[Berlin: c. 1846].
A remarkable survival, this is the autograph working manuscript of a paper on the application of mathematical analysis to problems in physics by “one of the most important mathematicians of the 19th century … the man who, more than any other, brought rigour to mathematical analysis” (Gray). “The great advances of mathematics in Germany during the first half of the nineteenth century are to a predominantly large extent associated with the pioneering work of C. F. Gauss, C. G. J. Jacobi, and G. Lejeune Dirichlet” (Elstrodt)..
A remarkable survival, this is the autograph working manuscript of a paper on the application of mathematical analysis to problems in physics by “one of the most important mathematicians of the 19th century … the man who, more than any other, brought rigour to mathematical analysis” (Gray, pp. 143-144). It was later published in August Crelle’s Journal für die reine und angewandte Mathematik, Bd. 32 (1846), pp. 80-84; the numerous corrections in the manuscript were all incorporated into the published version. “The great advances of mathematics in Germany during the first half of the nineteenth century are to a predominantly large extent associated with the pioneering work of C. F. Gauss (1777–1855), C. G. J. Jacobi (1804–1851), and G. Lejeune Dirichlet (1805–1859) … The leading role of German mathematics in the second half of the nineteenth century and even up to the fateful year 1933 would have been unthinkable without the foundations laid by Gauss, Jacobi, and Dirichlet” (Elstrodt, p. 2). When Alexander von Humboldt asked Gauss in 1845 for a proposal of a candidate for the order ‘pour le mérite’, Gauss wrote of Dirichlet: “his individual memoirs do not yet comprise a big volume. But they are jewels, and one does not weigh jewels on a grocer’s scales” (ibid., p. 18). “Dirichlet wrote some fundamental papers on the theory of Fourier series and mathematical physics … Dirichlet exerted a considerable influence on these branches of mathematics and even on the entire development of mathematics … To this we should add that by the style of his thinking, by the level of rigor and exactness, Dirichlet was probably the foremost representative of those new trends which were gradually establishing themselves in 19th century mathematics. Jacobi wrote in one of his letters to A. von Humboldt: ‘Dirichlet alone – rather than I, Cauchy or Gauss – knows what is a completely rigorous mathematical proof; we have learned it first from him. When Gauss claims to have proved something then I know that it is very likely so. When Cauchy says it then one can bet either way. But when Dirichlet says it then it is beyond doubt’” (Kolmogorov & Yushkevich, pp. 173-174). The present manuscript concerns ‘potential theory’, the problem of finding solutions to ‘Laplace’s equation’, or more generally ‘Poisson’s equation’, in some region of space with prescribed values at the boundary of the region. This problem occurs widely in mathematical physics, notably in gravitation, heat conduction, and electromagnetism. “Dirichlet was inspired by Gauss’s studies on the potential, a subject that came to be viewed as a developing discipline within mathematics. He developed it further, and he was one of the first to give special lectures on it” (Jungnickel & McCormmach, p. 173). In the present paper, Dirichlet proves the uniqueness of solutions of Poisson’s equation under certain conditions. As he points out, this can sometimes be used to actually find a solution, if for example a candidate solution can be found by physical analogy or some other method. Dirichlet gave arguments elsewhere which he claimed established that solutions always exist, based on the famous ‘Dirichlet’s principle’ (see below). This was used by his greatest student Bernhard Riemann in his work on Riemann surfaces, but it was later criticised by Karl Weierstrass and only rehabilitated at the beginning of the 20th century by David Hilbert. It is exceptionally rare for a working manuscript by a mathematician of Dirichlet’s importance to appear on the market.
