable to take the oath demanded of him. Being, in consequence, deprived of his positions, he went into voluntary exile. At Fribourg in Switzerland, Cauchy resumed his studies, and in 1831 was induced by the king of Piedmont to ​accept the chair of mathematical physics, especially created for him at the university of Turin. In 1833 he obeyed the call of his exiled king, Charles X., to undertake the education of a grandson, the Duke of Bordeaux. This gave Cauchy an opportunity to visit various parts of Europe, and to learn how extensively his works were being read. Charles X. bestowed upon him the title of Baron. On his return to Paris in 1838, a chair in the College de France was offered to him, but the oath demanded of him prevented his acceptance. He was nominated member of the Bureau of Longitude, but declared ineligible by the ruling power. During the political events of 1848 the oath was suspended, and Cauchy at last became professor at the Polytechnic School. On the establishment of the second empire, the oath was re-instated, but Cauchy and Arago were exempt from it. Cauchy was a man of great piety, and in two of his publications staunchly defended the Jesuits.
Cauchy was a prolific and profound mathematician. By a prompt publication of his results, and the preparation of standard text-books, he exercised a more immediate and beneficial influence upon the great mass of mathematicians than any contemporary writer. He was one of the leaders in infusing rigour into analysis. His researches extended over the field of series, of imaginaries, theory of numbers, differential equations, theory of substitutions, theory of functions, determinants, mathematical astronomy, light, elasticity, etc.,—covering pretty much the whole realm of mathematics, pure and applied.
Encouraged by Laplace and Poisson, Cauchy published in 1821 his Cours d'Analyse de l'École Royale Polytechnique, a work of great merit. Had it been studied more diligently by writers of text-books in England and the United States, many a lax and loose method of analysis hardly as yet eradicated ​from elementary text-books would have been discarded over half a century ago. Cauchy was the first to publish a rigorous proof of Taylor's theorem. He greatly improved the exposition of fundamental principles of the differential calculus by his mode of considering limits and his new theory on the continuity of functions. The method of Cauchy and Duhamel was accepted with favour by Hoüel and others. In England special attention to the clear exposition of fundamental principles was given by De Morgan. Recent American treatises on the calculus introduce time as an independent variable, and the allied notions of velocity and acceleration—thus virtually returning to the method of fluxions.
Cauchy made some researches on the calculus of variations. This subject is now in its essential principles the same as when it came from the hands of Lagrange. Recent studies pertain to the variation of a double integral when the limits are also variable, and to variations of multiple integrals in general. Memoirs were published by Gauss in 1829, Poisson in 1831, and Ostrogradsky of St. Petersburg in 1834, without, however, determining in a general manner the number and form of the equations which must subsist at the limits in case of a double or triple integral. In 1837 Jacobi published a memoir, showing that the difficult integrations demanded by the discussion of the second variation, by which the existence of a maximum or minimum can be ascertained, are included in the integrations of the first variation, and thus are superfluous. This important theorem, presented with great brevity by Jacobi, was elucidated and extended by V. A. Lebesgue, C. E. Delaunay, Eisenlohr, S. Spitzer, Hesse, and Clebsch. An important memoir by Sarrus on the question of determining the limiting equations which must be combined with the indefinite equations in order to determine completely the maxima and minima of multiple integrals, was awarded a ​prize by the French Academy in 1845, honourable mention being made of a paper by Delaunay. Sarrus's method was simplified by Cauchy. In 1852 G. Mainardi attempted to exhibit a new method of discriminating maxima and minima, and extended Jacobi's theorem to double integrals. Mainardi and F. Brioschi showed the value of determinants in exhibiting the terms of the second variation. In 1861 Isaac Todhunter (1820-1884) of St. John's College, Cambridge, published his valuable work on the History of the Progress of the Calculus of Variations, which contains researches of his own. In 1866 he published a most important research, developing the theory of discontinuous solutions (discussed in particular cases by Legendre), and doing for this subject what Sarrus had done for multiple integrals.
