Paris: Courcier, 1812-25.
First edition, very rare, accompanied by all four supplements (the fourth supplement is especially rare), of “the most influential book on probability and statistics ever written” (Anders Hald), which John Herschel called “the ne plus ultra of mathematical skill and power” (Essays from the Edinburgh and Quarterly Reviews (1857), p. 382). “In the Théorie Laplace gave a new level of mathematical foundation and development both to probability theory and to mathematical statistics. … [It] emerged from a long series of slow processes and once established, loomed over the landscape for a century or more.” (Stephen Stigler, Landmark Writings in Western Mathematics, pp. 329-30). “It was the first full–scale study completely devoted to a new specialty, … [and came] to have the same sort of relation to the later development of probability that, for example, Newton’s Principia Mathematica had to the later science of mechanics” (DSB). “The Théorie analytique des Probabilités contains, besides an introduction, two books and four supplements: Book I. Du calcul des Fonctions génératrices; Book II. Théorie générale des Probabilités; first supplement, composed in 1816. Sur l’Application du calcul des Probabilités à la philosophie naturelle; second supplement, composed in 1817. Sur l’Application du calcul des Probabilités aux opérations géodésiques, et sur la Probabilité des résultats déduits d’un grand nombre d’observations; third supplement, composed in 1819. Application des formules géodésiques de Probabilité à la Méridienne de France; [fourth supplement, composed in 1825]” (Hoefer, Nouvelle Biographie Générale, 547). ABPC/RBH list only one copy of this first edition sold at auction since Honeyman, and that copy lacked the fourth supplement (Sotheby’s, 11 December 2007, $25,000). The Honeyman copy had only the first supplement.
“The first edition contains a brief introduction that was eliminated in the second and third in favor of the Essai philosophique. It is worth notice, nevertheless, for the interest that Laplace claimed for the work in bringing it before the public:
“I am particularly concerned to determine the probability of causes and results, as exhibited in events that occur in large numbers, and to investigate the laws according to which that probability approaches a limit in proportion to the repetition of events. That investigation deserves the attention of mathematicians because of the analysis required. It is primarily there that the approximation of formulas that are functions of large numbers has its most important applications. The investigation will benefit observers in identifying the mean to be chosen among the results of their observations and the probability of the errors still to be apprehended. Lastly, the investigation is one that deserves the attention of philosophers in showing how in the final analysis there is a regularity underlying the very things that seem to us to pertain entirely to chance, and in unveiling the hidden but constant causes on which that regularity depends. It is on the regularity of the mean outcomes of events taken in large numbers that various institutions depend, such as annuities, tontines, and insurance policies. Questions about those subjects, as well as about inoculation with vaccine and decisions of electoral assemblies, present no further difficulty in the light of my theory. I limit myself here to resolving the most general of them, but the importance of these concerns in civil life, the moral considerations that complicate them, and the voluminous data that they presuppose require a separate work.” Laplace never wrote that separate work, although the thought of it may well have been what led him to expand his old lecture for the École Normale into the Essai philosophique two years later.
“The general subtitle of Book I is “Calcul des fonctions génératrices.” It consists almost entirely of a republication, with some revision, of the two cardinal mathematical investigations of the early 1780’s. The “Mémoire sur les suites”, on generating functions themselves, has now become the basis of its first part, and that on the approximation by definite integrals of formulas containing very large numbers the basis of its second part. The introduction reiterates what Laplace had first stated in the memoir of the preceding year, that the two theories are branches of a single calculus, the one concerned with solving the difference equations in which problems of chance events are formulated, the other with evaluating the expressions that result when events are repeated many times.
