Pierre-Simon, marqués de Laplace ( / l ə p l ɑː s / ; Francés: [pjɛʁ simɔ Laplas] ; 23 marzo 1749-5 marzo 1827) fue un erudito francés y polifacético cuyo trabajo era importante para el desarrollo de la ingeniería , matemáticas , estadística , física , astronomía y filosofía . Resumió y amplió el trabajo de sus predecesores en su Mécanique Céleste ( Mecánica celeste) (1799-1825). Este trabajo tradujo el estudio geométrico de la mecánica clásica a uno basado en el cálculo , abriendo una gama más amplia de problemas. En estadística, la interpretación bayesiana de la probabilidad fue desarrollada principalmente por Laplace. [2]
Pierre-Simon Laplace | |
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Nació | Beaumont-en-Auge , Normandía, Reino de Francia | 23 de marzo de 1749
Fallecido | 5 de marzo de 1827 | (77 años)
Nacionalidad | francés |
alma mater | Universidad de Caen |
Conocido por |
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Carrera científica | |
Campos | Astronomía y Matemáticas |
Instituciones | École Militaire (1769-1776) |
Asesores académicos | Jean d'Alembert Christophe Gadbled Pierre Le Canu |
Estudiantes notables | Siméon Denis Poisson Napoleón Bonaparte |
Firma | |
Laplace formuló la ecuación de Laplace y fue pionero en la transformada de Laplace que aparece en muchas ramas de la física matemática , un campo en el que asumió un papel principal en la formación. El operador diferencial laplaciano , ampliamente utilizado en matemáticas, también lleva su nombre. Reiteró y desarrolló la hipótesis nebular del origen del Sistema Solar y fue uno de los primeros científicos en postular la existencia de agujeros negros y la noción de colapso gravitacional .
Laplace es recordado como uno de los más grandes científicos de todos los tiempos. A veces conocido como el Newton francés o el Newton de Francia , se le ha descrito como poseedor de una facultad matemática natural fenomenal superior a la de cualquiera de sus contemporáneos. [3] Fue examinador de Napoleón cuando Napoleón asistió a la École Militaire de París en 1784. Laplace se convirtió en conde del Imperio en 1806 y fue nombrado marqués en 1817, después de la Restauración borbónica .
Primeros años
Se desconocen algunos detalles de la vida de Laplace, ya que los registros de la misma se quemaron en 1925 con el castillo familiar en Saint Julien de Mailloc , cerca de Lisieux , la casa de su tataranieto el conde de Colbert-Laplace. Otros habían sido destruidos antes, cuando su casa en Arcueil, cerca de París, fue saqueada en 1871. [4]
Laplace nació en Beaumont-en-Auge , Normandía, el 23 de marzo de 1749, un pueblo a seis kilómetros al oeste de Pont l'Évêque . Según WW Rouse Ball , [5] su padre, Pierre de Laplace, era propietario y cultivaba las pequeñas propiedades de Maarquis. Su tío abuelo, el maitre Oliver de Laplace, ostentaba el título de Chirurgien Royal. Parece que de alumno se convirtió en acomodador de la escuela de Beaumont; pero, habiendo conseguido una carta de presentación para D'Alembert , fue a París para adelantar su fortuna. Sin embargo, Karl Pearson [4] es mordaz sobre las inexactitudes en el relato de Rouse Ball y afirma:
De hecho, Caen fue probablemente en la época de Laplace la más activa intelectualmente de todas las ciudades de Normandía. Fue aquí donde Laplace se educó y fue provisionalmente profesor. Fue aquí donde escribió su primer artículo publicado en las Mélanges de la Real Sociedad de Turín, Tomo IV. 1766-1769, al menos dos años antes de ir a los 22 o 23 años a París en 1771. Así, antes de los 20, estuvo en contacto con Lagrange en Turín . ¡No fue a París como un joven campesino autodidacta con solo antecedentes campesinos! En 1765, a la edad de dieciséis años, Laplace dejó la "Escuela del Duque de Orleans" en Beaumont y fue a la Universidad de Caen , donde parece haber estudiado durante cinco años y fue miembro de la Esfinge. La ' École Militaire ' de Beaumont no reemplazó a la vieja escuela hasta 1776.
Sus padres, Pierre Laplace y Marie-Anne Sochon, procedían de familias cómodas. La familia Laplace estuvo involucrada en la agricultura hasta al menos 1750, pero Pierre Laplace padre también era un comerciante de sidra y síndico de la ciudad de Beaumont.
Pierre Simon Laplace asistió a una escuela en el pueblo administrada en un priorato benedictino , su padre tenía la intención de que fuera ordenado en la Iglesia Católica Romana . A los dieciséis años, para promover la intención de su padre, fue enviado a la Universidad de Caen para leer teología. [6]
En la universidad, fue asesorado por dos entusiastas profesores de matemáticas, Christophe Gadbled y Pierre Le Canu, quienes despertaron su celo por la asignatura. Aquí se reconoció rápidamente la brillantez de Laplace como matemático y, mientras estaba en Caen, escribió un libro de memorias Sur le Calcul integral aux difference infiniment petites et aux difference finies . Esto proporcionó la primera relación entre Laplace y Lagrange. Lagrange era mayor por trece años, y recientemente había fundado en su ciudad natal Turín una revista llamada Miscellanea Taurinensia , en la que se imprimieron muchas de sus primeras obras y fue en el cuarto volumen de esta serie donde apareció el artículo de Laplace. Por esa época, reconociendo que no tenía vocación por el sacerdocio, decidió convertirse en matemático profesional. Algunas fuentes afirman que luego rompió con la iglesia y se convirtió en ateo. [ cita requerida ] Laplace no se graduó en teología, pero se fue a París con una carta de presentación de Le Canu a Jean le Rond d'Alembert, quien en ese momento era supremo en los círculos científicos. [6] [7]
Según su tataranieto, [4] d'Alembert lo recibió bastante mal, y para deshacerse de él le dio un grueso libro de matemáticas, diciendo que regresara cuando lo hubiera leído. Cuando Laplace regresó unos días después, d'Alembert se mostró aún menos amigable y no ocultó su opinión de que era imposible que Laplace pudiera haber leído y entendido el libro. Pero al interrogarlo, se dio cuenta de que era cierto, y desde ese momento tomó a Laplace bajo su cuidado.
Otro relato es que Laplace resolvió de la noche a la mañana un problema que d'Alembert le puso para que lo presentara la semana siguiente y luego resolvió un problema más difícil la noche siguiente. D'Alembert quedó impresionado y lo recomendó para un puesto de enseñanza en la École Militaire . [8]
Con un ingreso seguro y una enseñanza poco exigente, Laplace ahora se dedicó a la investigación original y durante los siguientes diecisiete años, 1771-1787, produjo gran parte de su trabajo original en astronomía. [9]
De 1780 a 1784, Laplace y el químico francés Antoine Lavoisier colaboraron en varias investigaciones experimentales, diseñando su propio equipo para la tarea. [10] En 1783 publicaron su artículo conjunto, Memoir on Heat , en el que discutían la teoría cinética del movimiento molecular. [11] En sus experimentos midieron el calor específico de varios cuerpos y la expansión de los metales al aumentar la temperatura. También midieron los puntos de ebullición del etanol y el éter bajo presión.