The following are the more important authors of systematic treatises on the calculus of variations, and the dates of publication: Robert Woodhouse, Fellow of Caius College, Cambridge, 1810; Richard Abbatt in London, 1837; John Hewitt Jellett (1817-1888), once Provost of Trinity College, Dublin, 1850; G. W. Strauch in Zurich, 1849; Moigno and Lindelöf, 1861; Lewis Buffett Carll of Flushing in New York, 1881.
The lectures on definite integrals, delivered by Dirichlet in 1858, have been elaborated into a standard work by G. F. Meyer. The subject has been treated most exhaustively by D. Bierens de Haan of Leiden in his Exposé de la théorie des intégrals définies, Amsterdam, 1862.
The history of infinite series illustrates vividly the salient feature of the new era which analysis entered upon during the first quarter of this century. Newton and Leibniz felt the necessity of inquiring into the convergence of infinite series, but they had no proper criteria, excepting the test advanced by Leibniz for alternating series. By Euler and his contemporaries the formal treatment of series was greatly extended, ​while the necessity for determining the convergence was generally lost sight of. Euler reached some very pretty results on infinite series, now well known, and also some very absurd results, now quite forgotten. The faults of his time found their culmination in the Combinatorial School in Germany, which has now passed into deserved oblivion. At the beginning of the period now under consideration, the doubtful, or plainly absurd, results obtained from infinite series stimulated profounder inquiries into the validity of operations with them. Their actual contents came to be the primary, form a secondary, consideration. The first important and strictly rigorous investigation of series was made by Gauss in connection with the hypergeometric series. The criterion developed by him settles the question of convergence in every case which it is intended to cover, and thus bears the stamp of generality so characteristic of Gauss's writings. Owing to the strangeness of treatment and unusual rigour. Gauss's paper excited little interest among the mathematicians of that time.
More fortunate in reaching the public was Cauchy, whose Analyse Algébrique of 1821 contains a rigorous treatment of series. All series whose sum does not approach a fixed limit as the number of terms increases indefinitely are called divergent. Like Gauss, he institutes comparisons with geometric series, and finds that series with positive terms are convergent or not, according as the $\scriptstyle {n}$th root of the $\scriptstyle {n}$th term, or the ratio of the $\scriptstyle {(n+1)}$th term and the $\scriptstyle {n}$th term, is ultimately less or greater than unity. To reach some of the cases where these expressions become ultimately unity and fail, Cauchy established two other tests. He showed that series with negative terms converge when the absolute values of the terms converge, and then deduces Leibniz's test for alternating series. The product of two convergent series was not found to be necessarily convergent. Cauchy's theorem that the ​product of two absolutely convergent series converges to the product of the sums of the two series was shown half a century later by F. Mertens of Graz to be still true if, of the two convergent series to be multiplied together, only one is absolutely convergent.
The most outspoken critic of the old methods in series was Abel. His letter to his friend Holmboe (1826) contains severe criticisms. It is very interesting reading, even to modern students. In his demonstration of the binomial theorem he established the theorem that if two series and their product series are all convergent, then the product series will converge towards the product of the sums of the two given series. This remarkable result would dispose of the whole problem of multiplication of series if we had a universal practical criterion of convergency for semi-convergent series. Since we do not possess such a criterion, theorems have been recently established by A. Pringsheim of Munich and A. Voss of Würzburg which remove in certain cases the necessity of applying tests of convergency to the product series by the application of tests to easier related expressions. Pringsheim reaches the following interesting conclusions: The product of two semi-convergent series can never converge absolutely, but a semi-convergent series, or even a divergent series, multiplied by an absolutely convergent series, may yield an absolutely convergent product.
The researches of Abel and Cauchy caused a considerable stir. We are told that after a scientific meeting in which Cauchy had presented his first researches on series, Laplace hastened home and remained there in seclusion until he had examined the series in his Mécanique Céleste. Luckily, every one was found to be convergent! We must not conclude, however, that the new ideas at once displaced the old. On the contrary, the new views were generally accepted only after a ​severe and long struggle. As late as 1844 De Morgan began a paper on "divergent series" in this style: "I believe it will be generally admitted that the heading of this paper describes the only subject yet remaining, of an elementary character, on which a serious schism exists among mathematicians as to the absolute correctness or incorrectness of results."
