피에르시몽 드 라플라스 후작

피에르시몽 라플라스

틀:포털/수학 피에르시몽 마르키 드 라플라스(프랑스어: Pierre-Simon, Marquis de Laplace, 1749년 3월 23일~1827년 3월 5일)는 프랑스 수학자이다. 그의 저서 《천체역학(총 5권)》에서는 고전역학에서 뉴턴이 택했던 방식인, 기하학적 접근방식에 대한 번역을 실어, 당시 물리학을 집대성하고 확장한 것으로 평가 받는다. 더불어《확률론의 해석이론》등의 명저를 남겼으며, 수리 물리학 발전에 엄청난 공헌을 했다. 라플라스 변환, 라플라스 방정식등에 그의 이름이 남아있다.

귀족 집안에서 태어났으며, 1765년부터 칸의 한 예수회 계열 대학에서 공부했다. 1771년부터는 파리 군관학교에서 교편을 잡았다. 나폴레옹 보나파르트가 그의 제자 중 한명이다. 1773년 파리 아카데미의 회원이 되며, 1788년 마리 샤를로트와 결혼한다. 1799년엔 내무부 장관으로 발탁된다.

그는 nebular hypothesis 를 다시 진술하고 발전시켰다. 이는 블랙홀과 중력 붕괴에 대한 최초의 이론적 예측이다.

《천체역학》에서는 강체 또는 유체의 운동에서부터, 지구의 모양, 조석이론까지 논하고 있다. 수학적으로는 이 문제들은 여러 미분방정식을 푸는 것으로 귀착되지만, 방법론적으로 그가 새롭게 제시하여 발전시킨 부분도 있어 특히 오차평가 등의 방법은 그 자신의 확률론의 응용이기도 하다.

라플라스 변환의 발견자이며, 결정론자로 잘 알려져 있어 결정론적 세계관을 라플라스 세계관이라고도 한다. 결정론적 세계관이란 지금부터 일어날 모든 현상은 현재까지 일어났던 과거의 일들이 원인이란 생각이다. 어떤 특정 시간의 우주의 모든 입자의 운동상태를 알 수 있다면, 그때부터 일어날 모든 현상을 미분방정식을 풀어 계산해 낼 수 있다(라플라스의 악마 참조). 그러나 라플라스 사후 양자역학의 성립으로 이러한 생각은 옳지 않다는 것이 반증되었다.

He is remembered as one of the greatest scientists of all time, sometimes referred to as a French Newton or Newton of France, with a phenomenal natural mathematical faculty possessed by none of his contemporaries.

He became a count of the First French Empire in 1806 and was named a marquis in 1817, after the Bourbon Restoration.

Analysis, probability and astronomical stability

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.

Stability of the solar system

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 critical 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 in practice 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.[1] 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.

On the figure of the Earth

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 Méchanique 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 the potential, a name first used by George Green in 1828.

Spherical harmonics

Spherical harmonics

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:

${\frac {1}{d}}={\frac {1}{r'}}\left[1-2\cos(\theta '-\theta ){\frac {r}{r'}}+\left({\frac {r}{r'}}\right)^{2}\right]^{-{\tfrac {1}{2}}}.$

This expression can be expanded in powers of r/r ' using Newton's generalized binomial theorem to give:

${\frac {1}{d}}={\frac {1}{r'}}\sum _{k=0}^{\infty }P_{k}^{0}(\cos(\theta '-\theta ))\left({\frac {r}{r'}}\right)^{k}.$

The sequence of functions P⁰_k(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 now in common use. Every function of the points on a sphere can be expanded as a series of them.

Potential theory

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 Clairault had first suggested the idea in 1743 while working on a similar problem though he was using Newtonian-type geometric reasoning. Laplace described Clairault'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".

Laplace applied the language of calculus to the potential function and shows that it 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 gravitational attraction was based. The quantity $\nabla ^{2}V$ 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, 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 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 centre 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.

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.

