LAPLACE, Pierre Simon, Marquis de (1749-1827), one of the greatest mathematicians and physical astronomers who ever lived, was born at Beaumont-en-Auge in Normandy, March 28, 1749. His early years have remained in the obscurity with which poverty and some ignoble shame of poverty combined to cover them. It is known, however, that his father was a small farmer, and that he owed his education to the interest excited by his lively parts in some persons of position. His first distinctions are said, singularly enough, to have been gained in theological controversy, but at an early age he became mathematical teacher in the military school of Beaumont, the classes of which he had attended as an extern. He was not more than eighteen when, armed with letters of recommendation, he approached D’Alembert, then at the height of his fame and influence, in the hope of finding a career in Paris. The letters remained unnoticed, but Laplace was not a man to be crushed by the first rebuff of fortune. He wrote to the great geometer a letter on the principles of mechanics, which evoked an immediate and enthusiastic response. “You,” said D’Alembert to him, “needed no introduction; you have recommended yourself; my support is your due.” He accordingly obtained for him an appointment as professor of mathematics in the Ecole Militaire of Paris, and continued to forward his interests with zeal and constancy.
The future of the young mathematician was now assured, and his scientific vocation finally determined. He had not yet completed his twenty-fourth year when he entered upon the course of discovery which has earned him the title of “the Newton of France.” Having, in his first published paper, shown his mastery of analysis, he immediately proceeded to apply the powerful instrument at his command to the great outstanding problems in the application of the law of gravitation to the celestial motions. Of.these the most conspicuous was offered by the opposite inequalities of Jupiter and Saturn, which the emulous efforts of Euler and Lagrange had failed to bring within the bounds of theory. The discordance of their results incited Laplace to a searching examination of the whole subject of planetary perturbations, and his maiden effort was rewarded with a discovery which constituted, when developed and completely demonstrated by his own further labours and those of his illustrious rival Lagrange, the most important advance made in physical astronomy since the time of Newton. In a paper read before the Academy of Sciences, February 10, 1773 (Mém. présentes par divers Savaηs, tom. vii., 1776), Laplace announced his celebrated conclusion of the invariability of planetary mean motions, carrying the proof as far as the cubes of the eccentricities and inclinations. This was the first and most important step in the establishment of the stability of the solar system. It was followed up by a series of profound investigations, in which Lagrange and Laplace alternately surpassed and supplemented each other in assigning limits of variation to the several elements of the planetary orbits. The analytical tournament closed with the communication to the Academy by Laplace, in 1787, of an entire group of remarkable discoveries. It would be difficult, in the whole range of scientific literature, to point to a memoir of equal brilliancy with that published (divided into three parts) in the volumes of the Academy for 1784, 1785, and 1786. The long-sought cause of the “great inequality” of Jupiter and Saturn was found in the near approach to commensurability of their mean motions; it was demonstrated in two elegant theorems (see Astronomy, vol. ii. p. 781), independently of any except the most general considerations as to mass, that the mutual action of the planets could never largely affect the eccentricities and inclinations of their orbits; and the singular peculiarities detected by him in the Jovian system were expressed in the so-called “laws of Laplace” (Astronomy, p. 810). He completed the theory of these bodies in a treatise contained amongst the Paris Memoirs for 1788 and 1789; and the tables computed by Delambre from the data there supplied served, by their striking superiority to those hitherto available, to mark the profit derived from the investigation by practical astronomy. The year 1787 was rendered further memorable by Laplace’s announcement, November 19 (Memoirs, 1786), of the dependence of lunar acceleration upon the secular changes in the eccentricity of the earth’s orbit. The last apparent anomaly, and the last threat of instability, thus disappeared from the solar system.
With these brilliant performances the first period of Laplace’s scientific career may be said to have closed. I f he made no more striking discoveries in celestial mechanics, it was rather their subject matter than his powers that failed. The general working of the great machine was now laid bare, and it needed a further advance of knowledge to render a fresh set of problems accessible to investigation. The time had come when the results obtained in the development and application of the law of gravitation by three generations of illustrious mathematicians might be collected in a single work, and presented from a single point of view. It was to this task that the second period of Laplace’s activity was devoted. As a monument of mathematical genius applied to the celestial revolutions the Mécanique Celeste ranks second only to the Principia of Newton.