First in time in the evolution of more delicate criteria of convergence and divergence come the researches of Josef Ludwig Raabe (Crelle, Vol. IX.); then follow those of De Morgan as given in his calculus. De Morgan established the logarithmic criteria which were discovered in part independently by J. Bertrand. The forms of these criteria, as given by Bertrand and by Ossian Bonnet, are more convenient than De Morgan's. It appears from Abel's posthumous papers that he had anticipated the above-named writers in establishing logarithmic criteria. It was the opinion of Bonnet that the logarithmic criteria never fail; but Du Bois-Reymond and Pringsheim have each discovered series demonstrably convergent in which these criteria fail to determine the convergence. The criteria thus far alluded to have been called by Pringsheim special criteria, because they all depend upon a comparison of the $\scriptstyle {n}$th term of the series with special functions $\scriptstyle {a^{n}}$, $\scriptstyle {n^{n}}$, $\scriptstyle {n(\log n)^{n}}$, etc. Among the first to suggest general criteria, and to consider the subject from a still wider point of view, culminating in a regular mathematical theory, was Kummer. He established a theorem yielding a test consisting of two parts, the first part of which was afterwards found to be superfluous. The study of general criteria was continued by U. Dini of Pisa, Paul Du Bois-Reymond, G. Kohn of Minden, and Pringsheim. Du Bois-Reymond divides criteria into two classes: criteria of the first kind and criteria of the second kind, according as the general $\scriptstyle {n}$th term, or the ratio of the $\scriptstyle {(n+1)}$th term and ​the $\scriptstyle {n}$th term, is made the basis of research. Kummer's is a criterion of the second kind. A criterion of the first kind, analogous to this, was invented by Pringsheim. From the general criteria established by Du Bois-Reymond and Pringsheim respectively, all the special criteria can be derived. The theory of Pringsheim is very complete, and offers, in addition to the criteria of the first kind and second kind, entirely new criteria of a third kind, and also generalised criteria of the second kind, which apply, however, only to series with never increasing terms. Those of the third kind rest mainly on the consideration of the limit of the difference either of consecutive terms or of their reciprocals. In the generalised criteria of the second kind he does not consider the ratio of two consecutive terms, but the ratio of any two terms however far apart, and deduces, among others, two criteria previously given by Kohn and Ermakoff respectively.
Difficult questions arose in the study of Fourier's series. Cauchy was the first who felt the necessity of inquiring into its convergence. But his mode of proceeding was found by Dirichlet to be unsatisfactory. Dirichlet made the first thorough researches on this subject (Crelle, Vol. IV.). They culminate in the result that whenever the function does not become infinite, does not have an infinite number of discontinuities, and does not possess an infinite number of maxima and minima, then Fourier's series converges toward the value of that function at all places, except points of discontinuity, and there it converges toward the mean of the two boundary values. Schläfli of Bern and Du Bois-Reymond expressed doubts as to the correctness of the mean value, which were, however, not well founded. Dirichlet's conditions are sufficient, but not necessary. Lipschitz, of Bonn, proved that Fourier's series still represents the function when the number of discontinuities is infinite, and ​established a condition on which it represents a function having an infinite number of maxima and minima. Dirichlet's belief that all continuous functions can be represented by Fourier's series at all points was shared by Riemann and H. Hankel, but was proved to be false by Du Bois-Reymond and H. A. Schwarz.
