Dirichlet's Contributions to Mathematical Probability Theory

Home , Carl Wilhelm Borchardt, Johann Franz Encke, Rudolf Clausius

HISTORIA MATHEMATICA 21 (1994), 39-63

HANS FISCHER

Polkostrafle 41, D-81245 Muiichen, Germany

Only a few short papers on probability and error theory by Peter Gustav Lejeune Dirichlet are printed in his Werke. However, during his Berlin period, Dirichlet quite frequently gave courses on probability theory or the method of least squares. Unpublished lecture notes reveal that he presented original methods, especially when deriving probabilistic limit theorems; e.g., the use of his discontinuity-factor. The following article discusses some central ideas in Dirichlet's printed papers and unpublished lectures on probability and error theory. These include his deduction of the approximately normal distribution of medians connected with a criticism of least squares as well as his improvement of Laplace's method of approximations relating not only to Stirling's formula but also to the treatment of the central limit theorem. Moreover, the study attempts to place the methods Dirichlet used in probability and error calculus within the broader context of his work in analysis. © 1994 Academic Press, Inc. Nur einige kurze Artikel von Peter Gustav Lejeune Dirichlet zur Wahrscheinlichkeits- und Fehlerrechnung sind in seinen Werken gedruckt. Dirichlet hat aber w~ihrend seiner Berliner Zeit recht h~iufig Vorlesungen ~iber Wahrscheinlichkeitsrechnung oder Methode der kleinsten Quadrate gehalten. Aus unver6ffentlichten Vorlesungsmitschriften geht hervor, dab er gerade bei der Herleitung von Grenzwerts~itzen eigene und neue Methoden, z.B. die Verwendung seines Diskontinuit~itsfaktors vorgestellt hat. In folgender Arbeit werden Schwerpunkte der gedruckten Arbeiten und der Vorlesungen yon Dirichlet fiber Wahrscheinlichkeitsrechnung beschrieben: Grenzwertsatz for Mediane und Kritik der Methode der kleinsten Quadrate, Fortentwicklung der Laplaceschen Methode for Approxi- mationen im Hinblick auf die Stirlingsche Formel und die Behandlung des zentralen Grenzwertsatzes. Auberdem wird versucht, die von Dirichlet in Wahrscheinlichkeits- und Fehlerrechnung verwendeten Methoden in den Zusammenhang seiner analytischen Ar- beiten zu stellen. © 1994 AcademicPress, lnc. Les oeuvres publi6es de Peter Gustav Lejeune Dirichlet ne contiennent que quelques rares et courts articles sur le calcul des probabilit6s et les erreurs. Pourtant, alors qu'il 6tait ~t Berlin, ce math6maticien avait assez fr6quement donn6 des cours sur les probabilit6s ou la m6thode des moindres carr6s: des manuscrits in6dits de ses cours en t6moignent et permettent de voir qu'h l'occasion de th6or6mes sur les limites, il introduisit des m6thodes nouvelles et originales, telle, par exemple, celle qui repose sur l'utilisation du facteur de discontinuit6 qui porte maintenant son nom. Darts le pr6sent article, nous nous proposons de pr6senter certains aspects fondamentaux des travaux de Dirichlet publi6s ou non et relatifs au calcul des probabilit6s, notamment: --la question de la d6termination de la distribution approximative des m6dianes, associ6e b. une critique de la m6thode des moindres carr6s; --l'am61ioration de la m6thode d'approximation de Laplace relativement b. la formule de Stirling et au traitement du th6or~me limite central. Nous nous efforcerons aussi de resituer les m6thodes probabilistes de Dirichlet dans le contexte de ses travaux analytiques. © 1994 AcademicPress, Inc.

MSC 1991 subject classifications: 01A55, 41-03, 60-03, 62-03

KEY WORDS: method of least squares, method of least absolute values, median, Laplace's method of approximations, Stirling's formula, central limit theorem

Peter Gustav Lejeune Dirichlet (1805-1859) is well known for his profound and seminal contributions to Fourier series, mathematical physics, and number theory. After his studies in Paris (1822-1826, in particular with Sylvestre Francois Lacroix, Joseph Fourier, and Sim6on Denis Poisson), he became in 1827 a Privatdozent in Breslau. In Berlin, as a professor of mathematics at the Military Academy and at the University from 1828 to 1855, he produced the main part of his mathematical work. Dirichlet was also very engaged there in questions of mathematical teaching with the purpose of introducing more sophisticated subjects and stimulating students to learn more actively. In 1855 he became the successor of Carl Friedrich Gauss in G6ttingen. Dirichlet is now regarded as one of the most important figures in the transition towards new methods and views in analysis and number theory during the first half of the 19th century. In the field of probability calculus and its applications to error theory, however, one can find only a few notes in his Werke ([Dirichlet 1836; Dirichlet 1897a,b; Dirichlet & Encke 1832/1897]). At first glance, on the basis of these short papers, Dirichlet's activities in probability theory might appear unworthy of more detailed investigation, although his work has been addressed occasionally by historical articles in probability or statistics [Sheynin 1973, 293; 1979, 37; Stigler 1973b, 875; Harter 1974 I, 160 f.; 1983a, 36; 1983b, 595f.].

UNPUBLISHED SOURCES In 1885 Paul Du Bois-Reymond wrote a letter in response to an inquiry from Leopold Kronecker, who had asked for suggestions concerning a complete edition of Dirichlet's collected papers: "Would it not be possible to add the lectures on probability calculus--in an appendix, perhaps?" [Du Bois-Reymond 1885]. Du Bois-Reymond promised to meet Kronecker and discuss the matter, but there exists no further correspondence regarding this suggestion and we can only specu- late as to why the lectures did not appear in the final edition of Dirichlet's Werke. The unpublished notes [Dirichlet (b); (c)] on original papers by Abraham de Moivre, Joseph Louis Lagrange, Pierre Simon Laplace, and Leonhard Euler, written in French, show that Dirichlet, as it seems already during his studies in Paris, had dealt thoroughly with problems of probability theory. At Berlin, Dirichlet gave nine courses of lectures on probability theory or its applications to error theory, in particular to the probabilistic foundation of the method of least squares, between 1829 and 1850 (see Appendix). Thus, probability theory was one of his favorite disciplines after partial differential equations, definite integrals, and number theory. Only from unpublished lecture notes on courses in SS (summer semester) 1838, WS (winter semester) 1841/1842, and SS 1846 can one see the full background, context, and importance of Dirichlet's work in mathematical probability theory. It is well known that Dirichlet also brought forth new HM 21 DIRICHLET'S WORK IN PROBABILITY 41 and unpublished results in his courses. As we will see, this is particularly true of Dirichlet's courses on probability and error theory. The four sets of lecture notes on probability calculus [Dirichlet 1838b; 1838c; 1841/1842a; 1846], on which this study is based (for a detailed description see Bibliography), can be found in three locations: [Dirichlet 1838b] is in the Nachlass of Carl Wilhelm Borchardt (1817-1880), who studied in Berlin from 1836 to 1839; [Dirichlet 1838c] was found in the Nachlass of Rudolf Clausius (1822-1888, in Berlin from 1840 to 1844) [1]; and [Dirichlet 1841/1842a; 1846] are located in the Nachlass of Ludwig Philipp Seidel (1821-1896, in Berlin from 1840 to 1842). Several probabilistic problems solved in Seidel's Aufgaben aus Dirichlet's mathematischen Seminar [Dirichlet & Seidel 1840/1841] convey additional information about Dirichlet's methods. From [Dirichlet 1838c] and [1846] one can conclude that lecture notes on probability and error theory circulated between several students. This fact is an indica- tion of the reputation of Dirichlet's respective courses. [Dirichlet 1838c] is a copy of unknown lecture notes of 1838 made by Clausius in 1857, and [Dirichlet 1846], in the possession of Seidel, was written up by an unknown individual. There is also the example of Paul Du Bois-Reymond, mentioned above, who apparently knew of lecture notes on probability theory, despite the fact that he had only attended Dirichlet's course on partial differential equations (cf. [Dirichlet & Arendt 1904, 386]).

MAIN TOPICS OF DIRICHLET'S PUBLISHED WORK AND COURSES ON PROBABILITY THEORY Dirichlet published only one paper [1836] on probability theory in his lifetime. This was mainly a brief criticism of Laplace's foundation of least squares, a contribution which exemplifies Dirichlet's total neglect of the practical aspects of error theory. [Dirichlet 1897a], found in Dirichlet's Nachlass and published in Dirichlet's Werke, is an enlarged, rather popular version of [Dirichlet 1836]. The short passage [Dirichlet & Encke 1832/1897], originally published in Johann Franz Encke's survey on error theory [Encke 1832], deals with Dirichlet's proof of a formula by Gauss. Dirichlet had communicated his deduction of Gauss's formula to Encke, but he never published it himself. In this proof, a probabilistic limit theorem for medians is deduced. Finally, the note [Dirichlet 1897b], found in Dirichlet's Nachlass and published only posthumously in his Werke, contains a proof of the de Moivre-Laplace theorem, which uses the main idea of Poisson in his proof of the universal central limit theorem (e.g., [Poisson 1841, 215-231]) in a modified form. Deeper insights into Dirichlet's approach to probability theory as well as his analytical methods, however, can be gained from the unpublished notes on his lectures (which also dealt with all the problems treated in the papers published in Dirichlet's Werke). The main topics in these lecture notes are the following:

