Pierre-Simon Laplace

Overview

Mathematician

A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....

and astronomer

Astronomer

An astronomer is a scientist who studies celestial bodies such as planets, stars and galaxies.Historically, astronomy was more concerned with the classification and description of phenomena in the sky, while astrophysics attempted to explain these phenomena and the differences between them using...

whose work was pivotal to the development of mathematical astronomy

Astronomy

Astronomy is a natural science that deals with the study of celestial objects and phenomena that originate outside the atmosphere of Earth...

and statistics

Statistics

Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....

. He summarized and extended the work of his predecessors in his five volume

Celestial mechanics

Celestial mechanics is the branch of astronomy that deals with the motions of celestial objects. The field applies principles of physics, historically classical mechanics, to astronomical objects such as stars and planets to produce ephemeris data. Orbital mechanics is a subfield which focuses on...

) (1799–1825). This work translated the geometric study of classical mechanics

Classical mechanics

In physics, classical mechanics is one of the two major sub-fields of mechanics, which is concerned with the set of physical laws describing the motion of bodies under the action of a system of forces...

to one based on calculus

Calculus

Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...

, opening up a broader range of problems. In statistics, the so-called Bayesian interpretation

Bayesian probability

Bayesian probability is one of the different interpretations of the concept of probability and belongs to the category of evidential probabilities. The Bayesian interpretation of probability can be seen as an extension of logic that enables reasoning with propositions, whose truth or falsity is...

of probability was mainly developed by Laplace.

He formulated Laplace's equation

Laplace's equation

In mathematics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace who first studied its properties. This is often written as:where ∆ = ∇² is the Laplace operator and \varphi is a scalar function...

, and pioneered the Laplace transform which appears in many branches of mathematical physics

Mathematical physics

Mathematical physics refers to development of mathematical methods for application to problems in physics. The Journal of Mathematical Physics defines this area as: "the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and...

, a field that he took a leading role in forming.

Discussion

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Quotations

The exchange is reported by Victor Hugo (who in turn was citing Arago) as:

"[Sire,] je n'avais pas besoin de cette hypothèse-là."

Translation: "I did not need to make such an assumption."

"Les questions les plus importantes de la vie ne sont en effet, pour la plupart, que des problèmes de probabilité."

Translation: "Life's most important questions are, for the most part, nothing but probability problems."
----

"La dernière chose que nous attendions de vous, Général, est une leçon de géométrie !"

Translation: "The last thing we expect of you, General, is a lesson in geometry!"

"?"

Translation: "What we know is not much. What we do not know is immense."

Encyclopedia

Mathematician

A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....

and astronomer

Astronomer

An astronomer is a scientist who studies celestial bodies such as planets, stars and galaxies.Historically, astronomy was more concerned with the classification and description of phenomena in the sky, while astrophysics attempted to explain these phenomena and the differences between them using...

whose work was pivotal to the development of mathematical astronomy

Astronomy

Astronomy is a natural science that deals with the study of celestial objects and phenomena that originate outside the atmosphere of Earth...

and statistics

Statistics

Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....

. He summarized and extended the work of his predecessors in his five volume

Celestial mechanics

Celestial mechanics is the branch of astronomy that deals with the motions of celestial objects. The field applies principles of physics, historically classical mechanics, to astronomical objects such as stars and planets to produce ephemeris data. Orbital mechanics is a subfield which focuses on...

) (1799–1825). This work translated the geometric study of classical mechanics

Classical mechanics

In physics, classical mechanics is one of the two major sub-fields of mechanics, which is concerned with the set of physical laws describing the motion of bodies under the action of a system of forces...

to one based on calculus

Calculus

Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...

, opening up a broader range of problems. In statistics, the so-called Bayesian interpretation

Bayesian probability

Bayesian probability is one of the different interpretations of the concept of probability and belongs to the category of evidential probabilities. The Bayesian interpretation of probability can be seen as an extension of logic that enables reasoning with propositions, whose truth or falsity is...

of probability was mainly developed by Laplace.

He formulated Laplace's equation

Laplace's equation

In mathematics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace who first studied its properties. This is often written as:where ∆ = ∇² is the Laplace operator and \varphi is a scalar function...

, and pioneered the Laplace transform which appears in many branches of mathematical physics

Mathematical physics

Mathematical physics refers to development of mathematical methods for application to problems in physics. The Journal of Mathematical Physics defines this area as: "the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and...

, a field that he took a leading role in forming. The Laplacian differential operator

Laplace operator

In mathematics the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a function on Euclidean space. It is usually denoted by the symbols ∇·∇, ∇2 or Δ...

, widely used in mathematics, is also named after him.

He restated and developed the nebular hypothesis of the origin of the solar system and was one of the first scientists to postulate the existence of black hole

Black hole

A black hole is a region of spacetime from which nothing, not even light, can escape. The theory of general relativity predicts that a sufficiently compact mass will deform spacetime to form a black hole. Around a black hole there is a mathematically defined surface called an event horizon that...

s and the notion of gravitational collapse

Gravitational collapse

Gravitational collapse is the inward fall of a body due to the influence of its own gravity. In any stable body, this gravitational force is counterbalanced by the internal pressure of the body, in the opposite direction to the force of gravity...

.

He is remembered as one of the greatest scientists of all time, sometimes referred to as a

Isaac Newton

Sir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...