“Laplace says in the introduction that he is now presenting these theories in a more general manner than he had done thirty years before. The chief difference in principle is that the new calculus is held to have emerged along the main line of evolution of the analytical treatment of exponential quantities. In an opening historical chapter, Laplace traces the lineage back through the work of Lagrange, Leibniz, Newton, and Wallis to Descartes’s invention of numerical indices for denoting the operations of squaring, cubing, and raising magnitudes to higher integral powers. The principal difference in practice between the two earlier memoirs and their revision in Book I is that Laplace omitted certain passages that had come to appear extraneous in the interval and gave greater prominence to others that now appeared strategic. The most important omission is three articles from the “Mémoire sur les suites” on the solution of second-order partial differential equations, which were important for problems of physics but not for theory of games of chance. On the other hand, Laplace gave greater emphasis than in the earlier memoirs to the passage from the finite to the infinitesimal and also from real to imaginary quantity. He now develops as an argument what he had merely asserted in the immediately preceding memoir on definite integrals, namely that rigor is not impaired by the necessity of neglecting in appropriate circumstances infinitesimal quantities relative to finite quantities and higher-order infinitesimals relative to those of lower order. It is in support of this proposition that he brings in the solution to the problem of vibrating strings. It affords a convincing example that discontinuous solutions of partial differential equations are possible under conditions that he specifies. He attached even greater importance to the fertility that he increasingly found in the process of passing from real to imaginary quantities and discussed those procedures in the transitional section between the first and second parts …
“In Book II, subtitled “Théorie générale des probabilités,” Laplace turns from the calculus to probability itself. Indeed, it is fair to say that he constituted the subject, drawing together the main types of problems from the theory of chance already treated by many mathematicians, including himself, in a somewhat haphazard manner, and re–handling them in tandem with problems from the new areas of application in philosophy of science, astronomy, geodesy, instrumentation, error, population, and the procedures of judicial panels and electoral bodies. Unlike the two parts of Book I …, Book II is more than a republication of earlier memoirs with minor and incidental revision. Material from earlier work is incorporated in it, to be sure, but it is revised mathematically and fortified with new material … The first chapter gives the general principles and opens with the famous characterization of probability as a branch of knowledge both required by the limitation of the human intelligence and serving to repair its deficiencies in part. The subject is relative, therefore, both to our knowledge and to our ignorance of the laws of a determined universe. After stating the definition of probability itself and the rule for multiplying the probabilities of independent events, Laplace includes as the third basic principle a verbal statement of his theorem on the probability of cause, still without mentioning Bayes. Thereupon, he takes the example of the unsuspected asymmetries of a coin to consider the effect of unequal prior probabilities mistakenly taken for equal. Finally, he distinguishes between mathematical and moral expectation …
“The actual problems discussed in the early chapters also consist in part of examples reworked from … early papers on theory of chance. Laplace solved them by using generating functions and arranged them, not for their own sake, but to illustrate the typology of problems in probability at large, interspersing new subject matter where the methodology made it appropriate. Chapter 2, which is concerned with the probability of compound events composed of simple events of known probability, is much the most considerable, occupying about one-quarter of Book II. In a discussion of the old problem of determining the probability that all n numbers in a lottery will turn up at least once in i draws when r slips are chosen on each draw, he adduced the case of the French national lottery, composed of ninety numbers drawn five at a time. Laplace went on to other classic problems in direct probability: of odds and evens in extracting balls from an urn, of extracting given numbers of balls of a particular color from mixtures in several urns, of order and sequence in the retrieval of numbered balls, of the division of stakes and of the ruin or victory of one of a pair of gamblers in standard games …
“We shall not follow Laplace into yet another calculation (his last on this phenomenon) that the orbital arrangement of the planets results from a single cause and that the comets escape its compass, nor from that back to a variation on the original problem, in which any number of balls in the urn may be designated by the same integer, nor even into his derivation by the same method that the sum of the errors in a series of observations will be contained within given limits. Suffice it to indicate the sequence and the virtuosity it bespeaks. But we must notice his earliest venture, this late in life, into judicial probability. Imagine a number i of points along a straight line, at each of which an ordinate is erected. The first ordinate must be at least equal to the second, the second at least equal to the third, and so on; and the sum of these i ordinates is s. The problem is to determine, among all the values that each ordinate can assume, the mean value … Suppose now, however, that an event is produced by one of the i causes A, B, C, …, and that a panel of judges is to reach a verdict on which of the causes was responsible. Each member of the panel might write on a ballot the various letters in the order that appeared most probable to him … If all members of the tribunal follow that procedure, and the values for each cause are summed, the largest sum will point to the most probable cause in the view of that panel. Laplace hastened to add that since electors, unlike judges, are not constrained to decide for or against a candidate but impute to him all degrees of merit in making their choices, the above procedure may not be applied to elections. He proceeded to outline a probabilistic scheme for a preferential ballot that would produce the most mathematically exact expression of electoral will. Unfortunately, however, electors would not in fact make their choices on the basis of merit but would rank lowest the candidate who presented the greatest threat to their own man. In practice, therefore, preferential ballots favor mediocrity and had been abandoned wherever tried.