(1) Problems found in Chapter 2 of the second book of Laplace's Th~orie 42 HANS FISCHER HM 21 analytique des probabilit~s (TAP), above all the problem of duration of play, beginning with a concise outline of the theory of linear difference equations [Dirich- let 1838b, 6-20, 35-49, 60-63; 1841/1842a, 5-33, 36-54], (2) Stirling's formula [Dirichlet 1838b, 64-69; 1841/1842a, 55-60], (3) the probabilities of hypotheses as handled by Laplace [Dirichlet 1838b, I01-110], (4) Bernoulli's theorem, mostly treated in connection with the limit theorem of de Moivre-Laplace [Dirichlet 1838b, 84-100; 1841/1842a, 61-69; 1846, 6-16], (5) the central limit theorem within the scope of least squares [Dirichlet 1838b, 141-156; 1838c, 4-27; 1846, 25-37], (6) the deduction of the approximate distribution of medians, accompanied by a critical view of the method of least squares [Dirichlet 1838b, 163-167; 1838c, 27-35; 1846, 21-25], and (7) problems of geometrical probabilities, concerning a line divided into several parts taken at random [Dirichlet 1838b, 110-118] [2]. Obviously, Dirichlet was not much concerned with philosophical questions regarding probability. In the lecture notes, the notion of"probability" is explained only briefly and, as might be expected, entirely in the sense of Laplace [Dirichlet 1838b, 134-140; 1838c, 1-3; 1841/1842a, 33-35; 1846, 3-6]. One finds no detailed discussion of the general meaning of the theory for the natural or moral sciences, although it is possible that such problems were treated more thoroughly in Dirich- let's introductory courses on probability theory, from which lecture notes are no longer extant. Nevertheless, from Dirichlet's point of view, such questions apparently did not belong to the more advanced aspects of mathematical probability theory.

THE ROLE OF PROBABILITY THEORY IN DIRICHLET'S TEACHING As already mentioned, Dirichlet was very interested in questions concerning the teaching of mathematics. Thus, he participated from 1828 to 1835 in developing plans to establish a special "polytechnic" school in Berlin for the training of teachers of mathematics [Lorey 1916, 42-51; Schubring 1981, 173-186] and, in particular, he elaborated a mathematics curriculum [Dirichlet (d)] [3]. At this school, modeled after the polytechnics in France, mathematics, mainly pure mathematics, was to have been taught at a very high level. All these plans were never realized, however, so Dirichlet tried to pursue some of these basic ideas by organizing privately a Seminar in which gifted students could take their first steps in mathematical research [Biermann 1988, 46] [4]. According to his proposed mathematics curriculum for the Berlin polytechnic, the second year would have contained, as a part of analysis, "probability calculus with its application to the determination of the best results (les resultats les plus avantageux) from among a larger number of observations," taught in a 40-hr course. This proposal also called for 120 hours in the second year devoted to an advanced course on calculus and partial differential equations, and 40 hours for the theory of curves and sur- HM 21 DIRICHLET'S WORK IN PROBABILITY 43 faces. This suggests how much weight Dirichlet allocated to probability theory including the foundation of least squares, which should, as evidenced by the wording "plus avantageux," be done in a Laplacian manner, as a secondary but nevertheless important discipline for the training of mathematicians. The fact that many of Dirichlet's courses in this field consisted only of 1-hr lectures [5] does not at all detract from their significance, for they could still be taught on a high level, as the lecture notes from WS 1841/1842 and SS 1846 reveal. It must also be noted that, aside from Dirichlet's courses, lectures given on this topic at Berlin were rather infrequent [6], a situation quite common at German universities during the first half of the 19th century. Typically, probability theory was treated merely as ancillary to courses on least squares, such as those taught by Gauss (for a contemporary account see [Dedekind 1901]), or it was studied in connection with elementary combinatorical considerations (see [Schneider 1974, 144]).

DIRICHLET'S APPROACH TO PROBABILITY Probability theory, for Dirichlet, was in the first place an application of integral calculus, as it aids in the "representation of the end result" and especially in dealing with "a very large number of events" [Dirichlet 1841/1842a, 3f.]. Thus, he offered his (advanced) courses on probability theory to complement his courses on definite integrals. Indicative of this circumstance, in his course on probability theory in 1838, Dirichlet also taught nonprobabilistic topics which could be treated by similar analytical methods [Dirichlet 1838b, 69-83; 118-126]. On the other hand, he only indicated the principle of generating functions for the solution of special problems ("Princip der Reihenentwicklung" [Dirichlet 1838b, 24-29, 32-34]), and he omitted the calculus of generating functions completely in his treatment of difference equations. Thus, Dirichlet followed a didactic trend of his time in simplifying important contents of Laplace's TAP without incurring any losses at the basic level (see [Schneider 1987, 205]). Dirichlet's interest in problems of probability calculus stemmed from their analytical character rather than their practical importance. He showed a preference for analytically demanding problems, especially those in which one or more param- eters tend to infinity, with few connections to practice, as, for example, in the problem of determining the duration of play when an infinite stake is held by one gambler. This kind of interest reflects a movement toward abstraction in pure mathematics, similar to the contemporary situation in mathematical physics. As I will show below, in his courses Dirichlet presented new analytical methods for the treatment of probabilistic limit theorems, but his mode of reasoning in this connection was not always compatible with the style of his published work on analysis. In his lectures on least squares, Dirichlet did not teach the practical treatment of observations. At Berlin University, this topic was covered in courses given by the astronomer Encke (see note [6]). Dirichlet was only concerned with the discussion of the probabilistic fundamentals of Laplacian error theory and the 44 HANS FISCHER HM 21 analytical methods used in this connection. His purely theoretical attitude toward error theory is also characteristic of his article of 1836.

DIRICHLET'S CRITICISM OF THE METHOD OF LEAST SQUARES The article [Dirichlet 1836] is a brief summary of a lecture given at the Berlin Academy on July 28, 1836. In this paper, Dirichlet criticized the Laplacian foundation of least squares. For brevity, I will discuss the main ideas in modern termi- nology. If there are two systems of physical quantities g = (~:1..... ~:t) and 6 = (61, .... 6s), connected linearly by gA = 8 (A E ~"s being a given t by s matrix), the basic problem is to obtain an estimate ~ E ~l,t of the true system of "elements" from a system of observed values d = (dl ..... ds), which are afflicted by observational errors and therefore deviate from their respective true values (61, .... 6s). Usually, one tries to base this estimate on a number of observations s as large as possible, with the consequence that s > t. The estimate ~ is then to serve as an approximate solution of the overdetermined set of equations xA = d, which, in general, has no exact solution. This was an essential problem in mathematical astronomy during the 18th and 19th centuries. One idea (among others) was to combine the set of s equations xA = d linearly to obtain a new set with a solution ~; that is, to find a system of multipliers B ~ N~,t such that TAB = riB. In order to produce an "optimal" solution one could employ the method of least squares, which demanded that IxA - dl 2 should become a minimum among all possible estimates, a condition which is equivalent to the choice B = A ~. In the simplest case one looks for a mean value ~ of direct observations di; then the overdetermined set of equations has the form

X= dl,x= d2 ..... x= d s. According to the method of least squares, the mean X is identical with the arithmetic mean (• di)/s. It was Gauss's and Laplace's aim to give probabilistic arguments in favor of the method of least squares [7]. According to Laplace's approach, one considers a certain set of allowable methods and defines that procedure as the "most advantageous" for which the probability of exceeding any fixed deviation between an estimated value X and the respective true value Xtrue is the least possible [Laplace 1820/1886, LXII], in other words, for which P(IX - Xtruel -> a) becomes a minimum for any a > 0. Laplace claimed to have proved that the method of least squares would be the most advantageous, at least among those methods that combine the observational equations linearly and that utilize a large number of observations. Laplace believed that, in general, usable methods could be based only on linear combinations of the, commonly numerous, observational equations [Laplace 1820/ 1886, 348]. Thus, there was sufficient reason for Laplace to designate the least squares method as the absolutely "most advantageous" method possible. HM 21 DIRICHLET'S WORK IN PROBABILITY 45

In the second supplement to the third edition of his TAP, first published sepa- rately in 1818, Laplace [1820/1886, 559-571] repeated that the method of least squares would give the best estimates among all methods based on multipliers. But then, with the intention of obtaining an estimate 2 of one "element" from the overdetermined set of equations with the form

alx = dl, a2x = d2 ..... asx = ds, (1) he discussed the mdthode de situation, which he had used at earlier stages in his astronomical work, versus that of least squares [8]. Using the former method one obtains an estimate ~ from the condition that Y~ [aix - di[ becomes a minimum if x = ~. In the special case of an odd number of direct observations (a; = I), this method takes the median of the sample d 1..... ds as a "mean value" instead of using its arithmetic mean. Laplace was convinced that the mdthode de situation could not be based on linear combinations. Presupposing a large number s of observations, Laplace derived a new asymptotic formula for the probability of a given difference between the true value Xtme and its estimate x', obtained from (1) by the mdthode de situation. This formula was equivalent to

