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Pierre-Simon, marquis de Laplace (23 March 1749 – 5 March 1827) was a French mathematician and astronomer whose work was pivotal to the development of mathematical astronomy and statistics. He summarized and extended the work of his predecessors in his five volume Mécanique Céleste (Celestial Mechanics) (1799 – 1825). This work translated the geometric study of classical mechanics to one based on calculus, opening up a broader range of problems. In statistics, the so called Bayesian interpretation of probability was mainly developed by Laplace. Laplace formulated Laplace's equation, and pioneered the Laplace transform which appears in many branches of mathematical physics, a field that he took a leading role in forming. The Laplacian differential operator, widely used in mathematics, is also named after him. He restated and developed the nebular hypothesis of the origin of the solar system and was one of the first scientists to postulate the existence of black holes and the notion of gravitational collapse. Laplace is remembered as one of the greatest scientists of all time. Sometimes referred to as the French Newton or Newton of France, he possessed a phenomenal natural mathematical faculty superior to that of any of his contemporaries. Laplace became a count of the First French Empire in 1806
and was named a marquis in 1817, after the Bourbon
Restoration. Many details of the life of Laplace were lost when the family château burned in 1925. Laplace was born in Beaumont - en - Auge, Normandy, in 1749. According to W. W. Rouse Ball, he was the son of a small cottager or perhaps a farm laborer, and owed his education to the interest excited in some wealthy neighbors by his abilities and engaging presence. Very little is known of his early years. 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 push his fortune. However, Karl Pearson is scathing about the accuracies 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. Pierre Simon Laplace attended a school in the village run at a Benedictine priory, his father intending that he be ordained in the Roman Catholic Church. At sixteen, to further his father's intention, he was sent to the University of Caen to read theology. At the university, he was mentored by two enthusiastic teachers of mathematics, Christophe Gadbled and Pierre Le Canu, who awoke his zeal for the subject. Laplace did not graduate in theology but left for Paris with a letter of introduction from Le Canu to Jean le Rond d'Alembert. According to his great - great - grandson, d'Alembert received him rather poorly, and to get rid of him gave him a thick mathematics book, saying to come back when he had read it. When Laplace came back a few days later, d'Alembert was even less friendly and did not hide his opinion that it was impossible that Laplace could have read and understood the book. But upon questioning him, he realized that it was true, and from that time he took Laplace under his care. Another version is that Laplace solved overnight a problem that d'Alembert set him for submission the following week, then solved a harder problem the following night. D'Alembert was impressed and recommended him for a teaching place in the École Militaire. With a secure income and undemanding teaching, Laplace now threw himself into original research and in the next seventeen years, 1771 – 1787, he produced much of his original work in astronomy. Laplace further impressed the Marquis de Condorcet, and already in 1771 Laplace felt that he was entitled to membership of the French Academy of Sciences. However, in that year, admission went to Alexandre - Théophile Vandermonde and in 1772 to Jacques Antoine Joseph Cousin. Laplace was disgruntled, and at the beginning of 1773, d'Alembert wrote to Lagrange in Berlin to ask if a position could be found for Laplace there. However, Condorcet became permanent secretary of the Académie in February and Laplace was elected associate member on 31 March, at age 24. He married Marie - Charlotte de Courty de Romanges in his late thirties and the couple had a daughter, Sophie, and a son, Charles - Émile (b. 1789). 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. 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 now understood to be chaotic, although it actually appears to be fairly stable. 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 had a wide knowledge of all sciences and dominated all discussions in the Académie. 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.
During the years 1784 – 1787 he published some
memoirs of exceptional power. 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.
