Pierre-Simon Laplace

Sir Isaac Newton had published his Philosophiae Naturalis Principia Mathematica in 1687 in which he gave a derivation of Kepler's laws, which describe the motion of the planets, from his laws of motion and his law of universal gravitation. However, though Newton had privately developed the methods of calculus, all his published work used cumbersome geometric reasoning, unsuitable to account for the more subtle higher-order effects of interactions between the planets. Newton himself had doubted the possibility of a mathematical solution to the whole, even concluding that periodic Divine intervention was necessary to guarantee the stability of the Solar System. Dispensing with the hypothesis of Divine intervention would be a major activity of Laplace's scientific life. It is now generally regarded that Laplace's methods on their own, though vital to the development of the theory, are not sufficiently precise to demonstrate the stability of the Solar System, and indeed, the Solar System is understood to be chaotic, although it happens to be fairly stable.

Alexis Clairaut had first suggested the idea in 1743 while working on a similar Problem though he was using Newtonian-type geometric reasoning. Laplace described Clairaut's work as being "in the class of the most beautiful mathematical productions". However, Rouse Ball alleges that the idea "was appropriated from Joseph Louis Lagrange, who had used it in his memoirs of 1773, 1777 and 1780". The term "potential" itself was due to Daniel Bernoulli, who introduced it in his 1738 memoire Hydrodynamica. However, according to Rouse Ball, the term "potential function" was not actually used (to refer to a function V of the coordinates of space in Laplace's sense) until George Green's 1828 An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism.

As early as 1744, Euler, followed by Lagrange, had started looking for solutions of differential equations in the form:

One particular Problem from observational astronomy was the apparent instability whereby Jupiter's orbit appeared to be shrinking while that of Saturn was expanding. The Problem had been tackled by Leonhard Euler in 1748 and Joseph Louis Lagrange in 1763 but without success. In 1776, Laplace published a memoir in which he first explored the possible influences of a purported luminiferous ether or of a law of gravitation that did not act instantaneously. He ultimately returned to an intellectual investment in Newtonian gravity. Euler and Lagrange had made a practical approximation by ignoring small terms in the equations of motion. Laplace noted that though the terms themselves were small, when integrated over time they could become important. Laplace carried his analysis into the higher-order terms, up to and including the cubic. Using this more exact analysis, Laplace concluded that any two planets and the sun must be in mutual equilibrium and thereby launched his work on the stability of the Solar System. Gerald James Whitrow described the achievement as "the most important advance in physical astronomy since Newton".

Laplace was born in Beaumont-en-Auge, Normandy on 23 March 1749, a village four miles west of Pont l'Eveque. According to W. W. Rouse Ball, his Father, Pierre de Laplace, owned and farmed the small estates of Maarquis. His great-uncle, Maitre Oliver de Laplace, had held the title of Chirurgien Royal. It would seem that from a pupil he became an usher in the school at Beaumont; but, having procured a letter of introduction to d'Alembert, he went to Paris to advance his fortune. However, Karl Pearson is scathing about the inaccuracies in Rouse Ball's account and states:

His parents were from comfortable families. His Father was Pierre Laplace, and his mother was Marie-Anne Sochon. The Laplace family was involved in agriculture until at least 1750, but Pierre Laplace senior was also a cider merchant and syndic of the town of Beaumont.

As mentioned, the idea of the nebular hypothesis had been outlined by Immanuel Kant in 1755, and he had also suggested "meteoric aggregations" and tidal friction as causes affecting the formation of the Solar System. Laplace was probably aware of this, but, like many Writers of his time, he generally did not reference the work of others.

Even though Laplace is known as the first to express such ideas about causal determinism, his view is very similar to the one proposed by Boscovich as early as 1763 in his book Theoria philosophiae naturalis.

Laplace's early published work in 1771 started with differential equations and finite differences but he was already starting to think about the mathematical and philosophical concepts of probability and statistics. However, before his election to the Académie in 1773, he had already drafted two papers that would establish his reputation. The first, Mémoire sur la probabilité des causes par les événements was ultimately published in 1774 while the second paper, published in 1776, further elaborated his statistical thinking and also began his systematic work on celestial mechanics and the stability of the Solar System. The two disciplines would always be interlinked in his mind. "Laplace took probability as an instrument for repairing defects in knowledge." Laplace's work on probability and statistics is discussed below with his mature work on the analytic theory of probabilities.

