View source for Pierre-Simon Laplace

{{short description|French polymath (1749–1827)}}
{{Redirect|Laplace}}
{{EngvarB|date=July 2017}}
{{Use dmy dates|date=March 2024}}
{{Infobox scientist
| name = Pierre-Simon Laplace
| image = Laplace, Pierre-Simon, marquis de.jpg
| image_size = 225
| caption = Pierre-Simon Laplace as chancellor of the [[Senate]] under the [[First French Empire]]
| birth_date = {{birth date|df=yes|1749|3|23}}
| birth_place = [[Beaumont-en-Auge]], Normandy, [[Kingdom of France]]
| death_date = {{death date and age|df=yes|1827|3|5|1749|3|23}}
| death_place = [[Paris]], [[Bourbon Restoration in France|Kingdom of France]]
| fields     = [[Astronomy]] and [[Mathematics]]
| workplaces = ''[[École Militaire]]'' (1769–1776)
| alma_mater = [[University of Caen]]
| doctoral_advisor = <!--there were no PhDs in France before 1808-->
| academic_advisors = [[Jean d'Alembert]]<br />[[Christophe Gadbled]]<br />Pierre Le Canu
| doctoral_students = <!--there were no PhDs in France before 1808-->
| notable_students = [[Siméon Denis Poisson]]<br />[[Napoleon|Napoleon Bonaparte]]
| known_for = {{collapsible list|title=&nbsp;|Work in [[celestial mechanics]]<br />Predicting the existence of [[black holes]]<ref name=HE73p364>[[S.W. Hawking]] and [[George F.R. Ellis]], ''[[The Large Scale Structure of Space-Time]]'', Cambridge University Press, 1973, p. 364.</ref><br />[[Bayesian inference]]<br />[[Bayesian probability]]<br />[[Laplace's equation]]<br />[[Laplace operator|Laplacian]]<br />[[Laplace transform]]<br />[[Inverse Laplace transform]]<br />[[Laplace distribution]]<br />[[Laplace's demon]]<br />[[Young–Laplace equation]]<br />[[Laplace number]]<br />[[Laplace limit]]<br />[[Laplace invariant]]<br />[[Laplace principle (large deviations theory)|Laplace principle]]<br />[[Principle of indifference|Laplace's principle of insufficient reason]]<br />[[Laplace's method]]<br />[[Laplace force]]<br />[[Laplace filter]]<br />[[Laplace functional]]<br />[[Laplacian matrix]]<br />[[Variance gamma process|Laplace motion]]<br />[[Laplace plane]]<br />[[Laplace pressure]]<br />[[Laplace resonance]]<br />[[Spherical harmonics|Laplace's spherical harmonics]]<br />[[Additive smoothing|Laplace smoothing]]<br />[[Laplace expansion]]<br />[[Laplace expansion (potential)|Laplace expansion]]<br />[[Rule of succession|Laplace-Bayes estimator]]<br />[[Laplace–Stieltjes transform]]<br />[[Laplace–Runge–Lenz vector]]<br />[[Nebular hypothesis]]}}
| awards =
| signature = Pierre-Simon Laplace signature.svg
| module            = {{Infobox officeholder
| embed=yes|
| office              = [[Minister of the Interior (France)|Minister of the Interior]]
| term_start          = 12 November 1799
| term_end            = 25 December 1799
| primeminister       = [[Napoleon Bonaparte]] (as [[French Consulate|First Consul]])
| predecessor         = [[Nicolas Marie Quinette]]
| successor           = [[Lucien Bonaparte]]
}}
}}
'''Pierre-Simon, Marquis de Laplace''' ({{IPAc-en|l|ə|ˈ|p|l|ɑ:|s}}; {{IPA|fr|pjɛʁ simɔ̃ laplas|lang}}; 23 March 1749&nbsp;– 5 March 1827) was a French [[scholar]] whose work was important to the development of [[engineering]], [[mathematics]], [[statistics]], [[physics]], [[astronomy]], and [[philosophy]]. He summarized and extended the work of his predecessors in his five-volume [[Traité de mécanique céleste|''Mécanique céleste'']] (''Celestial Mechanics'') (1799–1825). This work translated the geometric study of [[classical mechanics]] to one based on [[calculus]], opening up a broader range of problems. Laplace also popularized and further confirmed [[Isaac Newton|Sir Isaac Newton]]'s work.<ref>{{Cite web |last=Forrister |first=Thomas |date=2018-03-28 |title=Happy Birthday, Pierre-Simon Laplace |url=https://www.comsol.com/blogs/happy-birthday-pierre-simon-laplace |access-date=2024-12-04 |website=COMSOL |language=en}}</ref> In statistics, the [[Bayesian probability|Bayesian interpretation]] of probability was developed mainly by Laplace.<ref>Stigler, Stephen M. (1986). ''The History of Statistics: The Measurement of Uncertainty before 1900''. Harvard University Press, Chapter 3.</ref>

Laplace formulated [[Laplace's equation]], and pioneered the [[Laplace transform]] which appears in many branches of [[mathematical physics]], a field that he took a leading role in forming. The [[Laplace operator|Laplacian differential operator]], widely used in mathematics, is also named after him. He restated and developed the [[nebular hypothesis]] of the [[origin of the Solar System]] and was one of the first scientists to suggest an idea similar to that of a [[black hole]],<ref>{{Cite journal |title=Michell, Laplace and the Origin of the Black Hole Concept |url=https://www.sciengine.com/JAHH/doi/10.3724/SP.J.1440-2807.2009.02.01;JSESSIONID=cca4504f-bcd6-490a-ad9b-bfa868ab7e30 |journal=Journal of Astronomical History and Heritage |language=en-US |pages=90–96 |doi=10.3724/SP.J.1440-2807.2009.02.01 |issn=1440-2807 | date=2009 | last1=Montgomery | first1=Colin | last2=Orchiston | first2=Wayne | last3=Whittingham | first3=Ian | volume=12 | issue=2 | s2cid=55890996 }}</ref> with [[Stephen Hawking]] stating that "Laplace essentially predicted the existence of black holes".<ref name="HE73p364" /> He originated [[Laplace's demon]], which is a hypothetical all-predicting intellect. He also refined Newton's calculation of the [[speed of sound]] to derive a more accurate measurement.<ref>{{Cite journal |last=Finn |first=Bernard S. |date=1964 |title=Laplace and the Speed of Sound |url=https://www.journals.uchicago.edu/doi/10.1086/349791 |journal=Isis |language=en |volume=55 |issue=1 |pages=7–19 |doi=10.1086/349791 |issn=0021-1753}}</ref>

Laplace is regarded as one of the greatest scientists of all time. Sometimes referred to as the ''French [[Isaac Newton|Newton]]'' or ''Newton of France'', he has been described as possessing a phenomenal natural mathematical faculty superior to that of almost all of his contemporaries.<ref name="eb1911">{{cite EB1911 |wstitle=Laplace, Pierre Simon |volume=16 |pages=200–202 |first=Agnes Mary |last=Clerke|author-link= Agnes Mary Clerke|short=1}}</ref> He was Napoleon's examiner when [[Napoleon]] graduated from the ''[[École Militaire]]'' in Paris in 1785.<ref>{{cite journal|first=Thomas L.|last=Hankins|title=Pierre Simon Laplace, 1749–1827: A Determined Scientist (Book review)|journal=Physics Today|volume=59|issue=9|pages=62–64|year=2006|doi=10.1063/1.2364251|doi-access=free}}</ref>  Laplace became a count of the [[First French Empire|Empire]] in 1806 and was named a [[marquess|marquis]] in 1817, after the [[Bourbon Restoration in France|Bourbon Restoration]].