He became a count of the First French Empire

First French Empire

The First French Empire , also known as the Greater French Empire or Napoleonic Empire, was the empire of Napoleon I of France...

in 1806 and was named a marquis

Marquis

Marquis is a French and Scottish title of nobility. The English equivalent is Marquess, while in German, it is Markgraf.It may also refer to:Persons:...

in 1817, after the Bourbon Restoration

Bourbon Restoration

The Bourbon Restoration is the name given to the period following the successive events of the French Revolution , the end of the First Republic , and then the forcible end of the First French Empire under Napoleon – when a coalition of European powers restored by arms the monarchy to the...

.

Many details of the life of Laplace were lost when the family château

Château

A château is a manor house or residence of the lord of the manor or a country house of nobility or gentry, with or without fortifications, originally—and still most frequently—in French-speaking regions...

burned in 1925.

Laplace was born in Beaumont-en-Auge

Beaumont-en-Auge

Beaumont-en-Auge is a commune in the Calvados department in the Basse-Normandie region in northwestern France. The city hosts one of the last kaleidoscope manufacturers in France.-Population:-Personalities:...

, Normandy

Normandy

Normandy is a geographical region corresponding to the former Duchy of Normandy. It is in France.The continental territory covers 30,627 km² and forms the preponderant part of Normandy and roughly 5% of the territory of France. It is divided for administrative purposes into two régions:...

in 1749.

According to W. W. Rouse Ball

W. W. Rouse Ball

-External links:*...

(

Jean le Rond d'Alembert

Jean-Baptiste le Rond d'Alembert was a French mathematician, mechanician, physicist, philosopher, and music theorist. He was also co-editor with Denis Diderot of the Encyclopédie...

, he went to Paris to push his fortune. However, Karl Pearson

Karl Pearson

Karl Pearson FRS was an influential English mathematician who has been credited for establishing the disciplineof mathematical statistics....

is scathing about the accuracies in Rouse Ball's account and states,

Indeed CaenCaenCaen is a commune in northwestern France. It is the prefecture of the Calvados department and the capital of the Basse-Normandie region. It is located inland from the English Channel....

was probably in Laplace's day the most intellectually active of all the towns of Normandy. It was here that Laplace was educated and was provisionally a professor. It was here he wrote his first paper published in theMélangesof the Royal Society of Turin, Tome iv. 1766–1769, at least two years before he went at 22 or 23 to Paris in 1771. Thus before he was 20 he was in touch with LagrangeJoseph Louis LagrangeJoseph-Louis Lagrange , born Giuseppe Lodovico Lagrangia, was a mathematician and astronomer, who was born in Turin, Piedmont, lived part of his life in Prussia and part in France, making significant contributions to all fields of analysis, to number theory, and to classical and celestial mechanics...

in TurinTurinTurin is a city and major business and cultural centre in northern Italy, capital of the Piedmont region, located mainly on the left bank of the Po River and surrounded by the Alpine arch. The population of the city proper is 909,193 while the population of the urban area is estimated by Eurostat...

. He did not go to Paris a raw self-taught country lad with only a peasant background! In 1765 at the age of sixteen Laplace left the "School of the Duke of Orleans" in Beaumont and went to the University of Caen, where he appears to have studied for five years. The 'École MilitaireÉcole MilitaireThe École Militaire is a vast complex of buildings housing various military training facilities located in the 7th arrondissement of Paris, France, southeast of the Champ de Mars....

' of Beaumont did not replace the old school until 1776.

His parents were from comfortable families. His father was Pierre Laplace, and his mother was Marie-Anne Sochon. The Laplace family was involved in agriculture until at least 1750, but Pierre Laplace senior was also a cider

Cider

Cider or cyder is a fermented alcoholic beverage made from apple juice. Cider varies in alcohol content from 2% abv to 8.5% abv or more in traditional English ciders. In some regions, such as Germany and America, cider may be termed "apple wine"...

merchant and

Syndic

Syndic , a term applied in certain countries to an officer of government with varying powers, and secondly to a representative or delegate of a university, institution or other corporation, entrusted with special functions or powers.The meaning which underlies both applications is that of...

Pierre Simon Laplace attended a school in the village run at a Benedictine

Benedictine

Benedictine refers to the spirituality and consecrated life in accordance with the Rule of St Benedict, written by Benedict of Nursia in the sixth century for the cenobitic communities he founded in central Italy. The most notable of these is Monte Cassino, the first monastery founded by Benedict...

priory

Priory

A priory is a house of men or women under religious vows that is headed by a prior or prioress. Priories may be houses of mendicant friars or religious sisters , or monasteries of monks or nuns .The Benedictines and their offshoots , the Premonstratensians, and the...

, his father intending that he would be ordained in the Roman Catholic Church

Roman Catholic Church

The Catholic Church, also known as the Roman Catholic Church, is the world's largest Christian church, with over a billion members. Led by the Pope, it defines its mission as spreading the gospel of Jesus Christ, administering the sacraments and exercising charity...

, and at sixteen he was sent to further his father's intention at the University of Caen, reading theology.

At the university, he was mentored by two enthusiastic teachers of mathematics, Christophe Gadbled and Pierre Le Canu, who awoke his zeal for the subject. Laplace did not graduate in theology but left for Paris with a letter of introduction from Le Canu to Jean le Rond d'Alembert

Jean le Rond d'Alembert

Jean-Baptiste le Rond d'Alembert was a French mathematician, mechanician, physicist, philosopher, and music theorist. He was also co-editor with Denis Diderot of the Encyclopédie...

.

According to his great-great-grandson, d'Alembert received him rather poorly, and to get rid of him gave him a thick mathematics book, saying to come back when he had read it. When Laplace came back a few days later, d'Alembert was even less friendly and did not hide his opinion that it was impossible that Laplace could have read and understood the book. But upon questioning him, he realized that it was true, and from that time he took Laplace under his care.

