Pierre-Simon Laplace

Pierre-Simon, Marquis de Laplace
French mathematician & astronomer
Born
March 23, 1749 Beaumont-en-Auge, Normandy
Died
March 5, 1827 (Age 78) Paris, France

Pierre-Simon, Marquis de Laplace (March 23, 1749 – March 5, 1827) was a French mathematician and astronomer who summarized and extended the work of his predecessors in his five volume Mécanique Céleste (Celestial Mechanics) (1799-1825). In this work he gave evidence of the stabiity of the solar system, of which at the time there was much question. Laplace also broke important ground in the field of probability, and made many important discoveries in calculus. He is often considered, as he was during his lifetime, the greatest mathematician of his age.

Biography

Pierre Simon Laplace was born in Beaumont-en-Auge, Normandy, the son of a small cottager or perhaps a farm-laborer, and owed his education to the interest excited in some wealthy neighbours by his abilities and engaging presence. It is interesting to note that Laplace was at first engaged in the study of theology, and was particularly adept at argumentation in that field. But his interests soon turned to mathematics, in which he found himself unusually proficient. By the time he was 18, he was given a teaching position in a college in his hometown, but, having procured a letter of introduction to famed French mathematician Jean le Rond d'Alembert, he went to Paris to pursue his fortune. D'Alembert, however, did not take kindly to Laplace's impositions, and at first rebuffed them. But Laplace, not to be defeated so easily, wrote again to D'Alembert discussing the principles of mechanics. This so impressed d'Alembert that he reversed his original judgment. "You see I pay but little respect to letters of recommendation," D'Alembert wrote back. "You, however, have no need of them. You have made yourself known to me in a more appropriate manner, and my support is your due." <<<American Quarterly Review, 1830, 257>>> D'Alembert used his influence to secure a position for Laplace as professor of mathematics in the Military School of Paris.

When he was 24, Laplace was admitted to the French Academy of Sciences, after which he 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 to a higher degree of accuracy than had yet been attained. This was followed by several papers on points in the integral calculus, finite differences, differential equations, and astronomy.

From 1780 to 1784, Laplace and famed French chemist Antoine Lavoisier collaborated on several experimental investigations, designing their own equipment for the task. In 1880, the two scientists published a paper, Memoir on Heat, in which they discussed the kinetic theory of molecular motion. They measured the specific heat of various bodies, and the expansion of metals with increasing temperature. They also measured the boiling points of alcohol and ether under pressure.

During the years 1784-1787 Laplace 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 the potential, a useful mathematical concept of broad applicability to the physical sciences.

Planetary inequalities

Laplace produced a memoir presented in three sections in 1784, 1785, and 1786. Laplace showed by general considerations that the mutual action of Jupiter and Saturn 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.

The year 1787 was rendered memorable by Laplace's explanation and analysis of the relation between the lunar acceleration and certain 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 under mutual gravitational attraction that move in a vacuum.

The French Revolution

Laplace took an interest in the affairs of the French Revolution. He was appointed to a general committee of weights and measures, consisting of a roster of scientific luminaries including Lagrange and Lavoisier, which in 1791 recommended a standard of length equal to one ten millionth of the length of a quarter meridian (The distance between the north pole and the equator along the earth's surface). The committee was dismissed after Robespierre assumed power in 1793. In 1795, Laplace was reinstated in a reconstituted committee, minus Laplace's former research partner, Lavoisier, who a year earlier had met a sad end at the guillotine. This committee assisted in the implementation of the standard meter based on its previous recommendations.<<<Bowring, John. 1854. The Decimal System in Numbers, Coins and Accounts. London: Nathaniel Cooke. 97-99.>>>

The same year, Laplace presented a copy of Exposition du Système du Monde to the Council of 500, the lower house of the legislative body of the French government. He was an instructor at the Ecole Normal, a short-lived teacher training school instituted by the revolutionary government, and went on to teach at the Polytechnic School, established in 1794. When Napoleon assumed power, he appealed for and received a position as interior minister, but, his personality not being up to the administrative and diplomatic tasks the position entailed, was soon dismissed.

Although Laplace was removed from office it was desirable to retain his allegiance. He was accordingly installed in the senate, and was later raised to the position of vice chancellor, and then, president, of that body. In 1806 he was given the title of Count of the Empire. To the third volume of the Mécanique céleste he prefixed a reference to Napoleon as the peacemaker of Europe, but in copies sold after the restoration this was struck out. In 1814 it was evident that the empire under Napoleon was falling; Laplace hastened to tender his services to the Bourbons, and on the restoration was rewarded with the title of marquis and appointed a seat in the Chamber of Peers.

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.

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 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.

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's assistance

Jean-Baptiste Biot (1774-1862) assisted Laplace in revising the manuscript for the press. Biot tells an interesting story. In 1803 he requested by letter a copy of pages of the unfinished manuscript. Laplace, in response, said that he much preferred the work to be known by the public in its entirety. Biot, in rebuttal, said that he was not the general public, but a mathematical specialist, and that he was much interested in the abstruse mathematical details that the manuscript must contain. Biot offered to check the manuscript for the press as he otherwise examined it. Laplace granted Biot's wish, and forarded the pages to Biot, often meeting with him, discussing the changes along with other topics that Biot introduced in conversation.