Laplace impresionó aún más al marqués de Condorcet , y ya en 1771 Laplace se sentía con derecho a ser miembro de la Academia de Ciencias de Francia . Sin embargo, ese año la admisión fue para Alexandre-Théophile Vandermonde y en 1772 para Jacques Antoine Joseph Cousin. Laplace estaba descontento y, a principios de 1773, d'Alembert escribió a Lagrange en Berlín para preguntarle si se podía encontrar un puesto para Laplace allí. Sin embargo, Condorcet se convirtió en secretario permanente de la Académie en febrero y Laplace fue elegido miembro asociado el 31 de marzo, a los 24 años. [12] En 1773 Laplace leyó su artículo sobre la invariabilidad del movimiento planetario frente a la Academia de Ciencias. Ese marzo fue elegido miembro de la academia, un lugar donde dirigió la mayor parte de su ciencia. [13]
El 15 de marzo de 1788, [14] [4] a la edad de treinta y nueve años, Laplace se casó con Marie-Charlotte de Courty de Romanges, una mujer de dieciocho años de una "buena" familia en Besançon . [15] La boda se celebró en Saint-Sulpice, París . La pareja tuvo un hijo, Charles-Émile (1789–1874) y una hija, Sophie-Suzanne (1792–1813). [16] [17]
Análisis, probabilidad y estabilidad astronómica
El primer trabajo publicado por Laplace en 1771 comenzó con ecuaciones diferenciales y diferencias finitas, pero ya estaba comenzando a pensar en los conceptos matemáticos y filosóficos de probabilidad y estadística. [18] Sin embargo, antes de su elección a la Academia en 1773, ya había redactado dos artículos que establecerían su reputación. El primero, Mémoire sur la probabilité des cause par les événements se publicó finalmente en 1774, mientras que el segundo artículo, publicado en 1776, profundizó su pensamiento estadístico y también comenzó su trabajo sistemático sobre la mecánica celeste y la estabilidad del Sistema Solar. Las dos disciplinas siempre estarán interrelacionadas en su mente. "Laplace tomó la probabilidad como un instrumento para reparar defectos en el conocimiento". [19] El trabajo de Laplace sobre probabilidad y estadística se analiza a continuación con su trabajo maduro sobre la teoría analítica de probabilidades.
Estabilidad del sistema solar
Sir Isaac Newton había publicado su Philosophiae Naturalis Principia Mathematica en 1687 en la que dio una derivación de las leyes de Kepler , que describen el movimiento de los planetas, a partir de sus leyes de movimiento y su ley de gravitación universal . Sin embargo, aunque Newton había desarrollado en forma privada los métodos de cálculo, todo su trabajo publicado utilizó un razonamiento geométrico engorroso, inadecuado para explicar los efectos más sutiles de orden superior de las interacciones entre los planetas. El propio Newton había dudado de la posibilidad de una solución matemática del conjunto, llegando incluso a concluir que era necesaria la intervención divina periódica para garantizar la estabilidad del Sistema Solar. Prescindir de la hipótesis de la intervención divina sería una de las principales actividades de la vida científica de Laplace. [20] Actualmente se considera generalmente que los métodos de Laplace por sí mismos, aunque vitales para el desarrollo de la teoría, no son lo suficientemente precisos para demostrar la estabilidad del Sistema Solar , [21] y, de hecho, se entiende que el Sistema Solar es caótico , aunque resulta ser bastante estable.
Un problema particular de la astronomía observacional fue la aparente inestabilidad por la cual la órbita de Júpiter parecía encogerse mientras que la de Saturno se expandía. El problema había sido abordado por Leonhard Euler en 1748 y Joseph Louis Lagrange en 1763, pero sin éxito. [22] En 1776, Laplace publicó una memoria en la que primero exploró las posibles influencias de un supuesto éter luminífero o de una ley de gravitación que no actuaba instantáneamente. Finalmente regresó a una inversión intelectual en la gravedad newtoniana. [23] Euler y Lagrange habían hecho una aproximación práctica ignorando términos pequeños en las ecuaciones de movimiento. Laplace señaló que, aunque los términos en sí eran pequeños, cuando se integraban con el tiempo, podían volverse importantes. Laplace llevó su análisis a los términos de orden superior, hasta e incluyendo el cúbico . Usando este análisis más exacto, Laplace concluyó que dos planetas cualesquiera y el Sol deben estar en equilibrio mutuo y, por lo tanto, lanzó su trabajo sobre la estabilidad del Sistema Solar. [24] Gerald James Whitrow describió el logro como "el avance más importante en astronomía física desde Newton". [20]
Laplace tenía un amplio conocimiento de todas las ciencias y dominó todas las discusiones en la Académie . [25] Laplace parece haber considerado el análisis simplemente como un medio para atacar los problemas físicos, aunque la habilidad con la que inventó el análisis necesario es casi fenomenal. Mientras sus resultados fueran ciertos, se tomó muy poco trabajo para explicar los pasos por los que llegó a ellos; nunca estudió la elegancia o la simetría en sus procesos, y le bastaba con poder resolver por cualquier medio la cuestión particular que estaba discutiendo. [9]
Dinámica de las mareas
Teoría dinámica de las mareas
Mientras Newton explicó las mareas mediante la descripción de las fuerzas generadoras de mareas y Bernoulli dio una descripción de la reacción estática de las aguas en la Tierra para el potencial de las mareas, la teoría dinámica de las mareas , desarrollado por Laplace en 1775, [26] describe real de la océano reacción a las fuerzas de las mareas . [27] La teoría de Laplace de las mareas oceánicas tuvo en cuenta la fricción , la resonancia y los períodos naturales de las cuencas oceánicas. Predijo los grandes sistemas anfidrómicos en las cuencas oceánicas del mundo y explica las mareas oceánicas que realmente se observan. [28] [29]
La teoría del equilibrio, basada en el gradiente gravitacional del Sol y la Luna pero ignorando la rotación de la Tierra, los efectos de los continentes y otros efectos importantes, no pudo explicar las mareas reales del océano. [30] [31] [32] [28] [33] [34] [35] [36] [37]
Dado que las mediciones han confirmado la teoría, muchas cosas ahora tienen posibles explicaciones, como cómo las mareas interactúan con las dorsales del mar profundo y las cadenas de montañas submarinas dan lugar a profundos remolinos que transportan nutrientes desde las profundidades a la superficie. [38] La teoría de la marea de equilibrio calcula la altura de la marea de menos de medio metro, mientras que la teoría dinámica explica por qué las mareas son de hasta 15 metros. [39] Las observaciones satelitales confirman la precisión de la teoría dinámica, y las mareas en todo el mundo ahora se miden dentro de unos pocos centímetros. [40] [41] Las mediciones del satélite CHAMP se asemejan mucho a los modelos basados en los datos de TOPEX . [42] [43] [44] Los modelos precisos de las mareas en todo el mundo son esenciales para la investigación, ya que las variaciones debidas a las mareas deben eliminarse de las mediciones al calcular la gravedad y los cambios en el nivel del mar. [45]
Ecuaciones de mareas de Laplace
En 1776, Laplace formuló un solo conjunto de ecuaciones diferenciales parciales lineales , para el flujo de marea descrito como un flujo laminar bidimensional barotrópico . Se introducen los efectos de Coriolis y el forzamiento lateral por gravedad. Laplace obtuvo estas ecuaciones simplificando las ecuaciones de dinámica de fluidos . Pero también pueden derivarse de integrales de energía a través de la ecuación de Lagrange .