It had been observed that since ancient times the Moon's overall orbital speed was changing over time. In 1693, Edmond Halley had shown that the rate of the drift in position was increasing, an effect known as the secular acceleration of the Moon. Laplace gave an explanation in 1787 in terms of changes in the eccentricity of the Earth's orbit. However, in 1853, John Couch Adams went on to show that Laplace had only considered the radial force on the moon and not the tangential, and hence had failed to explain more than half of the drift. The other half was subsequently shown to be due to tidal acceleration. However, Laplace was still able to use his result to complete his "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.

Celestial mechanics

틀:Classical 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 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 honour 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 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 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.

파일:Laplace house Arcueil.jpg

Laplace's house at Arcueil

Black holes

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.

Analytic theory of probabilities

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.

Probability-generating function

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 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.

Least squares

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 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.

Inductive probability

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 (1814), 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 transforms

틀:Mainarticle As early as 1744, Euler, followed by Lagrange, had started looking for solutions of differential equations in the form:

$z=\int X(x)e^{ax}dx$ and $z=\int X(x)x^{a}dx.$

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.

Other discoveries and accomplishments

Mathematics

Amongst the other discoveries of Laplace in pure and applicable mathematics are:

• Discussion, contemporaneously with Alexandre-Théophile Vandermonde, of the general theory of determinants, (1772);
• Proof that every equation of an even degree must have at least one real quadratic factor;
• Solution of the linear partial differential equation of the second order;
• He was 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; and
• In his theory of probabilities:
  • Evaluation of several common definite integrals; and
  • General proof of the Lagrange reversion theorem.

Surface tension

틀:Mainarticle Laplace built upon the qualitative work of Thomas Young to develop the theory of capillary action and the Young-Laplace equation.

Speed of sound

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 that 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.

Honours

• Asteroid 4628 Laplace is named for him.
• He is one of only seventy-two people to have their names on the Eiffel Tower.

See also

• Timeline of hydrogen technologies

Bibliography

By Laplace

• Œuvres complètes de Laplace, 14 vol. (1878–1912), Paris: Gauthier-Villars (in French) (PDF copy from Gallica)
• Théorie du movement et de la figure elliptique des planètes (1784) Paris (not in Œuvres complètes)
• Précis de l'histoire de l'astronomie

English translations

• Bowditch, N. (trans.) (1829-1839) Mécanique céleste, 4 vols, Boston
  • New edition by Reprint Services ISBN 078122022X
• — [1829-1839] (1966-1969) Celestial Mechanics, 5 vols, including the original French
• Pound, J. (trans.) (1809) The System of the World, 2 vols, London: Richard Phillips
• _ The system of the world (v. 1)
• _ The system of the World (v.2)
• — [1809] (2007) The System of the World, vol.1, Kessinger, ISBN 1432653679
• Toplis, J. (trans.) (1814) A treatise upon analytical mechanics Nottingham: H. Barnett

• Truscott, F. W. & Emory, F. L. (trans.) (2007) [1902]. 《A Philosophical Essay on Probabilities》. ISBN 1602063281.  CS1 관리 - 여러 이름 (링크), translated from the French 6th ed. (1840)