==Early years==
[[File:Pierre-Simon de Laplace by Johann Ernst Heinsius (1775).jpg|thumb|right|Portrait of Pierre-Simon Laplace by [[Johann Ernst Heinsius]] (1775)]]
Some details of Laplace's life are not known, as records of it were burned in 1925 with the family [[château]] in [[Saint Julien de Mailloc]], near [[Lisieux]], the home of his great-great-grandson the Comte de Colbert-Laplace. Others had been destroyed earlier, when his house at [[Arcueil]] near Paris was looted in 1871.<ref name=Pearson>"Laplace, being Extracts from Lectures delivered by [[Karl Pearson]]", ''[[Biometrika]]'', vol. 21, December 1929, pp. 202–216.</ref>

Laplace was born in [[Beaumont-en-Auge]], Normandy on 23 March 1749, a village four miles west of [[Pont-l'Évêque, Calvados|Pont l'Évêque]]. According to [[W. W. Rouse Ball]],<ref>[[W.W. Rouse Ball]] ''A Short Account of the History of Mathematics'', 4th edition, 1908.</ref> his father, Pierre de Laplace, owned and farmed the small estates of Maarquis. His great-uncle, Maitre Oliver de Laplace, had held the title of Chirurgien Royal. It would seem that from a pupil he became an usher in the school at Beaumont; but, having procured a letter of introduction to [[Jean le Rond d'Alembert|d'Alembert]], he went to Paris to advance his fortune. However, [[Karl Pearson]]<ref name=Pearson/> is scathing about the inaccuracies in Rouse Ball's account and states:
{{blockquote|Indeed [[Caen]] 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 the ''Mélanges'' of 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 [[Joseph Louis Lagrange|Lagrange]] in [[Turin]]. 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 and was a member of the Sphinx. The ''[[École Militaire]]'' of Beaumont did not replace the old school until 1776.}}

His parents, Pierre Laplace and Marie-Anne Sochon, were from comfortable families. The Laplace family was involved in agriculture until at least 1750, but Pierre Laplace senior was also a [[cider]] merchant and ''[[syndic]]'' of the town of Beaumont.

Pierre Simon Laplace attended a school in the village run at a [[Benedictine]] [[priory]], his father intending that he be ordained in the [[Roman Catholic Church]]. At sixteen, to further his father's intention, he was sent to the [[University of Caen]] to read theology.<ref name="mactutor">*{{MacTutor Biography|id=Laplace}}. Retrieved 25 August 2007</ref>

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. Here Laplace's brilliance as a mathematician was quickly recognised and while still at Caen he wrote a memoir ''Sur le Calcul integral aux differences infiniment petites et aux differences finies''. This provided the first correspondence between Laplace and Lagrange. Lagrange was the senior by thirteen years, and had recently founded in his native city [[Turin]] a journal named ''Miscellanea Taurinensia'', in which many of his early works were printed and it was in the fourth volume of this series that Laplace's paper appeared. About this time, recognising that he had no vocation for the priesthood, he resolved to become a professional mathematician. Some sources state that he then broke with the church and became an atheist.{{Citation needed|date=August 2016}}  Laplace did not graduate in theology but left for Paris with a letter of introduction from Le Canu to [[Jean le Rond d'Alembert]] who at that time was supreme in scientific circles.<ref name="mactutor"/><ref>{{cite journal | url=https://www.jstor.org/stable/3608408 | jstor=3608408 | doi=10.2307/3608408 | title=Laplace | last1=Whittaker | first1=Edmund | journal=The Mathematical Gazette | date=1949 | volume=33 | issue=303 | pages=1–12 }}</ref>

According to his great-great-grandson,<ref name=Pearson/> 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 realised that it was true, and from that time he took Laplace under his care.

Another account 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]]''.<ref name="gillsipie3n4">Gillispie (1997), pp. 3–4</ref>

With a secure income and undemanding teaching, Laplace now threw himself into original research and for the next seventeen years, 1771–1787, he produced much of his original work in astronomy.<ref name="ball">Rouse Ball (1908).</ref>

[[File:Calorimeter of Lavoisier and La Place plate xi the s1784m49j 3f462600t dl full size.jpg|thumb|right|The Calorimeter of Lavoisier and La Place, ''Encyclopaedia Londinensis'', 1801]]
From 1780 to 1784, Laplace and French chemist [[Antoine Lavoisier]] collaborated on several experimental investigations, designing their own equipment for the task.<ref name="ACS1999">{{cite web |title=The Chemical Revolution of Antoine-Laurent Lavoisier International Historic Chemical Landmark |url=https://www.acs.org/content/acs/en/education/whatischemistry/landmarks/lavoisier.html |website=American Chemical Society|date=8 June 1999}}</ref> 
In 1783 they published their joint paper, ''Memoir on Heat'', in which they discussed the kinetic theory of molecular motion.<ref name="Golinski">{{cite journal |last1=Golinski |first1=Jan V. |title=Antoine Laurent Lavoisier, Pierre Simon, Marquis de Laplace, Henry Guerlac |journal=Isis |date=June 1983 |volume=74 |issue=2 |pages=288–289 |doi=10.1086/353288}}</ref>
In their experiments they measured the [[specific heat]] of various bodies, and the expansion of metals with increasing temperature. They also measured the boiling points of [[ethanol]] and [[ethyl ether|ether]] under pressure.

Laplace further impressed the [[Marquis de Condorcet]], and already by 1771 Laplace felt entitled to membership in the [[French Academy of Sciences]]. However, that year admission went to [[Alexandre-Théophile Vandermonde]] and in 1772 to Jacques Antoine Joseph Cousin. Laplace was disgruntled, and early in 1773 d'Alembert wrote to [[Lagrange]] in Berlin to ask if a position could be found for Laplace there. However, Condorcet became permanent secretary of the ''Académie'' in February and Laplace was elected associate member on 31 March, at age&nbsp;24.<ref name="gillsipie5">Gillispie (1997), p. 5</ref> In 1773 Laplace read his paper on the invariability of planetary motion in front of the Academy des Sciences. That March he was elected to the academy, a place where he conducted the majority of his science.<ref>[https://sites.google.com/a/wisc.edu/ils202fall11/home/student-wikis/group5/ "Effects of the Scientific Community on Laplace"] Retrieved on 10 January 2018</ref>

On 15 March 1788,<ref>Hahn (2005), p. 99. However, Gillispie (1997), p. 67, gives the month of the marriage as May.</ref><!--Several sources give the date of marriage as 1788, but Pearson quotes the great-great-grandson as saying that Sophie (his great-grandmother) was born in 1787.--><ref name=Pearson/> at the age of thirty-nine, Laplace married Marie-Charlotte de Courty de Romanges, an eighteen-year-old girl from a "good" family in [[Besançon]].<ref>Hahn (2005), pp. 99–100</ref> The wedding was celebrated at [[Saint-Sulpice, Paris]]. The couple had a son, Charles-Émile (1789–1874), and a daughter, Sophie-Suzanne (1792–1813).<ref>Gillispie (1997), p. 67</ref><ref>Hahn (2005), p. 101</ref>

==Analysis, probability, and astronomical stability==
Laplace's early published work in 1771 started with [[differential equations]] and [[finite differences]] but he was already starting to think about the mathematical and philosophical concepts of probability and statistics.<ref name="gillispiech2">Gillispie (1989), pp. 7–12</ref> However, before his election to the ''Académie'' in 1773, he had already drafted two papers that would establish his reputation. The first, ''Mémoire sur la probabilité des causes par les événements'' was ultimately published in 1774 while the second paper, published in 1776, further elaborated his statistical thinking and also began his systematic work on [[celestial mechanics]] and the stability of the [[Solar System]]. The two disciplines would always be interlinked in his mind. "Laplace took probability as an instrument for repairing defects in knowledge."<ref name="gillispie14n15">Gillispie (1989). pp. 14–15</ref> Laplace's work on probability and statistics is discussed below with his mature work on the analytic theory of probabilities.