In this work, Laplace 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, and also for the
development of the use of what we would now call the
gravitational potential in celestial mechanics. 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: This expression can be expanded in powers of r/r ' using Newton's generalized binomial theorem to give: The sequence of functions P0k(cosф) is the set of so-called "associated Legendre functions" and their usefulness arises from the fact that every function of the points on a circle can be expanded as a series of them. Laplace, with scant regard for credit to Legendre, made
the non - trivial extension of the result to three
dimensions to yield a more general set of functions, the spherical
harmonics or Laplace coefficients. The
latter term is not in common use now. This paper is also remarkable for the development of the idea of the scalar potential. The gravitational force acting on a body is, in modern language, a vector, having magnitude and direction. A potential function is a scalar function that defines how the vectors will behave. A scalar function is computationally and conceptually easier to deal with than a vector function. 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. Laplace applied the language of calculus to the potential function and showed that it always satisfies the differential equation: An analogous result for the velocity potential of a fluid had been obtained some years previously by Leonard Euler. Laplace's subsequent work on gravitational attraction was based on this result. The quantity ∇2V 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 neighborhood of the point. Laplace's equation, a special case of Poisson's equation, appears ubiquitously in mathematical physics. The concept of a potential occurs in fluid dynamics, electromagnetism and other areas. Rouse Ball speculated that it might be seen as "the outward sign" of one of the "prior forms" in Kant's theory of perception. The spherical harmonics turn out to be critical to practical solutions of Laplace's equation. Laplace's equation in spherical coordinates, such as are used for mapping the sky, can be simplified, using the method of separation of variables into a radial part, depending solely on distance from the center point, and an angular or spherical part. The solution to the spherical part of the equation can be expressed as a series of Laplace's spherical harmonics, simplifying practical computation. 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. (Commensurability, in this context, means related
by ratios of small whole numbers. Two periods of Saturn's
orbit around the Sun almost equal five of Jupiter's. The
corresponding difference between multiples of the mean
motions, (2nJ − 5nS),
corresponds to a period of nearly 900 years, and it occurs
as a small divisor in the integration of a very small
perturbing force with this same period. As a result, the
integrated perturbations with this period are
disproportionately large, about 0.8° degrees of arc in
orbital longitude for Saturn and about 0.3° for Jupiter.)
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. Laplace also produced an analytical solution (as it turned out later, a partial solution), to a significant problem regarding the motion of the Moon. Edmond Halley had been the first to suggest, in 1695, that the mean motion of the Moon was apparently getting faster, by comparison with ancient eclipse observations, but he gave no data. (It was not yet known in Halley's or Laplace's times that what is actually occurring includes a slowing down of the Earth's rate of rotation. When measured as a function of mean solar time rather than uniform time, the effect appears as a positive acceleration.) In 1749, Richard Dunthorne confirmed Halley's suspicion after re-examining ancient records, and produced the first quantitative estimate for the size of this apparent effect: a centurial rate of +10" (arcseconds) in lunar longitude (a surprisingly good result for its time, not far different from values assessed later, e.g., in 1786 by de Lalande, and to compare with values from about 10" to nearly 13" being derived about century later.) The effect became known as the secular acceleration of the Moon, but until Laplace, its cause remained unknown. Laplace gave an explanation of the effect in 1787, showing how an acceleration arises from changes (a secular reduction) in the eccentricity of the Earth's orbit, which in turn is one of the effects of planetary perturbations on the Earth. Laplace's initial computation accounted for the whole effect, thus seeming to tie up the theory neatly with both modern and ancient observations.However, in 1853, J.C. Adams caused the question to be re-opened by finding an error in Laplace's computations: it turned out that only about half of the Moon's apparent acceleration could be accounted for on Laplace's basis by the change in the Earth's orbital eccentricity. (Adams showed that Laplace had in effect only considered the radial force on the moon and not the tangential, and the partial result hence had overestimated the acceleration, the remaining (negative), terms when accounted for, showed that Laplace's cause could not explain more than about half of the acceleration. The other half was subsequently shown to be due to tidal acceleration.) Laplace used his results concerning the lunar acceleration when completing his attempted "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 Académie
des sciences, and they are printed in the Mémoires
présentés par divers savants. 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 Mécanique 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 honor 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. Laplace developed the nebular hypothesis of the formation of the solar system, first suggested by Emanuel Swedenborg and expanded by Immanuel Kant, a hypothesis that continues to dominate accounts of the origin of planetary systems. According to Laplace's description of the hypothesis, the solar system had evolved from a globular mass of incandescent gas rotating around an axis through its center 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 represented the central core which was still left. On this view, Laplace predicted that the more distant planets would be older than those nearer the sun. 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. 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 other 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. 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. 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 (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 Edwin Hubble's major discovery 100 years in advance. In 1806, Laplace bought a house in Arcueil, then a
village and not yet absorbed into the Paris conurbation.
