LAPLACE, Pierre Simon, Marquis de.

Théorie Analytique des Probabilités. Paris: Courcier, 1812. [With:] Supplément a la Théorie Analytique des Probabilités, Paris: Courcier, n.d. [1816]; Deuxième Supplément …, n.p., February 1818; Troisième Supplément …, Paris: Courcier, n.d. [1819]; Quatrième Supplément … Paris: Huzard-Courcier, n.d. [1825].

Paris: Courcier; Huzard-Courcier, 1812-[1825].

First edition, very rare — especially when accompanied, as here, by all four supplements, the fourth of which is of extreme rarity — of the most influential work on probability and mathematical statistics ever written. Pierre-Simon Laplace’s Théorie analytique des probabilités, published at Paris by Veuve Courcier in 1812, codified into a single mathematical framework the principal results of probability theory from Jakob Bernoulli’s Ars Conjectandi of 1713 through Laplace’s own four decades of research, and established the analytical methods — generating functions, characteristic functions, asymptotic approximations, the method of least squares, the central limit theorem — that governed the subject for the next century and beyond. John Herschel called it ‘the ne plus ultra of mathematical skill and power’. The DSB observes that it came to have the same relation to the later development of probability that Newton’s Principia had to the later science of mechanics. This is the finest copy we have seen. RBH lists no copy with all four supplements, and only two copies with the first three (Gonnelli 2024, €35,280; Sotheby’s 2007, $25,000); the Honeyman copy had only the first supplement.

The work emerged from a long series of memoirs. Laplace had been publishing on probability since the early 1770s, but the results were scattered across the publications of the Académie Royale des Sciences and the Institut de France, and no single treatise had brought them together. The Théorie analytique is that treatise, and Laplace’s introduction to the first edition — eliminated in the second and third editions in favour of the Essai philosophique — sets out the programme in his own terms: he is concerned, he writes, 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. The investigation, he adds, will benefit observers in identifying the mean to be chosen among the results of their observations and the probability of errors still to be apprehended, and it will interest philosophers in showing how there is a regularity underlying the very things that seem to pertain entirely to chance. He closes by noting that the same analysis can in principle be applied to annuities, tontines, insurance, inoculation with vaccine, and the decisions of electoral assemblies. He never wrote the separate work on civil applications he here promises, but the thought of it may have been what led him to expand his old École Normale lecture into the Essai philosophique sur les probabilités two years later — the non-mathematical companion that contains the celebrated passage on Laplace’s demon and the philosophical argument that probability is the rational response to ignorance, not a measure of it.

Book I, subtitled Calcul des fonctions génératrices, develops the algebraic machinery of the treatise. Generating functions — power series whose coefficients encode the probabilities of interest — transform combinatorial and probabilistic problems into problems of analysis, making them accessible to the methods of the calculus. The first part presents the theory of generating functions as a branch of the calculus of finite differences; the second develops the method of approximating by definite integrals the values of expressions containing very large numbers, the technique that yields the asymptotic formulas on which all the practical results of the later chapters depend. Laplace traces the historical lineage of these methods through Lagrange, Leibniz, Newton, and Wallis back to Descartes’s invention of numerical exponents, situating his own work as the natural continuation of two centuries of analytical development. The most important revision from his earlier memoirs of the 1780s is the omission of material on partial differential equations that was relevant to physics but not to games of chance, and a correspondingly greater emphasis on the passage from finite to infinitesimal quantities and from real to imaginary (complex) numbers — two processes that Laplace found increasingly fertile as his command of the subject deepened.

Book II, Théorie générale des probabilités, constitutes the subject. It is more than a republication of earlier memoirs: Laplace drew together the main types of problems from the theory of chance, already treated by many mathematicians in a somewhat haphazard manner, and re-handled them in tandem with problems from the new areas of application in the philosophy of science, astronomy, geodesy, instrumentation, error, population, and the procedures of judicial panels and electoral bodies. The first chapter lays down the general principles: the definition of probability, the rule for multiplying the probabilities of independent events, and a verbal statement of Laplace’s theorem on the probability of causes (inverse probability, now called Bayesian inference) — still without mentioning Bayes by name. Chapter 2, the most considerable, occupying about a quarter of Book II, treats the probability of compound events: lotteries (Laplace adduces the French national lottery, with its ninety numbers drawn five at a time), the extraction of balls from urns, order and sequence in the retrieval of numbered objects, the division of stakes, and the ruin or survival of a gambler — the classical problems of the subject, now solved systematically by generating functions and arranged not for their own sake but to illustrate a typology of problems in probability at large.