P(-r

b - rN/-~-aa2 (3) - Xtrue < r) = e -t" dt, b o-X/-2 ' where o-2 is the dispersion of the law of error f. From these asymptotic formulas he concluded that the m~thode de situation should be preferred if 2(f(0)) > 1/o-. (4) Assuming a Gaussian law of error, on the other hand, the method of least squares was superior [Laplace 1820/1886, 571-577]. Laplace did not comment on this result in his second supplement, but in the Introduction of the third edition of his TAP he added a passage which touched on the use of the median as an approximate value as opposed to the arithmetic mean, and he concluded this section by asserting that "the result obtained by the most advantageous method [the method of least squares] is still to be preferred" [Laplace 1820/1886, LXIII], without giving any justification for this remark. Dirichlet [1836; 1897a] emphasized how, in his treatment of least squares, La- place had compared among all possible systems of multipliers only those which do not depend on the observed values in any way. If one dropped this restriction, 46 HANS FISCHER HM 21 however, there might exist "best" fittings based on systems of multipliers other than those obtained by least squares. As Dirichlet pointed out, the choice of the median rn as a mean for an odd number 2n + I of direct observations dl ..... d2n+~, for example, corresponds to a linear combination of these observed values. If

do <- . . . <- dk <- . . . do , then m =dkand

..... d2n+ 1) (bl), m = (d 1 where b k = 1 and bi = 0 (i ~ k). \b2n+l/

Thus, the multipliers b; are, in a certain sense, determined according to the observations. They do not depend, however, on the magnitude of the observed values but only on their order. Presupposing a large number of observations, Dirichlet completed his criticism by the statement that, for any fixed probability level, the lengths of the symmetrical confidence intervals (Fehlergrenzen) for the median and the arithmetic mean are in the ratio of

I 2~xZf(x)dx, (4') where f is the symmetrical probability density of the errors with values within I-a; a]. The statement (4') is, in the case of direct observations, equivalent to (4). Dirichlet concluded that without knowledge of the special law of errors, one cannot determine which method would be superior. Only in the enlarged version [Dirichlet 1897a, 350f.] did he explain, without giving any mathematical details, that his comparison (4') of the quality of the median and the arithmetic mean was based on laws equivalent to (2) and (3). He did not mention, however, Laplace's second supplement to the TAP. Apparently, as one can assume from the wording of [Dirichlet 1897a] and from his usually careful mode of citation, Dirichlet was unaware of this supplement [9]. Moreover, Dirichlet's mode of reasoning and his intentions in discussing this problem were quite different from Laplace's [10]. Laplace aimed to confirm the method of least squares in the case of a large number of observations by probabilistic arguments. Least squares were most advantageous only under special mathematical presuppositions. But in the background of Laplace's probability theory there were also criteria like universal practical applicability or computational simplicity. Thus, the linear combination of the observational equations by a priori constant coefficients, which did not depend on the observations, was according to "common sense" and not arbitrary. Although Dirichlet did not attack the basic idea of assuming linear combinations, HM 21 DIRICHLET'S WORK IN PROBABILITY 47 from his point of view, Laplace's reasoning was based on an arbitrary restriction to constant coefficients and therefore it was not valid. Dirichlet presented, if only in a special case, a procedure, also based on linear combinations, but very different from the method of least squares, though not inferior to the latter. In principle, Dirichlet conceded the same legitimacy to the method of least absolute values as to the method of least squares. Thus, Dirichlet's mathematical rigor went beyond the typical mode of reasoning of classical probability [11]. A similar attitude towards rigor was later held by Augustin Louis Cauchy in his 1853 controversy with Ir6nn6e Jules Bienaym6 concerning least squares (see [Heyde & Seneta 1977, 71-96; Schneider 1987, 200-202, 207-210]). For Dirichlet, there was sufficient reason to criticize Laplace's justification for least squares in the case of a large number of observations if one could construct a special counterexample (regarding direct observations). For Laplace, on the other hand, sufficient reason existed to champion least squares because this method was superior on the basis of"natural" presuppositions (constant multipliers). In his courses on error theory, Dirichlet discussed Laplace's treatment of the method of least squares at considerable length and he also presented the median as a possible alternative to the arithmetic mean. In contrast to Laplace, Dirichlet also dealt carefully with the case of an even number of observations, in which case the median is not uniquely determined. The deduction of the asymptotic laws (2) and (3) was a favorite topic in Dirichlet's courses. Unlike the usual German approach to teaching least squares, Dirichlet totally neglected Gauss's foundations of least squares. Nor did he prove to be a friend of a "natural" Gaussian error law, as evidenced by his saying: "The error law of any single observation can be represented by many different functions" [Dirich- let 1846, 25]. Regarding this question, he opposed his colleague at the Berlin Academy, Encke, a disciple of Gauss. Encke focused considerable attention on the problem of the arithmetic mean as a "best" value and, in this context, on the problem of the "true" probability law of errors. Unfortunately, there seem to be no published or private references to any discussions between Encke and Dirichlet regarding this issue [ 12]. Nevertheless, there was a certain amount of collaboration on error theory, as shown by Encke's publication of Dirichlet's deduction that the median of the absolute values of errors approaches a Gaussian distribution asymptotically [Dirichlet & Encke 1832/1897]. From this easily accessible text one can follow Dirichlet's proof of (2), which is, however, in principle analogous to Laplace's deduction, already published in 1818 [13]. As we have seen, Dirichlet approached error theory in a quite theoretical and abstract way which was, in a certain sense, already different from Laplace's. Yet one finds the same trend in Dirichlet's analytical methods. In his lectures on probability and error theory, Dirichlet revealed a deep knowledge and understanding of Laplace's analytical procedures and their modifications by Poisson [14]. He did not aim merely to reproduce this analysis but also introduced new tools and methods, as, for example, is evidenced by his treatment of Stirling's formula and the central limit theorem. 48 HANS FISCHER HM 21

STIRLING'S FORMULA The Laplacian idea for deducing Stirling's formula, that is, to compute an asymptotic term for F(s + 1), is based on a special representation of the integrand in F(s + 1) = ff e-Xx" dx = e-(Z+'~(z+ s)" dz. The function to be integrated attains its maximum M = e-'s" when x = s or z = 0. Laplace sets the integrand e-'e-Z(z + s)" equal to Me -t2(z), where t 2 = -log(e-Z(1 + z/s)') is generated as a series of powers in z, and then, vice versa, z can be expanded as a series of powers in t. Thus, one obtains after the transformation of the variable of integration from z to t [Laplace 1785, 258f.; 1820/1886, 128-131]

F(s + I) = M f+~ e-'2V~s(l + 4t + --t2 + • • ) dt 3~/~s 6s = s'+'/2e-'V'2-~rTr (11 + ~s + ~ 1 + ... 1

Laplace generally did not consider whether all operations carried out were permis- sible. For example, in [Laplace 1820/1886, 393], he failed to take into account the nonconvergence of the series, in concluding that F(s + 1) = sS+~/2e-'V2-~ for very large s. Dirichlet used these arguments almost without change in his course of 1838. At least he indicated the law for the series expansion--Cauchy [ 1844,258] considered it to be unknown--and mentioned that it is divergent, but nevertheless he executed the limit operation s ~ ~, referring only in a very dubious way to the velocity of the divergence [Dirichlet 1838b, 68]. In his course of 1841/1842, however, he employed a method of reasoning which avoided the use of a semiconvergent expansion, but which retained the essentials of Laplace's procedure. Laplace had already explicated, in the M(moire sur la probabilit~ des causes par les ev(nements 1774, his basic ideas on the approximation of integrals whose integrand depends on a very large number. In the first part of this article, Laplace searched for an approximate value of the inverse probability f p/s +to I~ E = Jp/s-to Xp(1 -- X)q dx _ pPqqs-s 3-to (1 + sz/p)P(1 - sz/q) q dz olxP(1 -x)qdx p!q!/(p + q + 1)! ' that in the case of s Bernoulli trials with p successes and q = s - p failures, the probability for success is contained within [p/s - to; p/s + to], if p and q are "infinitely large," and if to is very small number. Laplace's essential trick was to set the integrand (1 + sz/p)P(1 - sz/q) q = e -u2(z) and to expand u 2 = -log((1 + sZ/p)P(1 -- sz/q) q) in powers of z. He proved that E comes infinitely close to 1, if HM 21 DIRICHLET'S WORK IN PROBABILITY 49 to is "infinitely smaller" than S -1/3 and simultaneously "infinitely larger" than s-l/Z; for example, ifo~ is equal to s -a, where 1/3 < a < 1/2 [Laplace 1774, 33-36]. It follows immediately that the value of both integrals over xP(1 - x)q (in the first fraction) is essentially concentrated in an infinitely small range around p/s (the x- value corresponding to the maximum of the integrand), where the length of this range depends on s. In his later work, Laplace discussed similar problems of approximation only in the rather formal manner which I have described above with the example of Stirling's formula. Dirichlet, however, who in his course of 1838 quoted Laplace's procedure of 1774 for the calculation of the inverse probability E, might later have deduced therefrom his discussion of integrals depending in a similar mode on an infinitely large number s. His idea was to decompose the range of integration of such integrals in several intervals whose limits vary with s. The "main" interval should contain this value of the variable of integration which corresponds to the maximum of the integrand. The integral over this main interval was to be calculated for infinitely large s by suitable series expansions after describing the integrand as a bell shaped function whose maximum point coincides with the maximum point of the integrand. Additionally it had to be shown that the other integrals tend to 0 as s becomes infinitely large. In his proof of Stirling's formula from his course of 1841/1842, Dirichlet started in the same way as Laplace. He separated, however, the whole integral

se-: 1+ dz= ydz=F(s+ 1)eSs -s into the sum

f~sSmy dz -{- fs_sm m y dz + fs~ m Y dz = I l + 12 + 13, where 1/2 < m < 2/3. Setting y(z) = e -t2~z) and with the help of estimates for the expansion of log(y (z)) around z = 0 (corresponding to the maximum of y), Dirichlet showed that I~ and 13 both tend to 0 as s grows, while

s_ f s"-'12/V'2 u 2 l~uX/-~s~ I 2 = f Smsm e-zZ/Zse R(z~ dz = V~s j_s m_,,2/~/-~e- e du, where IR(z)l < $3m-2/[3( 1 -- sm-1)] and therefore I2/~s ~ f~ e -//z du (s ~ oo). • oe /12 Since f_= e- du = V~--~, the required result for F(s + 1) with infinitely large s was obtained [Dirichlet 1841/1842a, 56-61]. Neither in [Dirichlet 1838b], nor in [1841/1842a], nor in Seidel's summary added in 1870 to the latter, is the fact pointed out that