Para una lámina de fluido de espesor medio D , la elevación de marea vertical ζ , así como los componentes de velocidad horizontal u y v (en las direcciones de latitud φ y longitud λ , respectivamente) satisfacen las ecuaciones de marea de Laplace : [46]
donde Ω es la frecuencia angular de la rotación del planeta, g es la aceleración gravitacional del planeta en la superficie media del océano, a es el radio planetario y U es el potencial de fuerza de marea gravitacional externa .
William Thomson (Lord Kelvin) reescribió los términos del momento de Laplace usando el rizo para encontrar una ecuación de vorticidad . Bajo ciertas condiciones, esto se puede reescribir aún más como una conservación de la vorticidad.
Sobre la figura de la tierra
Durante los años 1784-1787 publicó algunas memorias de excepcional poder. Entre ellos destaca uno leído en 1783, reimpreso como Parte II de Théorie du Mouvement et de la figure elliptique des planètes en 1784, y en el tercer volumen de la Mécanique céleste . En este trabajo, Laplace determinó por completo la atracción de un esferoide sobre una partícula fuera de él. Esto es memorable por la introducción al análisis de armónicos esféricos o coeficientes de Laplace , y también por el desarrollo del uso de lo que ahora llamaríamos el potencial gravitacional en mecánica celeste .
Armónicos esféricos
En 1783, en un artículo enviado a la Academia , Adrien-Marie Legendre había introducido lo que ahora se conoce como funciones asociadas de Legendre . [9] Si dos puntos en un plano tienen coordenadas polares ( r , θ) y ( r ', θ'), donde r '≥ r , entonces, por manipulación elemental, el recíproco de la distancia entre los puntos, d , Se puede escribir como:
Esta expresión se puede expandir en potencias de r / r 'usando el teorema binomial generalizado de Newton para dar:
La secuencia de funciones P 0 k (cos φ) es el conjunto de las llamadas "funciones de Legendre asociadas" y su utilidad surge del hecho de que cada función de los puntos de un círculo puede expandirse como una serie de ellos. [9]
Laplace, con escasa consideración por el crédito de Legendre, hizo la extensión no trivial del resultado a tres dimensiones para producir un conjunto más general de funciones, los armónicos esféricos o coeficientes de Laplace . El último término no es de uso común ahora. [9]
Teoría potencial
This paper is also remarkable for the development of the idea of the scalar potential.[9] 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".[47] 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".[9] 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.[48][49]
Laplace applied the language of calculus to the potential function and showed that it always satisfies the differential equation:[9]
An analogous result for the velocity potential of a fluid had been obtained some years previously by Leonhard Euler.[50][51]
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 neighbourhood of the point.[52] 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 a priori forms in Kant's theory of perception.[9]
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.
Desigualdades planetarias y lunares
Jupiter–Saturn great inequality
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.[3][53]
In this context commensurability means that the ratio of the two planets' mean motions is very nearly equal to a ratio between a pair 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.[9]
Books
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."[3] The result is embodied in the Exposition du système du monde and the Mécanique céleste.[9]
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.[9]
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.[9][54]
As mentioned, the idea of the nebular hypothesis had been outlined by Immanuel Kant in 1755,[54] 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.[4]
Laplace's analytical discussion of the Solar System is given in his Mécanique 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.[3] 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 organised result of a century of patient toil — are frequently mentioned as if they were due to Laplace.[9]
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.[9] In the years 1784–1787, Laplace produced some memoirs of exceptional power. The significant among these was one issued in 1784, and reprinted in the third volume of the Méchanique céleste.[citation needed] In this work he completely determined the attraction of a spheroid on a particle outside it. This is known for the introduction into analysis of the potential, a useful mathematical concept of broad applicability to the physical sciences.
Agujeros negros
Laplace also came close to propounding the concept of the black hole. He suggested that there could be massive stars whose gravity is so great that not even light could escape from their surface (see escape velocity).[55][1][56][57] However, this insight was so far ahead of its time that it played no role in the history of scientific development.[58]
Arcueil
In 1806, Laplace bought a house in Arcueil, then a village and not yet absorbed into the Paris conurbation. The chemist Claude Louis Berthollet was a neighbour – their gardens were not separated[59] – 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.[60] In 1806, Laplace was also elected a foreign member of the Royal Swedish Academy of Sciences.
Teoría analítica de probabilidades
In 1812, Laplace issued his Théorie analytique des probabilités in which he laid down many fundamental results in statistics. The first half of this treatise was concerned with probability methods and problems, the second half with statistical methods and applications. Laplace's proofs are not always rigorous according to the standards of a later day, and his perspective slides back and forth between the Bayesian and non-Bayesian views with an ease that makes some of his investigations difficult to follow, but his conclusions remain basically sound even in those few situations where his analysis goes astray.[61] 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.[9] In its emphasis on the analytical importance of probabilistic problems, especially in the context of the "approximation of formula functions of large numbers," Laplace's work goes beyond the contemporary view which almost exclusively considered aspects of practical applicability.[62] Laplace's Théorie analytique remained the most influential book of mathematical probability theory to the end of the 19th century. The general relevance for statistics of Laplacian error theory was appreciated only by the end of the 19th century. However, it influenced the further development of a largely analytically oriented probability theory.
Inductive probability
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. He begins the text with a series of principles of probability, the first six being:
- Probability is the ratio of the "favored events" to the total possible events.
- The first principle assumes equal probabilities for all events. When this is not true, we must first determine the probabilities of each event. Then, the probability is the sum of the probabilities of all possible favoured events.
- For independent events, the probability of the occurrence of all is the probability of each multiplied together.
- For events not independent, the probability of event B following event A (or event A causing B) is the probability of A multiplied by the probability that, given A, B will occur.
- The probability that A will occur, given that B has occurred, is the probability of A and B occurring divided by the probability of B.
- Three corollaries are given for the sixth principle, which amount to Bayesian probability. Where event Ai ∈ {A1, A2, ... An} exhausts the list of possible causes for event B, Pr(B) = Pr(A1, A2, ..., An). Then
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, labelled "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 have only 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."[63]
Probability-generating function
The method of estimating the ratio of the number of favourable 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.[3] The latter is therefore called the probability-generating function of the former.[3] 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.[9]
Least squares and central limit theorem
The fourth chapter of this treatise includes an exposition of the method of least squares, a remarkable testimony to Laplace's command over the processes of analysis. In 1805 Legendre had published the method of least squares, making no attempt to tie it to the theory of probability. In 1809 Gauss had derived the normal distribution from the principle that the arithmetic mean of observations gives the most probable value for the quantity measured; then, turning this argument back upon itself, he showed that, if the errors of observation are normally distributed, the least squares estimates give the most probable values for the coefficients in regression situations. These two works seem to have spurred Laplace to complete work toward a treatise on probability he had contemplated as early as 1783.[61]
In two important papers in 1810 and 1811, Laplace first developed the characteristic function as a tool for large-sample theory and proved the first general central limit theorem. Then in a supplement to his 1810 paper written after he had seen Gauss's work, he showed that the central limit theorem provided a Bayesian justification for least squares: if one were combining observations, each one of which was itself the mean of a large number of independent observations, then the least squares estimates would not only maximise the likelihood function, considered as a posterior distribution, but also minimise the expected posterior error, all this without any assumption as to the error distribution or a circular appeal to the principle of the arithmetic mean.[61] In 1811 Laplace took a different non-Bayesian tack. Considering a linear regression problem, he restricted his attention to linear unbiased estimators of the linear coefficients. After showing that members of this class were approximately normally distributed if the number of observations was large, he argued that least squares provided the "best" linear estimators. Here it is "best" in the sense that it minimised the asymptotic variance and thus both minimised the expected absolute value of the error, and maximised the probability that the estimate would lie in any symmetric interval about the unknown coefficient, no matter what the error distribution. His derivation included the joint limiting distribution of the least squares estimators of two parameters.[61]
Demonio de Laplace
In 1814, Laplace published what may have been the first scientific articulation of causal determinism:[64]
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.