The declared aim of the author was to offer a complete solution of the great mechanical problem presented by the solar system, and to bring theory to coincide so closely with observation that empirical equations should no longer find a place in astronomical tables. His success in both respects fell but little short of his lofty ideal. The first part of the work (2 vols. 4to, Paris, 1799) contains methods for calculating the movements of translation and rotation of the heavenly bodies, for determining their figures, and resolving tidal problems; the second, especially dedicated to the improvement of tables, exhibits in the third and fourth volumes (1802 and 1805) the application of these formulae; while a fifth volume, published in three instalments, 1823-25, comprises the results of Laplace’s latest researches, together with a valuable history of progress in each separate branch of his subject. In the delicate task of apportioning his own large share of merit, he certainly does not err on the side of modesty; but it would perhaps be as difficult to produce an instance of injustice, as of generosity, in his estimate of others. Far more serious blame attaches to his all but total suppression in the body of the work—and the fault pervades the whole of his writings—of the names of his predecessors and contemporaries. Theorems and formulae are appropriated wholesale without acknowledgment, and a production which may be described as the organized result of a century of patient toil presents itself to the world as the offspring of a single brain. The Mécanique Céleste is, even to those most conversant with analytical methods, by no means easy reading. The late Μ. Biot, who had the privilege of assisting in the correction of its proof sheets, remarks that it would have extended, had the demonstrations been fully developed, to eight or ten instead of five volumes; and he saw at times the author himself obliged to devote an hour’s labour to recovering the dropped links in the chain of reasoning covered by the recurring formula, “Il est aisé à voir.’’
The Exposition du Système du Monde (Paris, 1796) has been styled by Arago “the Mécanique Celeste disembarrassed of its analytical paraphernalia.” Not only the conclusions reached by geometers are stated, but the methods followed by them are indicated. The integuments, so to speak, of a popular dissertation clothe and conceal the skeleton of an analytical treatise. The style is lucid and masterly, and the summary of astronomical history with which it terminates has been reckoned amongst the masterpieces of the language. To this linguistic excellence the writer owed the place accorded to him in 1816 amongst the “forty” of the French Academy, of which institution he became president in the following year. The famous “nebular hypothesis” of Laplace makes its appearance in the Système du Monde. Although relegated to a note (vii,), and propounded “Avec la défiance que doit inspirer tout ce qui n’est point un résultat de l’observation ou du calcul,” it is plain, from the complacency with which he recurs to it at the lapse of above a quarter of a century, that he regarded the speculation with considerable interest. That it formed the starting-point, and has remained the model, of thought on the subject of planetary origin is due to the simplicity of its assumptions, and the clearness of the mechanical principles involved, rather than to any cogent evidence of its truth. It is curious that Laplace, while bestowing more attention than they deserved on the crude conjectures of Buffon, seems to have been unaware that he had been, to some extent, anticipated by Kant, who had put forward in 1755, in his Allgemeine Naturgeschichte i a true nebular cosmogony, though one in which the primitive reign of chaos was little likely to terminate.
The career of Laplace was one of scarcely interrupted prosperity. Admitted to the Academy of Sciences as an associate in 1773, he became a member in 1785, having, about a year previously, succeeded Bezout as examiner to the royal artillery. During a temporary access of revolutionary suspicion, he was removed from the commission of weights and measures; but the slight was quickly effaced by new honours. He was one of the first members, and became president, of the Bureau of Longitudes, took a prominent place at the Institute (founded in 1796), professed analysis at the École Normale, and aided in the organization of the decimal system. The publication of the Mécanique Céleste gained him world-wide celebrity, and his name appeared on the lists of all the principal scientific associations of Europe, including the Royal Society. But merely scientific distinctions by no means satisfied his ambition. He aspired to the rôle of a politician, and has left a memorable example of genius degraded to servility for the sake of a riband and a title. The ardour of his republican principles gave place, after the 18th Brumaire, to devotion towards the first consul, a sentiment promptly rewarded with the post of minister of the interior. His incapacity for affairs was, however, so flagrant that it became necessary to supersede him at the end of six weeks, when Lucien Bonaparte became his successor. “He brought into the administration,” according to the dictum of the future emperor, “the spirit of the infinitesimals.” His failure was consoled by elevation to the senate, of which body he became chancellor in September 1803. He was at the same time named grand officer of the Legion of Honour, and obtained in 1813 the same rank in the new order of Reunion. The title of count he had previously acquired on the creation of the empire. Nevertheless he cheerfully gave his voice in 1814 for the dethronement of his patron, and his “suppleness” merited a seat in the chamber of peers, and, in 1817, the dignity of a marquisate. The memory of these tergiversations is perpetuated in his writings. The first edition of the Système du Monde was inscribed to the Council of Five Hundred; to the third volume of the Mécanique Céleste (1802) was prefixed the declaration that, of all the truths contained in the work, that most precious to the author was the expression of his gratitude and devotion towards the “pacificator of Europe”; upon which noteworthy protestation the suppression, in the editions of the Théorie des Probabilités subsequent to the restoration, of the original dedication to the emperor formed a fitting commentary.
