Pierre-Simon Laplace

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Pierre-Simon, Marquis de Laplace (March 23 1749, Beaumont-en-Auge, Normandy – March 5 1827, Paris) was a French mathematician and astronomer who put the final capstone on mathematical astronomy by summarizing and extending the work of his predecessors in his five volume Mécanique Céleste (Celestial Mechanics) (1799-1825). This masterpiece translated the geometrical study of mechanics used by Isaac Newton to one based on calculus, known as physical mechanics.

He is also the discoverer of Laplace's equation. Although the Laplace transform is named in honor of Laplace, who used the transform in his work on probability theory, the transform was discovered originally by Leonhard Euler, the prolific eighteenth-century Swiss mathematician. The Laplace transform appears in all branches of mathematical physics &mdash; a field he took a leading role in forming. The Laplacian differential operator, much relied-upon in applied mathematics, is likewise named after him.

He became count of the Empire in 1806 and was named a marquis in 1817 after the restoration of the Bourbons.

Biography
Pierre Simon Laplace was the son of a small cottager or perhaps a farm-labourer, and owed his education to the interest excited in some wealthy neighbours by his abilities and engaging presence. It would seem 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 push his fortune. A paper on the principles of mechanics excited D'Alembert's interest, and on his recommendation a place in the military school was offered to Laplace.

Secure of a competency, Laplace now threw himself into original research, and in the next seventeen years, 1771-1787, he produced much of his original work in astronomy. This commenced with a memoir, read before the French Academy in 1773, in which he showed that the planetary motions were stable, and carried the proof as far as the cubes of the eccentricities and inclinations. This was followed by several papers on points in the integral calculus, finite differences, differential equations, and astronomy.

Laplace had a wide knowledge of all sciences and dominated all discussions in the Académie. Quite uniquely for a mathematical prodigy of his skill, Laplace viewed mathematics as nothing in itself but a tool to be called upon in the investigation of a scientific or practical inquiry.

Laplace spent much of his life working on mathematical astronomy that culminated in his masterpiece on the proof of the dynamic stability of the solar system with the assumption that it consists of a collection of rigid bodies moving in a vacuum. He independently formulated the nebular hypothesis and was one of the first scientists to postulate the existence of black holes and the notion of gravitational collapse.

He is remembered as one of the greatest scientists of all time (sometimes referred to as a French Newton) with a natural phenomenal mathematical faculty possessed by none of his contemporaries. It does appear that Laplace was not modest about his abilities and achievements, and he probably failed to recognise the effect of his attitude on his colleagues. Lexell visited the Académie des Sciences in Paris in 1780-81 and reported that Laplace let it be known widely that he considered himself the best mathematician in France. The effect on his colleagues would have been only mildly eased by the fact that Laplace was very likely right. 

Probability theory
While he conducted much research in physics, another major theme of his life's endeavors was probability theory. In his Essai philosophique sur les probabilités, Laplace set out a mathematical system of inductive reasoning based on probability, which we would today recognise as Bayesian. One well-known formula arising from his system is the rule of succession. Suppose that some trial has only two possible outcomes, labeled "success" and "failure". Under the assumption that little or nothing is known a priori about the relative plausibilities of the outcomes, Laplace derived a formula for the probability that the next trial will be a success.


 * $$\Pr(\mbox{next outcome is success}) = \frac{s+1}{n+2}$$,

where s is the number of previously observed successes and n is the total number of observed trials. It is still used as an estimator for the probability of an event if we know the event space, but only have a small number of samples.

The rule of succession has been subject to much criticism, partly due to the example which Laplace chose to illustrate it. He calculated that the probability that the sun will rise tomorrow, given that it has never failed to in the past, was


 * $$\Pr(\mbox{sun will rise tomorrow}) = \frac{d+1}{d+2},$$

where d is the number of times the sun has risen in the past. This result has been derided as absurd, and some authors have concluded that all applications of the Rule of Succession are absurd by extension. 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."