Another version is that Laplace solved overnight a problem that d'Alembert set him for submission the following week, then solved a harder problem the following night. D'Alembert was impressed and recommended him for a teaching place in the

École Militaire

The École Militaire is a vast complex of buildings housing various military training facilities located in the 7th arrondissement of Paris, France, southeast of the Champ de Mars....

With a secure income and undemanding teaching, Laplace now threw himself into original research and, in the next seventeen years, 1771–1787, he produced much of his original work in astronomy.

Laplace further impressed the Marquis de Condorcet

Marquis de Condorcet

Marie Jean Antoine Nicolas de Caritat, marquis de Condorcet , known as Nicolas de Condorcet, was a French philosopher, mathematician, and early political scientist whose Condorcet method in voting tally selects the candidate who would beat each of the other candidates in a run-off election...

, and even in 1771 Laplace felt that he was entitled to membership in the French Academy of Sciences

French Academy of Sciences

The French Academy of Sciences is a learned society, founded in 1666 by Louis XIV at the suggestion of Jean-Baptiste Colbert, to encourage and protect the spirit of French scientific research...

. However, in that year, admission went to Alexandre-Théophile Vandermonde

Alexandre-Théophile Vandermonde

Alexandre-Théophile Vandermonde was a French musician, mathematician and chemist who worked with Bézout and Lavoisier; his name is now principally associated with determinant theory in mathematics. He was born in Paris, and died there.Vandermonde was a violinist, and became engaged with...

and in 1772 to Jacques Antoine Joseph Cousin. Laplace was disgruntled, and at the beginning of 1773, d'Alembert wrote to Lagrange

Lagrange

La Grange literally means the barn in French. Lagrange may refer to:- People :* Charles Varlet de La Grange , French actor* Georges Lagrange , translator to and writer in Esperanto...

in Berlin to ask if a position could be found for Laplace there. However, Condorcet became permanent secretary of the

He married Marie-Charlotte de Courty de Romanges in his late thirties and the couple had a daughter, Sophie, and a son, Charles-Émile (b. 1789).

Laplace's early published work in 1771 started with differential equations and finite differences but he was already starting to think about the mathematical and philosophical concepts of probability and statistics. However, before his election to the

Celestial mechanics

Celestial mechanics is the branch of astronomy that deals with the motions of celestial objects. The field applies principles of physics, historically classical mechanics, to astronomical objects such as stars and planets to produce ephemeris data. Orbital mechanics is a subfield which focuses on...

and the stability of the solar system. The two disciplines would always be interlinked in his mind. "Laplace took probability as an instrument for repairing defects in knowledge." Laplace's work on probability and statistics is discussed below with his mature work on the Analytic theory of probabilities.

Sir Isaac Newton

Isaac Newton

Sir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...

had published his

Philosophiae Naturalis Principia Mathematica

Philosophiæ Naturalis Principia Mathematica, Latin for "Mathematical Principles of Natural Philosophy", often referred to as simply the Principia, is a work in three books by Sir Isaac Newton, first published 5 July 1687. Newton also published two further editions, in 1713 and 1726...

Newton's laws of motion

Newton's laws of motion are three physical laws that form the basis for classical mechanics. They describe the relationship between the forces acting on a body and its motion due to those forces...

and his law of universal gravitation

Newton's law of universal gravitation

Newton's law of universal gravitation states that every point mass in the universe attracts every other point mass with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them...

. 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

Miracle

A miracle often denotes an event attributed to divine intervention. Alternatively, it may be an event attributed to a miracle worker, saint, or religious leader. A miracle is sometimes thought of as a perceptible interruption of the laws of nature. Others suggest that a god may work with the laws...

was necessary to guarantee the stability of the solar system. Dispensing with the hypothesis of divine intervention would be a major activity of Laplace's scientific life. It is now generally regarded that Laplace's methods on their own, though vital to the development of the theory, are not sufficiently precise

Accuracy and precision

In the fields of science, engineering, industry and statistics, the accuracy of a measurement system is the degree of closeness of measurements of a quantity to that quantity's actual value. The precision of a measurement system, also called reproducibility or repeatability, is the degree to which...

to demonstrate the stability of the Solar System

Stability of the Solar System

The stability of the Solar System is a subject of much inquiry in astronomy. Though the planets have been stable historically, and will be in the short term, their weak gravitational effects on one another can add up in unpredictable ways....

, and indeed, the Solar System is now understood to be chaotic, although it actually appears to be fairly stable.

One particular problem from observational astronomy

Observational astronomy

Observational astronomy is a division of the astronomical science that is concerned with getting data, in contrast with theoretical astrophysics which is mainly concerned with finding out the measurable implications of physical models...

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

Leonhard Euler

Leonhard Euler was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion...

in 1748 and Joseph Louis Lagrange

Joseph Louis Lagrange

Joseph-Louis Lagrange , born Giuseppe Lodovico Lagrangia, was a mathematician and astronomer, who was born in Turin, Piedmont, lived part of his life in Prussia and part in France, making significant contributions to all fields of analysis, to number theory, and to classical and celestial mechanics...

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

Integral

Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...

over time they could become important. Laplace carried his analysis into the higher-order terms, up to and including the cubic

Cubic function

In mathematics, a cubic function is a function of the formf=ax^3+bx^2+cx+d,\,where a is nonzero; or in other words, a polynomial of degree three. The derivative of a cubic function is a quadratic function...