===Stability of the Solar System===
Sir [[Isaac Newton]] had published his ''[[Philosophiæ Naturalis Principia Mathematica]]'' in 1687 in which he gave a derivation of [[Kepler's laws]], which describe the motion of the planets, from his [[Newton's laws of motion|laws of motion]] and his [[Newton's law of universal gravitation|law of universal gravitation]]. However, though Newton had privately developed the methods of calculus, all his published work used cumbersome geometric reasoning, unsuitable to account for the more subtle higher-order effects of interactions between the planets. Newton himself had doubted the possibility of a mathematical solution to the whole, even concluding that periodic [[miracle|divine intervention]] was necessary to guarantee the stability of the Solar System. Dispensing with the hypothesis of divine intervention would be a major activity of Laplace's scientific life.<ref name=eb2001>{{harvp|Whitrow|2001}}</ref> It is now generally regarded that Laplace's methods on their own, though vital to the development of the theory, are not sufficiently [[accuracy and precision|precise]] to demonstrate the [[stability of the Solar System]]; today the Solar System is understood to be generally [[chaos theory|chaotic]] at fine scales, although currently fairly stable on coarse scale.<ref>{{cite book | last1=Celletti | first1=A. | last2=Perozzi |first2=E. | name-list-style=amp | year=2007 | title=Celestial Mechanics: The waltz of the planets | place=Berlin, DE | publisher=Springer | isbn=978-0-387-30777-0 | bibcode=2006cmwp.book.....C }}</ref>{{rp|83,93}}

One particular problem from [[observational astronomy]] was the apparent instability whereby Jupiter's orbit appeared to be shrinking while that of Saturn was expanding. The problem had been tackled by [[Leonhard Euler]] in 1748, and [[Joseph Louis Lagrange]] in 1763, but without success.<ref name=whittakerb>{{harvp|Whittaker|1949b}}</ref> 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.<ref name=gillispie29to35>{{harvp|Gillispie|1997|pp=29–35}}</ref> 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 [[Integral|integrated]] over time they could become important. Laplace carried his analysis into the higher-order terms, up to and including the [[Cubic function|cubic]]. Using this more exact analysis, Laplace concluded that any two planets and the Sun must be in mutual equilibrium and thereby launched his work on the stability of the Solar System.<ref name=gillispie35n36>{{harvp|Gillispie|1997|pp=35–36}}</ref> [[Gerald James Whitrow]] described the achievement as "the most important advance in physical astronomy since Newton".<ref name=eb2001/>

Laplace had a wide knowledge of all sciences and dominated all discussions in the ''Académie''.<ref>{{cite web |title=Laplace |department=biographies |series=School of Mathematics and Statistics |publisher=[[University of St. Andrews]] |place=[[St Andrews|St. Andrews]], [[Scotland]] |url=http://www-history.mcs.st-andrews.ac.uk/Biographies/Laplace.html }}</ref> Laplace seems to have regarded analysis merely as a means of attacking physical problems, though the ability with which he invented the necessary analysis is almost phenomenal. As long as his results were true he took but little trouble to explain the steps by which he arrived at them; he never studied elegance or symmetry in his processes, and it was sufficient for him if he could by any means solve the particular question he was discussing.<ref name=ball/>

==Tidal dynamics==
{{main|Theory of tides}}

===Dynamic theory of tides===
While [[Isaac Newton|Newton]] explained the tides by describing the tide-generating forces and [[Daniel Bernoulli|Bernoulli]] gave a description of the static reaction of the waters on Earth to the tidal potential, the ''dynamic theory of tides'', developed by Laplace in 1775,<ref>{{cite web| url=http://www.preservearticles.com/2011112017524/short-notes-on-the-dynamical-theory-of-laplace.html| title=Short notes on the Dynamical theory of Laplace| date=20 November 2011| access-date=28 October 2017| archive-date=2 April 2015| archive-url=https://web.archive.org/web/20150402155415/http://www.preservearticles.com/2011112017524/short-notes-on-the-dynamical-theory-of-laplace.html| url-status=dead}}</ref> describes the ocean's real reaction to [[tidal force]]s.<ref>{{cite web |last1=Hautala |first1=Susan |last2=Kelly |first2=Kathryn |last3=Thompson |first3=LuAnne |author3-link=LuAnne Thompson |title=Tide Dynamics |url=http://faculty.washington.edu/luanne/pages/ocean420/notes/tidedynamics.pdf |date=2005}}</ref> Laplace's theory of ocean tides took into account [[friction]], [[resonance]] and natural periods of ocean basins. It predicted the large [[amphidromic]] systems in the world's ocean basins and explains the oceanic tides that are actually observed.<ref name="pearsonhighered.com">{{cite web | url=http://www.pearsonhighered.com/samplechapter/0132401223.pdf | title=Higher Education}}</ref><ref>{{cite web |last1=Ahn |first1=Kyungjin |title=An Astronomer's View on the Current College-Level Textbook Descriptions of Tides |url=http://ocean.kisti.re.kr/downfile/volume/kess/JGGHBA/2009/v30n5/JGGHBA_2009_v30n5_671.pdf |publisher=Korean Earth Science Society |date=September 2009}}</ref>
The equilibrium theory, based on the gravitational gradient from the Sun and Moon but ignoring the Earth's rotation, the effects of continents, and other important effects, could not explain the real ocean tides.<ref>[http://www.sanho.co.za/tides/tide_theory.pdf Tidal theory] {{Webarchive|url=https://web.archive.org/web/20170822175040/http://www.sanho.co.za/tides/tide_theory.PDF |date=22 August 2017 }} website South African Navy Hydrographic Office</ref><ref>{{cite web|url=http://www.oberlin.edu/faculty/swojtal/SFWpage/161Stuff/161Lect17/sld012.htm |title=Dynamic theory for tides |publisher=Oberlin.edu |access-date=2 June 2012}}</ref><ref>{{cite web|url=   http://ffden-2.phys.uaf.edu/645fall2003_web.dir/ellie_boyce/dynamic.htm|title=  Dynamic Theory of Tides}}</ref><ref name="pearsonhighered.com"/><ref>{{cite web |url=http://web.vims.edu/physical/research/TCTutorial/dynamic.htm |title=Dynamic Tides – In contrast to "static" theory, the dynamic theory of tides recognizes that water covers only three-quarters o |publisher=Web.vims.edu |access-date=2 June 2012 |url-status=dead |archive-url=https://web.archive.org/web/20130113022202/http://web.vims.edu/physical/research/TCTutorial/dynamic.htm |archive-date=13 January 2013 }}</ref><ref>{{cite web |url=http://www.coa.edu/stodd/oceanweb/oceanography/Oceanlectures02/Lecture8/sld014.htm |title=The Dynamic Theory of Tides |publisher=Coa.edu |access-date=2 June 2012 |url-status=dead |archive-url=https://web.archive.org/web/20131219161027/http://www.coa.edu/stodd/oceanweb/oceanography/Oceanlectures02/Lecture8/sld014.htm |archive-date=19 December 2013 }}</ref><ref>{{cite web |url=https://beacon.salemstate.edu/~lhanson/gls214/gls214_tides |title=Tides|website=beacon.salemstate.edu |access-date=3 February 2022 |archive-url=https://archive.today/20121214223022/https://beacon.salemstate.edu/~lhanson/gls214/gls214_tides |archive-date=14 December 2012 |url-status=dead}}</ref><ref>{{cite web|url=http://www.waterencyclopedia.com/St-Ts/Tides.html |title=Tides – building, river, sea, depth, oceans, effects, important, largest, system, wave, effect, marine, Pacific |publisher=Waterencyclopedia.com |date=27 June 2010}}</ref><ref>{{cite web |url=http://ocean.tamu.edu/wormuth/tidestxt.html |title=TIDES |publisher=Ocean.tamu.edu |access-date=2 June 2012 |archive-url=https://web.archive.org/web/20130616163843/http://ocean.tamu.edu/wormuth/tidestxt.html |archive-date=16 June 2013 |url-status=dead }}</ref> [[File:Newton's three-body diagram.PNG|400px|right|Newton's three-body model]]
Since measurements have confirmed the theory, many things have possible explanations now, like how the tides interact with deep sea ridges and chains of seamounts give rise to deep eddies that transport nutrients from the deep to the surface.<ref>{{cite web|author=Floor Anthoni |url=http://www.seafriends.org.nz/oceano/tides.htm |title=Tides |publisher=Seafriends.org.nz |access-date=2 June 2012}}</ref> The equilibrium tide theory calculates the height of the tide wave of less than half a meter, while the dynamic theory explains why tides are up to 15 meters.<ref>{{cite web|url=   http://www.linz.govt.nz/hydro/tidal-info/tidal-intro/cause-nature|title=  The Cause & Nature of Tides}}</ref> Satellite observations confirm the accuracy of the dynamic theory, and the tides worldwide are now measured to within a few centimeters.<ref>{{cite web|url=http://svs.gsfc.nasa.gov/stories/topex/tides.html |title=Scientific Visualization Studio TOPEX/Poseidon images |publisher=Svs.gsfc.nasa.gov |access-date=2 June 2012}}</ref><ref>{{cite web|url=https://archive.org/details/SVS-1333 |title=TOPEX/Poseidon Western Hemisphere: Tide Height Model : NASA/Goddard Space Flight Center Scientific Visualization Studio : Free Download & Streaming : Internet Archive|date=15 June 2000 }}</ref> Measurements from the [[CHAMP (satellite)|CHAMP]] satellite closely match the models based on the [[TOPEX]] data.<ref>TOPEX data used to model actual tides for 15 days from the year 2000 [http://svs.gsfc.nasa.gov/vis/a000000/a001300/a001332/ TOPEX/Poseidon Flat Earth Tide Height Model] {{Webarchive|url=https://web.archive.org/web/20150918082009/http://svs.gsfc.nasa.gov/vis/a000000/a001300/a001332/ |date=18 September 2015 }}</ref><ref>http://www.geomag.us/info/Ocean/m2_CHAMP+longwave_SSH.swf {{Dead link|date=November 2023|fix-attempted=yes}}</ref><ref>{{cite web |url=http://volkov.oce.orst.edu/tides/ |archive-url=https://wayback.archive-it.org/all/20121022041453/http://volkov.oce.orst.edu/tides/ |url-status=dead |archive-date=22 October 2012 |title=OSU Tidal Data Inversion |publisher=Volkov.oce.orst.edu |access-date=2 June 2012 }}</ref> Accurate models of tides worldwide are essential for research since the variations due to tides must be removed from measurements when calculating gravity and changes in sea levels.<ref>{{cite web|url=   http://www.dgfi.tum.de/en/news/baroclinic-tides/|title=   Dynamic and residual ocean tide analysis for improved GRACE de-aliasing (DAROTA)|url-status=dead|archive-url=   https://web.archive.org/web/20150402194935/http://www.dgfi.tum.de/en/news/baroclinic-tides/|archive-date=   2 April 2015|df=   dmy-all}}</ref>

