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Before the 1793 Reign of Terror Laplace together with his wife and two children left Paris and lived 50 km southeast of Paris. He did not return to Paris until after July 1794. Although Laplace managed to avoid the fate of some of his colleagues during the Revolution, such as Lavoisier who was guillotined in May 1794 while Laplace was out of Paris, he did have some difficult times. He was consulted, together with  Lagrange and Laland, over the new calendar for the Revolution. Laplace knew well that the proposed scheme did not really work because the length of the proposed year did not fit with the astronomical data. However he was wise enough not to try to overrule political dogma with scientific facts. He also conformed, perhaps more happily, to the decisions regarding the metric division of angles into 100 subdivisions.

In 1795 the École Normale was founded with the aim of training school teachers and Laplace taught courses there including one on probability which he gave in 1795. The École Normale survived for only four months for the 1200 pupils, who were training to become school teachers, found the level of teaching well beyond them. This is entirely understandable. Later Laplace wrote up the lectures of his course at the École Normale as Essai philosophique sur les probabilités published in 1814. A review of the Essai states:-

... after a general introduction concerning the principles of probability theory, one finds a discussion of a host of applications, including those to games of chance, natural philosophy, the moral sciences, testimony, judicial decisions and mortality.

In 1795 the Académie des Sciences was reopened as the Institut National des Sciences et des Arts. Also in 1795 the Bureau des Longitudes was founded with  Lagrange and Laplace as the mathematicians among its founding members and Laplace went on to lead the Bureau and the Paris Observatory. However although some considered he did a fine job in these posts others criticised him for being too theoretical.  Delambre wrote some years later:-

... never should one put a geometer at the head of an observatory; he will neglect all the observations except those needed for his formulas.

 Delambre also wrote concerning Laplace's leadership of the Bureau des Longitudes:-

One can reproach [Laplace] with the fact that in more than 20 years of existence the Bureau des Longitudes has not determined the position of a single star, or undertaken the preparation of the smallest catalogue.

Laplace presented his famous nebular hypothesis in 1796 in Exposition du systeme du monde, which viewed the solar system as originating from the contracting and cooling of a large, flattened, and slowly rotating cloud of incandescent gas. The Exposition consisted of five books: the first was on the apparent motions of the celestial bodies, the motion of the sea, and also atmospheric  refraction; the second was on the actual motion of the celestial bodies; the third was on force and momentum; the fourth was on the theory of universal gravitation and included an account of the motion of the sea and the shape of the Earth; the final book gave an historical account of astronomy and included his famous nebular hypothesis. Laplace states his philosophy of science in the Exposition as follows:-

If man were restricted to collecting facts the sciences were only a sterile nomenclature and he would never have known the great laws of nature. It is in comparing the phenomena with each other, in seeking to grasp their relationships, that he is led to discover these laws...

In view of modern theories of impacts of comets on the Earth it is particularly interesting to see Laplace's remarkably modern view of this:-

... the small probability of collision of the Earth and a comet can become very great in adding over a long sequence of centuries. It is easy to picture the effects of this impact on the Earth. The axis and the motion of rotation have changed, the seas abandoning their old position..., a large part of men and animals drowned in this universal deluge, or destroyed by the violent tremor imparted to the terrestrial globe.

Exposition du systeme du monde was written as a non-mathematical introduction to Laplace's most important work Traité du Mécanique Céleste whose first volume appeared three years later. Laplace had already discovered the invariability of planetary mean motions. In 1786 he had proved that the eccentricities and inclinations of planetary orbits to each other always remain small, constant, and self-correcting. These and many other of his earlier results formed the basis for his great work the Traité du Mécanique Céleste published in 5 volumes, the first two in 1799.