Riemann inquired what properties a function must have, so that there may be a trigonometric series which, whenever it is convergent, converges toward the value of the function. He found necessary and sufficient conditions for this. They do not decide, however, whether such a series actually represents the function or not. Riemann rejected Cauchy's definition of a definite integral on account of its arbitrariness, gave a new definition, and then inquired when a function has an integral. His researches brought to light the fact that continuous functions need not always have a differential coefficient. But this property, which was shown by Weierstrass to belong to large classes of functions, was not found necessarily to exclude them from being represented by Fourier's series. Doubts on some of the conclusions about Fourier's series were thrown by the observation, made by Weierstrass, that the integral of an infinite series can be shown to be equal to the sum of the integrals of the separate terms only when the series converges uniformly within the region in question. The subject of uniform convergence was investigated by Philipp Ludwig Seidel (1848) and G. G. Stokes (1847), and has assumed great importance in Weierstrass' theory of functions. It became necessary to prove that a trigonometric series representing a continuous function converges uniformly. This was done by Heinrich Eduard Heine (1821–1881), of Halle. Later researches on Fourier's series were made by G. Cantor and Du Bois-Reymond.
As compared with the vast development of other ​mathematical branches, the theory of probability has made very insignificant progress since the time of Laplace. Improvements and simplications in the mode of exposition have been made by A. De Morgan, G. Boole, A. Meyer (edited by E. Czuber), J. Bertrand. Cournot's and Westergaard's treatment of insurance and the theory of life-tables are classical. Applications of the calculus to statistics have been made by L. A. J. Quetelet (1796–1874), director of the observatory at Brussels; by Lexis; Harald Westergaard, of Copenhagen; and Düsing.
Worthy of note is the rejection of inverse probability by the best authorities of our time. This branch of probability had been worked out by Thomas Bayes (died 1761) and by Laplace (Bk. II., Ch. VI. of his Théorie Analytique). By it some logicians have explained induction. For example, if a man, who has never heard of the tides, were to go to the shore of the Atlantic Ocean and witness on $\scriptstyle {m}$ successive days the rise of the sea, then, says Quetelet, he would be entitled to conclude that there was a probability equal to $\scriptstyle {\frac {m+1}{m+2}}$ that the sea would rise next day. Putting $\scriptstyle {m=0}$, it is seen that this view rests upon the unwarrantable assumption that the probability of a totally unknown event is $\scriptstyle {\frac {1}{2}}$, or that of all theories proposed for investigation one-half are true. W. S. Jevons in his Principles of Science founds induction upon the theory of inverse probability, and F. Y. Edgeworth also accepts it in his Mathematical Psychics.
The only noteworthy recent addition to probability is the subject of "local probability," developed by several English and a few American and French mathematicians. The earliest problem on this subject dates back to the time of Buffon, the naturalist, who proposed the problem, solved by himself and Laplace, to determine the probability that a short needle, thrown at random upon a floor ruled with equidistant parallel ​lines, will fall on one of the lines. Then came Sylvester's four-point problem: to find the probability that four points, taken at random within a given boundary, shall form a re-entrant quadrilateral. Local probability has been studied in England by A. R. Clarke, H. McColl, S. Watson, J. Wolstenholme, but with greatest success by M, W, Crofton of the military school at Woolwich. It was pursued in America by E. B. Seitz; in France by C. Jordan, E. Lemoine, E. Barbier, and others. Through considerations of local probability, Crofton was led to the evaluation of certain definite integrals.