===Laplace's tidal equations===
{| border="0" cellpadding="1" cellspacing="0" style="float:right;"
|- align=center
|[[Image:Tideforcenw.jpg|thumb|150px| ''A.'' Lunar gravitational potential: this depicts the Moon directly over 30° N (or 30° S) viewed from above the Northern Hemisphere.]]
|[[Image:Tideforcese.jpg|thumb|150px| ''B.'' This view shows same potential from 180° from view ''A''. Viewed from above the Northern Hemisphere. Red up, blue down.]]
|}

In 1776, Laplace formulated a single set of linear [[partial differential equation]]s, for tidal flow described as a [[barotropic]] two-dimensional sheet flow. [[Coriolis effect]]s are introduced as well as lateral forcing by gravity. Laplace obtained these equations by simplifying the [[fluid dynamic]] equations. But they can also be derived from energy integrals via [[Lagrange's equation]].

For a fluid sheet of [[average]] thickness ''D'', the vertical tidal elevation ''ζ'', as well as the horizontal velocity components ''u'' and ''v'' (in the [[latitude]] ''φ'' and [[longitude]] ''λ'' directions, respectively) satisfy '''Laplace's tidal equations''':<ref>{{cite web|url=  http://kiwi.atmos.colostate.edu/group/dave/pdf/LTE.frame.pdf|title=  The Laplace Tidal Equations and Atmospheric Tides|access-date=  28 October 2017|archive-url=  https://web.archive.org/web/20190411204849/http://kiwi.atmos.colostate.edu/group/dave/pdf/LTE.frame.pdf|archive-date=  11 April 2019|url-status=  dead}}</ref>
:<math>
  \begin{align}
    \frac{\partial \zeta}{\partial t}
    &+ \frac{1}{a \cos( \varphi )} \left[
           \frac{\partial}{\partial \lambda} (uD)
         + \frac{\partial}{\partial \varphi} \left(vD \cos( \varphi )\right) 
       \right]
     = 0,
     \\[2ex]
   \frac{\partial u}{\partial t}
   &- v \left( 2 \Omega \sin( \varphi ) \right)
    + \frac{1}{a \cos( \varphi )} \frac{\partial}{\partial \lambda} \left( g \zeta + U \right)
    =0
    \qquad \text{and} \\[2ex]
   \frac{\partial v}{\partial t}
   &+ u \left( 2 \Omega \sin( \varphi ) \right)
    + \frac{1}{a} \frac{\partial}{\partial \varphi} \left( g \zeta + U \right)
    =0,
  \end{align}
</math>
where ''Ω'' is the [[angular frequency]] of the planet's rotation, ''g'' is the planet's gravitational acceleration at the mean ocean surface, ''a'' is the planetary radius, and ''U'' is the external gravitational tidal-forcing [[potential]].

[[William Thomson, 1st Baron Kelvin|William Thomson (Lord Kelvin)]] rewrote Laplace's momentum terms using the [[Curl (mathematics)|curl]] to find an equation for [[vorticity]]. Under certain conditions this can be further rewritten as a conservation of vorticity.

==On the figure of the Earth==
During the years 1784–1787 he published some papers of exceptional power. Prominent among these is one read in 1783, reprinted as Part II of ''Théorie du Mouvement et de la figure elliptique des planètes'' in 1784, and in the third volume of the ''Mécanique céleste''. In this work, Laplace completely determined the attraction of a [[spheroid]] on a particle outside it. This is memorable for the introduction into analysis of [[spherical harmonics]] or '''Laplace's coefficients''', and also for the development of the use of what we would now call the [[gravitational potential]] in [[celestial mechanics]].

===Spherical harmonics===
[[File:Rotating spherical harmonics.gif|frame|right|Spherical harmonics.]]
In 1783, in a paper sent to the ''Académie'', [[Adrien-Marie Legendre]] had introduced what are now known as [[associated Legendre function]]s.<ref name="ball"/> If two points in a [[Plane (mathematics)|plane]] have [[polar coordinates]] (''r'', θ) and (''r''<nowiki> '</nowiki>, θ'), where ''r''<nowiki> '</nowiki> ≥ ''r'', then, by elementary manipulation, the reciprocal of the distance between the points, ''d'', can be written as:

:<math>\frac{1}{d} = \frac{1}{r'} \left [ 1 - 2 \cos (\theta' - \theta) \frac{r}{r'} + \left ( \frac{r}{r'} \right ) ^2 \right ] ^{- \tfrac{1}{2}}.</math>

This expression can be [[power series|expanded in powers]] of ''r''/''r''<nowiki> '</nowiki> using [[Negative binomial theorem|Newton's generalised binomial theorem]] to give:

:<math>\frac{1}{d} = \frac{1}{r'} \sum_{k=0}^\infty P^0_k ( \cos ( \theta' - \theta ) ) \left ( \frac{r}{r'} \right ) ^k.</math>

The [[sequence]] of functions ''P''<sup>0</sup><sub>''k''</sub>(cos φ) is the set of so-called "associated Legendre functions" and their usefulness arises from the fact that every [[function (mathematics)|function]] of the points on a circle can be expanded as a [[series (mathematics)|series]] of them.<ref name="ball"/>

Laplace, with scant regard for credit to Legendre, made the non-trivial extension of the result to [[Three-dimensional space|three dimensions]] to yield a more general set of functions, the '''[[spherical harmonics]]''' or '''Laplace coefficients'''. The latter term is not in common use now.<ref name="ball"/>
{{rp|p. 340ff}}

===Potential theory===
This paper is also remarkable for the development of the idea of the [[scalar potential]].<ref name="ball"/> The gravitational [[force (physics)|force]] acting on a body is, in modern language, a [[vector (geometry)|vector]], having magnitude and direction. A potential function is a [[scalar (physics)|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".<ref>{{cite book | title=Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences | author=Grattan-Guinness, I. | date=2003 | location=Baltimore | publisher=Johns Hopkins University Press | isbn=978-0-8018-7396-6 | pages=1097–1098 | url=https://books.google.com/books?id=f5FqsDPVQ2MC&q=laplace+potential+1784&pg=PA1098 }}</ref> 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".<ref name="ball"/> 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 (mathematician)|George Green]]'s 1828 ''[[An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism]]''.<ref name=Ball>[http://www.maths.tcd.ie/pub/HistMath/People/Clairaut/RouseBall/RB_Clairaut.html W.W. Rouse Ball ''A Short Account of the History of Mathematics'' (4th edition, 1908)]</ref><ref>{{cite book
| author = Green, G.
| title = An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism
| publisher = Nottingham | date = 1828 | arxiv = 0807.0088|bibcode = 2008arXiv0807.0088G }}</ref>

Laplace applied the language of calculus to the potential function and showed that it always satisfies the [[differential equation]]:<ref name="ball"/>