. 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

Gerald James Whitrow

Gerald James Whitrow was a British mathematician, cosmologist and science historian.After completing school at Christ's Hospital, he obtained a scholarship at Christ Church, Oxford in 1930, earning his first degree in 1933, the MA in 1937, and the PhD in 1939...

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

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

Spheroid

A spheroid, or ellipsoid of revolution is a quadric surface obtained by rotating an ellipse about one of its principal axes; in other words, an ellipsoid with two equal semi-diameters....

on a particle outside it. This is memorable for the introduction into analysis of spherical harmonics

Spherical harmonics

In mathematics, spherical harmonics are the angular portion of a set of solutions to Laplace's equation. Represented in a system of spherical coordinates, Laplace's spherical harmonics Y_\ell^m are a specific set of spherical harmonics that forms an orthogonal system, first introduced by Pierre...

or

Celestial mechanics

.

In 1783, in a paper sent to the

Adrien-Marie Legendre

Adrien-Marie Legendre was a French mathematician.The Moon crater Legendre is named after him.- Life :...

had introduced what are now known as associated Legendre functions. If two points in a plane

Plane (mathematics)

In mathematics, a plane is a flat, two-dimensional surface. A plane is the two dimensional analogue of a point , a line and a space...

have polar co-ordinates (

This expression can be expanded in powers of

The sequence

Sequence

In mathematics, a sequence is an ordered list of objects . Like a set, it contains members , and the number of terms is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence...

of functions

Function (mathematics)

In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

of the points on a circle can be expanded as a series

Series (mathematics)

A series is the sum of the terms of a sequence. Finite sequences and series have defined first and last terms, whereas infinite sequences and series continue indefinitely....

of them.

Laplace, with scant regard for credit to Legendre, made the non-trivial extension of the result to three dimensions

Three-dimensional space

Three-dimensional space is a geometric 3-parameters model of the physical universe in which we live. These three dimensions are commonly called length, width, and depth , although any three directions can be chosen, provided that they do not lie in the same plane.In physics and mathematics, a...

to yield a more general set of functions, the

Spherical harmonics

In mathematics, spherical harmonics are the angular portion of a set of solutions to Laplace's equation. Represented in a system of spherical coordinates, Laplace's spherical harmonics Y_\ell^m are a specific set of spherical harmonics that forms an orthogonal system, first introduced by Pierre...

This paper is also remarkable for the development of the idea of the scalar potential

Scalar potential

A scalar potential is a fundamental concept in vector analysis and physics . The scalar potential is an example of a scalar field...

. The gravitational force acting on a body is, in modern language, a vector, having magnitude and direction. A potential function is a scalar

Scalar (physics)

In physics, a scalar is a simple physical quantity that is not changed by coordinate system rotations or translations , or by Lorentz transformations or space-time translations . This is in contrast to a vector...

function that defines how the vectors will behave. A scalar function is computationally and conceptually easier to deal with than a vector function.

Alexis Clairaut had first suggested the idea in 1743 while working on a similar problem though he was using Newtonian-type geometric reasoning. Laplace described Clairaut's work as being "in the class of the most beautiful mathematical productions". However, Rouse Ball alleges that the idea "was appropriated from Joseph Louis Lagrange

Joseph Louis Lagrange

Joseph-Louis Lagrange , born Giuseppe Lodovico Lagrangia, was a mathematician and astronomer, who was born in Turin, Piedmont, lived part of his life in Prussia and part in France, making significant contributions to all fields of analysis, to number theory, and to classical and celestial mechanics...

, who had used it in his memoirs of 1773, 1777 and 1780". The term "potential" itself was due to Daniel Bernoulli

Daniel Bernoulli

Daniel Bernoulli was a Dutch-Swiss mathematician and was one of the many prominent mathematicians in the Bernoulli family. He is particularly remembered for his applications of mathematics to mechanics, especially fluid mechanics, and for his pioneering work in probability and statistics...

, who introduced it in his 1738 memoire

George Green

George Green was a British mathematical physicist who wrote An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism...

's 1828 An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism

An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism

An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism, by George Green, extended the work of Siméon Denis Poisson concerning electricity and magnetism. The work's theorem of pure analysis is of the very greatest importance in all branches of physical...

.

Laplace applied the language of calculus to the potential function and showed that it always satisfies the differential equation

Differential equation

A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...

:

An analogous result for the velocity potential of a fluid had been obtained some years previously by Leonard Euler.

Laplace's subsequent work on gravitational attraction was based on this result. The quantity ∇

Laplace's equation

In mathematics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace who first studied its properties. This is often written as:where ∆ = ∇² is the Laplace operator and \varphi is a scalar function...

, a special case of Poisson's equation

Poisson's equation

In mathematics, Poisson's equation is a partial differential equation of elliptic type with broad utility in electrostatics, mechanical engineering and theoretical physics...

, appears ubiquitously in mathematical physics. The concept of a potential occurs in fluid dynamics

Fluid dynamics

In physics, fluid dynamics is a sub-discipline of fluid mechanics that deals with fluid flow—the natural science of fluids in motion. It has several subdisciplines itself, including aerodynamics and hydrodynamics...

, electromagnetism

Electromagnetism

Electromagnetism is one of the four fundamental interactions in nature. The other three are the strong interaction, the weak interaction and gravitation...

and other areas. Rouse Ball speculated that it might be seen as "the outward sign" of one the "

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

Separation of variables

In mathematics, separation of variables is any of several methods for solving ordinary and partial differential equations, in which algebra allows one to rewrite an equation so that each of two variables occurs on a different side of the equation....

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.