About Laplace and his work

• Andoyer, H. (1922). 《L'œuvre scientifique de Laplace》. Paris: Payot.  (in French)
• Bigourdan, G. (1931). “La jeunesse de P.-S. Laplace”. 《La Science moderne》 (프랑스어) 9: 377–384. 
• Crosland, M. (1967). 《The Society of Arcueil: A View of French Science at the Time of Napoleon I》. Cambridge MA: Harvard University Press. ISBN 043554201X. 
• Dale, A. I. (1982). “Bayes or Laplace? an examination of the origin and early application of Bayes' theorem”. 《Archive for the History of the Exact Sciences》 27: 23–47. 
• David, F. N. (1965) "Some notes on Laplace", in Neyman, J. & LeCam, L. M. (eds) Bernoulli, Bayes and Laplace, Berlin, pp30-44
• Deakin, M. A. B. (1981). “The development of the Laplace transform”. 《Archive for the History of the Exact Sciences》 25: 343–390. doi:10.1007/BF01395660. 
• — (1982). “The development of the Laplace transform”. 《Archive for the History of the Exact Sciences》 26: 351–381. doi:10.1007/BF00418754. 
• Dhombres, J. (1989). “La théorie de la capillarité selon Laplace: mathématisation superficielle ou étendue”. 《Revue d'Histoire des sciences et de leurs applications》 (프랑스어) 62: 43–70. 
• Duveen, D. & Hahn, R. (1957). “Laplace's succession to Bezout's post of Examinateur des élèves de l'artillerie”. 《Isis》 48: 416–427. doi:10.1086/348608.  CS1 관리 - 여러 이름 (링크)
• Finn, B. S. (1964). “Laplace and the speed of sound”. 《Isis》 55: 7–19. doi:10.1086/349791. 
• Fourier, J. B. J. (1827). “Éloge historique de M. le Marquis de Laplace”. 《Mémoires de l'Académie Royale des Sciences》 10: lxxxi–cii. , delivered 15 June 1829, published in 1831. (in French)
• Gillispie, C. C. (1972). “Probability and politics: Laplace, Condorcet, and Turgot”. 《Proceedings of the American Philosophical Society》. 116(1): 1–20. 
• — (1997) Pierre Simon Laplace 1749-1827: A Life in Exact Science, Princeton: Princeton University Press, ISBN 0-691-01185-0
• Grattan-Guiness, I., 2005, "'Exposition du système du monde' and 'Traité de méchanique céleste'" in his Landmark Writings in Western Mathematics. Elsevier: 242-57.
• Hahn, R. (1955). “Laplace's religious views”. 《Archives internationales d'histoire des sciences》 8: 38–40. 
• — (1982). 《Calendar of the Correspondence of Pierre Simon Laplace》 Berkeley Papers ine Hiory of Science, vol.8판. Berkeley, CA: University of California. 
• — (1994). 《New Calendar of the Correspondence of Pierre Simon Laplace》 Berkeley Papers ine Hiory of Science, vol.16판. Berkeley, CA: University of California. 
• — (2005) Pierre Simon Laplace 1749-1827: A Determined Scientist, Cambridge, MA: Harvard University Press, ISBN 0-674-01892-3
• Israel, Werner (1987), 〈Dark stars: the evolution of an idea〉, Hawking, Stephen W.; Israel, Werner, 《300 Years of Gravitation》, Cambridge University Press, 199–276쪽 
• O’Connor, John J.; Robertson, Edmund F. “피에르시몽 드 라플라스 후작”. 《MacTutor History of Mathematics Archive》 (영어). 세인트앤드루스 대학교.  (1999)
• Rouse Ball, W. W. [1908] (2003) "Pierre Simon Laplace (1749 - 1827)", in A Short Account of the History of Mathematics, 4th ed., Dover, ISBN 0486206300
• Stigler, S. M. (1975). “Napoleonic statistics: the work of Laplace”. 《Biometrika》 62: 503–517. doi:10.2307/2335393. 
• — (1978). “Laplace's early work: chronology and citations”. 《Isis》. 69(2): 234–254. 
• Whitrow, G. J. (2001) "Laplace, Pierre-Simon, marquis de", Encyclopaedia Britannica, Deluxe CDROM edition
• Whittaker, E. T. (1949a). “Laplace”. 《Mathematical Gazette》 33: 1–12. doi:10.2307/3608408. 
• — (1949b). “Laplace”. 《American Mathematical Monthly》. 56(6): 369–372. 
• Wilson, C. (1985). “The Great Inequality of Jupiter and Saturn: from Kepler to Laplace”. 《Archive for the History of the Exact Sciences》. 33(1-3): 15–290. doi:10.1007/BF00328048. 
• Young, T. (1821). 《Elementary Illustrations of the Celestial Mechanics of Laplace: Part the First, Comprehending the First Book》. London: John Murray.  (available from Google Books)

External links

• “Laplace, Pierre (1749-1827)”. 《Eric Weisstein's World of Scientific Biography》. Wolfram Research. 2007년 8월 24일에 확인함. 
• "Pierre-Simon Laplace" in the MacTutor History of Mathematics archive.
• “Bowditch's English translation of Laplace's preface”. 《Méchanique Céleste》. The MacTutor History of Mathematics archive. 2007년 9월 4일에 확인함.