:<math>\nabla^2V={\partial^2V\over \partial x^2 } +
{\partial^2V\over \partial y^2 } +
{\partial^2V\over \partial z^2 } = 0.
</math>

An analogous result for the [[velocity potential]] of a fluid had been obtained some years previously by [[Leonhard Euler]].<ref>{{cite book|first=Morris|last=Kline|title=Mathematical thought from ancient to modern times|volume=2|publisher=Oxford University Press|date=1972|pages=[https://archive.org/details/mathematicalthou00klin/page/524 524–525]|isbn=978-0-19-506136-9|url-access=registration|url=https://archive.org/details/mathematicalthou00klin/page/524}}</ref><ref>{{cite journal|first=Leonhard|last=Euler|author-link=Leonhard Euler|title=General principles of the motion of fluids|year=1757|journal=Novi. Comm. Acad. Sci. Petrop.|pages=271–311}}</ref>

Laplace's subsequent work on gravitational attraction was based on this result. The quantity ∇<sup>2</sup>''V'' has been termed the '''concentration''' of ''V'' and its value at any point indicates the "excess" of the value of ''V'' there over its mean value in the neighbourhood of the point.<ref>{{cite book|first=James|last=Maxwell|title=A Treatise on Electricity and Magnetism|date=1881|page=29|url=http://www.aproged.pt/biblioteca/MaxwellI.pdf}}</ref> [[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 and a posteriori|a priori]]'' forms in [[Immanuel Kant#Kant's theory of perception|Kant's theory of perception]].<ref name="ball"/>

The spherical harmonics turn out to be critical to practical solutions of Laplace's equation. Laplace's equation in [[spherical coordinates]], such as are used for mapping the sky, can be simplified, using the method of [[separation of variables]] into a radial part, depending solely on distance from the centre point, and an angular or spherical part. The solution to the spherical part of the equation can be expressed as a series of Laplace's spherical harmonics, simplifying practical computation.

==Planetary and lunar inequalities==
[[File:Delambre-3.jpg|left|thumb|248x248px|Title page of an 1817 copy of [[Jean Baptiste Joseph Delambre|Delambre's]] "''Tables écliptiques des satellites de Jupiter''," which references Laplace's contributions in its title.]]
[[File:Delambre-7.jpg|thumb|253x253px|Tables in an 1817 copy of [[Jean Baptiste Joseph Delambre|Delambre's]] "''Tables écliptiques des satellites de Jupiter''" – these calculations were influenced by Laplace's previous discoveries.]]

===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 [[perturbation (astronomy)|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.<ref name=eb1911/><ref name="Arago1874">{{cite book|last=Arago|first=François|translator-first=Baden | translator-last=Powell |translator-link=Baden Powell (mathematician)|title=Laplace: Eulogy|url=https://books.google.com/books?id=zeBDDwAAQBAJ&pg=PP5|access-date=21 March 2018|year=1874|publisher=Smithsonian Institution|page=5}}</ref>

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, {{nowrap|(2''n<sub>J</sub>'' − 5''n<sub>S</sub>'')}}, corresponds to a period of nearly 900&nbsp;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 [[Jean Baptiste Joseph Delambre|Delambre]] computed his astronomical tables.<ref name="ball"/>

===Books===
{{Classical mechanics|cTopic=Scientists}}
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."<ref name=eb1911/> The result is embodied in the ''Exposition du système du monde'' and the ''Mécanique céleste''.<ref name="ball"/>

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.<ref name="ball"/>

Laplace developed the [[nebular hypothesis]] of the formation of the Solar System, first suggested by [[Emanuel Swedenborg]] and expanded by [[Immanuel Kant]]. This hypothesis remains the most widely accepted model in the study of the origin of planetary systems. According to Laplace's description of the hypothesis, the Solar System evolved from a globular mass of [[incandescence|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.<ref name="ball"/><ref name="ebsolar">Owen, T. C. (2001) "Solar system: origin of the solar system", ''[[Encyclopædia Britannica]]'', Deluxe CDROM edition</ref>

As mentioned, the idea of the nebular hypothesis had been outlined by [[Immanuel Kant]] in 1755,<ref name="ebsolar"/> who 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.<ref name=Pearson/>

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.<ref name=eb1911/> 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. The ''Mécanique céleste'' contains numerous of Laplace's own investigations but many results are appropriated from other writers with little or no acknowledgement. The volume's conclusions, which are described by historians as the organised result of a century of work by other writers as well as Laplace, are presented by Laplace if they were his discoveries alone.<ref name="ball"/>
[[File:Laplace-33.jpg|alt=First pages to Exposition du Système du Monde (1799)|thumb|150x150px|First pages to ''Exposition du Système du Monde'' (1799)]]
[[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 phrase, "''Il est aisé à voir que...''" ("It is easy to see that..."). The ''Mécanique céleste'' is not only the translation of Newton's ''[[Philosophiæ Naturalis Principia Mathematica|Principia Mathematica]]'' into the language of [[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 remains a standard authority.<ref name="ball"/>
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 ''Mécanique céleste''.{{citation needed|date=May 2018}} 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.

==Optics==
Laplace was a supporter of the [[corpuscle theory of light]] of Newton. In the fourth edition of ''Mécanique Céleste'', Laplace assumed that short-ranged molecular forces were responsible for refraction of the corpuscles of light.<ref>{{Cite journal |last=Fox |first=Robert |date=1974 |title=The Rise and Fall of Laplacian Physics |url=https://www.jstor.org/stable/27757328 |journal=Historical Studies in the Physical Sciences |volume=4 |pages=89–136 |doi=10.2307/27757328 |jstor=27757328 |issn=0073-2672}}</ref> Laplace and [[Étienne-Louis Malus]] also showed that [[Huygens principle of double refraction]] could be recovered from the [[principle of least action]] on light particles.<ref name=":0">{{Cite book |last=Darrigol |first=Olivier |url=https://books.google.com/books?id=ImM62wvWE_cC&q=laplace+optics+corpuscular+theory+refraction |title=A History of Optics from Greek Antiquity to the Nineteenth Century |date=2012-01-26 |publisher=OUP Oxford |isbn=978-0-19-162745-3 |language=en}}</ref>

However in 1815, [[Augustin-Jean Fresnel]] presented a new wave theory for diffraction to a commission of the French Academy with the help of [[François Arago]]. Laplace was one of the commission members and they ultimately awarded a prize to Fresnel for his new approach.<ref>{{Cite book |last=Whittaker |first=E. T. |title=A history of the theories of aether & electricity |date=1989 |publisher=Dover Publications |isbn=978-0-486-26126-3 |location=New York}}</ref>{{rp|I.108}}

===Influence of gravity on light===
Using corpuscular theory, Laplace also came close to propounding the concept of the [[black hole]]. He suggested that gravity could influence light and that there could be massive stars whose gravity is so great that not even light could escape from their surface (see [[escape velocity]]).<ref>Laplace, P.-S. (1799). ''Allgemeine geographische Ephemeriden herausgegeben von [[Franz Xaver von Zach|F. von Zach]]''. IV. Band, I. Stück, I. Abhandlung, Weimar; translation in English: {{cite book |last1=Hawking |first1=Stephen W. |last2=Ellis |first2=George F.R. |title=The Large Scale Structure of Space-Time  |year=1973 |publisher=Cambridge University Press |isbn=978-0-521-09906-6 |pages=365ff|title-link=The Large Scale Structure of Space-Time }}</ref><ref name=HE73p364/><ref>Colin Montgomery, Wayne Orchiston and Ian Whittingham, [http://www.narit.or.th/en/files/2009JAHHvol12/2009JAHH...12...90M.pdf "Michell, Laplace and the origin of the Black Hole Concept"] {{Webarchive|url=https://web.archive.org/web/20140502005017/http://www.narit.or.th/en/files/2009JAHHvol12/2009JAHH...12...90M.pdf |date=2 May 2014 }}, ''Journal of Astronomical History and Heritage'', '''12'''(2), 90–96 (2009).</ref><ref>See Israel (1987), sec. 7.2.</ref> However, this insight was so far ahead of its time that it played no role in the history of scientific development.<ref>Gribbin, 299</ref>

==Arcueil==
[[File:Laplace house Arcueil.jpg|thumbnail|Laplace's house at [[Arcueil]] to the south of Paris.]]
{{main|Society of 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<ref name="Fourier">Fourier (1829).</ref> – 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]].<ref>Crosland (1967), p. 1</ref> In 1806, Laplace was also elected a foreign member of the [[Royal Swedish Academy of Sciences]].