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

Perturbation (astronomy)

Perturbation is a term used in astronomy in connection with descriptions of the complex motion of a massive body which is subject to appreciable gravitational effects from more than one other massive body....

now known as the "great Jupiter–Saturn inequality". Laplace solved a longstanding problem in the study and prediction of the movements of these planets. He showed by general considerations, first, that the mutual action of two planets could never cause large changes in the eccentricities and inclinations of their orbits; but then, even more importantly, that peculiarities arose in the Jupiter–Saturn system because of the near approach to commensurability of the mean motions of Jupiter and Saturn. (Commensurability, in this context, means related by ratios of small whole numbers. Two periods of Saturn's orbit around the Sun almost equal five of Jupiter's. The corresponding difference between multiples of the mean motions, , corresponds to a period of nearly 900 years, and it occurs as a small divisor in the integration of a very small perturbing force with this same period. As a result, the integrated perturbations with this period are disproportionately large, about 0.8° degrees of arc in orbital longitude for Saturn and about 0.3° for Jupiter.) Further developments of these theorems on planetary motion were given in his two memoirs of 1788 and 1789, but with the aid of Laplace's discoveries, the tables of the motions of Jupiter and Saturn could at last be made much more accurate. It was on the basis of Laplace's theory that Delambre computed his astronomical tables.

Laplace also produced an analytical solution (as it turned out later, a partial solution), to a significant problem regarding the motion of the Moon. Edmond Halley

Edmond Halley

Edmond Halley FRS was an English astronomer, geophysicist, mathematician, meteorologist, and physicist who is best known for computing the orbit of the eponymous Halley's Comet. He was the second Astronomer Royal in Britain, following in the footsteps of John Flamsteed.-Biography and career:Halley...

had been the first to suggest, in 1695, that the mean motion of the Moon was apparently getting faster, by comparison with ancient eclipse observations, but he gave no data. (It was not yet known in Halley's or Laplace's times that what is actually occurring includes a slowing-down of the Earth's rate of rotation: see also Ephemeris time - History. When measured as a function of mean solar time rather than uniform time, the effect appears as a positive acceleration.) In 1749 Richard Dunthorne

Richard Dunthorne

Richard Dunthorne was an English astronomer and surveyor, who worked in Cambridge as astronomical and scientific assistant to Roger Long , and also concurrently for many years as surveyor to the Bedford Level Corporation.-Life and work:There are short biographical notes of Dunthorne, one in...

confirmed Halley's suspicion after re-examining ancient records, and produced the first quantitative estimate for the size of this apparent effect: a centurial rate of +10" (arcseconds) in lunar longitude (a surprisingly good result for its time, not far different from values assessed later, e.g. in 1786 by de Lalande, and to compare with values from about 10" to nearly 13" being derived about century later.) The effect became known as the

Laplace gave an explanation of the effect in 1787, showing how an acceleration arises from changes (a secular reduction) in the eccentricity of the Earth's orbit, which in turn is one of the effects of planetary perturbations

Perturbation (astronomy)

Perturbation is a term used in astronomy in connection with descriptions of the complex motion of a massive body which is subject to appreciable gravitational effects from more than one other massive body....

on the Earth. Laplace's initial computation accounted for the whole effect, thus seeming to tie up the theory neatly with both modern and ancient observations. However, in 1853, J C Adams

John Couch Adams

John Couch Adams was a British mathematician and astronomer. Adams was born in Laneast, near Launceston, Cornwall, and died in Cambridge. The Cornish name Couch is pronounced "cooch"....

caused the question to be re-opened by finding an error in Laplace's computations: it turned out that only about half of the Moon's apparent acceleration could be accounted for on Laplace's basis by the change in the Earth's orbital eccentricity.

(Adams showed that Laplace had in effect only considered the radial force on the moon and not the tangential, and the partial result hence had overstimated the acceleration, the remaining (negative), terms when accounted for, showed that Laplace's cause could not explain more than about half of the acceleration. The other half was subsequently shown to be due to tidal acceleration

Tidal acceleration

Tidal acceleration is an effect of the tidal forces between an orbiting natural satellite , and the primary planet that it orbits . The "acceleration" is usually negative, as it causes a gradual slowing and recession of a satellite in a prograde orbit away from the primary, and a corresponding...

.)

Laplace used his results concerning the lunar acceleration when completing his attempted "proof" of the stability of the whole solar system

Stability of the Solar System

The stability of the Solar System is a subject of much inquiry in astronomy. Though the planets have been stable historically, and will be in the short term, their weak gravitational effects on one another can add up in unpredictable ways....

on the assumption that it consists of a collection of rigid bodies

Rigid body

In physics, a rigid body is an idealization of a solid body of finite size in which deformation is neglected. In other words, the distance between any two given points of a rigid body remains constant in time regardless of external forces exerted on it...

moving in a vacuum.

All the memoirs above alluded to were presented to the

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

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

Emanuel Swedenborg

was a Swedish scientist, philosopher, and theologian. He has been termed a Christian mystic by some sources, including the Encyclopædia Britannica online version, and the Encyclopedia of Religion , which starts its article with the description that he was a "Swedish scientist and mystic." Others...

and expanded by Immanuel Kant

Immanuel Kant

Immanuel Kant was a German philosopher from Königsberg , researching, lecturing and writing on philosophy and anthropology at the end of the 18th Century Enlightenment....

, 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

Incandescence

Incandescence is the emission of light from a hot body as a result of its temperature. The term derives from the Latin verb incandescere, to glow white....

gas

Gas

Gas is one of the three classical states of matter . Near absolute zero, a substance exists as a solid. As heat is added to this substance it melts into a liquid at its melting point , boils into a gas at its boiling point, and if heated high enough would enter a plasma state in which the electrons...