Laplace's determinism
Laplace strongly believed in causal determinism, which is expressed in the following quote from the introduction to the Essai:


 * "We may regard the present state of the universe as the effect of its past and the cause of its future. An intellect which at a certain moment would know all forces that set nature in motion, and all positions of all items of which nature is composed, if this intellect were also vast enough to submit these data to analysis, it would embrace in a single formula the movements of the greatest bodies of the universe and those of the tiniest atom; for such an intellect nothing would be uncertain and the future just like the past would be present before its eyes."

This intellect is often referred to as Laplace's demon (in the same vein as Maxwell's demon). Note that the description of the hypothetical intellect described above by Laplace as a demon does not come from Laplace, but from later biographers: Laplace saw himself as a scientist that hoped that humanity would progress in a better scientific understanding of the world, which, if and when eventually completed, would still need a tremendous calculating power to compute it all in a single instant. While Laplace saw foremost practical problems for mankind to reach this ultimate stage of knowledge and computation, later interpretations of quantum mechanics, which were adopted by philosophers defending the existence of free will, also leave the theoretical possibility of such an "intellect" contested.

There has recently been proposed a limit on the computational power of the universe, ie the ability of Laplace's Demon to process an infinite amount of information. The limit is based on the maximum entropy of the universe, the speed of light, and the minimum amount of time taken to move information across the Planck length, and the figure turns out to be 2130 bits. Accordingly, anything that requires more than this amount of data cannot be computed in the amount of time that has lapsed so far in the universe. (An actual theory of everything might find an exception to this limit, of course.)

Spherical harmonics or Laplace's coefficients
During the years 1784-1787 he produced some memoirs of exceptional power. Prominent among these is one read in 1784, and reprinted in the third volume of the Méchanique céleste, in which he completely determined the attraction of a spheroid on a particle outside it. This is memorable for the introduction into analysis of spherical harmonics or Laplace's coefficients, as also for the development of the use of the potential - a name first given by Green in 1828.

If the co-ordinates of two points be (r,μ,ω) and (r',μ',ω'), and if r' ≥ r, then the reciprocal of the distance between them can be expanded in powers of r/r', and the respective coefficients are Laplace's coefficients. Their utility arises from the fact that every function of the co-ordinates of a point on the sphere can be expanded in a series of them. It should be stated that the similar coefficients for space of two dimensions, together with some of their properties, had been previously given by Legendre in a paper sent to the French Academy in 1783. Legendre had good reason to complain of the way in which he was treated in this matter.

This paper is also remarkable for the development of the idea of the potential, which was appropriated from Lagrange, who had used it in his memoirs of 1773, 1777 and 1780. Laplace showed that the potential always satisfies the differential equation
 * $$\nabla^2V={\partial^2V\over \partial x^2 } +

{\partial^2V\over \partial y^2 } + {\partial^2V\over \partial z^2 } = 0, $$ and on this result his subsequent work on attractions was based. The quantity $$\nabla^2V= 0 $$ has been termed the concentration of $$V\,$$ and its value at any point indicates the excess of the value of $$V\,$$ there over its mean value in the neighbourhood of the point. Laplace's equation, or the more general form $$\nabla^2 V=- 4 \pi \rho$$, appears in all branches of mathematical physics. According to some writers this follows at once from the fact that $$\nabla^2$$ is a scalar operator; or the equation may represent analytically some general law of nature which has not been yet reduced to words; or possibly it might be regarded by a Kantian as the outward sign of one of the necessary forms through which all phenomena are perceived.

Planetary inequalities
This memoir was followed by another on planetary inequalities, which was presented in three sections in 1784, 1785, and 1786. This deals mainly with the explanation of the "great inequality" of Jupiter and Saturn. Laplace showed by general considerations that the mutual action of two planets could never largely affect the eccentricities and inclinations of their orbits; and that the peculiarities of the Jovian system were due to the near approach to commensurability of the mean motions of Jupiter and Saturn: further developments of these theorems on planetary motion were given in his two memoirs of 1788 and 1789. It was on these data that Delambre computed his astronomical tables.