F(s + 1) -~ 1, while ]F(s + I) - ss+VZe-sX/~] ~ ~. ss+ l/2e-S~/2-~ 50 HANS FISCHER HM 21

Yet Dirichlet distinguished strictly between these two sorts of limits, as we can see from [Dirichlet 1838a, 353] and [Dirichlet & Arendt 1904, 423], where he discussed the meaning of the adjective "asymptotic•"

CENTRAL LIMIT THEOREM The point of the universal central limit theorem within the Laplacian theory of least squares is as follows: There are independent errors x~, x2 ..... x~ of a very large number s of observations with respective symmetrical probability densities fl, f2 ..... f~ vanishing beyond the interval [-a; a]; al, a2 ..... a s are given constants. The problem is to find an approximate expression for the probability

P that the value of the linear term otix I -t- ot2x2 -I- .... + a~x~ lies between -h' and +k', where k' = kV~s (k being a given constant). Treating this problem in his course on least squares of 1838, Dirichlet followed exactly the procedure employed by Poisson in the Connaissance des terns for 1827 [Poisson 1824]. In 1846, however, there were essential innovations regarding the deduction of the main formulae and the discussion of the approximations, where the latter was now handled analogously to his treatment of Stirling's formula. Dirichlet had presented "his" factor

2 ~sin~cosk~d~= ~ if-I

in three papers of 1839 [Dirichlet 1839a,b,c], using it to deal with certain problems, especially in potential theory and the calculation of spatial volumes. In his course of 1846, he applied this tool to the representation of the above defined probability

P fcfl(xO . . . f~(x~) dxl . . . dx~,

where G = {x C ~s I-h' < alxl + a2x2 + "'" + a~x~ < h'}. Thus, with the aid of Dirichlet's factor,

P = -~ f[-a;a]s I<,'O.~~ (xl)"f~(x~) sin+,p ~P cos[(alXl +"" OtsXs)~,] d~dxl'" dxs. (5)

From (5) follows (the interchange of the order of integration is not discussed)

• ' a a / p:2foSmk,p[r-~ ~Lafl(Xl)e a,x,¢V-I dXl"" f2afs(Xs)ea,x,¢ v~-i dx s~ d~

_2f;=sinkV~s,p(f:~71" ~ afl(Xl)COS(OtlXl¢fl) dXl"" f~,afs(Xs) coS(asXs~) dxs ) d¢ (6) [Dirichlet 1846, 25-27] [15].

The integral in (6) is divided into three parts, so that HM 21 DIRICHLET'S WORK IN PROBABILITY 51

2 f8 sinhV'ss~p ..... 2 inhV~s~o 2(= sirth~/-ss¢ P = ~r J0 F Ht~p)a~o ± --7r f a s II(~p) d~ + 7raa ~o II(9) d9 = I~ + I2 + I3, where : f . . . cos o.x.

The quantities 8 and A have to satisfy the following conditions:

~$1/4 ~ 0 (S ~ 00) (7a) ~S1/2----> 00 (S""> 00) (7b) A proportional to s ~ (a > 0). (7c) The conditions (7a) and (7b) are met, for example, if 8 = s -1/3. In the first of these three integrals 11, the expression II(~), which takes its maximal value 1 when = 0, is described as

2 2 H(~o) = e-Xk,%~ e m~) (kv= ~l f~a x2f~(x)dx' v= 1. . s).

On the basis of the expansion of

log( f a_af~(x~)COS(a~x~p)dx,,) in powers of ~o, R is estimated by IR( )I < sL8 4 + sM8 6 + '''. (8) Here L, M .... are constants depending on the functions f, and coefficients oe, (v = 1 .. s). A closer specification of these constants cannot be found in the lecture notes. Referring to (8) and (7a), Dirichlet concluded that R can be neglected as it grows. After the transformation ~Tss~0 = x, the upper limit of the first integral 11 equals 8~/-ss. Because of (7b) this term tends to oo. Hence "for a very large number of observations" the first integral becomes

2 fo SinxXX e_(X2xko~2v,/,dx. (9)

With the help of elementary calculations, Dirichlet showed that (9) is equal to the well known expression

2 [x/~z~) e ~ dt [Dirichlet 1846, 27-30]. X/_~ ~0 The fact that the integrals I 2 and 13 both tend to 0 is finally explained only by brief and incomplete, but in principle sound, arguments based on (7a) and (7c) [Dirichlet 1846, 30 f.]. 52 HANS FISCHER HM 21

Dirichlet's reasoning reminds one of the proof given by Cauchy in 1853 for the central limit theorem [Cauchy 1853] (for a description see [Heyde & Seneta 1977, 93-96]), and even of the proof by Aleksandr Mikhailovich Lyapunov [Liapounoff 1900, 1901]. Altogether, Dirichlet's outline leaves a less modern impression, for instance, in his use of infinite series expansions instead of Taylor's theorem followed by estimates of the remainder term. Some additional, but within error calculus natural, presuppositions needed to carry out his idea for the proof in a satisfactory manner are missing in the lecture notes. The wording of the text suggests that Dirichlet used an analytical mixture, somewhat strange for the modern reader, consisting of estimates of the absolute (real) values of the respective integrals and calculations with infinitely small values. Moreover, there is some lack of clarity, which may have been due to the transcriber of [Dirichlet 1846], who certainly did not render all arguments entirely correctly. Yet it must be stated that the idea of a more rigorous application of Laplace's method of approximation by splitting the respective integrals into several parts, which Dirichlet developed apparently in the 1840s, was an important step away from a simply intuitive understanding of the limits to a stricter estimation of the approximations.

GENERAL CONTEXT OF DIRICHLET'S CONTRIBUTIONS TO PROBABILITY THEORY AND HIS ANALYTICAL WORK In Dirichlet's courses and papers on probability theory one finds neither a continual method of analysis nor a uniform standard of rigor• Yet especially in relation to mathematical rigor, Dirichlet enjoyed an excellent reputation, as indicated by Carl Gustav Jacobi's well known saying: "Dirichlet alone, neither I nor Cauchy, nor Gauss knows what a perfectly rigorous mathematical proof is like •.." (cf. [Biermann 1988, 46]). In his memorial speech on Dirichlet, Ernst Eduard Kummer pointed out the "... clarity of his mind, which enabled him to know and understand the most difficult topics in their plain truth and how to present them" [Kummer 1860, 342]. For Dedekind it was "the greatest delight to listen to his extremely impressive lectures . ." (cf. [Lorey 1916, 82f.]), and Wilhelm Lorey stated somewhat later: "The more silently acting Dirichlet.. has produced downright classic patterns in his courses" [Lorey 1916, 71]. Certainly Dirichlet far exceeded the common standards in his lectures, and without doubt this fact contributed to his reputation. In his lectures on partial differential equations, Dirichlet's treatment of the convergence of Fourier series exemplifies his analytical rigor [Dirichlet 1840/1841, 92-128]. Yet due to the contemporary state of analysis and the multitude of subjects covered, Dirichlet could not generally maintain the same standards in his lectures as he did in his publications• In his lectures Dirichlet obviously had no reservations about giving dubious deductions, even by contemporary standards, as, for example, in his courses of 1838 treating Stirling's formula or the central limit theorem. This is true as well of the problems discussed in the Seminar with particularly gifted students. Here he gave a direct solution of the problem of duration of play when one gambler holds an infinite stake. This solution is not based on the formula for finite properties, but HM 21 DIRICHLET'S WORK IN PROBABILITY 53

rather uses a somewhat arbitrary transition from a sum to a definite integral [Dirichlet & Seidel 1840/1841, 27-30]. Such loose reasoning was not restricted to probability theory. Fourier's integral theorem, discussed in the course on partial differential equations of WS 1840/1841 (Seidel's notes), was only derived by a transition from a Fourier series with infinite period to a Fourier integral [Dirichlet 1840/1841, 131-133]. In this context the 20-year-old student Seidel wrote in his notes: "Exact methods are of course out of the question." [Dirichlet 1840/1841, 133f.]. This most probably was a comment by Dirichlet himself. In the historical assessment of analytical proofs, there is quite frequently a failure to differentiate between deductions which were already considered unrigorous from a contemporary point of view and those forms of mathematical reasoning which only later appeared to be unrigorous because of intermediate changes in methodology, style, or standards of representation. One finds examples of these sorts of misjudgements in connection with Dirichlet's contributions to probability theory. Dirichlet's deduction of the approximately normal distribution of medians, which also forms the basis of Encke's report [Dirichlet & Encke 1832/1897], made strong use of infinitesimals. The following sentence, in an 1885 letter from Georg Hettner to Kronecker dealing with this passage in Encke's article, is characteristic of the methodological changes that took place during the preceeding 50 years as well as of the later universal recognition accorded Dirichlet's analysis: "The wording of the proof is surely due to Encke, at least Dirichlet would never have written for mathematicians in such a manner" [Hettner 1885]. In his lectures, however, Dirichlet [1838b, 164-166; 1838c, 32-35; 1846, 21-25] followed exactly the same line of argument as reported by Encke. Thus, the "unmathematical" style of this proof was in fact due to Dirichlet. Lazarus Fuchs, the editor of the second volume of Dirichlet's Werke, added several notes to the proof of the theorem of de Moivre-Laplace [Dirichlet 1897b], which was found in Dirichlet's Nachlass. Fuchs' notes show that 50 years later the understanding of some ideas and methods used by Dirichlet had become lost. A letter [1897] concerning this proof by Richard Dedekind, who contributed to the edition of the second volume in an advisory capacity, shows the attitude toward mathematical rigor of the post-Weierstrassian era and emphasizes criteria, for example referring to the use of expansions by omitting higher powers, which had apparently been of less importance for Dirichlet here. Dirichlet used calculations with infinitesimals as an analytical tool not only in the context of probability theory. This is evidenced, for example, by his proof that, for a modulus k infinitely close to unity, the elliptic integral

fj dx = log(4) (1 + p), where p is infinitely small V'(1 - x2)(1 - kZx)