The first full scientific treatment of differential equations was given by Lagrange and Laplace. This remark is especially true of partial differential equations. The latter were investigated in more recent time by Monge, Pfaff, Jacobi, Émile Bour (1831–1866) of Paris, A. Weiler, Clebsch, A. N. Korkine of St. Petersburg, G. Boole, A. Meyer, Cauchy, Serret, Sophus Lie, and others. In 1873 their reseaches, on partial differential equations of the first order, were presented in text-book form by Paul Mansion, of the University of Gand. The keen researches of Johann Friedrich Pfaff (1795–1825) marked a decided advance. He was an intimate friend of young Gauss at Göttingen. Afterwards he was with the astronomer Bode. Later he became professor at Helmstädt, then at Halle. By a peculiar method, Pfaff found the general integration of partial differential equations of the first order for any number of variables. Starting from the theory of ordinary differential equations of the first order in $\scriptstyle {n}$ variables, he gives first their general integration, and then considers the integration of the partial differential equations as a particular case of the former, assuming, however, as known, the general integration of differential equations of any order between two variables. His researches led Jacobi to introduce the name "Pfaffian problem." From the connection, observed by Hamilton, between ​a system of ordinary differential equations (in analytical mechanics) and a partial differential equation, Jacobi drew the conclusion that, of the series of systems whose successive integration Pfaff's method demanded, all but the first system were entirely superfluous. Clebsch considered Pfaff's problem from a new point of view, and reduced it to systems of simultaneous linear partial differential equations, which can be established independently of each other without any integration. Jacobi materially advanced the theory of differential equations of the first order. The problem to determine unknown functions in such a way that an integral containing these functions and their differential coefficients, in a prescribed manner, shall reach a maximum or minimum value, demands, in the first place, the vanishing of the first variation of the integral. This condition leads to differential equations, the integration of which determines the functions. To ascertain whether the value is a maximum or a minimum, the second variation must be examined. This leads to new and difficult differential equations, the integration of which, for the simpler cases, was ingeniously deduced by Jacobi from the integration of the differential equations of the first variation. Jacobi's solution was perfected by Hesse, while Clebsch extended to the general case Jacobi's results on the second variation. Cauchy gave a method of solving partial differential equations of the first order having any number of variables, which was corrected and extended by Serret, J. Bertrand, O. Bonnet in France, and Imschenetzky in Russia. Fundamental is the proposition of Cauchy that every ordinary differential equation admits in the vicinity of any non-singular point of an integral, which is synectic within a certain circle of convergence, and is developable by Taylor's theorem. Allied to the point of view indicated by this theorem is that of Riemann, who regards a function of a single variable as ​defined by the position and nature of its singularities, and who has applied this conception to that linear differential equation of the second order, which is satisfied by the hypergeometric series. This equation was studied also by Gauss and Kummer. Its general theory, when no restriction is imposed upon the value of the variable, has been considered by J. Tannery, of Paris, who employed Fuchs' method of linear differential equations and found all of Kummer's twenty-four integrals of this equation. This study has been continued by Édouard Goursat of Paris.
A standard text-book on Differential Equations, including original matter on integrating factors, singular solutions, and especially on symbolical methods, was prepared in 1859 by George Boole (1815–1864), at one time professor in Queen's University, Cork, Ireland. He was a native of Lincoln, and a self-educated mathematician of great power. His treatise on Finite Differences (1860) and his Laws of Thought (1854) are works of high merit.
The fertility of the conceptions of Cauchy and Riemann with regard to differential equations is attested by the researches to which they have given rise on the part of Lazarus Fuchs of Berlin (born 1835), Felix Klein of Göttingen (born 1849), Henri Poincaré of Paris (born 1854), and others. The study of linear differential equations entered a new period with the publication of Fuchs' memoirs of 1866 and 1868. Before this, linear equations with constant coefficients were almost the only ones for which general methods of integration were known. While the general theory of these equations has recently been presented in a new light by Hermite, Darboux, and Jordan, Fuchs began the study from the more general standpoint of the linear differential equations whose coefficients are not constant. He directed his attention mainly to those whose integrals are all regular. ​If the variable be made to describe all possible paths enclosing one or more of the critical points of the equation, we have a certain substitution corresponding to each of the paths; the aggregate of all these substitutions being called a group. The forms of integrals of such equations were examined by Fuchs and by G. Frobenius by independent methods. Logarithms generally appear in the integrals of a group, and Fuchs and Frobenius investigated the conditions under which no logarithms shall appear. Through the study of groups the reducibility or irreducibility of linear differential equations has been examined by Frobenius and Leo Königsberger. The subject of linear differential equations, not all of whose integrals are regular, has been attacked by G. Frobenius of Berlin, W. Thomé of Greifswald (born 1841), and Poincaré, but the resulting theory of irregular integrals is as yet in very incomplete form.
The theory of invariants associated with linear differential equations has been developed by Halphen and by A. R. Forsyth.