During the later years of his life, Laplace lived much at Arcueil, where he had a country-place adjoining that of his friend Berthollet. With his co-operation the Société d’Arcueil was formed, and he occasionally contributed to its Memoirs. In this peaceful retirement he pursued his studies with unabated ardour, and received with uniform courtesy distinguished visitors from all parts of the world. Here, too, he died, attended to the last by his physician Dr Majendie, and his mathematical coadjutor Bouvard, March 5, 1827, having nearly completed his seventy-eighth year. His last words were: “Ce que nous connaissons est peu de chose, ce que nous ignorons est immense.”
Although commonly believed to have held atheistical opinions, Laplace refrained from giving any direct expression to them in his writings. His character, notwithstanding the vanity and egotism by which it was disfigured, had an amiable and engaging side. Young men of science found in him an active benefactor. His relations with these “adopted children of his thought” possessed a singular charm of affectionate simplicity; their intellectual progress and material interests were objects of equal solicitude to him, and he demanded in return only diligence in the pursuit of knowledge. M. Biot relates that, when he himself was beginning his career, Laplace introduced him at the Institute for the purpose of explaining his supposed discovery of equations of mixed differences, and afterwards showed him, under a strict pledge of secrecy, the papers, then yellow with age, in which he had long before obtained the same results, but which he had laid aside with a view to future development. This instance of abnegation is the more worthy of record that it formed a marked exception to Laplace’s usual course. Between him and Legendre there was a feeling of “more than coldness,” owing to his appropriation, with scant acknowledgment, of the fruits of the other’s labours; and our celebrated countryman, Dr Thomas Young, counted himself, rightly or wrongly, amongst the number of those similarly aggrieved by him. With Lagrange, on the other hand, he always remained on the best of terms.
The extreme abstemiousness of his life, joined to a naturally good constitution, preserved Laplace from most of the infirmities incidental to old age. He was indeed obliged to use his eyes with precaution; but his powerful memory remained unimpaired, and it was not until within two years of his death that his health began to suffer from his severe application. He married a beautiful and amiable woman, and left a son, born in 1789, who succeeded to his title, and rose to the rank of general in the artillery.
It might be said that Laplace was a great mathematician by the original structure of his mind, and became a great discoverer through the sentiment which animated it. The regulated and persistent enthusiasm with which he regarded the system of nature was with him from first to last. It can be traced in his earliest essay, and it dictated the ravings of his final illness. By it his extraordinary analytical powers became strictly subordinated to physical investigations. To this lofty quality of mind he added a rare sagacity in perceiving analogies, and in detecting the new truths that lay concealed in his formulae, and a tenacity of mental grip, by which problems, once seized, were held fast, year after year, until they yielded up their solutions. In every branch of physical astronomy, accordingly, deep traces of his work are visible. “He would have completed the science of the skies,” Fourier remarks, “had the science been capable of completion.”