(k' = V'I -k 2) [Dirichlet 1842]. Thus, on the one hand, Dirichlet did not unequivocally speak the "language of limits;" on the other hand, however, he published only those papers in detail which 54 HANS FISCHER HM 21 were, to a great extent, conformable even to the standards of post-Weierstrassian analysis, although neither he nor his contemporaries were provided (from today's point of view) with adequate fundamental definitions. Referring to rigor and applied methodology, one must therefore differentiate between Dirichlet's "public" analysis, as presented in his publications, and his rather "private" analysis, which appears in his courses and in the few notes kept in his Nachlass. For that reason it becomes understandable that Dirichlet did not publish a more exhaustive paper on probability calculus, in spite of remarkable results, because he considered the reasoning employed therein part of his "private" mathematics, and he was either unwilling or unable to translate this work into his "public" style. One important tool for this translation would have been the mean value theorem of differential calculus. It seems that Dirichlet explicitly used this theorem only at a late period in his analytical work. In the elaboration of his course on definite integrals, made by Gustav Arendt in SS 1854 and published in [Dirichlet & Arendt 1904], Stirling's formula is deduced as shown above. The series expansions, however, are replaced by the use of the mean value theorem, and the limits are proved in elementary fashion by estimates [Dirichlet & Arendt 1904, 440-448]. If these lecture notes are actually authentic, as Arendt asserts in the preface and strives to support by citing all his own annotations, then, in fact, Dirichlet had taken large, but late steps toward an analytical style of which he is now considered to be one of the leading promulgators [16].

DIRICHLET'S PLACE IN 19TH CENTURY PROBABILITY THEORY Dirichlet's favorite problems, limit theorems and the discussion of least squares based on them, remained the only topics in classical probability theory which were studied with considerable intensity during the last third of the 19th and the beginning of the 20th centuries. Because the analytical methods used by Dirichlet were--at least in retrospect--promising, it is regrettable that his students scarcely took up the impulses, given also in the Seminar, for active research on probability calculus. One reason for this lack of interest may have been that other mathematical disciplines connected with Dirichlet's work were more popular; another is certainly the temporal concentration of his courses on probability theory around the end of the 1830s and the beginning of the 1840s. The only essential exception [17] to this general disinterest among his Berlin disciplesappears to be Seidel. Having been very strongly influenced in mathematics by Dirichlet, as one can already see from the many annotations in his lecture notes extending to the 1870s, Seidel initiated a fundamental reorganisation of mathematical teaching at Munich University together with Gustav Bauer, who also was Dirichlet's student (cf. [Gericke & Uebele 1972; Toepell 1992, Chap. 5.1 .]). Seidel made a name for himself as the author of papers not only on convergence of series and on continued fractions, but also on geometrical optics and on the photometry of stars. During almost 50 years, beginning in 1846, he gave many HM 21 DIRICHLET'S WORK IN PROBABILITY 55 courses on probability theory and least squares. His publications in this connection deal with statistics of epidemics [Seidel 1865, 1866] (for an appreciation see [Shey- nin 1982, 277f.]) and the critical examination of systematic influences on observations [1862], and he also wrote an article in which the problem of fitting the frequency of the appearance of comets to the Poisson distribution is broached but not elaborated [1876]. Dirichlet's lack of influence on his disciples in the field of probability theory corresponds with the fact that Dirichlet's scientific correspondence, as it seems, does not contain essential discussions on probabilistic problems [18]. The internal relevance of Dirichlet's lectures on probability theory, however, is evidenced by the fact that the courses given by Pafnutii Lvovich Chebyshev on probability theory (cf. [Ermaloeva 1987]) had the same structure in regard to the use of integrals as Dirichlet's courses. This reflects a similar interest of both scientists in a further development of specific analytical methods used in the deduction of probabilistic limit theorems. The relative quality of Dirichlet's lectures on probability theory can be assessed by a comparison with Lyapunov's lecture notes from Chebyshev's courses on probability theory in 1879/1880, which were published in 1936. In general, it can be stated that the overall level of Dirichlet's lectures on probability theory was by no means lower than Chebyshev's. From the point of view of mathematical rigor, Dirichlet's deduction of the central limit theorem in 1846 seems to be superior to Chebyshev's in his course of 1879/1880 ([Chebyshev 1936, 219-224]). The latter largely follows Laplace's original argument, and does not contain any analytical innovations. On the one hand, it is interesting to note that Dirichlet's approach to the teaching of probability calculus placed it within the framework of the application of definite integrals, and thus took an abstract turn which totally neglected "philosophical" and "moral" aspects of probability theory [19]. Dirichlet's mode of discussing probabilistic problems indicates to some extent a movement from classical probability toward a new point of view. His attempt to treat important topics of probability theory using more sophisticated analysis introduced a new factor which referred no longer only to the applications. But this process alone could not lead to the full autonomy of probability calculus as a purely mathematical theory. General notions, precise statements, and the willingness to weaken the presuppositions as far as possible were still lacking. There was a certain trend in this direction, however, in Dirichlet's criticism of least squares. On the other hand, the significance of Dirichlet's lectures and papers on probability calculus is due to several analytical innovations, although their presentation does not always meet the standards usually connected with his name. Even if proceeding cautiously, as is advisable to do with lecture notes, it turns out that Dirichlet's contributions to mathematical probability theory provide very good sources for a better methodological understanding of his analytical work. 56 HANS FISCHER HM 21

APPENDIX Courses on Probability Theory Announced by Dirichlet at Berlin University and Lecture Notes on Them Sources: [Biermann 1959, 34-39] and [Friedrich-Wilhelms-Universit/it 1811-].

WS 1829/1830 Wahrscheinlichkeitsrechnung. Without any information about the number of hours ("privatim"). WS 1831/1832 Die Elemente der Wahrscheinlichkeitsrechnung. 2 hours. SS 1836 Anfangsgriinde der Wahrscheinlichkeitsrechnung. 1 hour. Ausgewdhlte Kapitel der Integralrechnung mit Anwendungen, besonders auf die Bestimmung der Wahrscheinlichkeit. 4 hours. SS 1838 Die Methode der kleinsten Quadrate. 1 hour. [Dirichlet 1838b,c] [20] Ausgewdhlte Kapitel der Integralrechnung mit Anwendung auf die Probabilitdtslehre. 4 hours. [Dirichlet 1838b] WS 1841/1842 Olberverschiedene Anwendungen der Lehre yon den bestimmten Integralen (in fact treated probability theory). I hour. [Dirichlet 1841/1842a] [21] SS 1846 EinigeAnwendungenderlntegralrechnungaufWahrscheinlich- keitsbestimmung. 1 hour. [Dirichlet 1846] SS 1850 Die Anfangsgriinde der Wahrscheinlichkeitsrechnung. 1 hour. Dirichlet apparently did not offer courses on probability theory after 1850, either at Berlin or GSttingen (for a survey of Dirichlet's lecture courses in GOttingen see [Biermann 1959, 73f.]).

ACKNOWLEDGMENTS I thank Professor Ivo Schneider for very helpful and stimulating suggestions, and for giving me the chance to present an early version of this paper during the Oberwolfach meeting Geschichte der Mathematik in May 1990. To Professor Eberhard Knobloch I am very grateful for encouraging me to submit the article to Historia Mathematica. Professor David E. Rowe kindly gave many useful suggestions for the presentation and organization of this paper, and he corrected my English patiently. Professor Kurt-R. Biermann and Dr. Gert Schubring provided me with helpful information concerning unpublished lecture notes and Dirichlet's teaching activities. To the following persons and institutions I am indebted for allowing me to use unpublished material and providing me with copies: Professor Menso Folkerts (Institut fiir Geschichte der Naturwissenschaften der Universit~t Miinchen), Deutsches Museum Miinchen--Sondersammlung, Staatsbibliothek zu Berlin, PreuBischer Kulturbe- sitz--Handschriftenabteilung, Archiv der Humboldt-Universitat zu Berlin, and Akademie der Wis- senschaften zu Berlin--Akademiearchiv.