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

Immanuel Kant

Immanuel Kant was a German philosopher from Königsberg , researching, lecturing and writing on philosophy and anthropology at the end of the 18th Century Enlightenment....

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

Jean-Baptiste Biot

Jean-Baptiste Biot

Jean-Baptiste Biot was a French physicist, astronomer, and mathematician who established the reality of meteorites, made an early balloon flight, and studied the polarization of light.- Biography :...

, 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, "

Differential calculus

In mathematics, differential calculus is a subfield of calculus concerned with the study of the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral 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

Félix Tisserand

François Félix Tisserand was a French astronomer.Tisserand was born at Nuits-Saint-Georges, Côte-d'Or. In 1863 he entered the École Normale Supérieure, and on leaving he went for a month as professor at the lycée at Metz. Urbain Le Verrier offered him a post in the Paris Observatory, which he...

In 1806, Laplace bought a house in Arcueil

Arcueil

Arcueil is a commune in the Val-de-Marne department in the southern suburbs of Paris, France. It is located from the center of Paris.-Name:The name Arcueil was recorded for the first time in 1119 as Arcoloï, and later in the 12th century as Arcoïalum, meaning "place of the arches" , in...

, then a village and not yet absorbed into the Paris conurbation

Conurbation

A conurbation is a region comprising a number of cities, large towns, and other urban areas that, through population growth and physical expansion, have merged to form one continuous urban and industrially developed area...

. Claude Louis Berthollet

Claude Louis Berthollet

Claude Louis Berthollet was a Savoyard-French chemist who became vice president of the French Senate in 1804.-Biography:...

was a near neighbour 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

Patronage

Patronage is the support, encouragement, privilege, or financial aid that an organization or individual bestows to another. In the history of art, arts patronage refers to the support that kings or popes have provided to musicians, painters, and sculptors...

. In 1806, he was also elected a foreign member of the Royal Swedish Academy of Sciences

Royal Swedish Academy of Sciences

The Royal Swedish Academy of Sciences or Kungliga Vetenskapsakademien is one of the Royal Academies of Sweden. The Academy is an independent, non-governmental scientific organization which acts to promote the sciences, primarily the natural sciences and mathematics.The Academy was founded on 2...

.

An account of a famous interaction between Laplace and Napoleon is provided by Rouse Ball:

Laplace also came close to propounding the concept of the black hole

Black hole

A black hole is a region of spacetime from which nothing, not even light, can escape. The theory of general relativity predicts that a sufficiently compact mass will deform spacetime to form a black hole. Around a black hole there is a mathematically defined surface called an event horizon that...

. 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

Escape velocity

In physics, escape velocity is the speed at which the kinetic energy plus the gravitational potential energy of an object is zero gravitational potential energy is negative since gravity is an attractive force and the potential is defined to be zero at infinity...

). Laplace also speculated that some of the nebulae revealed by telescopes may not be part of the Milky Way and might actually be galaxies themselves. Thus, he anticipated Edwin Hubble's

Edwin Hubble

Edwin Powell Hubble was an American astronomer who profoundly changed the understanding of the universe by confirming the existence of galaxies other than the Milky Way - our own galaxy...

major discovery 100 years in advance.

In 1812, Laplace issued his

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

Function (mathematics)

In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

as the coefficients in the expansion of another function, with reference to a different variable. The latter is therefore called the probability-generating function

Probability-generating function

In probability theory, the probability-generating function of a discrete random variable is a power series representation of the probability mass function of the random variable...

of the former. Laplace then shows how, by means of interpolation

Interpolation

In the mathematical field of numerical analysis, interpolation is a method of constructing new data points within the range of a discrete set of known data points....

, these coefficients may be determined from the generating function. Next he attacks the converse problem, and from the coefficients he finds the generating function; this is effected by the solution of a finite difference equation.

This treatise includes an exposition of the method of least squares, a remarkable testimony to Laplace's command over the processes of analysis. The method of least squares for the combination of numerous observations had been given empirically by Carl Friedrich Gauss

Carl Friedrich Gauss

Johann Carl Friedrich Gauss was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy and optics.Sometimes referred to as the Princeps mathematicorum...

(around 1794) and Legendre (in 1805), but the fourth chapter of this work contains a formal proof of it, on which the whole of the theory of errors has been since based. This was effected only by a most intricate analysis specially invented for the purpose, but the form in which it is presented is so 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.

While he conducted much research in physics

Physics

Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

, another major theme of his life's endeavours was probability theory

Probability theory

Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...

. In his

Probability

Probability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...

, which we would today recognise as Bayesian

Bayesian probability

Bayesian probability is one of the different interpretations of the concept of probability and belongs to the category of evidential probabilities. The Bayesian interpretation of probability can be seen as an extension of logic that enables reasoning with propositions, whose truth or falsity is...

. He begins the text with a series of principles of probability, the first six being:

1) Probability is the ratio of the "favored events" to the total possible events.

2) 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 favored events.

3) For independent events, the probability of the occurrence of all is the probability of each multiplied together.

4) 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 A and B both occur.

5) The probability that

6) Three corollaries are given for the sixth principle, which amount to Bayesian probability. Where event } exhausts the list of possible causes for event B, . Then

One well-known formula arising from his system is the rule of succession

Rule of succession

In probability theory, the rule of succession is a formula introduced in the 18th century by Pierre-Simon Laplace in the course of treating the sunrise problem....