The concept of ‘potential’ arose first in the study of gravitational attraction: the potential is a function whose ‘gradient’, or rate of change, gives the gravitational force. “Lagrange introduced the idea of a potential function in 1773. His question was: how to determine the potential function of a given body, and thus its gravitational attraction? After Legendre had opened up the subject, Laplace showed how to find the potential function, V, for the gravitational force due to an arbitrary spheroid in 1782 by showing that it satisfied [the differential equation
ΔV = Vxx + Vyy + Vzz = 0,
where Vxx denotes the second partial derivative d2V/dx2, etc.], which is why we speak of Laplace’s equation today. In the 1780s Charles Coulomb, a French military engineer, proposed the inverse-square law in electrostatics, which differs from the law of gravitation only because the force [can be] one of repulsion … In his efforts to produce a mathematical theory of electrostatics, in 1813 Poisson was led to generalise Laplace’s equation to ΔV = – 4πρ” (Gray, p. 132), where ρ is the mass or charge density. At about the same time, Joseph Fourier showed that Poisson’s equation occurred in the theory of heat conduction. In the 1820s, Poisson’s equation was shown to apply also to problems in magnetism by André-Marie Ampère. Gauss’s interest in potential theory began with his work on geodesy, published in his paper Theoria attractionis corporum sphaeroidicorum ellipticorum homogeneorum methodus nova tractatus (1813), in which he calculated the gravity at different points on the surface of the earth, treated as a homogeneous ellipsoid (although Gauss did not use the term ‘potential’ at that time). Later, in the 1830s, working with the young physicist Wilhelm Weber, Gauss’s interests turned to magnetism, and this culminated in his great work Allgemeine Lehrsätze in Beziehung auf die im verkehrten Verhältnisse des Quadrats der Entfernung wirkenden Anziehungs- und Abstossungs-Kräfte (1840), in which Gauss gave rigorous proofs for many of the assertions about the potential made by his French predecessors.
“Dirichlet’s contributions to potential theory appear to have been prompted by the publication a few years previously of Gauss’s memoir [Allgemeine Lehrsätze] on the subject. The first of Dirichlet’s four memoirs [offered here], published in Crelle’s Journal, was titled ‘On a general means of verifying the expression of a potential relative to any mass, homogeneous or inhomogeneous.’ He introduced it as follows:
‘Given any mass, bounded in every sense whether forming a continuum or composed of several separate pieces, and having a finite density at each of its points, if one sums all the elements of this mass, each divided by its distance to some point m, one will have what geometers have agreed to call the potential of the mass with respect to this point. One knows that the triple integral thus defined, which is always a determined function of the rectangular coordinates x, y, z, of the point m, enjoys a great number of important properties.’
“Dirichlet [recalled] the following properties …
- The potential V and its first order differential coefficients Vx, Vy, Vz, which express the components of the attraction exerted by the mass at the point m, are finite and continuous functions of x, y, z in every extent of space.
- There always exist determined limits that the products xV, yV, zV, x2Vx, y2Vy, z2Vz should not exceed at any point of space.
- If one excepts special points about which the density varies abruptly, each of the three derivatives Vxx, Vyy, Vzz will always be finite and single-valued, and these simultaneous values will be such that ΔV = – 4πρ, where ρ designates the density at the point (x, y, z) which must be considered equal to zero when this point is outside the mass …
“What he wished to prove … was the statement that these properties completely characterize the potential V and that there exists no other function which has all of these properties. He proceeded to give a proof by contradiction. Having accomplished this, he noted that the result furnishes a general means of verifying the expression of a potential when the potential is given throughout the extent of the space considered. He repeated that for this to hold it is sufficient that all the enumerated conditions are satisfied. He pointed out that this procedure can be applied to the celebrated case of a homogeneous ellipsoid … He worked out some examples, also noting the applicability for an interior point as well as the necessary conditions for a point exterior to the ellipsoid. Dirichlet concluded by pointing out that the argument requires only minor modification if one wishes to obtain a similar result when the mass is distributed over one or more surface, rather than being considered in three dimensions …
“Dirichlet remarked that, aside from the usefulness of proving that the properties listed completely characterize the potential because it furnishes a way of verifying an expression for a potential obtained by some other means, the characterization is of even more essential interest. He stressed the identical nature of Gauss’s important assertion, according to which a mass can be distributed over given surfaces in such a way that the potential corresponding to such a distribution receives finite, continuously changing values, and the statement that in a homogeneous mass occupying the entire space, if at the outset there are vanishing temperatures at an infinite distance, then, under the influence of constant heat sources distributed over the surfaces, after an infinite period of time there will be a constant temperature everywhere. He commented that the identity of the two statements, and the new connection established thereby between two branches of mathematical physics already presenting such a major relationship, becomes immediately evident, since the known conditions that determine the permanent state of heat coincide with those whereby the potential corresponding to the desired distribution of the mass is characterized. It clearly pleased Dirichlet to provide another example of the close mathematical relationship between two different areas of study – in this case within physics” (Merzbach, pp. 186-188).