NOTES 1. The lecture notes [Dirichlet 1838c] are mentioned in [Schneider 1974, 155; Schneider 1975, 248; Butzer 1987, 70]. 2. For the case of a line divided in 3 parts x < y < z at random, Dirichlet calculates the probability that y < (1 + ~)x and z < (1 + fl)y, a and/3 given constants. The result is 6a/[(3+/3)(3+2c~+/3+a/3)]. In [Dirichlet & Seidel 1840/1841, 10-12] one can find a similar problem solved. 3. As Gert Schubring communicated to me (December 3, 1991), Dirichlet's curriculum was adopted HM 21 DIRICHLET'S WORK IN PROBABILITY 57 with only some minor changes to the plans of the Prussian Kultusministerium (see [Schubring 1981, 177]) for the polytechnic school in 1832. 4. An official mathematical Seminar was established at Berlin University only in 1861 by Ernst Eduard Kummer and Karl Weierstrass [Biermann 1988, 97-100]. 5. Originally, the one hour Publi¢um was intended to be an introduction for students of all disciplines, but it was not unusual to use it as a supplement to the large Privatvorlesung [Lorey 1916, 9f., 13]. 6. According to the Verzeiehnis der Vorlesungen [Friedrich-Wilhelms-Universit~it 1811-], prior to Dirichlet's, lectures on probability theory had been announced by Johann Georg Tralles (SS 1817), Johann Philipp Gruson (WS 1817/1818) and Enno Heeren Dirksen (SS 1822). During Dirichlet's Berlin period Ferdinand Minding read in SS 1840 on elementary probability calculus. Johann Franz Encke's courses on least squares mainly concerned practical applications and can only be regarded as courses on probability theory in a wider sense (cf. [Brnhns 1869, 158f, 172]). After Dirichlet, Kummer (SS 1864) was the first to give a 1-hour course on Principien der Wahrscheinlichkeitsrechnung. 7. There is a good deal of historical literature on least squares. Concise introductions to the fundamentals are [Harter 1983b] and [Heyde & Seneta 1977, 62-66]. A discussion of the most important issues of error theory can be found in [Stigler 1986]. Especially for Laplace see [Sheynin 1976, 1977]. The main aspects of the dispute over the foundations of error theory in the 19th century are discussed in [Knobloch 1992]. 8. For a description of Laplace's m~thode de situation, see, for example, [Stigler 1986, 39-55]; for an appreciation of Laplace's second supplement to the TAP, see [Stigler 1973a]. 9. Dirichlet [1897a, 348] only mentioned Laplace's statement regarding the median (see above) in the introduction to the third (!) edition of the TAP and he expressed his astonishment that Laplace had not discussed this issue at greater length. Moreover, Dirichlet [1897a, 350] quoted a verbal analogue to (3) from Laplace's TAP, but the respective analogue to (2) he claimed for himself. It seems that Dirichlet, despite having read evidently the third edition of the TAP, simply overlooked Laplace's second supplement at the end of this book. 10. Harter's summary [Harter 1974 I, 160f.; 1983a, 36; 1983b, 595f.] does not hit the focal point of Dirichlet's criticism in my opinion. 11. The term "classical probability" is used in just the same sense as explicated in [Daston 1989]. 12. In the archives of the Berlin Academy, minutes of the meeting from July 28, 1836, during which Dirichlet gave this lecture, do not exist. 13. [Dirichlet & Encke 1832/1897] refers to a formula which Gauss used without any proof in connection with his suggestions for estimating the probable error from the median of the absolute values of errors [Gauss 1816, 116f.]. Gauss showed with the help of the formula that this method was not practical, taking the hypothesis of a "Gaussian" error law as a basis (see [Sheynin 1979, 35-38; Stigler 1973b, 875]). Before Encke's publication of Dirichlet's proof, Carl Friedrich Hauber [1830, 304-306] had already deduced Gauss's formula in an article on error theory. It seems that Hauber was also unaware of Laplace's second supplement to the TAP. 14. Dirichlet's esteem for Poisson's papers on least squares, published in the Connaisance des tems for 1827 and 1832, which contain proofs of the central limit theorem, is evidenced by a letter to Encke in which Dirichlet asks to borrow the respective volumes for use by an interested student [Dirichlet (a)]. 15. James Whitbread Lee Glaisher [1872a, 195; 1872b, 98] was perhaps the first to publish the use of Dirichlet's factor in this way. Thereafter the trick was applied in this or other similar contexts by various authors, such as Matthieu Paul Hermann Laurent [1873, 163,175], Charles H. Kummell [1882, 181], and Rudolph Lipschitz [1890] (Dirichlet's disciple, see [Sheynin 1979, 37f.]), none of whom referred to Glaisher, unlike Emanuel C zuber [189 t, 253-257], who did cite him. Aleksandr Mikhailovich Lyapunov [1900, 360] mentioned Glaisher and Czuber, but, as he himself described [1900, 363f.], he only used the discontinuity-factor in a first idea of the proof. In his definitive proof he did not need Dirichlet's factor. 58 HANS FISCHER HM 21

16. In a methodologically similar way, Joseph Liouville [1846] gave a simple and short proof for the asymptotic behavior of the F-function. Dirichlet had published several papers in Liouville's journal and probably knew this publication. But Dirichlet's approach is close to his deduction of limit theorems for other Eulerian integrals [Dirichlet & Arendt 1904, 423-450], and suggests an estimate of the residuum, which however was not fully elaborated. 17. The brief article of Dirichlet's disciple Lipschitz, already mentioned in note [15], was the only contribution of this mathematician to probability theory. This paper deals with the application of Dirichlet's discontinuity-factor to power sums of errors. Regarding the essential point, discussion of the limit distribution, Lipschitz [1890, 165] only referred to the "route fray6e par Laplace," and he gave the end results without any further explanation. 18. Most of the unpublished scientific letters sent to Dirichlet are in the Staatsbibliothek zu Berlin, Preuflischer Kulturbesitz. For an outline see [Butzer et al. 1982, 49-50]. The only letter on probability in this collection was written by Anton Meyer [1856]. Meyer asks for an assessment of a paper refused by the Academy of Brussels. For an introduction to the Dirichlet Nachlass see [Schubring 1986]. Dirichlet's correspondence with Alexander von Humboldt [Biermann 1982] and parts of the correspondence with Gauss (mainly [Werke I, 373-387], for details see [Biermann 1959, 74]), and Leopold Kronecker [Werke II, 388-411] have been published. The edition of the correspondence between Dirichlet and Liouville [Tannery 1910] was completed by [Neuenschwander 1984, 87-104]. 19. The process of abstraction, which was essential for the survival of probability theory as a field of mathematical research in the 19th century, is described in [Schneider 1987, Sect. 4, especially 210f.]. 20. The Abgangszeugnis [Friedrich-Wilhelms-Universit~it 1839] of Borchardt, the author of the originally undated [Dirichlet 1838b], indicates that he only attended Dirichlet's courses on probability theory and least squares in SS 1838. Thus, it becomes evident that his lecture notes on probability calculus cannot refer to courses with similar titles in 1836. It is very probable that [Dirichlet 1838c] and the second part of [Dirichlet 1838b] derive from the same course on least squares. In fact, they agree to a large extent, but they also differ in a few minor points. 21. Only the contents of [Dirichlet 1841/1842a] make it evident that the l-hr course, Verschiedene Anwendungen der Integralrechnung, which Dirichlet announced together with a 4-hr course on definite integrals [Dirichlet 1841/1842b] for WS 1841/1842, was in fact on probability theory. Whether other courses with similar titles and without detailed specification of the applications also dealt with probabilistic problems cannot be definitively known from the few extant lecture notes. Several lecture notes on integrals, however, indicate that the 1-hr Publicum complementing the se lectures was often dedicated to special problems like the F-Function or certain expansions by series. Aside from [Dirichlet 1841/ 1842b], there exist lecture notes on integral calculus written by Ludwig Seidel for SS 1840 [Dirichlet 1840], and by Gustav Arendt for SS 1854 (edited in 1904 making use of the lectures given in SS 1852). From [Meyer 1871] one can discern the contents of the lectures given by Dirichlet in SS 1858.

REFERENCES Biermann, K.-R. 1959. J. P. G. Lejeune Dirichlet, Dokumente far sein Leben und Wirken. Berlin: Akademie-Verlag. -- (Ed.) 1982. Briefwechsel zwischen Alexander von Humboldt und Peter Gustav Lejeune Dirich- let. Berlin: Akademie-Verlag. -- 1988. Die Mathematik und ihre Dozenten an der Berliner Universitdt: 1810-1933. Berlin: Akademie-Verlag. Bruhns, C. 1869. Johann Franz Encke, sein Leben und Wirken, Leipzig: Gianther. Butzer, P. L. 1987. Dirichlet and his role in the founding of mathematical physics. Archives Interna- tionales d'Histoire des Sciences 37, 49-82. Butzer, P. L., Jansen, M., & Zilles, H. 1982. Johann Peter Gustav Lejeune Dirichlet (1805-1859). Genealogie und Werdegang. Diirener Geschichtsbl?itter 71, 31-56. HM 21 DIRICHLET'S WORK IN PROBABILITY 59

Cauchy, A. L. 1844. Note sur les intrgrales euleriennes. Comptes Rendus Hebdomaires des Sdances de l'Acad~mie des Sciences 19, 67-73. All references are to the reprint in Oeuvres completes (1) 8, Acadrmie des Sciences, Ed., pp. 258-264. Paris: Gauthier-Villars, 1893.

-- 1853. Mrmoire sur les rrsultats moyens d'un trrs-grand nombre des observations. Comptes Rendus Hebdomaires des S~ances de l'Acad~mie des Sciences 37, 381-385. All references are to the reprint in Oeuvres compl?tes (1) 12, Acadrmie des Sciences, Ed., pp. 125-130. Paris: Gauthier-Villars, 1900. Chebyshev, P. L. 1936. Teoriya oeroyatnostei. Moscow/Leningrad: Akademiya Nauk SSSR. Czuber, E. 1891. Theorie der Beobachtungsfehler. Leipzig: Teubner. Daston, L. 1989. Classical Probability in the Enlightenment. Princeton, NJ: Princeton Univ. Press. Dedekind, R. 1897. Unpublished letter to L.Fuchs. 5 pp., April 28, 1897. Akademie der Wissenschaften Berlin, Zentrales Archiv, Nachlal5 Dirichlet 65.