Claude Louis Berthollet was a near neighbor 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. A frequently cited but apocryphal interaction between Laplace and Napoleon purportedly concerns the existence of God. A typical version is provided by Rouse Ball:
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:
Laplace's younger colleague, the astronomer François Arago, who gave his eulogy before the French Academy in 1827, told Faye that the garbled version of Laplace's interaction with Napoleon was already in circulation towards the end of Laplace's life. Faye writes:
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." Since the only eyewitness account of Laplace's interaction with Napoleon, an entry in the diary of the British astronomer Sir William Herschel, makes no mention of the bon mot, "I had no need of that hypothesis," Daniel Johnson argues that "Laplace never used the words attributed to him." Arago's testimony, on the other hand, appears to imply that he did, only not in reference to the existence of God. In 1470 the humanist scholar Bartolomeo Platina wrote that Pope Callixtus III had asked for prayers for deliverance from the Turks during a 1456 appearance of Halley's Comet. Platina's account does not accord with Church records, which do not mention the comet. Laplace is alleged to have embellished the story by claiming the Pope had "excommunicated" Halley's comet. What Laplace actually said, in Exposition du Système du monde (1796), was that the Pope had ordered the comet to be "exorcized" (conjuré). It was Arago, in Des Comètes en général (1832), who first spoke of an excommunication. Neither the exorcism nor the excommunication can be regarded as anything but pure fiction. In 1812, Laplace issued his Théorie analytique des probabilités in which he laid down many fundamental results in statistics. 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. The method of estimating the ratio of the number of favorable 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.
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 Carl Friedrich Gauss (around 1794) and
Legendre (in 1805), 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
meager 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. 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 (1814), Laplace set out a mathematical system of inductive reasoning based on probability, which we would today recognize as Bayesian. He begins the text with a series of principles of probability, the first six being:
One well known formula arising from his system is the rule of succession, given as principle seven. 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. 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 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 published the first articulation of causal or scientific determinism:
This intellect is often referred to as Laplace's
demon (in the same vein as Maxwell's demon)
and sometimes Laplace's Superman (after Hans
Reichenbach). Laplace, himself, did not use the word
"demon", which was a later embellishment. As translated
into English above, he simply referred to: "Une
intelligence... Rien ne serait incertain pour elle, et
l'avenir comme le passé, serait présent à ses yeux." As early as 1744, Euler, followed by Lagrange, had started looking for solutions of differential equations in the form: In 1785, Laplace took the key forward step in using
integrals of this form in order 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. Amongst the other discoveries of Laplace in pure and applicable mathematics are:
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. According to W. W. Rouse Ball, as Napoleon's power increased Laplace begged him to give him the post of Minister of the Interior. However this is disputed by Pearson. Napoleon, who desired the support of men of science, did make him Minister of the Interior in November 1799, but a little less than six weeks saw the close of Laplace's political career. Napoleon later (in his Mémoires de Sainte Hélène) wrote of his dismissal as follows:
Lucien, Napoleon's brother, was given the post. 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 Bourbon Restoration this was struck out. (Pearson points out that the censor would not have allowed it anyway.) In 1814 it was evident that the empire was falling; Laplace hastened to tender his services to the Bourbons, and in 1817 during the Restoration he was rewarded with the title of marquis. According to Rouse Ball, the contempt that his more honest colleagues felt for his conduct in the matter may be read in the pages of Paul Louis Courier. His knowledge was useful on the numerous scientific commissions on which he served, and probably accounts for the manner in which his political insincerity was overlooked. He died in Paris in 1827. His brain was removed by his
physician, François Magendie, and kept for many years,
eventually being displayed in a roving anatomical museum
in Britain. It was reportedly smaller than the average
brain.
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