Later in his career, Biot showed Laplace a correction to a geometrical problem that remained unsolved by the famous mathematician Euler. Laplace examined Biot's manuscript, and immediately arranged to have it read before the French Academy. Napoleon, who was keenly interested in mathematics and was himself proficient in the subject, was at the reading, but his presence did not but for a short moment intimidate Biot, as none other than the famous Laplace had arranged for the presentation. After delivering the memoir, Biot was invited by Laplace to his study, and there the eminent mathematician uncovered unpublished works devoted to the very corrections that Biot had shared with the academy just a short time before. Biot said he observed a like generosity on the part of Laplace on many other occasions.<<<Laplace and Biot, 1853>>>.

Biot says that Laplace himself was frequently unable to recover the details in the chain of reasoning in his work, and, if satisfied that the conclusions were correct, he was content to insert the constantly recurring formula, "Il est aisé à voir." (It is easy to see). 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.

Laplace went in state to beg Napoleon to accept a copy of his work, who had heard that the book contained no mention 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" ("This hypothesis, Sire, does explain everything, but does not permit to predict anything. As a scholar, I must provide you with works permitting predictions." - quoted by Ian Stewart and Jack Cohen). Laplace thus defined science as a predicting tool.

Analytic theory of probabilities

In 1812, Laplace issued his Théorie analytique des probabilités. The method of estimating the ratio of the number of favourable cases, compared 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 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 leads most of the time to a normal probability distribution the so called Laplace-Gauss distribution.

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 affected 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.

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.

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.

One of Laplace's last acts was a controversial one. The French government had instituted a law that would penalize the press. In 1827 the French Academy of Sciences entertained a motion to oppose this law. The academy was deeply divided on the issue, and Laplace, who was its director at the time, voted against the motion, and resigned his post.

Laplace died in Paris in 1827.

Legacy

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. Anders Johan 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. [1]

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 (see escape velocity). 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.

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.

Probability theory

While he conducted much research in physics, another major theme of his life's endeavours 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 times. 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 demon

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. A physical implementation of Laplace's Demon has been referred to as a Laplace Computer.

There has recently been proposed a limit on the computational power of the universe, i.e., 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 elapsed 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

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

$\nabla ^{2}V={\partial ^{2}V \over \partial x^{2}}+{\partial ^{2}V \over \partial y^{2}}+{\partial ^{2}V \over \partial z^{2}}=0$

and on this result his subsequent work on attractions was based. The quantity $\nabla ^{2}V=0$ 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 possibly it might be regarded by a Kantian as the outward sign of one of the necessary forms through which all phenomena are perceived.

Minor discoveries and accomplishments

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.

Together with Thomas Young, Laplace is credited with describing the pressure across a curved surface, as set out in the Young-Laplace equation.

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.

Quotes

• What we know is not much. What we do not know is immense.
• I had no need of that hypothesis. ("Je n'avais pas besoin de cette hypothèse-là," 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)

See also

• Thomas Young
• William Thomson

References

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<<Whatever references we use, they need to be properly formatted.>>

• 1829. Laplace's Celestial Mechanics. Foreign Quarterly Review 3:146-148
• Bidwell, W.H. and J.H. Agnew, eds. 1853. Biot and Laplace. The Eclectic Magazine of Foreign Literature, Science and Art. 106-109.
• Ball, R.S. 2001. Laplace, in The Great Astronomers: The Essential Library Edition. The Essential Library. 153-162. ISBN 1401017851.
• 1856. Notes and Queries: a Medium of Inter-Communication for Literary Men, Artists, Antiquaries, Genealogists, Etc. 1:42.
• 1830. The Astronomy of Laplace. American Quarterly Review 7:255-279.
• Malkin, Arthur T. 1838. Distinguished Men of Modern Times London: Charles Knight & Co. 4:358 incl, priestly, dr. black, Lavoisier, Banks, Delambre, Wollaston
• A Short Account of the History of Mathematics (4th edition, 1908) by W. W. Rouse Ball.

• Gillispie, Charles Coulston (1997) Pierre Simon Laplace 1749-1827: A Life in Exact Science, Princeton: Princeton University Press, ISBN 0-691-01185-0

• Hahn, Roger (2005) Pierre Simon Laplace 1749-1827: A Determined Scientist, Cambridge, MA: Harvard University Press, ISBN 0-674-01892-3

External links

• John J. O'Connor and Edmund F. Robertson. Pierre-Simon Laplace at the MacTutor archive
• Biography of Pierre-Simon Laplace - copy of 1908 text
• The Antikythera Calculator (Italian and English versions)
• Pastore, Giovanni, Antikythera E I Regoli Calcolatori, Rome, 2006, privately published

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