However, Karl Pearson[8] is scathing about the inaccuracies in Rouse Ball's account and states: Indeed Caen was probably in Laplace's day the most intellectually active of all the towns of Normandy.

Lagrange was the senior by thirteen years, and had recently founded in his native city Turin a journal named Miscellanea Taurinensia, in which many of his early works were printed and it was in the fourth volume of this series that Laplace's paper appeared.

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.

As long as his results were true he took but little trouble to explain the steps by which he arrived at them; he never studied elegance or symmetry in his processes, and it was sufficient for him if he could by any means solve the particular question he was discussing.

For a fluid sheet of average thickness D, the vertical tidal elevation ζ, as well as the horizontal velocity components u and v (in the latitude φ and longitude λ directions, respectively) satisfy Laplace's tidal equations:[50] where Ω is the angular frequency of the planet's rotation, g is the planet's gravitational acceleration at the mean ocean surface, a is the planetary radius, and U is the external gravitational tidal-forcing potential.

Prominent among these is one read in 1783, reprinted as Part II of Théorie du Mouvement et de la figure elliptique des planètes in 1784, and in the third volume of the Mécanique céleste.

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.

[13] Laplace now set himself the task to write a work which should "offer a complete solution of the great mechanical problem presented by the Solar System, and bring theory to coincide so closely with observation that empirical equations should no longer find a place in astronomical tables.

[13][58] As mentioned, the idea of the nebular hypothesis had been outlined by Immanuel Kant in 1755,[58] who had also suggested "meteoric aggregations" and tidal friction as causes affecting the formation of the Solar System.

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 remains a standard authority.

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.

[13] 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.

In his Essai philosophique sur les probabilités (1814), Laplace set out a mathematical system of inductive reasoning based on probability, which we would today recognise as Bayesian.

However, Laplace was fully aware of the absurdity of the result; immediately following the example, he wrote, "But this number [i.e., the probability that the sun will rise tomorrow] is far greater for him who, seeing in the totality of phenomena the principle regulating the days and seasons, realizes that nothing at the present moment can arrest the course of it.

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.

Even though Laplace is generally credited with having first formulated the concept of causal determinism, in a philosophical context the idea was actually widespread at the time, and can be found as early as 1756 in Maupertuis' 'Sur la Divination'.

He was accordingly raised to the senate, and to the third volume of the Mécanique céleste he prefixed a note that of all the truths therein contained the most precious to the author was the declaration he thus made of his devotion towards the peacemaker of Europe.

[13] 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.

[79][80] 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".

Let us not offer fruitless and often injurious resistance to the inevitable benefits derived from the progress of enlightenment; but let us change our institutions and the usages that we have for a long time adopted only with extreme caution.

"[82] In the Essai philosophique, Laplace also illustrates the potential of probabilities in political studies by applying the law of large numbers to justify the candidates’ integer-valued ranks used in the Borda method of voting, with which the new members of the Academy of Sciences were elected.

[8] Laplace was buried at Père Lachaise in Paris but in 1888 his remains were moved to Saint Julien de Mailloc in the canton of Orbec and reinterred on the family estate.

A typical version is provided by Rouse Ball:[13] Laplace went in state to Napoleon to present a copy of his work, and the following account of the interview is well authenticated, and so characteristic of all the parties concerned that I quote it in full.

Laplace, who, though the most supple of politicians, was as stiff as a martyr on every point of his philosophy, drew himself up and answered bluntly, Je n'avais pas besoin de cette hypothèse-là.

")In 1884, however, the astronomer Hervé Faye[88][89] affirmed that this account of Laplace's exchange with Napoleon presented a "strangely transformed" (étrangement transformée) or garbled version of what had actually happened.

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:[92] The first Consul then asked a few questions relating to Astronomy and the construction of the heavens to which I made such answers as seemed to give him great satisfaction.

[103] Laplace cites with approval Leibniz's criticism of Newton's invocation of divine intervention to restore order to the Solar System: "This is to have very narrow ideas about the wisdom and the power of God.

"[105] In a group of manuscripts, preserved in relative secrecy in a black envelope in the library of the Académie des sciences and published for the first time by Hahn, Laplace mounted a deist critique of Christianity.