For a fuller account of the results achieved by him, the article Astronomy, vol. ii. p. 761, may be consulted; it need only be added that he first examined the conditions of stability of the system formed by Saturn's rings, pointed out the necessity for their rotation, and fixed for it a period (10 h 33 m ) differing by little more than a minute from that established by the observations of Herschel; that he detected the existence in the solar system of an invariable plane such that the sum of the products of the planetary masses by the projections upon it of the areas described by their radii vectores in a given time is always a maximum, made notable advances in the theory of astronomical refraction (J7ec. Cél., tom. iv. p. 258), and constructed formulae, agreeing remarkably with observation, for the barometrical determination of heights {Méc. Cél., tom. iv. p. 324). His removal of the considerable discrepancy between the actual and Newtonian velocities of sound, by taking into account the increase of elasticity due to the heat of compression, would alone have sufficed to illustrate a lesser name. Molecular physics also engaged a large share of his attention, and he announced in 1824 his purpose of treating the subject in a separate work. With Lavoisier he made an important series of experiments on specific heat (1782-84), in the course of which the “ice calorimeter” was discovered; and they contributed jointly to the Memoirs of the Academy (1781) a paper on the development of electricity by evaporation. Laplace was, moreover, the first to offer a complete analysis of capillary action based upon a definite hypothesis—that of forces ‘ ‘ sensible only at insensible distances”; and he made strenuous but unsuccessful efforts to explain the phenomena of light on an identical principle. It was a favourite idea of his that chemical affinity and capillary attraction would eventually be included under the same law, and it was perhaps as much because it threatened an inroad on a cherished generalization as because it seemed to him little capable of mathematical treatment that the undulatory theory of light was distasteful to him.
The investigation of the figure of equilibrium of a rotating fluid mass engaged the attention of Laplace during the greater part of his long life. His first memoir was communicated to the Academy in 1773, when he was only twenty-four years of age, his last in 1817, when he was sixty-eight. The results of his many papers on this subject—characterized by him as “un des points les plus intéressans du système du monde ”—are embodied in the Mécanique Céleste, and furnish one of the most remarkable proofs of his analytical genius. Maclaurin, Legendre, and D’Alembert had furnished partial solutions of the problem, confining their attention to the possible figures which would satisfy the conditions of equilibrium. Laplace treated the subject from the point of view of the gradual aggregation and cooling of a mass of matter, and demonstrated that the form which such a mass would ultimately assume must be an ellipsoid of revolution whose equator was determined by the primitive plane of maximum areas.
The honour of having brought almost to perfection the closely related problem of the attraction of spheroids must also be accorded to him. All the powers of analysis in the hands of its greatest masters replaced the old geometrical methods, and their superiority was soon evidenced by a succession of remarkable discoveries. Legendre, in 1783, extended Maclaurin’s theorem concerning ellipsoids of revolution to the case of any spheroid of revolution where the attracted point, instead of being limited to the axis or equator, occupied any position in space; and Laplace, in his treatise Théorie du Mouvement et de la Figure Elliptique des Planètes (pub lished in 1784), effected a still further generalization by proving, what had been suspected by Legendre, that the theorem was equally true for any confocal ellipsoids. Finally, in a celebrated memoir, Théorie des Attractions des Sphéroïdes et de la Figure des Planètes, published in 1785 among the Paris Memoirs for the year 1782, written, however, after the treatise of 1784, Laplace treated exhaustively the general problem of the attraction of any spheroid upon a particle situated outside or upon its surface.
The researches of Laplace and Legendre on the subject of attractions derive additional interest and importance from having introduced two powerful engines of analysis for the treatment of physical problems, Laplace’s Coefficients and the Potential Function. The expressions for the attraction of an ellipsoid involved integrations which presented insuperable difficulties; it was, therefore, with pardonable exultation that Laplace announced his discovery that the attracting force in any direction could be obtained by the direct process of differentiating a single function. He thereby translated the forces of nature into the language of analysis, and laid the foundations of the mathematical sciences of heat, electricity, and magnetism. This function, V, which received the name of potential from Green in 1828, and independently from Gauss in 1840, is defined as the sum of the masses of the molecules of the attracting body divided by their respective distances from the attracted point; or, in mathematical language—
V = ∕77Z pdxdydz
JJJ {(χ - α) 2 + (y - ∕3) 2 +(« - γ) ’} i ’
p being the density of the body at the point x, y, z; a, β, γ, the coordinates of the attracted point; and the limits of integration being determined by the form of the attracting mass. V is thus a function of a, β, γ, that is to say, depends for its value on the position of the point, and its several differentials with respect to these coordinates furnish the components of the attractive force. The integrations, however, could not in general be effected so as to express V in finite terms; but Laplace showed that V satisfied the partial differential equation d 2 V d 2 V d 2 V dd i+ dβ*½dy* ’
which is still known as Laplace’s equation. It is worthy of remark that it was not in this symmetrical form that the equation was discovered, but in the complicated shape which it assumes when expressed in polar coordinates :—
d (1 ~ M2 Cj 1 d 2 V d 2 (rV) dμ + l-μ*'dω i + dr* ’
where μ is substituted for cos θ. This differential equation forms the basis of all Laplace’s researches in attractions, and makes its appearance in every branch of physical science.