==Analytic theory of probabilities==
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.<ref name="stigler"/> 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 ''Mécanique céleste''.<ref name="ball"/> 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.<ref>{{cite web|title=Laplace, Pierre-Simon Marquis de – Encyclopedia of Mathematics|url=https://encyclopediaofmath.org/wiki/Laplace,_Pierre-Simon_Marquis_de|access-date=18 June 2021|website=encyclopediaofmath.org}}</ref> 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 [[Induction (philosophy)|inductive reasoning]] based on [[probability]], which we would today recognise as [[Bayesian probability|Bayesian]]. He begins the text with a series of principles of probability, the first seven 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.
# When two events ''A'' and ''B'' depend on each other, the probability of compound event 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&nbsp;''B''.
# Three corollaries are given for the sixth principle, which amount to Bayesian rule. Where event {{nowrap|''A<sub>i</sub>'' ∈ {''A''<sub>1</sub>, ''A''<sub>2</sub>, ... ''A<sub>n</sub>''}}} exhausts the list of possible causes for event ''B'', {{nowrap|Pr(''B'') {{=}} Pr(''A''<sub>1</sub>, ''A''<sub>2</sub>, ..., ''A''<sub>''n''</sub>)}}. Then <math>\Pr(A_i \mid B) = \Pr(A_i)\frac{\Pr(B\mid A_i)}{\sum_j \Pr(A_j)\Pr(B\mid A_j)}.</math>
# The probability of a future event ''C'' is the sum of the products of the probability of each causes ''B<sub>i</sub>'' drawn from the event observed ''A'', by the probability that, this cause existing, the future event will occur. Symbolically, <math>\Pr(C|A) = \sum_i \Pr(C|B_i) \Pr(B_i|A).</math>

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.

:<math>\Pr(\text{next outcome is success}) = \frac{s+1}{n+2}</math>

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

:<math>\Pr(\text{sun will rise tomorrow}) = \frac{d+1}{d+2}</math>

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."<ref>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.</ref>

===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 (mathematics)|function]] as the coefficients in the expansion of another function, with reference to a different variable.<ref name=eb1911/> The latter is therefore called the [[probability-generating function]] of the former.<ref name=eb1911/> 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]].<ref name="ball"/>

===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 [[Adrien-Marie Legendre|Legendre]] had published the method of least squares, making no attempt to tie it to the theory of probability. In 1809 [[Carl Friedrich Gauss|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.<ref name="stigler">Stigler, 1975</ref>

In two important papers in 1810 and 1811, Laplace first developed the [[characteristic function (probability theory)|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.<ref name="stigler"/> 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.<ref name="stigler"/>

==Laplace's demon==
{{Main|Laplace's demon}}
In 1814, Laplace published what may have been the first scientific articulation of [[causal determinism]]:<ref name=Hawking/>
{{blockquote| 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 the present to it.|Pierre Simon Laplace, ''[[A Philosophical Essay on Probabilities]]''<ref>Laplace, ''A Philosophical Essay'', New York, 1902, p. 4.<!--Nice translation, but it's not the one in this edition.--></ref>}}

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'.<ref>{{cite journal |last1=van Strien |first1=Marij |title=On the origins and foundations of Laplacian determinism |journal=Studies in History and Philosophy of Science |date=2014 |volume=45 |pages=24–31 |doi=10.1016/j.shpsa.2013.12.003 |pmid=24984446 |bibcode=2014SHPSA..45...24V |s2cid=19302364 |url=https://pubmed.ncbi.nlm.nih.gov/24984446/ |access-date=5 February 2021}}</ref> As well, [[Jesuits|Jesuit]] scientist [[Roger Joseph Boscovich|Boscovich]] first proposed a version of scientific determinism very similar to Laplace's in his 1758 book ''Theoria philosophiae naturalis''.<ref>{{cite book |last1=Cercignani |first1=Carlo |author-link1=Carlo Cercignani |title=Ludwig Boltzmann, The Man Who Trusted Atoms |date=1998 |publisher=Oxford University Press |isbn=978-0-19-850154-1 |chapter=Chapter 2: Physics before Boltzmann |page=[https://archive.org/details/ludwigboltzmannm0000cerc/page/55 55] |chapter-url=https://archive.org/details/ludwigboltzmannm0000cerc/page/55 }}</ref>

==Laplace transforms==
{{main|Laplace transform#History}}
As early as 1744, [[Euler]], followed by [[Lagrange]], had started looking for solutions of [[differential equation]]s in the form:<ref>[[Ivor Grattan-Guinness|Grattan-Guinness]], in Gillispie (1997), p. 260</ref>

:<math> z = \int X(x) e^{ax} \, dx\text{  and  }z = \int X(x) x^a \, dx.</math>
The Laplace transform has the form:
:<math> F(s) = \int f(t) e^{-st}\, dt</math>
This integral operator transforms a function of time (<math>t</math>) into a function of a complex variable (<math>s</math>), usually interpreted as [[complex frequency]].

==Other discoveries and accomplishments==
===Mathematics===
Among the other discoveries of Laplace in pure and applied mathematics are:
* Discussion, contemporaneously with [[Alexandre-Théophile Vandermonde]], of the general theory of [[determinant]]s, (1772);<ref name="ball"/>
* Proof that every equation of an odd degree must have at least one [[real number|real]] [[quadratic function|quadratic]] factor{{clarify| date=April 2020| reason=The quadratic factors are needed for conjugate complex roots, real roots generate linear factors. Might this claim have become distorted in quotation?}};<ref name="ball"/>
* [[Laplace's method]] for approximating integrals
* Solution of the [[linear partial differential equation]] of the second order;<ref name="ball"/>
* 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]];<ref name=eb1911/><ref name="ball"/>
* In his theory of probabilities:
** [[de Moivre–Laplace theorem]] that approximates binomial distribution with a normal distribution
** Evaluation of several common [[definite integral]]s;<ref name="ball"/>
** General proof of the [[Lagrange reversion theorem]].<ref name="ball"/>

===Surface tension===
{{main|Young–Laplace equation#History}}
Laplace built upon the qualitative work of [[Thomas Young (scientist)|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 process|adiabatic]] [[Gas compression|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.<ref name="ball"/>

==Politics==
===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.<ref>Crosland (2006), p. 30</ref>

In November 1799, immediately after seizing power in the coup of [[18 Brumaire]], Napoleon appointed Laplace to the post of [[Minister of the Interior (France)|Minister of the Interior]].<ref name=eb1911/> The appointment, however, lasted only six weeks, after which [[Lucien Bonaparte]], Napoleon's brother, was given the post.<ref name=eb1911/> Evidently, once Napoleon's grip on power was secure, there was no need for a prestigious but inexperienced scientist in the government.<ref name=GGp333>Grattan-Guinness (2005), p. 333</ref> Napoleon later (in his ''Mémoires de Sainte Hélène'') wrote of Laplace's dismissal as follows:<ref name="ball"/>

{{blockquote | 
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".<ref name=GGp333/>

===From Bonaparte to the Bourbons===
[[File:Pierre-Simon-Laplace (1749-1827).jpg|thumb|left|upright|Laplace.]]
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.<ref name=eb1911/> In copies sold after the [[Bourbon Restoration in France|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 [[Bourbon Restoration in France|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.<ref name="ball"/>

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 [[Battle of Leipzig|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.'"<ref>Hahn (2005), p. 191</ref>

===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".<ref>Laplace, ''A Philosophical Essay'', New York, 1902, p. 62. (Translation in this paragraph of article is from Hahn.)</ref><ref>Hahn (2005), p. 184</ref> 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".<ref>Laplace, ''A Philosophical Essay'', New York, 1902, p. 63. (Translation in this paragraph of article is from Hahn.)</ref><ref name=Hahnp185>Hahn (2005), p. 185</ref>

About the political upheavals he had witnessed, Laplace formulated a set of principles derived from physics to favour evolutionary over revolutionary change:
{{Blockquote|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.<ref>Laplace, ''A Philosophical Essay'', New York, 1902, pp. 107–108. (Translation in this paragraph of article is from Hahn.)</ref>}}

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."<ref name=Hahnp185/>

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 count|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.<ref name="Black1958">{{Cite book |title=The Theory of Committees and Elections|last=Black|first=Duncan|date=1987|publisher=Springer Science & Business Media|isbn=978-0-89838-189-4|language=en|orig-year=1958}}</ref><ref name="Tangian2020">{{Cite book|last=Tangian |first=Andranik |date=2020|title=Analytical Theory of Democracy. Vols. 1 and 2|series=Studies in Choice and Welfare |pages = 132ff|publisher=Springer |location=Cham, Switzerland |isbn=978-3-030-39690-9 |doi=10.1007/978-3-030-39691-6 |s2cid=216190330 }}</ref>

==Death==
[[File:Tomb of Pierre-Simon Laplace.JPG|thumb|right|Tomb of Pierre-Simon Laplace]]
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.<ref name=Pearson/> 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.<ref>Gillispie (1997), p. 278</ref> The tomb is situated on a hill overlooking the village of St Julien de Mailloc, Normandy, France.