-- 1901. Gaul5 in seiner Vorlesung ~iber die Methode der kleinsten Quadrate. In Festschrift zur Feier des hundertfiinfzigjgihrigen Bestehens der krniglichen Gesellschaft der Wissenschaften zu GOttingen, Beitrgige zur Gelehrtengeschichte Grttingens, pp. 45-59. Berlin: Weidmann. All references are to the reprint in Gesammelte mathematische Werke, R. Fricke, E. Noether, & Oy Ore, Eds., Vol. II, pp. 293-306. Braunschweig: Vieweg, 1931. Dirichlet, P. G. Lejeune 1889-1897. Werke, 2 vols., L. Kronecker, & L. Fuchs, Eds. Berlin: Georg Reimer. Vols. I and II reprinted together New York: Chelsea, 1969. -- 1836. Ueber die Methode der kleinsten Quadrate. Bericht iiber die Verhandlungen der Krnig- lich Preuflischen Akademie der Wissenschaften, 67f. All references are to the reprint in Werke, Vol. I, pp. 279-282.

-- 1838a. fiber die Bestimmung asymptotischer Gesetze in der Zahlentheorie. Bericht iiber die Verhandlungen der Krniglich Preuflischen Akademie der Wissenschaften, 13-15. All references are to the reprint in Werke, Vol. I, pp. 351-356.

-- 1838b. Wahrscheinlichkeitsrechnung nach Dirichlet. Unpublished lecture notes, divided into two parts, the first dealing mainly with general probability calculus, the second with least squares. 132 + 35 pp., undated, certainly SS 1838, written by C. W. Borchardt. Staatsbibliothek zu Berlin, Preul5ischer Kulturbesitz, Nachlal5 Borchardt 3. -- 1838c. Ueber die Methode der kleinsten Quadrate nach Lejeune-Dirichlet. Nach einem Hefte yon 1838, Ziirich 1857. Copy made by R. Clausius of unpublished lecture notes, evidently SS 1838, by an unknown author, 35 pp. Added 4 small pages with formulas on error theory. Deutsches Museum Mtinchen, Sondersammlung, Nachlal5 Clausius 6455.

-- 1839a. Sur une nouvelle mrthode pour la drtrrmination des intrgrales multiples. Comptes Rendus Hebdomaires des S~auces de l'Acaddmie des Sciences 8, 156-160. All references are to the reprint in Werke, Vol. I, pp. 375-380.

-- 1839b. Ueber eine neue Methode zur Bestimmung vielfacher Integrale. Bericht iiber die Ver- handlungen der KOniglich Preuflischen Akademie der Wissenschaften, 18-25. All references are to the reprint in Werke, Vol. I, pp. 381-390.

-- 1839c. Ueber eine neue Methode zur Bestimmung vielfacher Integrale. Abhandlungen der Krniglich Preuflischen Akademie der Wissenschaften, pp. 61-79. All references are to the reprint in Werke, Vol. I, pp. 391-410. -- 1840. Die Lehre yon den bestimmten lntegralen und einige Anwendungen davon. Zwei Colle- gien, nach Professor Dr. Lejeune-Dirichlet. Berlin, Sommersemester 1840. Philipp Ludwig Seidel, Stud. math. Unpublished lecture notes, written by L. Seidel, 238 pp. Von den bestimmten lntegralen, pp. 3-186.--Einige Anwendungen der bestimmten lntegrale, pp. 191-238. Institut for Ge- schichte der Naturwissenschaften der Universit/it Miinchen, Nachlal5 Seidel. -- 1840/1841. Theorie der partiellen Differentialgleichungen nebst Anwendungen davon auf physikalische Untersuchungen. Nach Professor Dr. Lejeune-Dirichlet. Ludwig Seidel, stud. math. Berlin, Wintersemester 1840/41. Unpublished lecture notes, written by L. Seidel, 378 pp., Institut for Geschichte der Naturwissenschaften der Universit/it Mfinchen, Nachlal5 Seidel. 60 HANS FISCHER HM 21

-- 1841/1842a. Anwendung der bestimmten Integrale auf die Wahrscheinlichkeitsrechnung. Nach Professor Dr. Lejeune-Dirichlet. Ludwig Seidel, Stud. math. Berlin, Wintersemester 1841/42. Unpublished lecture notes, 69 pp., pp. 47-60 shorthand, written by L. Seidel; added a summary by Seidel on Stirling's formula (dated 1870), 2 pp. Institut fiir Geschichte der Naturwissenschaften der Universit~it MiJnchen, NachlaB Seidel. -- 1841/1842b. Methoden zur Angabe bestimmter lntegrale nach Dr. Lejeune-Dirichlet. Berlin, Wintersemester 1841/42. (Als Ergginzung zu dem Hefte iiber best. Int. vom Sommer 1840). Nach- geschrieben in der Vorlesung. Ludwig Seidel. Unpublished lecture notes, written by Seidel, 111 pp. Institut for Geschichte der Naturwissenschaften der Universit~it Miinchen, NachlaB Seidel. 1842. Beweis dafl fiir einen der Einheit unendlich nahen Modul k das elliptische Integral

~o' dx - log(4)(l + p), wo o unendlich klein ist. ~/(1 - x-')(1 - k2x)

Unpublished note, part of an exercise book with title Mathematik, neueste Ordnung yon Gustav Lejeune Dirichlet. I ste Classe, Miirz 1842. Akademie der Wissenschaften zu Berlin, zentrales Archiv, NachlaB Dirichlet 27, BI. 3. -- 1846. Anwendungen der bestimmten Integrale auf Wahrscheinlichkeitsbestimmungen, besonders auf die Methode der kleinsten Quadrate. Lejeune Dirichlet. Unpublished lecture notes by an unknown author, 37 pp., undated, probably SS. 1846, in the possession of L. Seidel. Institut for Geschichte der Naturwissenschaften der Universit~it Mtmchen, NachlaB Seidel.

-- 1897a. Bemerkungen iiber die zweckm~iBigste Art, Beobachtungen zur Bestimmung unbe- kannter Elemente zu verbinden. Posthumously edited for the first time in Werke II, pp. 347-351.

-- 1897b. Demonstration du throrrme de Bernoulli. Posthumously edited for the first time in Werke II, pp. 354-357.

--(a). Unpublished Letter to Encke, undated, 1 p. Akademie der Wissenschaften zu Berlin, zentrales Archiv, NachlaB Encke 17.

-- (b). Unpublished notes on probability theory (duration of play, combinatorics). 7 pp., undated, written by Dirichlet. Added the beginning of a copy (1 p., evidently for the use of a possible edition in Werke). Akademie der Wissenschaften zu Berlin, zentrales Archiv, NachlaB Dirichlet 4.

-- (c). Unpublished notes on the rencontre problem. 2 pp., undated, written by Dirichlet. Added a copy (2 pp.) Akademie der Wissenschaften zu Berlin, zentrales Archiv, NachlaB Dirichlet 7.

-- (d). Unpublished plan of studies in analysis and geometry. 3 pp., undated, written by Dirichlet. Akademie der Wissenschaften zu Berlin, zentrales Archiv, NachlaB Dirichlet 22. --, & Arendt, G. 1904. G. Lejeune-Dirichlets Vorlesungen iiber die Lehre yon den einfachen und mehrfachen bestimmten Integralen. Braunschweig: Vieweg.

--, & Encke, F. 1832/1897. Ueber einen von Lejeune Dirichlet herrfihrenden Beweis aus der Wahrscheinlichkeitsrechnung. Originally published as a part (pp. 295-298) of [Encke 1832]. All references are to the reprint in Werke II, pp. 368-372. -- , & Seidel, L. 1840/1841. Aufgaben aus Dirichlet's mathematischen Seminar. WS 1840/41. Unpublished notes by Seidel, 57 pp. Added a draft Ober die vielfachen Puncte der Curven, 6 pp. Institut for Geschichte der Naturwissenschaften der Universit~it Miinchen, NachlaB Seidel. Du Bois-Reymond, P. 1885. Unpublished letter to Kronecker. 1 p., March 23, 1885. Akademie der Wissenschaften zu Berlin, zentrales Archiv, NachlaB Dirichlet 65. Encke, J. F. 1832. Ueber die Methode der kleinsten Quadrate. Berliner Astronomisches Jahrbuch, 1834, 249-304. Ermaloeva, N. S. 1987. Obodnom neopublikovannom kurse teorii veroyatnostei P. L. Chebysheva. Voprosy istorii estestvoznaniya i tekhniki No. 4, 106-112. HM 21 DIRICHLET'S WORK IN PROBABILITY 61

Friedrich-Wilhelms-Universit~t. 1811-. Verzeichnis der Vorlesungen, welche yon der Friedrich- Wil- helms-Universitiit zu Berlin gehalten werden. Berlin: Akademische Buchdruckerei.