— Pierre Simon Laplace, A Philosophical Essay on Probabilities[65]
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."
Even though Laplace is generally credited with having first formulated the concept of causal determinism, in a philosophical context the idea was actually widespread at the time, and can be found as early as 1756 in Maupertuis' 'Sur la Divination'.[66] Jesuit scientist Boscovich first proposed a version of scientific determinism very similar to Laplace's in his 1758 book Theoria philosophiae naturalis.[67]
Transformaciones de Laplace
As early as 1744, Euler, followed by Lagrange, had started looking for solutions of differential equations in the form:[68]
The Laplace transform has form:
This integral operator transforms a function of time (t) into a function of position or space (s).
In 1785, Laplace took the key forward step in using integrals of this form to transform a whole differential equation from a function of time into a lower order function of space. The transformed equation was easier to solve than the original because algebra could be used to manipulate the transformed differential equation into a simpler form. The inverse Laplace transform was then taken to convert the simplified function of space back into a function of time.[69][70]
Otros descubrimientos y logros
Mathematics
Amongst the other discoveries of Laplace in pure and applied mathematics are:
- Discussion, contemporaneously with Alexandre-Théophile Vandermonde, of the general theory of determinants, (1772);[9]
- Proof that every equation of an odd degree must have at least one real quadratic factor[clarification needed];[9]
- Laplace's method for approximating integrals
- Solution of the linear partial differential equation of the second order;[9]
- 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 always be obtained in the form of a continued fraction;[3][9]
- In his theory of probabilities:
- de Moivre–Laplace theorem that approximates binomial distribution with a normal distribution
- Evaluation of several common definite integrals;[9]
- General proof of the Lagrange reversion theorem.[9]
Surface tension
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 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.[9]
Política
Minister of the Interior
In his early years Laplace was careful never to become involved in politics, or indeed in life outside the Académie des sciences. He prudently withdrew from Paris during the most violent part of the Revolution.[71]
In November 1799, immediately after seizing power in the coup of 18 Brumaire, Napoleon appointed Laplace to the post of Minister of the Interior.[3] The appointment, however, lasted only six weeks, after which Lucien Bonaparte, Napoleon's brother, was given the post.[3] Evidently, once Napoleon's grip on power was secure, there was no need for a prestigious but inexperienced scientist in the government.[72] Napoleon later (in his Mémoires de Sainte Hélène) wrote of Laplace's dismissal as follows:[9]
Géomètre de premier rang, Laplace ne tarda pas à se montrer administrateur plus que médiocre; dès son premier travail nous reconnûmes que nous nous étions trompé. Laplace ne saisissait aucune question sous son véritable point de vue: il cherchait des subtilités partout, n'avait que des idées problématiques, et portait enfin l'esprit des 'infiniment petits' jusque dans l'administration. (Geometrician of the first rank, Laplace was not long in showing himself a worse than average administrator; from his first actions in office we recognized our mistake. Laplace did not consider any question from the right angle: he sought subtleties everywhere, conceived only problems, and finally carried the spirit of "infinitesimals" into the administration.)
Grattan-Guinness, however, describes these remarks as "tendentious", since there seems to be no doubt that Laplace "was only appointed as a short-term figurehead, a place-holder while Napoleon consolidated power".[72]
From Bonaparte to the Bourbons
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.[3] 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, says Rouse Ball, probably accounts for the manner in which his political insincerity was overlooked.[9]
Roger Hahn in his 2005 biography disputes this portrayal of Laplace as an opportunist and turncoat, pointing out that, like many in France, he had followed the debacle of Napoleon's Russian campaign with serious misgivings. The Laplaces, whose only daughter Sophie had died in childbirth in September 1813, were in fear for the safety of their son Émile, who was on the eastern front with the emperor. Napoleon had originally come to power promising stability, but it was clear that he had overextended himself, putting the nation at peril. It was at this point that Laplace's loyalty began to weaken. Although he still had easy access to Napoleon, his personal relations with the emperor cooled considerably. As a grieving father, he was particularly cut to the quick by Napoleon's insensitivity in an exchange related by Jean-Antoine Chaptal: "On his return from the rout in Leipzig, he [Napoleon] accosted Mr Laplace: 'Oh! I see that you have grown thin—Sire, I have lost my daughter—Oh! that's not a reason for losing weight. You are a mathematician; put this event in an equation, and you will find that it adds up to zero.'"[73]
Political philosophy
In the second edition (1814) of the Essai philosophique, Laplace added some revealing comments on politics and governance. Since it is, he says, "the practice of the eternal principles of reason, justice and humanity that produce and preserve societies, there is a great advantage to adhere to these principles, and a great inadvisability to deviate from them".[74][75] Noting "the depths of misery into which peoples have been cast" when ambitious leaders disregard these principles, Laplace makes a veiled criticism of Napoleon's conduct: "Every time a great power intoxicated by the love of conquest aspires to universal domination, the sense of liberty among the unjustly threatened nations breeds a coalition to which it always succumbs." Laplace argues that "in the midst of the multiple causes that direct and restrain various states, natural limits" operate, within which it is "important for the stability as well as the prosperity of empires to remain". States that transgress these limits cannot avoid being "reverted" to them, "just as is the case when the waters of the seas whose floor has been lifted by violent tempests sink back to their level by the action of gravity".[76][77]
About the political upheavals he had witnessed, Laplace formulated a set of principles derived from physics to favour evolutionary over revolutionary change:
Let us apply to the political and moral sciences the method founded upon observation and calculation, which has served us so well in the natural sciences. Let us not offer fruitless and often injurious resistance to the inevitable benefits derived from the progress of enlightenment; but let us change our institutions and the usages that we have for a long time adopted only with extreme caution. We know from past experience the drawbacks they can cause, but we are unaware of the extent of ills that change may produce. In the face of this ignorance, the theory of probability instructs us to avoid all change, especially to avoid sudden changes which in the moral as well as the physical world never occur without a considerable loss of vital force.[78]
In these lines, Laplace expressed the views he had arrived at after experiencing the Revolution and the Empire. He believed that the stability of nature, as revealed through scientific findings, provided the model that best helped to preserve the human species. "Such views," Hahn comments, "were also of a piece with his steadfast character."[77]
In the Essai philosophique, Laplace also illustrates the potential of probabilities in political studies by applying the law of large numbers to justify the candidates’ integer-valued ranks used in the Borda method of voting, with which the new members of the Academy of Sciences were elected. Laplace’s verbal argument is so rigorous that it can easily be converted into a formal proof.[79][80]
Muerte
Laplace died in Paris on 5 March 1827, which was the same day Alessandro Volta died. 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.[4] Laplace was buried at Père Lachaise in Paris but in 1888 his remains were moved to Saint Julien de Mailloc in the canton of Orbec and reinterred on the family estate.[81] The tomb is situated on a hill overlooking the village of St Julien de Mailloc, Normandy, France.