While Newton explained the tides by describing the tide-generating forces and Bernoulli gave a description of the static reaction of the waters on Earth to the tidal potential, the dynamic theory of tides, developed by Laplace in 1775, describes the ocean's real reaction to tidal forces. Laplace's theory of ocean tides took into account friction, resonance and natural periods of ocean basins. It predicted the large amphidromic systems in the world's ocean basins and explains the oceanic tides that are actually observed.

In 1776, Laplace formulated a single set of linear partial differential equations, for tidal flow described as a barotropic two-dimensional sheet flow. Coriolis effects are introduced as well as lateral forcing by gravity. Laplace obtained these equations by simplifying the fluid dynamic equations. But they can also be derived from Energy integrals via Lagrange's equation.

The method of estimating the ratio of the number of favourable cases to the whole number of possible cases had been previously indicated by Laplace in a paper written in 1779. It consists of 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 probability-generating function of the former. Laplace then shows how, by means of interpolation, these coefficients may be determined from the generating function. Next he attacks the converse Problem, and from the coefficients he finds the generating function; this is effected by the solution of a finite difference equation.

From 1780–1784, Laplace and French Chemist Antoine Lavoisier collaborated on several experimental investigations, designing their own equipment for the task. In 1783 they published their joint paper, Memoir on Heat, in which they discussed the kinetic theory of molecular motion. In their experiments they measured the specific heat of various bodies, and the expansion of metals with increasing temperature. They also measured the boiling points of ethanol and ether under pressure.

In 1783, in a paper sent to the Académie, Adrien-Marie Legendre had introduced what are now known as associated Legendre functions. If two points in a plane have polar co-ordinates (r, θ) and (r ', θ'), where r ' ≥ r, then, by elementary manipulation, the reciprocal of the distance between the points, d, can be written as:

Laplace presented a memoir on planetary inequalities in three sections, in 1784, 1785, and 1786. This dealt mainly with the identification and explanation of the perturbations now known as the "great Jupiter–Saturn inequality". Laplace solved a longstanding Problem in the study and prediction of the movements of these planets. He showed by general considerations, first, that the mutual action of two planets could never cause large changes in the eccentricities and inclinations of their orbits; but then, even more importantly, that peculiarities arose in the Jupiter–Saturn system because of the near approach to commensurability of the mean motions of Jupiter and Saturn.

In 1785, Laplace took the key forward step in using integrals of this form to transform a whole difference equation, rather than simply as a form for the solution, and found that the transformed equation was easier to solve than the original.

Further developments of these theorems on planetary motion were given in his two memoirs of 1788 and 1789, but with the aid of Laplace's discoveries, the tables of the motions of Jupiter and Saturn could at last be made much more accurate. It was on the basis of Laplace's theory that Delambre computed his astronomical tables.

The former was published in 1796, and gives a general explanation of the phenomena, but omits all details. It contains a summary of the history of astronomy. This summary procured for its author the honour of admission to the forty of the French Academy and is commonly esteemed one of the masterpieces of French literature, though it is not altogether reliable for the later periods of which it treats.

In November 1799, immediately after seizing power in the coup of 18 Brumaire, Napoleon appointed Laplace to the post of Minister of the Interior. The appointment, however, lasted only six weeks, after which Lucien, Napoleon's brother, was given the post. Evidently, once Napoleon's grip on power was secure, there was no need for a prestigious but inexperienced scientist in the government. Napoleon later (in his Mémoires de Sainte Hélène) wrote of Laplace's dismissal as follows:

The only eyewitness account of Laplace's interaction with Napoleon is from the entry for 8 August 1802 in the diary of the British Astronomer Sir william Herschel:

The fourth chapter of this treatise includes an exposition of the method of least squares, a remarkable testimony to Laplace's command over the processes of analysis. In 1805 Legendre had published the method of least squares, making no attempt to tie it to the theory of probability. In 1809 Gauss had derived the normal distribution from the principle that the arithmetic mean of observations gives the most probable value for the quantity measured; then, turning this argument back upon itself, he showed that, if the errors of observation are normally distributed, the least squares estimates give the most probable values for the coefficients in regression situations. These two works seem to have spurred Laplace to complete work toward a treatise on probability he had contemplated as early as 1783.