The year 1787 was rendered memorable by Laplace's explanation and analysis of the relation between the lunar acceleration and the secular changes in the eccentricity of the earth's orbit: this investigation completed the proof of the stability of the whole solar system on the assumption that it consists of a collection of rigid bodies moving in a vacuum. All the memoirs above alluded to were presented to the French Academy, and they are printed in the Mémoires présentés par divers savants.

Celestial mechanics
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." The result is embodied in the Exposition du système du monde and the Méchanique céleste.

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; it is commonly esteemed one of the masterpieces of French literature, though it is not altogether reliable for the later periods of which it treats.

The nebular hypothesis was here enunciated. According to this hypothesis the solar system has been evolved from a globular mass of incandescent gas rotating around an axis through its centre of mass. As it cooled this mass contracted and successive rings broke off from its outer edge. These rings in their turn cooled, and finally condensed into the planets, while the sun represents the central core which is still left. On this view we should expect that the more distant planets would be older than those nearer the sun. The subject is one of great difficulty, and though it seems certain that the solar system has a common origin, there are various features which appear almost inexplicable on the nebular hypothesis as enunciated by Laplace.

Probably the best modern opinion inclines to the view that nebular condensation, meteoric condensation, tidal friction, and possibly other causes as yet unsuggested, have all played their part in the evolution of the system.

The idea of the nebular hypothesis had been outlined by Kant in 1755, and he had also suggested meteoric aggregations and tidal friction as causes affecting the formation of the solar system: it is probable that Laplace was not aware of this.

According to the rule published by Titius of Wittemberg in 1766-but generally known as Bode's Law, from the fact that attention was called to it by Johann Elert Bode in 1778 - the distances of the planets from the sun are nearly in the ratio of the numbers 0 + 4, 3 + 4, 6 + 4, 12+4, etc., the (n+2)th term being ( 3) + 4. It would be an interesting fact if this could be deduced from the nebular, meteoric, or any other hypotheses, but so far as I am aware only one writer has made any serious attempt to do so, and his conclusion seems to be that the law is not sufficiently exact to be more than a convenient means of remembering the general result.

Laplace's analytical discussion of the solar system is given in his Méchanique céleste published in five volumes. The first two volumes, published in 1799, contain methods for calculating the motions of the planets, determining their figures, and resolving tidal problems. The third and fourth volumes, published in 1802 and 1805, contain applications of these methods, and several astronomical tables. The fifth volume, published in 1825, is mainly historical, but it gives as appendices the results of Laplace's latest researches. Laplace's own investigations embodied in it are so numerous and valuable that it is regrettable to have to add that many results are appropriated from writers with scanty or no acknowledgement, and the conclusions - which have been described as the organized result of a century of patient toil - are frequently mentioned as if they were due to Laplace.

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." The Méchanique céleste is not only the translation of the Principia into the language of the differential calculus, but it completes parts of which Newton had been unable to fill in the details. F. F. Tisserand's recent work may be taken as the modern presentation of dynamical astronomy on classical lines, but Laplace's treatise will always remain a standard authority.

Science as prediction
Laplace went in state to beg Napoleon to accept 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. Someone had told Napoleon that the book contained no mention of the name of God; Napoleon, who was fond of putting embarrassing questions, received it with the remark, "M. Laplace, they tell me you have written this large book on the system of the universe, and have never even mentioned its Creator." 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." (I did not need to make such an assumption). Napoleon, greatly amused, told this reply to Lagrange, who exclaimed, "Ah! c'est une belle hypothèse; ça explique beaucoup de choses" (Ah! that is a beautiful assumption;  it explains many things). Laplace then declared: "Cette hypothèse, Sire, explique en effet tout, mais ne permet de prédire rien. En tant que savant, je me dois de vous fournir des travaux permettant des prédictions" (quoted by Ian Stewart and Jack Cohen &mdash; This hypothesis, Sir, does explains everything, but does not permit to predict anything. As a scholar, I must provide you with works permitting predictions."). Laplace thus defined science as a predicting tool.