-- 1839. Abgangszeugnis for C. W. Borchardt and Meldebogen. Humboldt-Universit~it Berlin, Archiv: Rektor und Senat, Abgangszeugnisse vom 21.3. bis 28.3.1839, BI. 158, 158R., 160, 160R., 161, 162. Gauss, C. F. 1816. Bestimmung der Genauigkeit von Beobachtungen. Zeitschriftfiir Astronomie und verwandte Wissenschaften 1, 185-197. All references are to the reprint in Werke, Vol. IV, zweiter Abdruck, Krnigliche Gesellschaft der Wissenschaften zu Grttingen, Ed., pp. 109-117, GOttingen, 1880. Gericke, H., & Uebele, H. 1972. Philipp Ludwig von Seidel und Gustav Bauer, zwei Erneuerer der Mathematik in MiJnchen. In Die Ludwig-Maximilians-Universitgit in ihren Fakultiiten, Vol. 1, L. Boehm, & J. SpOrl, Eds., pp. 380-393. Berlin: Duncker & Humblot. Glaisher, J. W. L. 1872a. Remarks on certain portions of Laplace's proof of the method of least squares. The London, Edinburgh and Dublin Philosophical Magazine and Journal of Science (4) 43, 194-201.

-- 1872b. On the law of facility of errors, and on the method of least squares. Memoirs of the Royal Astronomical Society (London) 39, 75-124. Harter, H. L. 1974-1976. The method of least squares and some alternatives. Parts I-VI. International Statistical Review 42, 147-174,235-264, 282; 43, 1-44, 125-190, 269-278; 44, 113-159. -- 1983a. The chronological annotated bibliography of order statistics, Vol. I: Pre-1950, 2nd ed. Columbus, Ohio: American Sciences Press. -- 1983b. Least squares. In Encyclopedia of statistical sciences, Vol. 4, N. L. Johnson, & S. Kotz, Eds., pp. 593-598. New York: Wiley. Hauber, C. F. 1830. Ober die Bestimmung der Genauigkeit der Beobachtungen. Zeitschriftfiir Physik und Mathematik 7, 286-314. Hettner, G. 1885. Unpublished letter to Kronecker. I p., May 23, 1885. Akademie der Wissenschaftenzu Berlin, Zentrales Archiv, NachlaB Dirichlet 65. Heyde, C. C., & Seneta, E. 1977. I. J. Bienaym~: Statistical theory anticipated. New York: Springer. Knobloch, E. 1992. Historical aspects of the foundations of error theory. In The space of mathematics, J. Echeverria, A. Ibarra, & Th. Mormann, Eds., pp. 253-279. Berlin/New York: de Gruyter. Kummell, Ch. H. 1882. On the composition of errors from single causes of error. Astronomische Nachrichten 103, 177-206. Kummer, E. E. 1860. Ged/ichtnisrede auf Gustav Peter Lejeune Dirichlet. Abhandlungen der K6niglich Preuflischen Akademie der Wissenschaften. 136. All references are to the reprint in [Dirichlet, Werke II, pp. 311-344]. Laplace, P. S. 1774. Mrmoire sur la probabilit6 des causes par les evrnrments. Mdmoires de l'Acad~mie royale des sciences de Paris (savants Otrangers) 6, 621-656. All references are to the reprint in Oeuvres completes de Laplace VIII, Acadrmie des sciences, Ed., pp. 27-65. Paris: Gauthier-Vil- lars, 1891.

-- 1785. Mrmoire sur les approximations des formules qui sont fonctions de trrs grands nombres. M~moires de l'Acad(mie royale des sciences de Paris, annre 1782, 1-88. All references are to the reprint in Oeuvres compldtes de Laplace X, Acadrmie des sciences, Ed., pp. 209-291. Paris: Gauthier-ViUars, 1894. -- 1820/1886. Th~orie analytique des probabilit~s. 3rd ed. with supplements. Paris: Courcier. (lst ed. 1812; 2nd ed. 1814). All references are to the reprint of the 3rd edition in Oeuvres complOtes de Laplace VII, Acadrmie des sciences, Ed. Paris: Gauthier-Villars, 1886. Laurent, H. 1873. Trait~ du calcul des probabilit~s. Paris: Gauthier-Villars. Liapounoff, A. 1900. Sur une proposition de la throrie des probabilitrs. Bulletin de l'Acad~mie lmp~riale des Sciences de St.-P~t~rsbourg (5)13, 359-386. 62 HANS FISCHER HM 21

-- 1901. Nouvelle forme du theor~me sur la limite de probabilit6. M~moires de l'Acad~mie lmp~riale des Sciences de St.-P~t~rsbourg VIII e S~rie, Classe Physico-Math~matique 12, 1-24. Liouville, J. 1846. Sur l'int6grale f; e-Xx n dx. Journal de Math~matiques Pures et Appliqu~es (I) U, 464f. Lipschitz, R. 1890. Sur la combinaison des observations. Comptes Rendus Hebdomaires des S~ances de l'Acaddmie des Sciences lU, 163-166. Lorey, W. 1916. Das Studium der Mathematik an den deutschen Universitiiten seit Anfang des 19. Jahrhunderts. Leipzig/Berlin: Teubner. Meyer, A. 1856. Unpublished letter to Dirichlet, 2 pp., July 12, 1856; added a handwritten essay TheorOrne de Bernoulli dans le calcul des probabilit~s, ~tendu au binome des factorielles, demontr~ par A. Meyer, 14 pp. Staatsbibliothek zu Berlin, PreuBischer Kulturbesitz, NachlaB Dirichlet. Meyer, G. F. 1871. Vorlesungen iiber die Theorie der bestimmten lntegrale zwischen reellen Grenzen. Leipzig: Teubner. Neuenschwander, E. 1984. Joseph Liouville (1809-1882): Correspondance in6dite et documents bio- graphiques provenant de diff6rentes archives parisiennes. Bollettino di Storia delle Science Mathe- matiche 4, 55-132. Poisson, S. D. 1824. Sur la probabilit6 des r6sultats moyens des observations. Connaissance des Tems pour l'an 1827, 273-302. German translation: [Poisson 1841, pp. 477-507].

-- 1829. Suite du m6moire sur la probabilit6 des r6sultats moyens des observations, ins6r6 dans la connaissance des tems de l'ann6e 1827. Connaissance des Terns pour l'an 1832, 3-22. German translation: [Poisson 1841, pp. 508-527]. -- 1837. Recherches sur la probabilit~ des jugements en matiOre criminelle et en matidre civile, precedes des rdgles g~n~ral~s du calcul des probabilit~s. Paris: Bachelier. --1841. Lehrbuch der Wahrscheinlichkeitsrechnung und deren wichtigsten Anwendungen. Deutsch iibersetzt und mit den nOthigen Zusdtzen versehen yon Dr. C. H. Schnuse. Braunschweig: Meyer. The main part of this book is the German translation of [Poisson 1837] by Christian Heinrich Schnuse. Schneider, I. 1974. Clausius' erste Anwendung der Wahrscheinlichkeitsrechnung im Rahmen der atmosph~rischen Lichtstreuung. Archive for History of Exact Sciences 14, (1974/1975), 143-158.

-- 1975. Rudolph Clausius' Beitrag zur Einfiihrung wahrscheinlichkeitstheoretischer Methoden in die Physik der Gase nach 1856. Archive for History of Exact Sciences 14 (1974/1975), 237-261.

-- 1987. Laplace and thereafter: The status of probability calculus in the nineteenth century. In The Probabilistic Revolution, Vol. 1, L. Kriiger, L. J. Daston, & M. Heidelberg, Eds., pp. 191-214. Cambridge, MA/London: MIT Press. Schubring, G. 1981. Mathematics and teacher training: Plans for a polytechnic in Berlin. Historical Studies in the Physical Sciences 12, 161-194. -- 1986. The three parts of the Dirichlet Nachlass. Historia Mathematica 13~ 52-56. Seidel, L. 1862. Ueber eine Anwendung der Wahrscheinlichkeits-Rechnung auf die Schwankungen in den Durchsichtigkeitsverhaltnissen der Luft. Sitzungsberichte der K6niglich Baierischen Akade- mie der Wissenschaften 2, 320-350.

-- 1865. Ueber den numerischen Zusammenhang, welcher zwischen der H~iufigkeit der Typhus- Erkrankungen und dem Stande des Grundwassers ... hervorgetreten ist. Zeitschrift far Biologie 1, 221-236.

-- 1866. Vergleichung der Schwankung der Regenmengen mit den Schwankungen in der H/iufigkeit des Typhus. Zeitschriftfiir Biologie 2, 145-177.

-- 1876. Ueber die Probabilit~iten solcher Ereignisse, welche nur selten vorkommen, obgleich sie unbeschr~nkt oft m6glich sind. Sitzungsberichte der K6niglich Baierischen Akademie der Wissenschaften, Mathematisch-Physikalische Klasse 6, 44-50. HM 21 DIRICHLET'S WORK IN PROBABILITY 63

Sheynin, O. B. 1973. Finite random sums (a historical essay). Archive for History of Exact Sciences 9, 275-305. -- 1976. P. S. Laplace's work on probability. Archive for History of Exact Sciences 16, 137-187. -- 1977. Laplace's theory of errors. Archive for History of Exact Sciences 17, 1-61. -- 1979. C. F. Gauss and the theory of errors. Archive for the History of Exact Sciences 20, 21-72.

-- 1982. On the history of medical statistics. Archive for History of Exact Sciences 26, 241-286. Stigler, S. M. 1973a. Laplace, Fisher, and the discovery of the concept of suftiency. Biometrika 60, 439-445.

-- 1973b. Simon Newcomb, Percy Daniell, and the history of robust estimation, 1885-1920. Journal of the American Statistical Association 68, 872-879.

-- 1986. History of statistics. Cambridge, MA/London: Belknap. Tannery, J. 1910. Correspondance entre Lejeune-Dirichlet et Liouville. Paris: Gauthier-Villars. Toepell, M. 1992. Mathematiker und Mathematik an der Universitiit Miinchen, 500 Jahre Lehre und Forschung (Habilitationsschrift). Mfinchen, in press.