, given as principle seven. Suppose that some trial has only two possible outcomes, labeled "success" and "failure". Under the assumption that little or nothing is known

where

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

Laplace published the first articulation of causal or scientific determinism:

This intellect is often referred to as

Maxwell's demon

In the philosophy of thermal and statistical physics, Maxwell's demon is a thought experiment created by the Scottish physicist James Clerk Maxwell to "show that the Second Law of Thermodynamics has only a statistical certainty." It demonstrates Maxwell's point by hypothetically describing how to...

Hans Reichenbach

Hans Reichenbach was a leading philosopher of science, educator and proponent of logical empiricism...

). Laplace, himself, did not use the word "demon", which was a later embellishment. As translated into English above, he simply referred to:

As early as 1744, Euler, followed by Lagrange

Lagrange

La Grange literally means the barn in French. Lagrange may refer to:- People :* Charles Varlet de La Grange , French actor* Georges Lagrange , translator to and writer in Esperanto...

, had started looking for solutions of differential equation

Differential equation

A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...

s in the form:

In 1785, Laplace took the key forward step in using integrals of this form in order to transform a whole difference equation, rather than simply as a form for the solution, and found that the transformed equation was easier to solve than the original.

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

- Discussion, contemporaneously with Alexandre-Théophile VandermondeAlexandre-Théophile VandermondeAlexandre-Théophile Vandermonde was a French musician, mathematician and chemist who worked with Bézout and Lavoisier; his name is now principally associated with determinant theory in mathematics. He was born in Paris, and died there.Vandermonde was a violinist, and became engaged with...

, of the general theory of determinantDeterminantIn linear algebra, the determinant is a value associated with a square matrix. It can be computed from the entries of the matrix by a specific arithmetic expression, while other ways to determine its value exist as well...

s, (1772); - Proof that every equation of an even degree must have at least one realReal numberIn mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

quadraticQuadraticIn mathematics, the term quadratic describes something that pertains to squares, to the operation of squaring, to terms of the second degree, or equations or formulas that involve such terms...

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 fractionContinued fractionIn mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer part and another reciprocal, and so on...

; and - In his theory of probabilities:
- Evaluation of several common definite integrals; and
- General proof of the Lagrange reversion theoremLagrange reversion theoremIn mathematics, the Lagrange reversion theorem gives series or formal power series expansions of certain implicitly defined functions; indeed, of compositions with such functions....

.

Laplace built upon the qualitative work of Thomas Young

Thomas Young (scientist)

Thomas Young was an English polymath. He is famous for having partly deciphered Egyptian hieroglyphics before Jean-François Champollion eventually expanded on his work...

to develop the theory of capillary action

Capillary action

Capillary action, or capilarity, is the ability of a liquid to flow against gravity where liquid spontanously rise in a narrow space such as between the hair of a paint-brush, in a thin tube, or in porous material such as paper or in some non-porous material such as liquified carbon fiber, or in a...

and the Young-Laplace equation.

Laplace in 1816 was the first to point out that the speed of sound

Speed of sound

The speed of sound is the distance travelled during a unit of time by a sound wave propagating through an elastic medium. In dry air at , the speed of sound is . This is , or about one kilometer in three seconds or approximately one mile in five seconds....

in air depends on the heat capacity ratio

Heat capacity ratio

The heat capacity ratio or adiabatic index or ratio of specific heats, is the ratio of the heat capacity at constant pressure to heat capacity at constant volume . It is sometimes also known as the isentropic expansion factor and is denoted by \gamma or \kappa . The latter symbol kappa is...

. Newton's original theory gave too low a value, because it does not take account of the adiabatic

Adiabatic process

In thermodynamics, an adiabatic process or an isocaloric process is a thermodynamic process in which the net heat transfer to or from the working fluid is zero. Such a process can occur if the container of the system has thermally-insulated walls or the process happens in an extremely short time,...

compression of the air which results in a local rise in temperature

Temperature

Temperature is a physical property of matter that quantitatively expresses the common notions of hot and cold. Objects of low temperature are cold, while various degrees of higher temperatures are referred to as warm or hot...

and pressure

Pressure

Pressure is the force per unit area applied in a direction perpendicular to the surface of an object. Gauge pressure is the pressure relative to the local atmospheric or ambient pressure.- Definition :...

. Laplace's investigations in practical physics were confined to those carried on by him jointly with Lavoisier in the years 1782 to 1784 on the specific heat of various bodies.

According to W. W. Rouse Ball

W. W. Rouse Ball

-External links:*...

, as Napoleon's power increased Laplace begged him to give him the post of Minister of the Interior

Minister of the Interior (France)

The Minister of the Interior in France is one of the most important governmental cabinet positions, responsible for the following:* The general interior security of the country, with respect to criminal acts or natural catastrophes...

. However this is disputed by Pearson

Karl Pearson

Karl Pearson FRS was an influential English mathematician who has been credited for establishing the disciplineof mathematical statistics....

. Napoleon, who desired the support of men of science, did make him Minister of the Interior in November 1799, but a little less than six weeks saw the close of Laplace's political career. Napoleon later (in his

Lucien, Napoleon's brother, was given the post. Although Laplace was removed from office, it was desirable to retain his allegiance. He was accordingly raised to the senate, and to the third volume of the

Bourbon Restoration

The Bourbon Restoration is the name given to the period following the successive events of the French Revolution , the end of the First Republic , and then the forcible end of the First French Empire under Napoleon – when a coalition of European powers restored by arms the monarchy to the...