==Religious opinions==
===''I had no need of that hypothesis''===
{{See also|God of the gaps}}
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:<ref name="ball"/>
{{Blockquote|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):<ref>p. 282, [https://books.google.com/books?id=88xZAAAAcAAJ ''Mémoires du docteur F. Antommarchi, ou les derniers momens de Napoléon''], vol. 1, 1825, Paris: Barrois L'Ainé</ref>
{{Blockquote|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]]<ref name=Faye>Faye, Hervé (1884), ''Sur l'origine du monde: théories cosmogoniques des anciens et des modernes''. Paris: Gauthier-Villars, pp. 109–111</ref><ref name="Pasquier">Pasquier, Ernest (1898). [http://www.persee.fr/web/revues/home/prescript/article/phlou_0776-5541_1898_num_5_18_1596 "Les hypothèses cosmogoniques (''suite'')"]. ''Revue néo-scholastique'', 5<sup>o</sup> année, N<sup>o</sup> 18, pp. 124–125, footnote 1.</ref> 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:
{{Blockquote|In fact Laplace never said that. Here, I believe, is what truly happened. Newton, believing that the [[secular phenomena|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,<ref>Arago, François (1827), ''Laplace: Eulogy before the French Academy'', translated by Prof. Baden Powell, ''Smithsonian Report'', 1874</ref> told Faye of an attempt by Laplace to keep the garbled version of his interaction with Napoleon out of circulation. Faye writes:<ref name=Faye/><ref name=Pasquier/>
{{Blockquote|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.<ref>Cajori, Florian (1893), ''A History of Mathematics''. Fifth edition (1991), reprinted by the [[American Mathematical Society]], 1999, p. 262. {{isbn|0-8218-2102-4}}</ref> [[Stephen Hawking]] said in 1999,<ref name="Hawking">{{cite web |last=Hawking |first=Stephen |year=1999 |title=Does God Play Dice? |url=https://www.hawking.org.uk/in-words/lectures/does-god-play-dice |archive-url=https://web.archive.org/web/20000708041816/http://www.hawking.org.uk/lectures/dice.html |archive-date=8 July 2000 |work=Public Lecture}}</ref> "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]]:<ref>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}}.</ref>

{{blockquote|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 (journalist)|Daniel Johnson]]<ref>Johnson, Daniel (18 June 2007), [http://www.commentarymagazine.com/2007/06/18/the-hypothetical-atheist "The Hypothetical Atheist"], ''[[Commentary (magazine)|Commentary]]''.</ref> 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",<ref name=Faye/> 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."<ref>''Talks of Napoleon at St. Helena with General Baron Gourgaud'', translated by Elizabeth Wormely Latimer. Chicago: A.C. McClurg & Co., 1903, p. 276.</ref> 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]]."<ref>Hahn (2005), p. 67.</ref> 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."<ref>Dumas, Jean-Baptiste (1885). ''Discours et éloges académiques'', Vol. II. Paris: Gauthier-Villars, p. 255.</ref><ref name="Kneller">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, [http://www.ebooksread.com/authors-eng/karl-alois-kneller/christianity-and-the-leaders-of-modern-science-a-contribution-to-the-history-of-hci/page-6-christianity-and-the-leaders-of-modern-science-a-contribution-to-the-history-of-hci.shtml pp. 73–74].</ref>

Hahn states: "Nowhere in his writings, either public or private, does Laplace deny God's existence."<ref>Hahn (1981), p. 95.</ref> Expressions occur in his private letters that appear inconsistent with atheism.<ref name="eb1911"/> 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]."<ref name=Pasquier/><ref>''Œuvres de Laplace''. Paris: Gauthier-Villars, 1878, Vol. I, pp. v–vi.</ref> Ian S. Glass, quoting Herschel's account of the celebrated exchange with Napoleon, writes that Laplace was "evidently a deist like Herschel".<ref>Glass, Ian S. (2006). ''Revolutionaries of the Cosmos: The Astrophysicists''. Cambridge University Press, p. 108. {{isbn|0-19-857099-6}}.</ref>

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."<ref>[[General Scholium]], from the end of Book III of the ''Principia''; first appeared in the second edition, 1713.</ref> 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."<ref>Laplace, ''[https://archive.org/details/expositiondusys05laplgoog Exposition du système du monde]'', 6th edition. Brussels, 1827, pp. 522–523.</ref> 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.<ref>Laplace, ''Exposition'', 1827, p. 523.</ref> 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."<ref>Leibniz to [[Antonio Schinella Conti|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.</ref> 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."<ref>Leibniz to Conti, 1715, in Alexander, ed., 1956, p. 185.</ref>

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."<ref>Hahn (2005), p. 220.</ref> 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."<ref>Hahn (2005), p. 223.</ref>

Laplace also ridiculed the use of probability in theology. Even following Pascal's reasoning presented in [[Pascal's wager]], it is not worth making a bet, for the hope of profit – equal to the product of the value of the testimonies (infinitely small) and the value of the happiness they promise (which is significant but finite) – must necessarily be infinitely small.<ref>{{citation |author=[[Jacques Attali]] |title=Pascal |date=2004 |page=368 |place=Warszawa}}</ref>

In old age, Laplace remained curious about the question of God<ref name="Hahnp202">Hahn (2005), p. 202.</ref> and frequently discussed Christianity with the Swiss astronomer Jean-Frédéric-Théodore Maurice.<ref>Hahn (2005), pp. 202, 233.</ref> 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.<ref name=Hahnp202/> After Laplace's death, [[Siméon Denis Poisson|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'']."<ref>[[Augustus De Morgan|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".</ref> These were his last words, interpreted by Maurice as a realisation of the ultimate "[[vanitas|vanity]]" of earthly pursuits.<ref name="Hahnp204">Hahn (2005), p. 204.</ref> Laplace received the [[last rites]] from the [[curé]] of the Missions Étrangères (in whose parish he was to be buried)<ref name=Kneller/> and the curé of Arcueil.<ref name=Hahnp204/>

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.<ref>{{cite book|title=Pierre Simon Laplace, 1749–1827: A Determined Scientist|date=2005|publisher=Harvard University Press|isbn=978-0-674-01892-1|author=Roger Hahn|page=204|quote=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.}}</ref> Laplace in his last years has been described as an agnostic.<ref>{{cite book|title=Pierre Simon Laplace, 1749–1827: A Determined Scientist|date=2005|publisher=Harvard University Press|isbn=978-0-674-01892-1|author=Roger Hahn|page=202|quote=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.}}</ref><ref>{{cite book|title=Mathematics and the Search for Knowledge|date=1986|publisher=Oxford University Press|isbn=978-0-19-504230-6|author=Morris Kline|page=214|quote=Lagrange and Laplace, though of Catholic parentage, were agnostics.}}</ref><ref>{{cite book|title=Mathematics and the Imagination|date=2001|publisher=Courier Dover Publications|isbn=978-0-486-41703-5|author=Edward Kasner|author2=James Newman|author3=James Roy Newman|page=253|quote=Modern physics, indeed all of modern science, is as humble as Lagrange, and as agnostic as Laplace.}}</ref>