The first volume of the Mécanique Céleste is divided into two books, the first on general laws of equilibrium and motion of solids and also fluids, while the second book is on the law of universal gravitation and the motions of the centres of gravity of the bodies in the solar system. The main mathematical approach here is the setting up of differential equations and solving them to describe the resulting motions. The second volume deals with mechanics applied to a study of the planets. In it Laplace included a study of the shape of the Earth which included a discussion of data obtained from several different expeditions, and Laplace applied his theory of errors to the results. Another topic studied here by Laplace was the theory of the tides but  Airy, giving his own results nearly 50 years later, wrote:-

It would be useless to offer this theory in the same shape in which Laplace has given it; for that part of the Mécanique Céleste which contains the theory of tides is perhaps on the whole more obscure than any other part...

In the Mécanique Céleste  Laplace's equation appears but although we now name this equation after Laplace, it was in fact known before the time of Laplace. The  Legendre functions also appear here and were known for many years as the Laplace coefficients. The Mécanique Céleste does not attribute many of the ideas to the work of others but Laplace was heavily influenced by  Lagrange and by  Legendre and used methods which they had developed with few references to the originators of the ideas.

Under Napoleon Laplace was a member, then chancellor, of the Senate, and received the Legion of Honour in 1805. However Napoleon, in his memoirs written on St Hélène, says he removed Laplace from the office of Minister of the Interior, which he held in 1799, after only six weeks:-

... because he brought the spirit of the infinitely small into the government.

Laplace became Count of the Empire in 1806 and he was named a marquis in 1817 after the restoration of the Bourbons.

The first edition of Laplace's Théorie Analytique des Probabilités was published in 1812. This first edition was dedicated to Napoleon-le-Grand but, for obvious reason, the dedication was removed in later editions! The work consisted of two books and a second edition two years later saw an increase in the material by about an extra 30 per cent.

The first book studies generating functions and also approximations to various expressions occurring in probability theory. The second book contains Laplace's definition of probability,  Bayes's rule (so named by  Poincaré many years later), and remarks on moral and mathematical expectation. The book continues with methods of finding probabilities of compound events when the probabilities of their simple components are known, then a discussion of the method of least squares,  Buffon's needle problem, and inverse probability. Applications to mortality, life expectancy and the length of marriages are given and finally Laplace looks at moral expectation and probability in legal matters.

Later editions of the Théorie Analytique des Probabilités also contains supplements which consider applications of probability to: errors in observations; the determination of the masses of Jupiter, Saturn and Uranus; triangulation methods in surveying; and problems of geodesy in particular the determination of the meridian of France. Much of this work was done by Laplace between 1817 and 1819 and appears in the 1820 edition of the Théorie Analytique. A rather less impressive fourth supplement, which returns to the first topic of generating functions, appeared with the 1825 edition. This final supplement was presented to the Institute by Laplace, who was 76 years old by this time, and by his son.

We mentioned briefly above Laplace's first work on physics in 1780 which was outside the area of mechanics in which he contributed so much. Around 1804 Laplace seems to have developed an approach to physics which would be highly influential for some years. This is best explained by Laplace himself:-

... I have sought to establish that the phenomena of nature can be reduced in the last analysis to actions at a distance between molecule and molecule, and that the consideration of these actions must serve as the basis of the mathematical theory of these phenomena.

This approach to physics, attempting to explain everything from the forces acting locally between molecules, already was used by him in the fourth volume of the Mécanique Céleste which appeared in 1805. This volume contains a study of pressure and density, astronomical refraction, barometric pressure and the transmission of gravity based on this new philosophy of physics. It is worth remarking that it was a new approach, not because theories of molecules were new, but rather because it was applied to a much wider range of problems than any previous theory and, typically of Laplace, it was much more mathematical than any previous theories.

Laplace's desire to take a leading role in physics led him to become a founder member of the Société d'Arcueil in around 1805. Together with the chemist Berthollet, he set up the Society which operated out of their homes in Arcueil which was south of Paris. Among the mathematicians who were members of this active group of scientists were  Biot and  Poisson. The group strongly advocated a mathematical approach to science with Laplace playing the leading role. This marks the height of Laplace's influence, dominant also in the Institute and having a powerful influence on the École Polytechnique and the courses that the students studied there.

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