In 1806, Laplace bought a house in Arcueil, then a village and not yet absorbed into the Paris conurbation. Claude Louis Berthollet was a neighbour—their gardens were not separated—and the pair formed the nucleus of an informal scientific circle, latterly known as the Society of Arcueil. Because of their closeness to Napoleon, Laplace and Berthollet effectively controlled advancement in the scientific establishment and admission to the more prestigious offices. The Society built up a complex pyramid of patronage. In 1806, Laplace was also elected a foreign member of the Royal Swedish Academy of Sciences.

Hahn states: "Nowhere in his writings, either public or private, does Laplace deny God's existence." Expressions occur in his private letters that appear inconsistent with atheism. On 17 June 1809, for instance, he wrote to his son, "Je prie Dieu qu'il veille sur tes jours. Aie-Le toujours présent à ta pensée, ainsi que ton père et ta mère [I pray that God watches over your days. Let Him be always present to your mind, as also your Father and your mother]." Ian S. Glass, quoting Herschel's account of the celebrated exchange with Napoleon, writes that Laplace was "evidently a deist like Herschel".

In two important papers in 1810 and 1811, Laplace first developed the characteristic function as a tool for large-sample theory and proved the first general central limit theorem. Then in a supplement to his 1810 paper written after he had seen Gauss's work, he showed that the central limit theorem provided a Bayesian justification for least squares: if one were combining observations, each one of which was itself the mean of a large number of independent observations, then the least squares estimates would not only maximise the likelihood function, considered as a posterior distribution, but also minimise the expected posterior error, all this without any assumption as to the error distribution or a circular appeal to the principle of the arithmetic mean. In 1811 Laplace took a different non-Bayesian tack. Considering a linear regression Problem, he restricted his attention to linear unbiased estimators of the linear coefficients. After showing that members of this class were approximately normally distributed if the number of observations was large, he argued that least squares provided the "best" linear estimators. Here it is "best" in the sense that it minimised the asymptotic variance and thus both minimised the expected absolute value of the error, and maximised the probability that the estimate would lie in any symmetric interval about the unknown coefficient, no matter what the error distribution. His derivation included the joint limiting distribution of the least squares estimators of two parameters.

In 1812, Laplace issued his Théorie analytique des probabilités in which he laid down many fundamental results in statistics. The first half of this treatise was concerned with probability methods and problems, the second half with statistical methods and applications. Laplace's proofs are not always rigorous according to the standards of a later day, and his perspective slides back and forth between the Bayesian and non-Bayesian views with an ease that makes some of his investigations difficult to follow, but his conclusions remain basically sound even in those few situations where his analysis goes astray. In 1819, he published a popular account of his work on probability. This book bears the same relation to the Théorie des probabilités that the Système du monde does to the Méchanique céleste. In its emphasis on the analytical importance of probabilistic problems, especially in the context of the "approximation of formula functions of large numbers," Laplace's work goes beyond the contemporary view which almost exclusively considered aspects of practical applicability. Laplace's Théorie analytique remained the most influential book of mathematical probability theory to the end of the 19th century. The general relevance for statistics of Laplacian error theory was appreciated only by the end of the 19th century. However, it influenced the further development of a largely analytically oriented probability theory.

In the second edition (1814) of the Essai philosophique, Laplace added some revealing comments on politics and governance. Since it is, he says, "the practice of the eternal principles of reason, justice and humanity that produce and preserve societies, there is a great advantage to adhere to these principles, and a great inadvisability to deviate from them". Noting "the depths of misery into which peoples have been cast" when ambitious Leaders disregard these principles, Laplace makes a veiled criticism of Napoleon's conduct: "Every time a great power intoxicated by the love of conquest aspires to universal domination, the sense of liberty among the unjustly threatened nations breeds a coalition to which it always succumbs." Laplace argues that "in the midst of the multiple causes that direct and restrain various states, natural limits" operate, within which it is "important for the stability as well as the prosperity of empires to remain". States that transgress these limits cannot avoid being "reverted" to them, "just as is the case when the waters of the seas whose floor has been lifted by violent tempests sink back to their level by the action of gravity".