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

Bourbon Restoration

The Bourbon Restoration is the name given to the period following the successive events of the French Revolution , the end of the First Republic , and then the forcible end of the First French Empire under Napoleon – when a coalition of European powers restored by arms the monarchy to the...

he was rewarded with the title of marquis

Marquis

Marquis is a French and Scottish title of nobility. The English equivalent is Marquess, while in German, it is Markgraf.It may also refer to:Persons:...

.

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

Paul Louis Courier

Paul Louis Courier , French Hellenist and political writer, was born in Paris.Brought up on his father's estate of Méré in Touraine, he conceived a bitter aversion for the nobility, which seemed to strengthen with time. He would never take the name "de Méré", to which he was entitled, lest he...

. His knowledge was useful on the numerous scientific commissions on which he served, and probably accounts for the manner in which his political insincerity was overlooked.

He died in Paris in 1827. His brain was removed by his physician, François Magendie

François Magendie

François Magendie was a French physiologist, considered a pioneer of experimental physiology. He is known for describing the foramen of Magendie. There is also a Magendie sign, a downward and inward rotation of the eye due to a lesion in the cerebellum...

, and kept for many years, eventually being displayed in a roving anatomical museum in Britain. It was reportedly smaller than the average brain.

- AsteroidAsteroidAsteroids are a class of small Solar System bodies in orbit around the Sun. They have also been called planetoids, especially the larger ones...

4628 Laplace4628 Laplace4628 Laplace is a main-belt asteroid discovered on September 7, 1986 by E. W. Elst at Rozhen....

is named for him. - He is one of only seventy-two people to have their name engraved on the Eiffel Tower.
- The European Space Agency'sEuropean Space AgencyThe European Space Agency , established in 1975, is an intergovernmental organisation dedicated to the exploration of space, currently with 18 member states...

working-title for the international Europa Jupiter System MissionEuropa Jupiter System MissionThe Europa Jupiter System Mission – Laplace was a proposed joint NASA/ESA unmanned space mission slated to launch around 2020 for the in-depth exploration of Jupiter's moons with a focus on Europa, Ganymede and Jupiter's magnetosphere...

is "Laplace".

- What we know is not much. What we do not know is immense. (attributed)
- I had no need of that hypothesis. ("Je n'avais pas besoin de cette hypothèse-là", as a reply to NapoleonNapoleon I of FranceNapoleon Bonaparte was a French military and political leader during the latter stages of the French Revolution.As Napoleon I, he was Emperor of the French from 1804 to 1815...

, who had asked why he hadn't mentioned God in his book on astronomyAstronomyAstronomy is a natural science that deals with the study of celestial objects and phenomena that originate outside the atmosphere of Earth...

.) - "It is therefore obvious that ..." (frequently used in the
*Celestial Mechanics*when he had proved something and mislaid the proof, or found it clumsy. Notorious as a signal for something true, but hard to prove.) - The weight of evidence for an extraordinary claim must be proportioned to its strangeness.
- "...(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."

*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*

- Bowditch, N.Nathaniel BowditchNathaniel Bowditch was an early American mathematician remembered for his work on ocean navigation. He is often credited as the founder of modern maritime navigation; his book The New American Practical Navigator, first published in 1802, is still carried on board every commissioned U.S...

(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

, translated from the French 6th ed. (1840)

- David, F. N. (1965) "Some notes on Laplace", in Neyman, J.Jerzy NeymanJerzy Neyman , born Jerzy Spława-Neyman, was a Polish American mathematician and statistician who spent most of his professional career at the University of California, Berkeley.-Life and career:...

& LeCam, L. M. (eds)*Bernoulli, Bayes and Laplace*, Berlin,*pp*30–44, delivered 15 June 1829, published in 1831. (in French) - — (1997)
*Pierre Simon Laplace 1749–1827: A Life in Exact Science*, Princeton: Princeton University Press, ISBN 0-691-01185-0 - Grattan-Guinness, I.Ivor Grattan-GuinnessIvor Grattan-Guinness, born 23 June 1941, in Bakewell, in England, is a historian of mathematics and logic.He gained his Bachelor degree as a Mathematics Scholar at Wadham College, Oxford, got an M.Sc in Mathematical Logic and the Philosophy of Science at the London School of Economics in 1966...

, 2005, "'Exposition du système du monde' and 'Traité de méchanique céleste'" in his*Landmark Writings in Western Mathematics*. Elsevier: 242–57. - — (2005)
*Pierre Simon Laplace 1749–1827: A Determined Scientist*, Cambridge, MA: Harvard University Press, ISBN 0-674-01892-3 (1999) - Rouse Ball, W. W.W. W. Rouse Ball-External links:*...

[1908] (2003) "Pierre Simon Laplace (1749–1827)", in*A Short Account of the History of Mathematics*, 4th ed., Dover, ISBN 0486206300 - Whitrow, G. J.Gerald James WhitrowGerald James Whitrow was a British mathematician, cosmologist and science historian.After completing school at Christ's Hospital, he obtained a scholarship at Christ Church, Oxford in 1930, earning his first degree in 1933, the MA in 1937, and the PhD in 1939...

(2001) "Laplace, Pierre-Simon, marquis de",*Encyclopaedia Britannica*, Deluxe CDROM edition (available from Google Books)

- "Pierre-Simon Laplace" in the MacTutor History of Mathematics archiveMacTutor History of Mathematics archiveThe MacTutor History of Mathematics archive is a website maintained by John J. O'Connor and Edmund F. Robertson and hosted by the University of St Andrews in Scotland...

. - Guide to the Pierre Simon Laplace Papers at The Bancroft Library
- English translation of a large part of Laplace's work in probability and statistics, provided by Richard Pulskamp