Opiniones religiosas
I had no need of that hypothesis
A frequently cited but potentially apocryphal interaction between Laplace and Napoleon purportedly concerns the existence of God. Although the conversation in question did occur, the exact words Laplace used and his intended meaning are not known. A typical version is provided by Rouse Ball:[9]
Laplace went in state to Napoleon to present a copy of his work, and the following account of the interview is well authenticated, and so characteristic of all the parties concerned that I quote it in full. 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 had no need of that hypothesis.") Napoleon, greatly amused, told this reply to Lagrange, who exclaimed, Ah! c'est une belle hypothèse; ça explique beaucoup de choses. ("Ah, it is a fine hypothesis; it explains many things.")
An earlier report, although without the mention of Laplace's name, is found in Antommarchi's The Last Moments of Napoleon (1825):[82]
Je m'entretenais avec L ..... je le félicitais d'un ouvrage qu'il venait de publier et lui demandais comment le nom de Dieu, qui se reproduisait sans cesse sous la plume de Lagrange, ne s'était pas présenté une seule fois sous la sienne. C'est, me répondit-il, que je n'ai pas eu besoin de cette hypothèse. ("While speaking with L ..... I congratulated him on a work which he had just published and asked him how the name of God, which appeared endlessly in the works of Lagrange, didn't occur even once in his. He replied that he had no need of that hypothesis.")
In 1884, however, the astronomer Hervé Faye[83][84] 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:
In fact Laplace never said that. Here, I believe, is what truly happened. Newton, believing that the secular perturbations which he had sketched out in his theory would in the long run end up destroying the Solar System, says somewhere that God was obliged to intervene from time to time to remedy the evil and somehow keep the system working properly. This, however, was a pure supposition suggested to Newton by an incomplete view of the conditions of the stability of our little world. Science was not yet advanced enough at that time to bring these conditions into full view. But Laplace, who had discovered them by a deep analysis, would have replied to the First Consul that Newton had wrongly invoked the intervention of God to adjust from time to time the machine of the world (la machine du monde) and that he, Laplace, had no need of such an assumption. It was not God, therefore, that Laplace treated as a hypothesis, but his intervention in a certain place.
Laplace's younger colleague, the astronomer François Arago, who gave his eulogy before the French Academy in 1827,[85] told Faye of an attempt by Laplace to keep the garbled version of his interaction with Napoleon out of circulation. Faye writes:[83][84]
I have it on the authority of M. Arago that Laplace, warned shortly before his death that that anecdote was about to be published in a biographical collection, had requested him [Arago] to demand its deletion by the publisher. It was necessary to either explain or delete it, and the second way was the easiest. But, unfortunately, it was neither deleted nor explained.
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.[86] Stephen Hawking said in 1999,[64] "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."
The only eyewitness account of Laplace's interaction with Napoleon is from the entry for 8 August 1802 in the diary of the British astronomer Sir William Herschel:[87]
The first Consul then asked a few questions relating to Astronomy and the construction of the heavens to which I made such answers as seemed to give him great satisfaction. He also addressed himself to Mr Laplace on the same subject, and held a considerable argument with him in which he differed from that eminent mathematician. The difference was occasioned by an exclamation of the first Consul, who asked in a tone of exclamation or admiration (when we were speaking of the extent of the sidereal heavens): 'And who is the author of all this!' Mons. De la Place wished to shew that a chain of natural causes would account for the construction and preservation of the wonderful system. This the first Consul rather opposed. Much may be said on the subject; by joining the arguments of both we shall be led to 'Nature and nature's God'.
Since this makes no mention of Laplace's saying, "I had no need of that hypothesis," Daniel Johnson[88] argues that "Laplace never used the words attributed to him." Arago's testimony, however, appears to imply that he did, only not in reference to the existence of God.
Views on God
Raised a Catholic, Laplace appears in adult life to have inclined to deism (presumably his considered position, since it is the only one found in his writings). However, some of his contemporaries thought he was an atheist, while a number of recent scholars have described him as agnostic.
Faye thought that Laplace "did not profess atheism",[83] but Napoleon, on Saint Helena, told General Gaspard Gourgaud, "I often asked Laplace what he thought of God. He owned that he was an atheist."[89] Roger Hahn, in his biography of Laplace, mentions a dinner party at which "the geologist Jean-Étienne Guettard was staggered by Laplace's bold denunciation of the existence of God". It appeared to Guettard that Laplace's atheism "was supported by a thoroughgoing materialism".[90] But the chemist Jean-Baptiste Dumas, who knew Laplace well in the 1820s, wrote that Laplace "provided materialists with their specious arguments, without sharing their convictions".[91][92]
Hahn states: "Nowhere in his writings, either public or private, does Laplace deny God's existence."[93] Expressions occur in his private letters that appear inconsistent with atheism.[3] On 17 June 1809, for instance, he wrote to his son, "Je prie Dieu qu'il veille sur tes jours. Aie-Le toujours présent à ta pensée, ainsi que ton père et ta mère [I pray that God watches over your days. Let Him be always present to your mind, as also your father and your mother]."[84][94] Ian S. Glass, quoting Herschel's account of the celebrated exchange with Napoleon, writes that Laplace was "evidently a deist like Herschel".[95]
In Exposition du système du monde, Laplace quotes Newton's assertion that "the wondrous disposition of the Sun, the planets and the comets, can only be the work of an all-powerful and intelligent Being".[96] This, says Laplace, is a "thought in which he [Newton] would be even more confirmed, if he had known what we have shown, namely that the conditions of the arrangement of the planets and their satellites are precisely those which ensure its stability".[97] By showing that the "remarkable" arrangement of the planets could be entirely explained by the laws of motion, Laplace had eliminated the need for the "supreme intelligence" to intervene, as Newton had "made" it do.[98] Laplace cites with approval Leibniz's criticism of Newton's invocation of divine intervention to restore order to the Solar System: "This is to have very narrow ideas about the wisdom and the power of God."[99] He evidently shared Leibniz's astonishment at Newton's belief "that God has made his machine so badly that unless he affects it by some extraordinary means, the watch will very soon cease to go".[100]
In a group of manuscripts, preserved in relative secrecy in a black envelope in the library of the Académie des sciences and published for the first time by Hahn, Laplace mounted a deist critique of Christianity. It is, he writes, the "first and most infallible of principles ... to reject miraculous facts as untrue".[101] As for the doctrine of transubstantiation, it "offends at the same time reason, experience, the testimony of all our senses, the eternal laws of nature, and the sublime ideas that we ought to form of the Supreme Being". It is the sheerest absurdity to suppose that "the sovereign lawgiver of the universe would suspend the laws that he has established, and which he seems to have maintained invariably".[102]
In old age, Laplace remained curious about the question of God[103] and frequently discussed Christianity with the Swiss astronomer Jean-Frédéric-Théodore Maurice.[104] He told Maurice that "Christianity is quite a beautiful thing" and praised its civilising influence. Maurice thought that the basis of Laplace's beliefs was, little by little, being modified, but that he held fast to his conviction that the invariability of the laws of nature did not permit of supernatural events.[103] After Laplace's death, Poisson told Maurice, "You know that I do not share your [religious] opinions, but my conscience forces me to recount something that will surely please you." When Poisson had complimented Laplace about his "brilliant discoveries", the dying man had fixed him with a pensive look and replied, "Ah! we chase after phantoms [chimères]."[105] These were his last words, interpreted by Maurice as a realisation of the ultimate "vanity" of earthly pursuits.[106] Laplace received the last rites from the curé of the Missions Étrangères (in whose parish he was to be buried)[92] and the curé of Arcueil.[106]
According to his biographer, Roger Hahn, it is "not credible" that Laplace "had a proper Catholic end", and he "remained a skeptic" to the very end of his life.[107] Laplace in his last years has been described as an agnostic.[108][109][110]
Excommunication of a comet
In 1470 the humanist scholar Bartolomeo Platina wrote[111] 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.[112] What Laplace actually said, in Exposition du système du monde (1796), was that the Pope had ordered the comet to be "exorcised" (conjuré). It was Arago, in Des Comètes en général (1832), who first spoke of an excommunication.[113][114][115]
Honores
- Correspondent of the Royal Institute of the Netherlands in 1809.[116]
- Foreign Honorary Member of the American Academy of Arts and Sciences in 1822.[117]
- The asteroid 4628 Laplace is named for Laplace.[118]
- A spur of the Montes Jura on the Moon is known as Promontorium Laplace.