The expressions which are known as Laplace’s coefficients, a name first given to them by Dr Whewell, occupy a distinguished place in modern analysis. They were first introduced in their generality by Laplace in the memoir on attractions, 1785, above referred to, which is, to a great extent, reprinted in the third book of the Mécanique Céleste; but Legendre, in a celebrated paper entitled Recherches sur l’attraction des Sphéroïdes homogènes, printed in the tenth volume of the Divers Savans, 1783, had previously made use of them, and proved some of their properties, in the simplified form which they assume with one instead of two variables. They may be defined as follows. If two points in space are determined by their polar coordinates, r, θ, ω, and r f , θ ’, ω ’, T the reciprocal of the distance between them is expressed in terms of those coordinates by p' 2 _ 2 r r ’ Iμμ! + *√i-μ 1 √l - μ* cos (ω -«') j∙ + r' 2 l ' j where μ and μ ’ are written for cos θ and cos β ’ respectively. This expression may be expanded in a series of the form
⅜[r.+½+≤ + · · · p ⅞ + · ] ·
where P o , P 1 . . . P l ∙ are Laplace’s coefficients of th e orde rs 0, 1 . . . i. They are rational integral functions of μ, √1 - μ* cos ω, and √1 - μ* sin ω, a nd are precisely the same functions of μ ’, √1 - μ* cos ω' and √1 - √ 2 sin ω,; or, in other words, of the rectangular co-ordinates of the two points divided by their distances from the origin. The general coefficient P,∙ is of i dimensions in these quantities, and its maximum value can be shown to be unity, so that the above written series will be convergent if √ is greater than r. It can be easily proved that T satisfies Laplace’s differential equation—
. ' ∕ι s∖ f * τ I
∣ 0 '' rf "* l 1 il + ,⅛τ). o .
clμ l-μ 2 ' dω 2 dr* ’
and if for T we substitute the expanded form, we obtain the general differential equation of which Laplace’s coefficients are particular integrals :—
dμ 1 - μ i dω 2 i
Expressions which satisfy this equation are referred to as Laplace’s functions; they include as a particular case the coefficients, which are, as we have seen, certain definite functions of the spherical surface coordinates of the two points. If
X = μf + √1 - μ 2 √1 - μ 2 COS (ω - ω'), the coefficients become functions of x alone, and it was in this form that Legendre first introduced them. One of the fundamental properties of Laplace’s functions, known as Laplace’s theorem, is that, if Y,∙ and Z∕ be two such functions, i and i' being whole numbers and not identical, then y rl ^∕" 2a γ i z i '⅛dω=o.
Again, if Y∕ is the same function of f and that Y< is of μ and ω, we have the important relation
2i+l∕- 1 ∕∙ 2π vι , , ,
1 i = —-— ∕ ∕ ΥiPidμdω .
4ιry-l√ 0
But the property on which their utility in physical researches chiefly depends is that every function of the coordinates of a point on a sphere can be expanded in a series of Laplace’s functions.
In the figure of the earth, the theory of attractions, and the sciences of electricity and magnetism this powerful calculus o<Scupies a prominent place. Gauss in particular has employed it in the calculation of the magnetic potential of the earth, and it has recently received new light from Professor Clerk Maxwell’s interpretation of harmonics with reference to poles on the sphere.
Laplace, always profound rather than elegant, nowhere displays the massiveness of his genius so conspicuously as in the theory of probabilities. The science which Pascal and Fermat had initiated he brought very nearly to perfection; but the demonstrations are so involved, and the omissions in the chain of reasoning so frequent, that the Théorie Analytique is to the best mathematicians a work requiring the most arduous study. The theory of probabilities, which Laplace describes as common sense expressed in mathematical language, first attracted his attention from its importance in physics ami astronomy; and he applies his theory, not only to the ordinary problems of chances, but also to the inquiry into the causes of phenomena, vital statistics, and future events.
The device known as the method of least squares, for reducing numerous equations of condition to the number of unknown quantities to be determined, had been adopted as a practically convenient rule by Gauss and Legendre; but Laplace first treated it as a problem in probabilities, and proved by an intricate and difficult course of reasoning that it was also the most advantageous, the mean of the probabilities of error in the determination of the elements being thereby reduced to a minimum.