“The third chapter deals with limits, in the sense in which the idea figures in the frequency definitions of the discipline of probability that have developed out of it. In his own terminology, his concern was with the laws of probability that result from the indefinite multiplication of events. No single passage is as clear and definite as his derivation of the central limit theorem in the memoir on approximating the values of formulas containing large numbers that had brought him back to probability several years before, but the examples he adduces are much more various … Laplace points out that two sorts of approximations are involved. The first is relative to the limits of the a priori probability (facilité) of the event A, and the second to the probability that the ratio of the occurrences of A to the total number of events will be contained within certain limits. As the events are repeated, the latter probability increases so long as the limits remain the same. On the other hand, so long as the probability remains the same, the limits grow closer together. When the number of events reaches infinity, the limits converge in a point and the probability becomes a certainty. Just as he had done in his earliest general memoir on probability, Laplace turned to birth records to illustrate how the ratio of boys to girls gives figures from experience for prior probabilities, or facilitiés.
“The latter part of the discussion contains another of the many passages scattered throughout his writings that have led modern readers to feel that Laplace must somehow have had an inkling (or perhaps a repressed belief) that random processes occur in nature itself and not merely as a function of our ignorance. The mathematical occasion here is the use … of partial differential equations in solving certain limit problems. The concluding example turns on a ring of urns, one containing only white and another only black balls, and the rest mixtures of very different proportions. Laplace proves that if a ball is drawn from any urn and placed in its neighbor, and if that urn is well shaken and a ball drawn from it and placed in the next further on, and so on an indefinite number of times around the circle, the ratio of white to black balls in each urn will eventually be the same as the ratio of white to black balls in all of them. But what Laplace really thought to show by such examples was the tendency of constant forces in nature to bring order into the most chaotic systems.
“By comparison to the early probabilistic memoirs of the 1770’s, and 1780’s, the fourth chapter on probability of error certainly represents the most significant development in the subject as a whole. It contains, of course, a derivation of the least–square law for taking the mean in a series of observations … The chapter opens with determinations that the sum of errors of a large number of errors – equivalent to the distribution of sums of random variables – will be contained within given limits, on the assumption of a known and equipossible law of errors. It continues with the probability that the sum of the errors (again amounting to random variables), all considered as positive, and of their squares and cubes, will be contained within given limits. This is equivalent to considering the distribution of the sum of moduli. That leads to the problem of correcting values known approximately by the results of a great number of observations, which is to say by least squares, first in the case of a single element and then of two or more elements. Mathematically, this involves a discussion of linearized equations with one unknown and with two unknowns, respectively. Laplace includes instructions on application of the analysis to the correction of astronomical data by comparison of the values given in a number of tables. He then considers the case in which the probability of positive and negative error is unequal, and derives the distribution that results. The next-to-last section deals with the statistical prediction of error and methods of allowing for it on the basis of experience. At least, that seems to be a fair statement of what Laplace had in mind in speaking of “the mean result of observations large in number and not yet made,” on the basis of the mean determined for past observations of which the respective departures from the mean are known. The chapter closes with a historical sketch of the methods used by astronomers to minimize error up to the formulation of least squares, in which account Laplace renders Legendre and Gauss each his due.
“In the fifth chapter, Laplace discussed the application of probability to the investigation of phenomena themselves and of their causes, wherein it might serve to establish the physical significance of data amid all the complexities of the world. The approach offers practical instances of his sense of the relativity of the subject to knowledge and to ignorance, to science and to nature. In the analysis of error, it is the phenomena that are considered certain, whereas here the existence and boundaries of the phenomena themselves are the object of the calculation. The main example is the daily variation of the barometer, which long and frequent observation shows to be normally at its highest at 9:00 a.m. and lowest at 4:00 p.m., after which it rises to a lower peak at 11:00 p.m. and sinks until 4:00 p.m. Laplace calculated the probability that this diurnal pattern is due to some regular cause, namely the action of the sun, and determined its mean extent. He then raised a further question which, for lack of data, he could not resolve mathematically here, but which is interesting since it came to occupy the very last calculation of his life. For in theory, atmospheric tides would constitute a second and independent cause contributing to the daily variations of barometric pressure … The detection of such a small effect was not yet possible, although Laplace expressed his confidence that observations would one day become sufficiently extensive and precise to permit its isolation.