Laplace in 1816 was the first to point out that the speed of sound in air depends on the heat capacity ratio. Newton's original theory gave too low a value, because it does not take account of the adiabatic compression of the air which results in a local rise in temperature and pressure. Laplace's investigations in practical physics were confined to those carried on by him jointly with Lavoisier in the years 1782 to 1784 on the specific heat of various bodies.

Faye thought that Laplace "did not profess atheism", but Napoleon, on Saint Helena, told General Gaspard Gourgaud, "I often asked Laplace what he thought of God. He owned that he was an atheist." Roger Hahn, in his biography of Laplace, mentions a dinner party at which "the Geologist Jean-Étienne Guettard was staggered by Laplace's bold denunciation of the existence of God". It appeared to Guettard that Laplace's atheism "was supported by a thoroughgoing materialism". But the Chemist Jean-Baptiste Dumas, who knew Laplace well in the 1820s, wrote that Laplace "provided Materialists with their specious arguments, without sharing their convictions".

An earlier report, although without the mention of Laplace's name, is found in Antommarchi's The Last Moments of Napoleon (1825):

Laplace's younger colleague, the Astronomer François Ara Go, who gave his eulogy before the French Academy in 1827, told Faye of an attempt by Laplace to keep the garbled version of his interaction with Napoleon out of circulation. Faye writes:

In 1884, however, the Astronomer Hervé Faye affirmed that this account of Laplace's exchange with Napoleon presented a "strangely transformed" (étrangement transformée) or garbled version of what had actually happened. It was not God that Laplace had treated as a hypothesis, but merely his intervention at a determinate point:

Jean-Baptiste Biot, who assisted Laplace in revising it for the press, says that Laplace himself was frequently unable to recover the details in the chain of reasoning, and, if satisfied that the conclusions were correct, he was content to insert the constantly recurring formula, "Il est aisé à voir que..." ("It is easy to see that..."). The Mécanique céleste is not only the translation of Newton's Principia into the language of the differential calculus, but it completes parts of which Newton had been unable to fill in the details. The work was carried forward in a more finely tuned form in Félix Tisserand's Traité de mécanique céleste (1889–1896), but Laplace's treatise will always remain a standard authority. In the years 1784–1787, Laplace produced some memoirs of exceptional power. The significant among these was one issued in 1784, and reprinted in the third volume of the Méchanique céleste. In this work he completely determined the attraction of a spheroid on a particle outside it. This is known for the introduction into analysis of the potential, a useful mathematical concept of broad applicability to the physical sciences.

The Swiss-American Historian of mathematics Florian Cajori appears to have been unaware of Faye's research, but in 1893 he came to a similar conclusion. Stephen Hawking said in 1999, "I don't think that Laplace was claiming that God does not exist. It's just that he doesn't intervene, to break the laws of Science."

Some records of Laplace's life were burned in 1925 with the family château in Saint Julien de Mailloc, near Lisieux, the home of his great-great-grandson the Comte de Colbert-Laplace. Others had been destroyed earlier, when his house at Arcueil near Paris was looted by house breakers in 1871.

Roger Hahn in his 2005 biography disputes this portrayal of Laplace as an opportunist and turncoat, pointing out that, like many in France, he had followed the debacle of Napoleon's Russian campaign with serious misgivings. The Laplaces, whose only daughter Sophie had died in childbirth in September 1813, were in fear for the safety of their son Émile, who was on the eastern front with the Emperor. Napoleon had originally come to power promising stability, but it was clear that he had overextended himself, putting the nation at peril. It was at this point that Laplace's loyalty began to weaken. Although he still had easy access to Napoleon, his personal relations with the Emperor cooled considerably. As a grieving Father, he was particularly cut to the quick by Napoleon's insensitivity in an exchange related by Jean-Antoine Chaptal: "On his return from the rout in Leipzig, he [Napoleon] accosted Mr Laplace: 'Oh! I see that you have grown thin—Sire, I have lost my daughter—Oh! that's not a reason for losing weight. You are a mathematician; put this event in an equation, and you will find that it adds up to zero.'"

Laplace built upon the qualitative work of Thomas Young to develop the theory of capillary action and the Young–Laplace equation.