The present manuscript corresponds to the printed version, except for the omission of the final paragraph (six lines) of the latter, in which Dirichlet writes that the same methods can be used to treat the case in which the mass is distributed over a two-dimensional surface instead of a three-dimensional body, but that some modifications are necessary. In place of this paragraph, the manuscript has at the end a note ‘here to be added’, and a correction to his ‘special address’ (‘besondere Adresse’) request. Dirichlet in fact treated the two-dimensional case in a brief article published later in the same year (‘Ueber die charakteristischen Eigenschaften des Potentials einer auf einer oder mehreren endlichen Flächen vertheilten Masse,’ Bericht über die zur Bekanntmachung geeigneten Verhandlungen der Königlichen Preussischen Akademie der Wissenschaften zu Berlin, 1846, pp. 211-212).
Between 1839 and 1858 Dirichlet gave eight courses of lectures on potential theory. “The first of these courses dealing with attraction was restricted to the attraction of ellipsoids, the topic Dirichlet once described as most celebrated, and that he had used as an example of his new techniques on a number of occasions in the late 1830s. Beginning with the winter term 1846/47, however, the expanded course was offered every other year while Dirichlet was in Berlin. It was presented twice more in Göttingen: in the summer of 1856 ‘with applications to electricity and magnetism’ and in the winter term 1857/58 ‘with applications to physical problems’” (Merzbach, p. 248). “It is from these lectures that Riemann became familiar with potential theory, a topic which was to play an important role in his later work. Klein, in his Development of Mathematics in the 19th Century, writes: ‘Dirichlet loved to make things clear to himself in an intuitive [manner]; along with this he would give acute, logical analyses of foundational questions and would avoid long computations as much as possible. His manner suited Riemann, who adopted it and worked according to Dirichlet’s methods.’ Riemann’s admiration for Dirichlet is expressed at several places of his writings, for instance in the third section of the historical part of his habilitation dissertation on trigonometric series” (Papadopoulos, p. 156).
In his lectures Dirichlet showed that the solution V to Laplace’s equation ΔV = 0 made the integral
∫∫∫ (Vx2 + Vy2 + Vz2) dxdydz
a minimum (among those functions with the prescribed values on the boundary of the region over which the integral is taken), and conversely a function which minimizes this integral solves Laplace’s equation. Dirichlet’s name was attributed to this principle by Riemann in his epoch-making memoir on Abelian functions (Theorie der Abel’schen Functionen, 1857), “‘because Professor Dirichlet informed me that he had been using this method in his lectures (since the beginning of the 1840’s if I’m not mistaken)’” (Elstrodt, p. 27). Riemann used the two-dimensional version of Dirichlet’s Principle to give existence proofs for certain functions without giving an analytic expression for them.
Dirichlet was born in Düren, Germany, on 13 February 1805. His grandfather Antoine Lejeune Dirichlet had been born in Verviers, Belgium, and settled in Düren; it was his father who first went under the name “Lejeune Dirichlet” (meaning “the young Dirichlet”) in order to differentiate him from his father, who had the same first name. The name “Dirichlet” (or “Derichelette”) means “from Richelette” after a small town in Belgium.
After graduating from a Jesuit College in Cologne at the early age of 16, Dirichlet went to study in Paris. In the summer of the following year 1823 he secured the position of tutor to the children of General Maximilien Fay, a national hero of the Napoleonic wars and then the liberal leader of the opposition in the Chamber of Deputies. Dirichlet was treated as a member of the family and met many of the most prominent figures in French intellectual life. Among the mathematicians, he was particularly attracted to Fourier, whose ideas had a strong influence upon his later works on trigonometric series and mathematical physics. But Dirichlet’s first interest in mathematics was number theory, which had been awakened through an early study of Gauss’s famous Disquisitiones arithmeticae (1801), until then not completely understood by mathematicians.