- His name is one of the 72 names inscribed on the Eiffel Tower.
- The tentative working name of the European Space Agency Europa Jupiter System Mission is the "Laplace" space probe.
- A train station in the RER B in Arcueil bears his name.
- A street in Verkhnetemernitsky (near Rostov-on-Don, Russia).
Citas
- I had no need of that hypothesis. ("Je n'avais pas besoin de cette hypothèse-là", allegedly as a reply to Napoleon, who had asked why he hadn't mentioned God in his book on astronomy.)[9]
- 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.)
- "We are so far from knowing all the agents of nature and their diverse modes of action that it would not be philosophical to deny phenomena solely because they are inexplicable in the actual state of our knowledge. But we ought to examine them with an attention all the more scrupulous as it appears more difficult to admit them."[119]
- This is restated in Theodore Flournoy's work From India to the Planet Mars as the Principle of Laplace or, "The weight of the evidence should be proportioned to the strangeness of the facts."[120]
- Most often repeated as "The weight of evidence for an extraordinary claim must be proportioned to its strangeness." (see also: Sagan standard)
- This simplicity of ratios will not appear astonishing if we consider that all the effects of nature are only mathematical results of a small number of immutable laws.[121]
- Infinitely varied in her effects, nature is only simple in her causes.[122]
- What we know is little, and what we are ignorant of is immense. (Fourier comments: "This was at least the meaning of his last words, which were articulated with difficulty.")[59]
- One sees in this essay that the theory of probabilities is basically only common sense reduced to a calculus. It makes one estimate accurately what right-minded people feel by a sort of instinct, often without being able to give a reason for it.[123]
Bibliografía
- Œuvres complètes de Laplace, 14 vol. (1878–1912), Paris: Gauthier-Villars (copy from Gallica in French)
- Théorie du movement et de la figure elliptique des planètes (1784) Paris (not in Œuvres complètes)
- Précis de l'histoire de l'astronomie
- Alphonse Rebière, Mathématiques et mathématiciens, 3rd edition Paris, Nony & Cie, 1898.
English translations
- Bowditch, N. (trans.) (1829–1839) Mécanique céleste, 4 vols, Boston
- New edition by Reprint Services ISBN 0-7812-2022-X
- – [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 1-4326-5367-9
- Toplis, J. (trans.) (1814) A treatise upon analytical mechanics Nottingham: H. Barnett
- Laplace, Pierre Simon Marquis De (2007) [1902]. A Philosophical Essay on Probabilities. Translated by Truscott, F.W. & Emory, F.L. ISBN 978-1-60206-328-0., translated from the French 6th ed. (1840)
- A Philosophical Essay on Probabilities (1902) at the Internet Archive
- Dale, Andrew I.; Laplace, Pierre-Simon (1995). Philosophical Essay on Probabilities. Sources in the History of Mathematics and Physical Sciences. 13. Translated by Andrew I. Dale. Springer. doi:10.1007/978-1-4612-4184-3. hdl:2027/coo1.ark:/13960/t3126f008. ISBN 978-1-4612-8689-9., translated from the French 5th ed. (1825)
Ver también
- History of the metre
- Laplace–Bayes estimator
- Ratio estimator
- Seconds pendulum
- List of things named after Pierre-Simon Laplace
Referencias
Citations
- ^ a b S.W. Hawking and George F.R. Ellis, The Large Scale Structure of Space-Time, Cambridge University Press, 1973, p. 364.
- ^ Stigler, Stephen M. (1986). The History of Statistics: The Measurement of Uncertainty before 1900. Harvard University Press, Chapter 3.
- ^ a b c d e f g h i j k Clerke, Agnes Mary (1911). . In Chisholm, Hugh (ed.). Encyclopædia Britannica. 16 (11th ed.). Cambridge University Press. pp. 200–202.
- ^ a b c d e f "Laplace, being Extracts from Lectures delivered by Karl Pearson", Biometrika, vol. 21, December 1929, pp. 202–216.
- ^ W.W. Rouse Ball A Short Account of the History of Mathematics, 4th edition, 1908.
- ^ a b * O'Connor, John J.; Robertson, Edmund F., "Pierre-Simon Laplace", MacTutor History of Mathematics archive, University of St Andrews.. Retrieved 25 August 2007
- ^ Edmund Whittaker (Vol. 33, No. 303 (Feb. 1949), pp. 1–12), "Laplace", The Mathematical Gazette.
- ^ Gillispie (1997), pp. 3–4
- ^ a b c d e f g h i j k l m n o p q r s t u v w x y z aa ab Rouse Ball (1908)
- ^ "The Chemical Revolution of Antoine-Laurent Lavoisier International Historic Chemical Landmark". American Chemical Society. 8 June 1999.
- ^ Golinski, Jan V. (June 1983). "Antoine Laurent Lavoisier , Pierre Simon , Marquis de Laplace , Henry Guerlac". Isis. 74 (2): 288–289. doi:10.1086/353288.
- ^ Gillispie (1997), p. 5
- ^ "Effects of the Scientific Community on Laplace" Retrieved on 10 January 2018
- ^ Hahn (2005), p. 99. However, Gillispie (1997), p. 67, gives the month of the marriage as May.
- ^ Hahn (2005), pp. 99–100
- ^ Gillispie (1997), p. 67
- ^ Hahn (2005), p. 101
- ^ Gillispie (1989), pp. 7–12
- ^ Gillispie (1989). pp. 14–15
- ^ a b Whitrow (2001)
- ^ Celletti, A. & Perozzi, E. (2007). Celestial Mechanics: The Waltz of the Planets. Celestial Mechanics – the Waltz of the Planets. Berlin: Springer. pp. 91–93. Bibcode:2006cmwp.book.....C. ISBN 978-0-387-30777-0.
- ^ Whittaker (1949b)
- ^ Gillispie (1989). pp. 29–35
- ^ Gillispie (1989), pp. 35–36
- ^ School of Mathematics and Statistics, University of St Andrews, Scotland.
- ^ "Short notes on the Dynamical theory of Laplace". 20 November 2011.
- ^ http://faculty.washington.edu/luanne/pages/ocean420/notes/tidedynamics.pdf
- ^ a b "Higher Education" (PDF).