Chapter 3 concerns the laws that result from the indefinite multiplication of events. It contains Laplace’s most general statement of the central limit theorem: that when the number of independent observations increases, the distribution of their mean tends toward a Gaussian form regardless of the distribution of the individual observations, and that the limits within which the mean will fall with a given probability grow closer together as the number of observations grows. The theorem is the mathematical foundation of modern statistics; it explains why the bell curve appears ubiquitously in nature and why averaging works. Laplace illustrates the theorem from birth records — the ratio of boys to girls, taken from the registries of Paris, London, and Naples, provides empirical figures for the prior probabilities that the theory then subjects to analysis.

Chapter 4, on the probability of error, represents the most significant development in the subject as a whole compared to Laplace’s earlier memoirs. It derives the least-squares law for combining observations — the method that Legendre had published in 1805 and that Gauss had claimed priority for — first for the case of a single unknown element and then for two or more, together with the probability that the resulting estimate will fall within given limits. The chapter includes instructions on the application of the analysis to the correction of astronomical tables and closes with a historical sketch in which Laplace renders Legendre and Gauss each his due in the development of the method.

Chapter 5 applies probability to the investigation of phenomena themselves — to establishing the physical significance of data amid the complexities of the world. The principal example is the daily variation of the barometric pressure, which long observation shows to be normally highest at nine in the morning and lowest at four in the afternoon. Laplace calculates the probability that this diurnal pattern is due to a regular cause (the action of the sun) rather than to chance, and determines its mean extent. He raises the further question of atmospheric tides as a second contributing cause — a question he could not then resolve for lack of data, but which became the subject of the last calculation he ever performed. The chapter closes with an application of probability to Buffon’s needle problem — the probability that a needle tossed onto a grid of parallel lines will cross a line, a problem Laplace adapts to a statistical method for approximating the value of π. Chapter 6 reworks Laplace’s early memoir on the probability of causes, applying inverse probability and the method of definite-integral approximation to population statistics; he now has figures from the partial census which, at his request, the French government had conducted in 1801.

The brief seventh chapter returns to the effect of unequal prior probabilities mistakenly assumed to be equal — a finding Laplace always considered among his most important, since it demonstrated the care required when mathematical calculations are applied to physical events. There are, he insists, no perfect symmetries in the real world, and allowance must be made for slight deviations of parameters from assumed values. Chapters 8 through 10 treat life expectancy, annuities, tontines, insurance, the effect of vaccination on the death rate (Laplace concludes that the eradication of smallpox would increase life expectancy by three years, provided the resulting population growth did not outrun the food supply), and moral expectation — Daniel Bernoulli’s principle that the prospective benefit of a gain is not its absolute amount but its ratio to the total wealth of the beneficiary. Laplace adopts the principle as a guide to prudent conduct and concludes that in the most mathematically advantageous games of chance the odds are always unfavourable over time, and that diversification is a prudent practice in the investment of wealth.

The four supplements, issued between 1816 and 1825, extend the theory into applied territory. The first (1816) applies inverse probability to the analysis of criminal proceedings — Laplace calculates the probability that a conviction is erroneous, given assumptions about the truthfulness of jurors, and concludes that the English jury system with its requirement for unanimity weights the odds too heavily against the security of society, while the French criminal code with its plurality rule across two courts is unjust to the accused. The second (1818) turns to geodesy, comparing Laplace’s method of combining equations of condition with the method of Boskovic; Stigler considers this discussion the earliest instance of a rigorous comparison of two well-elaborated methods of estimation for a general population, strikingly similar to the analysis that led R. A. Fisher to the concept of sufficiency in 1920. The third (1819) applies the method to the extension of the Delambre–Méchain survey of the meridian from Perpignan to Formentera; Laplace estimates the law of error on the basis of all 700 triangles in the original survey and calculates the probabilities of error in the resulting length of the meridian. The fourth (1825), the rarest, generalises the Problem of Points using the method of generating functions and includes corrections to earlier results in the main text.