“In short, it was calculations of this sort that Laplace had in mind when he claimed, as he here remarked again, that on the cosmic scale probability had permitted him to identify the great inequalities of Jupiter and Saturn, even as it had enabled him to detect the minuscule deviation from the vertical of a body falling toward a rotating earth. He even had hopes for its calculus in physiological investigations, imagining that application to a large number of observations might suffice to determine whether electrical or magnetic charges have detectable effects upon the nervous system, and whether animal magnetism reflects reality or suggestibility. In general – he felt confident – the same analysis could in principle be applied to medical and economic questions, and even to problems of morality, for the operations of causes many times repeated are as regular in those domains as in physics. Laplace had no examples to propose, however, and closed the chapter with a mathematical problem extraneous in subject matter but not in methodology. The problem had been imagined by Buffon in order to show the applicability of geometry to probability. It consists of tossing a needle onto a grid of parallel lines, and then onto a grid ruled in rectangles, of which the optimal dimensions relative to the length of the needle constitute the problem. Laplace adapted it to a probabilistic, or in this instance a statistical, method for approximating to the value of π. It would be possible, he points out, although not mathematically inviting, to apply a similar approach to the rectification of curves and squaring of surfaces in general.
“Chapter 6, “On the Probability of Causes and Future Events, Drawn From Observed Events,” is in effect concerned with problems of statistical inference. In practice, the material represents a reworking of his early memoir on probability of cause and of the application of inverse probability by means of approximations of definite integrals to calculations involving births of boys and girls and also to population problems at large. He now had figures for Naples as well as for Paris and London. In calculating the probable error in estimates of the population of France based on the available samples, he made use of the partial census which, at his request, the government had instituted in 1801.
“The next, very brief, chapter also starts with old material, to which he gave a new turn. He recurs to his own discovery of the effect of inequalities in the prior probabilities that are mistakenly supposed to be equal. He always attached great importance to that finding, so much so that he alluded to it in the opening chapter of Book II, where definitions were laid down. The significance was that it brought out the care that needed to be taken when mathematical calculations of probability were applied to physical events. In effect, there are no perfect symmetries in the real world, and what Laplace was saying was that allowance has to be made for slight deviations of parameters from assumed values in making predictions. He discussed the problem in the same connection in which he had started it, in relation to the unfairness to one of two players of unsuspected asymmetries in a coin to be tossed. That could be mitigated, he now suggested, by submitting the chance of asymmetry itself to calculation …
“In chapters 8, 9, and 10, all of them quite brief, Laplace took up life expectancy, annuities, insurance, and moral expectation (or prudence). We do not know where he obtained his information, but it is reasonable to suppose that some of it must have been derived from occasional service on commissions appointed to review writings in this area and various actuarial schemes submitted to the government. The procès–verbaux of the Academy in its last years and of the Institute contain record of his having thus been called on from time to time. Moreover, the “Notice sur les probabilités” (1800) was published as a rationale of the application that it was legitimate to make to tables of mortality. This piece was an expansion of his École Normale lecture of 1795. It was then further expanded to become the first edition of the Essai philosophique and concludes with a summons to governments to license and regulate underwriters of insurance, annuities, and tontines, and to encourage investment in soundly managed associations. For an insurance industry and a literature did exist, although Laplace does not refer either to actual practice or to authorities. Comparison of his chapters with both would be required before a judgment could be made of what his contribution may have been.
“Mathematically, his model for calculations of the “mean duration,” both of life and marriage, is error theory. Given the tables of mortality covering a large population, a value for the mean length of life may be taken and the probability calculated that the mean life of a sample of stated size will fall within given limits. Calculation of life expectancy at any age follows directly. Laplace also gave a calculation for estimating the effect of smallpox on the death rate and of vaccination on life expectancy. The conclusion is that the eradication of smallpox would increase life expectancy by three years, if the growth in the population did not diminish the improvement by outrunning the food supply. Laplace did not give the data or provide numerical examples here, as he had done in his population studies. The succeeding chapter on annuities and tontines is equally abstract and gives expressions for the capital required to create annuities on one or several lives, for the investment needed to build an estate of given size, and for the advantages to be expected from participation in mutual benefit societies.