- ^ http://ocean.kisti.re.kr/downfile/volume/kess/JGGHBA/2009/v30n5/JGGHBA_2009_v30n5_671.pdf
- ^ Tidal theory Archived 22 August 2017 at the Wayback Machine website South African Navy Hydrographic Office
- ^ "Dynamic theory for tides". Oberlin.edu. Retrieved 2 June 2012.
- ^ "Dynamic Theory of Tides".
- ^ "Dynamic Tides – In contrast to "static" theory, the dynamic theory of tides recognizes that water covers only three-quarters o". Web.vims.edu. Archived from the original on 13 January 2013. Retrieved 2 June 2012.
- ^ "The Dynamic Theory of Tides". Coa.edu. Archived from the original on 19 December 2013. Retrieved 2 June 2012.
- ^ [1]
- ^ "Tides – building, river, sea, depth, oceans, effects, important, largest, system, wave, effect, marine, Pacific". Waterencyclopedia.com. 27 June 2010.
- ^ "TIDES". Ocean.tamu.edu. Archived from the original on 16 June 2013. Retrieved 2 June 2012.
- ^ Floor Anthoni. "Tides". Seafriends.org.nz. Retrieved 2 June 2012.
- ^ "The Cause & Nature of Tides".
- ^ "Scientific Visualization Studio TOPEX/Poseidon images". Svs.gsfc.nasa.gov. Retrieved 2 June 2012.
- ^ "TOPEX/Poseidon Western Hemisphere: Tide Height Model : NASA/Goddard Space Flight Center Scientific Visualization Studio : Free Download & Streaming : Internet Archive". 15 June 2000.
- ^ TOPEX data used to model actual tides for 15 days from the year 2000 TOPEX/Poseidon Flat Earth Tide Height Model
- ^ http://www.geomag.us/info/Ocean/m2_CHAMP+longwave_SSH.swf
- ^ "OSU Tidal Data Inversion". Volkov.oce.orst.edu. Retrieved 2 June 2012.
- ^ "Dynamic and residual ocean tide analysis for improved GRACE de-aliasing (DAROTA)". Archived from the original on 2 April 2015.
- ^ "The Laplace Tidal Equations and Atmospheric Tides" (PDF). Archived from the original (PDF) on 11 April 2019. Retrieved 28 October 2017.
- ^ Grattan-Guinness, I. (2003). Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences. Baltimore: Johns Hopkins University Press. pp. 1097–1098. ISBN 978-0-8018-7396-6.
- ^ W.W. Rouse Ball A Short Account of the History of Mathematics (4th edition, 1908)
- ^ Green, G. (1828). An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism. Nottingham. arXiv:0807.0088. Bibcode:2008arXiv0807.0088G.
- ^ Kline, Morris (1972). Mathematical thought from ancient to modern times. 2. Oxford University Press. pp. 524–525. ISBN 978-0-19-506136-9.
- ^ Euler, Leonhard (1757). "General principles of the motion of fluids". Novi. Comm. Acad. Sci. Petrop.: 271–311.
- ^ Maxwell, James (1881). A Treatise on Electricity and Magnetism (PDF). p. 29.
- ^ Arago, François (1874). Laplace: Eulogy. Translated by Powell, Baden. Smithsonian Institution. p. 5. Retrieved 21 March 2018.
- ^ a b Owen, T. C. (2001) "Solar system: origin of the solar system", Encyclopædia Britannica, Deluxe CDROM edition
- ^ Laplace, P.-S. (1799). Allgemeine geographische Ephemeriden herausgegeben von F. von Zach. IV. Band, I. Stück, I. Abhandlung, Weimar; translation in English: Hawking, Stephen W.; Ellis, George F.R. (1973). The Large Scale Structure of Space-Time. Cambridge University Press. pp. 365ff. ISBN 978-0-521-09906-6..
- ^ Colin Montgomery, Wayne Orchiston and Ian Whittingham, "Michell, Laplace and the origin of the Black Hole Concept" Archived 2 May 2014 at the Wayback Machine, Journal of Astronomical History and Heritage, 12(2), 90–96 (2009).
- ^ See Israel (1987), sec. 7.2.
- ^ Gribbin, 299
- ^ a b Fourier (1829)
- ^ Crosland (1967), p. 1
- ^ a b c d Stigler, 1975
- ^ "Laplace, Pierre-Simon Marquis de"[permanent dead link] Retrieved on 10 January 2018
- ^ Laplace, Pierre Simon, A Philosophical Essay on Probabilities, translated from the 6th French edition by Frederick Wilson Truscott and Frederick Lincoln Emory. New York: John Wiley & Sons, 1902, p. 19. Dover Publications edition (New York, 1951) has same pagination.
- ^ a b Hawking, Stephen (1999). "Does God Play Dice?". Public Lecture. Archived from the original on 8 July 2000.
- ^ Laplace, A Philosophical Essay, New York, 1902, p. 4.
- ^ van Strien, Marij (2014). "On the origins and foundations of Laplacian determinism". Studies in History and Philosophy of Science. 45: 24–31. doi:10.1016/j.shpsa.2013.12.003. PMID 24984446. Retrieved 5 February 2021.
- ^ Cercignani, Carlo (1998). "Chapter 2: Physics before Boltzmann". Ludwig Boltzmann, The Man Who Trusted Atoms. Oxford University Press. p. 55. ISBN 978-0-19-850154-1.
- ^ Grattan-Guinness, in Gillispie (1997), p. 260
- ^ Grattan-Guinness, in Gillispie (1997), pp. 261–262
- ^ Deakin (1981)
- ^ Crosland (2006), p. 30
- ^ a b Grattan-Guinness (2005), p. 333
- ^ Hahn (2005), p. 191
- ^ Laplace, A Philosophical Essay, New York, 1902, p. 62. (Translation in this paragraph of article is from Hahn.)
- ^ Hahn (2005), p. 184
- ^ Laplace, A Philosophical Essay, New York, 1902, p. 63. (Translation in this paragraph of article is from Hahn.)
- ^ a b Hahn (2005), p. 185
- ^ Laplace, A Philosophical Essay, New York, 1902, pp. 107–108. (Translation in this paragraph of article is from Hahn.)
- ^ Black, Duncan (1987) [1958]. The Theory of Committees and Elections. Springer Science & Business Media. ISBN 978-0-89838-189-4.
- ^ Tangian, Andranik (2020). Analytical Theory of Democracy. Vols. 1 and 2. Studies in Choice and Welfare. Cham, Switzerland: Springer. pp. 132ff. doi:10.1007/978-3-030-39691-6. ISBN 978-3-030-39690-9.
- ^ Gillispie (1997), p. 278
- ^ p. 282, Mémoires du docteur F. Antommarchi, ou les derniers momens de Napoléon, vol. 1, 1825, Paris: Barrois L'Ainé
- ^ a b c Faye, Hervé (1884), Sur l'origine du monde: théories cosmogoniques des anciens et des modernes. Paris: Gauthier-Villars, pp. 109–111
- ^ a b c Pasquier, Ernest (1898). "Les hypothèses cosmogoniques (suite)". Revue néo-scholastique, 5o année, No 18, pp. 124–125, footnote 1
- ^ Arago, François (1827), Laplace: Eulogy before the French Academy, translated by Prof. Baden Powell, Smithsonian Report, 1874
- ^ Cajori, Florian (1893), A History of Mathematics. Fifth edition (1991), reprinted by the American Mathematical Society, 1999, p. 262. ISBN 0-8218-2102-4
- ^ William Herschel's diary of his trip to Paris, as quoted on p. 310 of The Herschel Chronicle, Constance A. Lubbock, Cambridge: Cambridge University Press, 2013, ISBN 1-107-65001-1.