The bibliographical context of the supplements compounds the difficulty of assembling a complete set. Each was issued separately by Veuve Courcier and her successor Huzard-Courcier, in print runs of unknown but evidently small size, and the four were never gathered together as a single bound publication during the early nineteenth century. Copies that left the publishing house with the supplements bound in were assembled by the binder copy by copy from sheets purchased separately; copies sold without binding rarely retain the supplements at all, since these were slim pamphlets without the binding of their parent and easily separated from it. The fourth supplement of 1825, the rarest of the four, was issued only two years before Laplace’s death in 1827 and saw no further reprinting in the parent author’s lifetime; it is the supplement most often missing from copies offered for sale. RBH records no copy of the work with all four supplements; the two copies recently in commerce with the first three lack the fourth, and the Honeyman copy lacked the second, third, and fourth. The probability of all four reaching a single contemporary owner, surviving together through two centuries of binding, rebinding, and dispersal, and arriving intact in a single contemporary cloth was, in any literal sense, vanishingly small.

Laplace’s analytical apparatus passed quickly into general use after his death. Adolphe Quetelet, the Belgian astronomer and statistician who had visited Laplace in Paris in the 1820s, applied the central limit theorem to social phenomena in his Sur l’homme of 1835, founding what he called physique sociale — the systematic statistical study of human populations. Francis Galton in the 1880s adapted Laplace’s framework to the study of inheritance, defining regression to the mean and the correlation coefficient. Karl Pearson at University College London, working from Galton, formalised the chi-squared test in 1900 and established the moment-generating function (Laplace’s generating function under another name) as the standard apparatus of the new mathematical statistics. R. A. Fisher’s 1922 paper On the Mathematical Foundations of Theoretical Statistics introduced the concepts of likelihood and sufficiency that defined the modern subject; Stephen Stigler’s 1986 study of the history of statistics observes that Fisher’s argument on sufficiency closely parallels the comparison Laplace had drawn in the second supplement of 1818 between his own method of combining equations of condition and that of Boskovic. Through Quetelet, Galton, Pearson, and Fisher, the analytical framework of the Théorie analytique survived intact into the twentieth century, and it continues to underlie every confidence interval, hypothesis test, and regression analysis performed in any field that depends on quantitative inference. The hidden continuity is part of what Laplace meant when he wrote, in the introduction here suppressed, that probability furnishes a regularity to the very things that seem to belong to chance.

Pierre-Simon Laplace (1749–1827), the son of a cider-farmer in Normandy, arrived in Paris in 1769 and within a decade had established himself as the foremost mathematical astronomer in Europe. His Mécanique céleste (1799–1825) is the most important work in mathematical astronomy since the Principia; the Théorie analytique des probabilités holds the same position in its own field. Between them the two works secured his reputation as — in the phrase that has clung to him since his own century — the Newton of France. He served Napoleon as Minister of the Interior for six weeks in 1799 (Napoleon, dismissing him, observed that he had brought the spirit of the infinitely small into the work of government); he was elected to the Académie française in 1816, made a count of the Empire under Napoleon in 1806, and elevated to the rank of marquis at the Bourbon Restoration in 1817. He died at Paris on 5 March 1827, on the eve of the issue of the fifth volume of the Mécanique céleste, which his son Charles-Émile saw through the press in his name. The 1812 Théorie analytique was the keystone of his late career and the work that consolidated mathematics as the apparatus of a quantitative science of human and natural events.

Provenance: Barthélémy Aoust (1814–1885), French mathematician, professor in the Faculty of Sciences at the University of Marseille from 1854 until his retirement. Aoust’s research interests centred on differential geometry and on the analysis of curves and surfaces — questions distant enough from probability theory for the present volume to have functioned in his library as a reference work rather than a working text, and the absence of marginalia in his hand suggests as much. His contemporary printed address label, pasted to the title page in the early 1850s when he established himself at Marseille, places this copy among the books he assembled for his teaching there and identifies it as part of his personal library.

References: DSB XV, 367–376 — Evans 12 — Grolier/Horblit 63a — Honeyman 1923 — Printing and the Mind of Man 252 — Stigler, ch. 24 in Grattan-Guinness (ed.), Landmark Writings in Western Mathematics 1640–1940, 2005 — Hald, A History of Mathematical Statistics from 1750 to 1930, 1998 — Stigler, The History of Statistics: The Measurement of Uncertainty before 1900 (Harvard, 1986).

Five parts bound in one vol., 4to (253 × 194 mm), pp. [vi], 464, [2: errata and blank]; 34; 50; 36; [2], 28. Contemporary green half-cloth over green marbled boards, spine with raised bands, compartments ruled in gilt and decorated in blind with a gilt lettering-piece, top edges yellow-speckled (spine very slightly rubbed, small paper label at tail). A fresh, well-margined copy, internally clean.

.

Item #6347

Price: $75,000.00