Black hole
Laplace also came close to propounding the concept of the black hole. He pointed out that there could be massive stars whose gravity is so great that not even light could escape from their surface. Laplace also speculated that some of the nebulae revealed by telescopes may not be part of the Milky Way and might actually be galaxies themselves. Thus, he anticipated the major discovery of Edwin Hubble, some 100 years before it happened.

Analytic theory of probabilities
In 1812 Laplace issued his Théorie analytique des probabilités. The theory is stated to be only common sense expressed in mathematical language. The method of estimating the ratio of the number of favourable cases to the whole number of possible cases had been indicated by Laplace in a paper written in 1779. It consists 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. 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 an equation in finite differences. The method is cumbersome, and in consequence of the increased power of analysis is now rarely used.

This treatise includes an exposition of the method of least squares, a remarkable testimony to Laplace's command over the processes of analysis. The method of least squares for the combination of numerous observations had been given empirically by Gauss and Legendre, but the fourth chapter of this work contains a formal proof of it, on which the whole of the theory of errors has been since based. This was effected only by a most intricate analysis specially invented for the purpose, but the form in which it is presented is so meagre and unsatisfactory that in spite of the uniform accuracy of the results it was at one time questioned whether Laplace had actually gone through the difficult work he so briefly and often incorrectly indicates.

In 1819 Laplace 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.

Minor discoveries
Amongst the minor discoveries of Laplace in pure mathematics is his discussion (simultaneously with Vandermonde) of the general theory of determinants in 1772; his proof that every equation of an even degree must have at least one real quadratic factor; his reduction of the solution of linear differential equations to definite integrals; and his solution of the linear partial differential equation of the second order. He was also the first to consider the difficult problems involved in equations of mixed differences, and to prove that the solution of an equation in finite differences of the first degree and the second order might be always obtained in the form of a continued fraction. Besides these original discoveries he determined, in his theory of probabilities, the values of a number of the more common definite integrals; and in the same book gave the general proof of the theorem enunciated by Lagrange for the development of any implicit function in a series by means of differential coefficients.

In theoretical physics the theory of capillary attraction is due to Laplace, who accepted the idea propounded by Hauksbee in the Philosophical Transactions for 1709, that the phenomenon was due to a force of attraction which was insensible at sensible distances. The part which deals with the action of a solid on a liquid and the mutual action of two liquids was not worked out thoroughly, but ultimately was completed by Gauss: Carl Neumann later filled in a few details. In 1862 Lord Kelvin (Sir William Thomson) showed that if the molecular constitution of matter is assumed, the laws of capillary attraction can be deduced from the Newtonian law of gravitation.

Laplace in 1816 was the first to point out explicitly why Newton's theory of vibratory motion gave an incorrect value for the velocity of sound. The actual velocity is greater than that calculated by Newton in consequence of the heat developed by the sudden compression of the air which increases the elasticity and therefore the velocity of the sound transmitted. 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.

Laplace seems to have regarded analysis merely as a means of attacking physical problems, though the ability with which he invented the necessary analysis is almost phenomenal. 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.

As Napoleon's power increased Laplace begged the first consul to give him the post of minister of the interior. Napoleon, who desired the support of men of science, agreed to the proposal; but a little less than six weeks saw the close of Laplace's political career.

Although Laplace was removed from office it was desirable to retain his allegiance. 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. In copies sold after the restoration this was struck out. In 1814 it was evident that the empire was falling; Laplace hastened to tender his services to the Bourbons, and on the restoration was rewarded with the title of marquis

Quotes

 * What we know is not much. What we do not know is immense.
 * I have no need of that hypothesis. ("Je n'ai pas besoin de cette hypothèse", as a reply to Napoleon, who had asked why he hadn't mentioned God in his book on astronomy)
 * "It is therefore obvious that..." (frequently used in the Celestial Mechanics when he had proved something and mislaid the proof, or found it clumsy. Notorious as a signal for something true, but hard to prove.)
 * The weight of evidence for an extraordinary claim must be proportioned to its strangeness. (known as the principle of Laplace)