- ^ Johnson, Daniel (18 June 2007), "The Hypothetical Atheist", Commentary.
- ^ Talks of Napoleon at St. Helena with General Baron Gourgaud, translated by Elizabeth Wormely Latimer. Chicago: A.C. McClurg & Co., 1903, p. 276.
- ^ Hahn (2005), p. 67.
- ^ Dumas, Jean-Baptiste (1885). Discours et éloges académiques, Vol. II. Paris: Gauthier-Villars, p. 255.
- ^ a b Kneller, Karl Alois. Christianity and the Leaders of Modern Science: A Contribution to the History of Culture in the Nineteenth Century, translated from the second German edition by T.M. Kettle. London: B. Herder, 1911, pp. 73–74
- ^ Hahn (1981), p. 95.
- ^ Œuvres de Laplace. Paris: Gauthier-Villars, 1878, Vol. I, pp. v–vi.
- ^ Glass, Ian S. (2006). Revolutionaries of the Cosmos: The Astrophysicists. Cambridge University Press, p. 108. ISBN 0-19-857099-6
- ^ General Scholium, from the end of Book III of the Principia; first appeared in the second edition, 1713.
- ^ Laplace, Exposition du système du monde, 6th edition. Brussels, 1827, pp. 522–523.
- ^ Laplace, Exposition, 1827, p. 523.
- ^ Leibniz to Conti, Nov. or Dec. 1715, in H.G. Alexander, ed., The Leibniz–Clarke Correspondence (Manchester University Press, 1956), Appendix B. 1: "Leibniz and Newton to Conti", p. 185 ISBN 0-7190-0669-4; cited in Laplace, Exposition, 1827, p. 524.
- ^ Leibniz to Conti, 1715, in Alexander, ed., 1956, p. 185.
- ^ Hahn (2005), p. 220
- ^ Hahn (2005), p. 223
- ^ a b Hahn (2005), p. 202
- ^ Hahn (2005), pp. 202, 233
- ^ De Morgan, Augustus (1872). A budget of paradoxes, Longmans, Green, and co, London, p. 3. Compare Edmund Burke's famous remark, occasioned by a parliamentary candidate's sudden death, about "what shadows we are, and what shadows we pursue".
- ^ a b Hahn (2005), p. 204
- ^ Roger Hahn (2005). Pierre Simon Laplace, 1749–1827: A Determined Scientist. Harvard University Press. p. 204. ISBN 978-0-674-01892-1.
The Catholic newspaper La Quotidienne [The Daily] announced that Laplace had died in the arms of two curés (priests), implying that he had a proper Catholic end, but this is not credible. To the end, he remained a skeptic, wedded to his deterministic creed and to an uncompromised ethos derived from his vast scientific experience.
- ^ Roger Hahn (2005). Pierre Simon Laplace, 1749–1827: A Determined Scientist. Harvard University Press. p. 202. ISBN 978-0-674-01892-1.
Publicly, Laplace maintained his agnostic beliefs, and even in his old age continued to be skeptical about any function God might play in a deterministic universe.
- ^ Morris Kline (1986). Mathematics and the Search for Knowledge. Oxford University Press. p. 214. ISBN 978-0-19-504230-6.
Lagrange and Laplace, though of Catholic parentage, were agnostics.
- ^ Edward Kasner; James Newman; James Roy Newman (2001). Mathematics and the Imagination. Courier Dover Publications. p. 253. ISBN 978-0-486-41703-5.
Modern physics, indeed all of modern science, is as humble as Lagrange, and as agnostic as Laplace.
- ^ E. Emerson (1910). Comet Lore. Schilling Press, New York. p. 83.
- ^ C.M. Botley (1971). "The Legend of 1P/Halley 1456". The Observatory. 91: 125–126. Bibcode:1971Obs....91..125B.
- ^ Hagen, John G. (1910). . In Herbermann, Charles (ed.). Catholic Encyclopedia. 8. New York: Robert Appleton Company.
- ^ Stein, John (1911). . In Herbermann, Charles (ed.). Catholic Encyclopedia. 12. New York: Robert Appleton Company.
- ^ Rigge, William F. (04/1910), "An Historical Examination of the Connection of Calixtus III with Halley's Comet", Popular Astronomy, Vol. 18, pp. 214–219
- ^ "P.S. de Laplace (1749–1827)". Royal Netherlands Academy of Arts and Sciences. Retrieved 19 July 2015.
- ^ "Book of Members, 1780–2010: Chapter L" (PDF). American Academy of Arts and Sciences. Retrieved 28 July 2014.
- ^ Schmadel, L.D. (2003). Dictionary of Minor Planet Names (5th rev. ed.). Berlin: Springer-Verlag. ISBN 978-3-540-00238-3.
- ^ Laplace, Pierre Simon (1814). "Essai philosophique sur les probabilités". Nature. 110 (2748): 50. Bibcode:1922Natur.110....6B. doi:10.1038/110006b0. S2CID 4099834.
- ^ Flournoy, Théodore (1899). Des Indes à la planète Mars: étude sur un cas de somnambulisme avec glossolalie. Slatkine. pp. 344–345. ISBN 978-2-05-100499-2.* Flournoy, Théodore (2007). From India to the Planet Mars: A Study of a Case of Somnambulism. Daniel D. Vermilye, trans. Cosimo, Inc. pp. 369–370. ISBN 978-1-60206-357-0.
- ^ Laplace, A Philosophical Essay, New York, 1902, p. 177.
- ^ Laplace, The System of the World, Dublin, 1830, p. 91.
- ^ Miller and Gelman, Joshua B; Andrew. "Laplace's theories of cognitive illusions, heuristics, and biases∗" (PDF). Columbia University. unpublished. Retrieved 17 January 2021.
General sources
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- Wilson, C. (1985). "The Great Inequality of Jupiter and Saturn: from Kepler to Laplace". Archive for History of Exact Sciences. 33 (1–3): 15–290. Bibcode:1985AHES...33...15W. doi:10.1007/BF00328048. S2CID 121751666.
- Young, T. (1821). Elementary Illustrations of the Celestial Mechanics of Laplace: Part the First, Comprehending the First Book. London, England: John Murray – via Internet Archive.
laplace.
enlaces externos
- "Laplace, Pierre (1749–1827)". Eric Weisstein's World of Scientific Biography. Wolfram Research. Retrieved 24 August 2007.
- "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. Retrieved 4 September 2007.
- Guide to the Pierre Simon Laplace Papers at The Bancroft Library
- Pierre-Simon Laplace at the Mathematics Genealogy Project
- English translation of a large part of Laplace's work in probability and statistics, provided by Richard Pulskamp
- Pierre-Simon Laplace – Œuvres complètes (last 7 volumes only) Gallica-Math
- "Sur le mouvement d'un corps qui tombe d'une grande hauteur" (Laplace 1803), online and analysed on BibNum (English).
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Preceded by Nicolas Marie Quinette | Minister of the Interior 12 November 1799 – 25 December 1799 | Succeeded by Lucien Bonaparte |