The researches above referred to are closely connected with the theory of functions and of groups. Endeavours have thus been made to determine the nature of the function defined by a differential equation from the differential equation itself, and not from any analytical expression of the function, obtained first by solving the differential equation. Instead of studying the properties of the integrals of a differential equation for all the values of the variable, investigators at first contented themselves with the study of the properties in the vicinity of a given point. The nature of the integrals at singular points and at ordinary points is entirely different. Albert Briot (1817–1882) and Jean Claude Bouquet (1819–1885), both of Paris, studied the case when, near a singular point, the differential equations take the form $\scriptstyle {(x-x_{0}){\frac {dy}{dx}}=\int (xy)}$. Fuchs ​gave the development in series of the integrals for the particular case of linear equations. Poincaré did the same for the case when the equations are not linear, as also for partial differential equations of the first order. The developments for ordinary points were given by Cauchy and Madame Kowalevsky.
The attempt to express the integrals by developments that are always convergent and not limited to particular points in a plane necessitates the introduction of new transcendents, for the old functions permit the integration of only a small number of differential equations. Poincaré tried this plan with linear equations, which were then the best known, having been studied in the vicinity of given points by Fuchs, Thomé, Frobenius, Schwarz, Klein, and Halphen. Confining himself to those with rational algebraical coefficients, Poincaré was able to integrate them by the use of functions named by him Fuchsians. He divided these equations into "families." If the integral of such an equation be subjected to a certain transformation, the result will be the integral of an equation belonging to the same family. The new transcendents have a great analogy to elliptic functions; while the region of the latter may be divided into parallelograms, each representing a group, the former may be divided into curvilinear polygons, so that the knowledge of the function inside of one polygon carries with it the knowledge of it inside the others. Thus Poincaré arrives at what he calls Fuchsian groups. He found, moreover, that Fuchsian functions can be expressed as the ratio of two transcendents (theta-fuchsians) in the same way that elliptic functions can be. If, instead of linear substitutions with real coefficients, as employed in the above groups, imaginary coefficients be used, then discontinuous groups are obtained, which he called Kleinians. The extension to non-linear equations of the method thus applied to linear equations has been begun by Fuchs and Poincaré.
​We have seen that among the earliest of the several kinds of "groups" are the finite discontinuous groups (groups in the theory of substitution), which since the time of Galois have become the leading concept in the theory of algebraic equations; that since 1876 Felix Klein, H. Poincaré, and others have applied the theory of finite and infinite discontinuous groups to the theory of functions and of differential equations. The finite continuous groups were first made the subject of general research in 1873 by Sophus Lie, now of Leipzig, and applied by him to the integration of ordinary linear partial differential equations.
Much interest attaches to the determination of those linear differential equations which can be integrated by simpler functions, such as algebraic, elliptic, or Abelian. This has been studied by C. Jordan, P. Appel of Paris (born 1858), and Poincaré.
The mode of integration above referred to, which makes known the properties of equations from the standpoint of the theory of functions, does not suffice in the application of differential equations to questions of mechanics. If we consider the function as defining a plane curve, then the general form of the curve does not appear from the above mode of investigation. It is, however, often desirable to construct the curves defined by differential equations. Studies having this end in view have been carried on by Briot and Bouquet, and by Poincaré.
The subject of singular solutions of differential equations has been materially advanced since the time of Boole by G. Darboux and Cayley. The papers prepared by these mathematicians point out a difficulty as yet unsurmounted: whereas a singular solution, from the point of view of the integrated equation, ought to be a phenomenon of universal, or at least of general occurrence, it is, on the other hand, a very special and ​exceptional phenomenon from the point of view of the differential equation. A geometrical theory of singular solutions resembling the one used by Cayley was previously employed by W. W. Johnson of Annapolis.
An advanced Treatise on Linear Differential Equations (1889) was brought out by Thomas Craig of the Johns Hopkins University. He chose the algebraic method of presentation followed by Hermite and Poincaré, instead of the geometric method preferred by Klein and Schwarz. A notable work, the Traité d'Analyse, is now being published by Émile Picard of Paris, the interest of which is made to centre in the subject of differential equations.