General Fay died in November 1825, and the next year Dirichlet decided to return to Germany, a plan strongly supported by Alexander von Humboldt, who worked for the strengthening of the natural sciences in Germany. Dirichlet was permitted to qualify for habilitation as Privatdozent at the University of Breslau; since he did not have the required doctorate, this was awarded honoris causa by the University of Cologne.
Dirichlet was appointed extraordinary professor in Breslau, but the conditions for scientific work were not inspiring. In 1828 he moved to Berlin, again with the assistance of Humboldt, to become a teacher of mathematics at the military academy. Shortly afterward, at the age of twenty-three, he was appointed extraordinary (later ordinary) professor at the University of Berlin. In 1831 he became a member of the Berlin Academy of Sciences, and in the same year he married Rebecca Mendelssohn-Bartholdy, granddaughter of the philosopher Moses Mendelssohn.
“Dirichlet spent twenty-seven years as a professor in Berlin and exerted a strong influence on the development of German mathematics through his lectures, through his many pupils, and through a series of scientific papers of the highest quality that he published during this period. He was an excellent teacher, always expressing himself with great clarity. His manner was modest; in his later years he was shy and at times reserved. He seldom spoke at meetings and was reluctant to make public appearances …
“In 1855, when Gauss died, the University of Göttingen – which had long enjoyed the reflection of his scientific fame – was anxious to seek a successor of great distinction, and the choice fell upon Dirichlet. His position in Berlin had been relatively modest and onerous, and the teaching schedule at the military academy was very heavy and without scientific appeal. Dirichlet wrote to his pupil Kronecker in 1853 that he had little time for correspondence, for he had thirteen lectures a week and many other duties to attend to. Dirichlet responded to the offer from Göttingen that he would accept unless he was relieved of the military instruction in Berlin. The authorities in Berlin seem not to have taken the threat very seriously, and only after it was too late did the Ministry of Education offer to improve his teaching load and salary.
“Dirichlet moved to Göttingen in the fall of 1855, bought a house with a garden, and seemed to enjoy the more quiet life of a prominent university in a small city. He had a number of excellent pupils and relished the increased leisure for research. His work in this period was centered on general problems of mechanics. This new life, however, was not to last long. In the summer of 1858 Dirichlet traveled to a meeting in Montreux, Switzerland, to deliver a memorial speech in honor of Gauss. While there, he suffered a heart attack and was barely able to return to his family in Göttingen. During his illness his wife died of a stroke, and Dirichlet himself died the following spring” (DSB).
Gray, The Real and the Complex: A History of Analysis in the 19th Century, 2015. Elstrodt, ‘The Life and Work of Gustav Lejeune Dirichlet (1805–1859),’ Clay Mathematics Proceedings 7 (2007), pp. 1-37. Jungnickel & McCormmach, Intellectual Mastery of Nature. Theoretical Physics from Ohm to Einstein, Vol. 1, 1986. Kolmogorov & Yushkevich,Mathematics of the 19th Century: Mathematical Logic, Algebra, Number Theory, 2001. Merzbach, Dirichlet: A Mathematical Biography, 2019. Papadopoulos, ‘Physics in Riemann’s mathematical papers,’ pp. 151-208 in: From Riemann to differential geometry and relativity (Ji, Papadopoulos & Yamada, eds.), 2017. The published version of Dirichlet’s article is digitized here: https://www.digizeitschriften.de/dms/img/?PID=GDZPPN002145219
Autograph manuscript, c. 1846, 2 leaves (single sheet folded once), numerous additions, deletions and corrections, all of which were incorporated into the published article. A few marginal annotations in different hands, and a couple of what seem to be printer’s ink smudges, which suggest that this was the copy of the manuscript sent to the printer.