“In the tenth chapter, with which Laplace concluded the first edition, he softened the asperity he had once expressed about Daniel Bernoulli’s calculation of a value for moral expectation in distinguishing that notion from mathematical expectation. Bernoulli had proposed that prospective benefits, in practice financial ones, may be quantified as the quotient of their amount divided by the total worth of the beneficiary. Laplace now adopted that principle as a useful guide to conduct–without attributing it this time to Bernoulli … Laplace concluded that in the most mathematically advantageous games of chance the odds are always unfavorable over time, and that diversification is a prudent practice in the investment of wealth …
“Laplace extended the application of inverse probability to the analysis of criminal procedures in a supplement to the Théorie analytique, the first of four, composed in 1816. For this purpose a condemnation is an event, and the probability is required that it was caused by the guilt of the accused. As always in the probability of cause, prior probabilities had to be known or assumed, and Laplace again supposed that the probability of a truthful juror or judge lies between 1/2 and 1. In a panel of eight members of whom five suffice to convict, the probability of error came to 65/256. He felt that the English jury system with its requirement for unanimity weighted the odds too heavily against the security of society, but that the French criminal code was unjust to the accused. By one provision, if a defendant was found guilty by a majority of 7 to 5 in a court of first instance, it required a vote of 4 to 1 to overturn the verdict in a court of appeal, since a majority of only 3 to 2 there still left a plurality against him in the two courts taken together. That rule was as offensive to common sense as to common humanity …
“Between 1817 and 1819, Laplace investigated the application of probability to sharpening the precision of geodesic data and gathered these studies for publication as the second and third supplements of the Théorie analytique. In recent years scholars concerned with the history of statistics have been especially interested in these two pieces, which Laplace himself saw in the context of his theory of error. His intention was to improve on the method of least squares in the minimization of instrumental and observational error. That had also been his motivation in the opening articles of the First Supplement, where before taking up judicial probabilities, he further developed what he called the “most advantageous” method of combining equations of condition formed from observations of a single element, like those exemplified in his initial justification of the least–squares method. He then applied the method to estimating the probable error in Bouvard’s recent, highly refined calculations of the masses of Uranus, Saturn, and Jupiter.
“In the Second Supplement, Laplace turned to geodesy and compared the results of his method with the so–called method of situation of Boskovic. Stigler considers that this discussion contains the earliest instance of a comparison of two well–elaborated methods of estimation for a general population, in which the conditions that make one of them preferable are specified. He is particularly enthusiastic about the growing statistical sophistication, as he sees it, of Laplace’s later work in probability and argues that the analysis here is strikingly similar to that which led R. A. Fisher to the discovery of the concept of sufficiency in 1920. In the Third Supplement, Laplace reports the result of applying his method to the extension of the revolutionary Delambre-Méchain survey of the meridian from a base in Perpignan to Formentera in the Balearic Islands. The data were the discrepancies between 180° and the sums of the angles measured for each triangle. There were only twenty-six triangles in the Perpignan-Formentera chain, however, and Laplace preferred to estimate the law of error on the basis of all 700 triangles in the original survey. He could then calculate the probabilities of error of various magnitudes in the length of the meridian by the formulas already developed for his modified, or most advantageous, method of least squares. In a further article on a general method for cases involving several sources of error, Laplace obtained a paradigm equation for formulating equations of condition that relate values for observed elements to the error distributions involved. The equation served him in his later investigation of variations of the barometer as evidence for lunar atmospheric tides, the last he ever undertook” (DSB).
The main part of the fourth Supplement consists of the solution of problems which may be considered as generalisations of the Problem of Points. In the simplest problem a player A draws a ball from an urn containing white balls and black balls; after the ball has been drawn it is replaced. Then a second player B draws a ball from a second urn containing white balls and black balls; after the ball has been drawn it is replaced. If a player draws a white ball he counts a point; if he draws a black ball he counts nothing. Suppose that A wants x points, and B wants x' points to complete an assigned set; required the probabilities in favour of each player. This is then generalized by allowing balls of more than two kinds, and allowing drawing without replacement. These problems are solved by the method of generating functions. This is followed by a section headed Remarque sur les fonctions génératrices which refers to some problems solved in sections 8 and 10 of the Traité, cautioning that the solutions given there cannot be replied upon; correct solutions by means of generating functions are now provided.
“Pierre-Simon Laplace (1749-1827), sometimes called "the Newton of France," was a mathematician and astronomer who made many important contributions to the fields of mathematical astronomy and probability. Laplace's Mécanique Céleste (Celestial Mechanics) was the most important work in mathematical astronomy since Isaac Newton. His Théorie Analytique des Probabilités (Analytical Theory of Probability) influenced work on statistical probability for most of the nineteenth century. These two works alone guaranteed Laplace's place among the great scientists of the age” (DSB).
DSB XV, 367-376; Evans, First Editions of Epochal Achievements 12; Landmark Writings in Western Mathematics 24; Honeyman 1923.
[Théorie:] 4to (255 x 190 mm), pp. [vi], 464, [2, errata and blank]. Contemporary half calf. [Suppléments:] 4to (258 x 203 mm), pp. 34; 50; 36; , 28. Nineteenth-century half-cloth and marbled boards.