The method of generating functions, the foundation of his theory of probabilities, Laplace published in 1779; and the first part of his Théorie Analytique is devoted to the exposition of its principles, which in their simplest form consist in treating the successive values of any function as the coefficients in the expansion of another function with reference to a different variable. The latter is therefore called the generating function of the former. A direct and an inverse calculus is thus created, the object of the former being to determine the coefficients from the generating function, of the latter to discover the generating function from the coefficients. The one is a problem of interpolation, the other a step towards the solution of an equation in finite differences. The method, how ever, is now obsolete from the more extended facilities afforded by the calculus of operations.
The first formal proof of Lagrange’s theorem for the development in a series of an implicit function was furnished by Laplace, who gave to it an extended generality. He also showed that every equation of an even degree must have at least one real quadratic factor, reduced the solution of linear differential equations to definite integrals, and furnished an elegant method by which the linear partial differential equation of the second order might be solved. He was also the first to consider the difficult problems involved in equations of mixed differences, and to prove that an equation in finite differences of the first degree and the second order might always be converted into a continued fraction.
In 1842, the works of Lapl·ice being nearly out of print, his widow was about to sell a farm in order to procure funds for a new impression, when the Government of Louis Philippe took the matter in hand. A grant of 40,000 francs having been obtained from the chamber, a national edition was issued in seven 4to vols., bearing the title Oeuvres de Laplace, 1843-47. The Mécanique Céleste with its four supplements occupies the first 5 vols., the 6th contains the Système du Monde, and the 7th the Th. des Probabilités, to which the more popular Essai Philosophique forms an introduction. Of the four supplements added by the author, 1816-25, he tells us that the problems in the last were contributed by his son. An enumeration of Laplace’s memoirs and papers (about one hundred in number) is rendered superfluous by their embodiment in his principal works. The Th. des Prob, was first published in 1812, the Essai in 1814; and both works as well as the Système dzι Monde went through repeated editions. Laplace’s first separate work, Théorie du Mouvement et de la Figure Elliptique des Planètes, 1784, was published at the expense of President Bochard de Saron. The Précis de V Histoire de l ’ Astronomie, 1821, formed the fifth book of the 5th edition of the Système du Monde. An English translation, with copious elucidatory notes, of the first 4 vols, of the Mécanique Céleste, by Dr Bowditch, was published at Boston, U.S., 1829-39, in 4 vols. 4to; a compendium of certain portions of the same work by Mrs Somerville appeared in 1831, and a German version of the first 2 vols, by Burckhardt at Berlin in 1801. English translations of the Système du Monde by Mr Pond and Mr Harte were published, the first in 1809, the second in 1830. An edition entitled Les Oeuvres Complètes de Laplace, 1878, &c., which is to include the whole of his memoirs, is now in course of publication under the auspices of the Academy of Sciences. The four 4to vols, which have already appeared comprise the first ten books of the Mécanique Céleste.
Scanty notices of Laplace’s life will be found in Fourier’s Éloge, June 15, 1829, in the funeral oration of Poisson, and Arago’s Report, 1842, translated amongst his Biographies by Admiral Smyth and Mr Grant. His astronomical work is treated of in Gautier’s Problème des trois Corps and Grant’s Hist, of Astronomy. For Laplace’s functions see Dr E. Heine, Handbuch der Kugelfunctionen, Berlin, 1861; John H. Pratt, A Treatise on Attractions, 1865; Todhunter’s Elementary Treatise on Laplace's Functions, 1875, and History of the Mathematical Theories of Attraction, 1873; N. Μ. Ferrers’s Elementary Treatise on Spherical Harmonics, 1877; and L. Sclιlafli, Die zwei Heine’sehen Kugelfunctionem, 1881. Consult also Thomson and Tait, Treatise on Natural Philosophy, 1879, p. 141; Clerk Maxwell, Treatise on Electricity, chap, ix; Professor Niven in Phil. Trans., 1879, p. 379; Dirichlet in Crelle, xvi. p. 35; and Jacobi, vol. ii. p. 223, xxvi. p. 82. Some of Laplace’s results in the theory of probabilities are simplified in Lacroix’s Traité élémentaire du Calcul des Probabilités and De Morgan’s Essay, published in Lardner’s Cabinet Cyclopaedia. For the history of the subject see A History of the Mathematical Theory of Probability, by Isaac Todhunter, 1865. (A. Μ. C.)