===Excommunication of a comet===
In 1470 the [[Renaissance humanism|humanist]] scholar [[Bartolomeo Platina]] wrote<ref>{{Cite book|title=Comet Lore|date=1910|author=E. Emerson|publisher=Schilling Press, New York|page=83}}</ref> 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 "[[excommunication|excommunicated]]" Halley's comet.<ref>{{Cite journal|title=The Legend of 1P/Halley 1456|author=C.M. Botley|journal=The Observatory|volume=91|year=1971|pages=125–126|bibcode=1971Obs....91..125B}}</ref> What Laplace actually said, in ''Exposition du système du monde'' (1796), was that the Pope had ordered the comet to be "[[exorcism|exorcised]]" (''conjuré''). It was Arago, in ''Des Comètes en général'' (1832), who first spoke of an excommunication.<ref>{{Cite Catholic Encyclopedia |last=Hagen |first=John G. |wstitle=Pierre-Simon Laplace |volume=8}}</ref><ref>{{Cite Catholic Encyclopedia |last=Stein |first=John |wstitle=Bartolomeo Platina|volume=12}}</ref><ref>Rigge, William F. (04/1910), [http://adsabs.harvard.edu/full/1910PA.....18..214R "An Historical Examination of the Connection of Calixtus III with Halley's Comet"], ''[[Popular Astronomy (US magazine)|Popular Astronomy]]'', Vol. 18, pp. 214–219</ref>

==Honors==
* Correspondent of the [[Royal Netherlands Academy of Arts and Sciences|Royal Institute of the Netherlands]] in 1809.<ref>{{cite web|url=http://www.dwc.knaw.nl/biografie/pmknaw/?pagetype=authorDetail&aId=PE00001498 |title=P.S. de Laplace (1749–1827) |publisher=Royal Netherlands Academy of Arts and Sciences |access-date=19 July 2015}}</ref>
* Foreign Honorary Member of the [[American Academy of Arts and Sciences]] in 1822.<ref name=AAAS>{{cite web|title=Book of Members, 1780–2010: Chapter L|url=http://www.amacad.org/publications/BookofMembers/ChapterL.pdf|publisher=American Academy of Arts and Sciences|access-date=28 July 2014}}</ref>
* The asteroid [[4628 Laplace]] is named for Laplace.<ref>{{cite book | author=Schmadel, L.D. | title=Dictionary of Minor Planet Names | edition=5th rev. | location=Berlin | publisher=Springer-Verlag | date=2003 | isbn=978-3-540-00238-3}}</ref>
* A spur of the [[Montes Jura]] on the Moon is known as [[Promontorium Laplace]].
* His name is one of the [[List of the 72 names on the Eiffel Tower|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]]).
* The [[Institute of Electrical and Electronics Engineers]] (IEEE) [[IEEE Signal Processing Society|Signal Processing Society]]'s Early Career Technical Achievement Award is named in his honor.<ref>https://signalprocessingsociety.org/sites/default/files/uploads/community_involvement/awards/Early_Career_Technical_Achievement_Award.pdf</ref>

==Quotations==
* I had no need of that hypothesis. ("Je n'avais pas besoin de cette hypothèse-là", allegedly as a reply to [[Napoleon I of France|Napoleon]], who had asked why he hadn't mentioned God in his book on [[astronomy]].)<ref name="ball"/>
* 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.)
* If we seek a cause wherever we perceive symmetry, it is not that we regard a symmetrical event as less possible than the others, but, since this event ought to be the effect of a regular cause or that of chance, the first of these suppositions is more probable than the second.<ref>Laplace, ''A Philosophical Essay on Probability'', New York, 1902, p. 16.</ref>
* The more extraordinary the event, the greater the need of its being supported by strong proofs.<ref>Laplace, ''A Philosophical Essay on Probability'', New York, 1902, p. 17.</ref>
* "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."<ref>{{cite journal|last=Laplace|first=Pierre Simon|title=Essai philosophique sur les probabilités|journal=Nature|volume=110|issue=2748|date=1814|page=50|url=https://books.google.com/books?id=rDUJAAAAIAAJ&pg=PA50|bibcode=1922Natur.110....6B|doi=10.1038/110006b0|s2cid=4099834}}</ref>
** 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."<ref>{{cite book|last=Flournoy|first=Théodore|title=Des Indes à la planète Mars: étude sur un cas de somnambulisme avec glossolalie|date=1899|publisher=Slatkine|pages=344–345|url=https://books.google.com/books?id=6xdk111WDScC&q=laplace&pg=PA344|isbn=978-2-05-100499-2}}*{{cite book|last=Flournoy|first=Théodore|title=From India to the Planet Mars: A Study of a Case of Somnambulism|date=2007|publisher=Cosimo, Inc.|isbn=978-1-60206-357-0|pages=369–370|url=https://books.google.com/books?id=Vnog3NS5aY4C&pg=PA369|others=Daniel D. Vermilye, trans.}}</ref>
** 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'''.<ref>Laplace, ''A Philosophical Essay on Probability'', New York, 1902, p. 177.</ref>
* Infinitely varied in her effects, nature is only simple in her causes.<ref>Laplace, ''The System of the World'', Dublin, 1830, p. 91.</ref>
* 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.")<ref name=Fourier/>
* 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.<ref>{{cite web |last1=Miller |last2= Gelman |first1=Joshua B |first2=Andrew |title=Laplace's theories of cognitive illusions, heuristics, and biases∗ |url=http://www.stat.columbia.edu/~gelman/research/unpublished/laplace16.pdf |website=Columbia University |publisher=unpublished |access-date=17 January 2021}}</ref>

== List of works ==
* {{Cite book|title=Traité de mécanique céleste|volume=1|publisher=Charles Crapelet|location=Paris|year=1799|language=fr|url=https://gutenberg.beic.it/webclient/DeliveryManager?pid=12217993}}
* {{Cite book|title=Traité de mécanique céleste|volume=2|publisher=Charles Crapelet|location=Paris|year=1799|language=fr|url=https://gutenberg.beic.it/webclient/DeliveryManager?pid=12219220}}
* {{Cite book|title=Traité de mécanique céleste|volume=3|publisher=Charles Crapelet|location=Paris|year=1802|language=fr|url=https://gutenberg.beic.it/webclient/DeliveryManager?pid=12220405}}
* {{Cite book|title=Traité de mécanique céleste|volume=4|publisher=Charles Crapelet|location=Paris|year=1805|language=fr|url=https://gutenberg.beic.it/webclient/DeliveryManager?pid=12221428}}
* {{Cite book|title=Traité de mécanique céleste|volume=5|publisher=Charles Louis Étienne Bachelier|location=Paris|year=1852|language=fr|url=https://gutenberg.beic.it/webclient/DeliveryManager?pid=12222637}}
* {{Cite book|title=Précis de l'histoire de l'astronomie|volume=|publisher=Angelo Stanislao Brambilla|location=Milano|year=1823|language=it|url=https://gutenberg.beic.it/webclient/DeliveryManager?pid=12414013}}
* {{Cite book|title=Exposition du système du monde|volume=|publisher=Charles Louis Étienne Bachelier|location=Paris|year=1824|language=fr|url=https://gutenberg.beic.it/webclient/DeliveryManager?pid=3903942}}
<gallery>
File:Laplace-1.jpg|Volumes 1-5 of Pierre-Simon Laplace's "[[Traité de mécanique céleste]]" (1799)
File:Laplace-2.jpg|Title page to Volume I of "[[Traité de mécanique céleste]]" (1799)
File:Laplace-3.jpg|Table of contents to Volume I of "[[Traité de mécanique céleste]]" (1799)
File:Laplace-4.jpg|First page of Volume I of "[[Traité de mécanique céleste]]" (1799)
</gallery>

==Bibliography==
* ''[http://gallica.bnf.fr/Search?ArianeWireIndex=index&lang=EN&q=oeuvres+completes+de+laplace&p=1&f_creator=Laplace%2C+Pierre+Simon+de+%281749-1827%29 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'')
* ''[https://archive.org/details/prcisdelhistoir00laplgoog Précis de l'histoire de l'astronomie]''
* [[Alphonse Rebière]], ''Mathématiques et mathématiciens'', 3rd edition Paris, Nony & Cie, 1898.

===English translations===
* [[File:Laplace-12.jpg|alt=Volumes 1 and 2 of "System of the World" (1809)|thumb|246x246px|Volumes 1 and 2 of "System of the World" (1809)]] [[Nathaniel Bowditch|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
* {{cite book | title=A Philosophical Essay on Probabilities | isbn=978-1-60206-328-0 | others=Translated by Truscott, F.W. & Emory, F.L. | date=2007 | orig-year=1902 | last1=Laplace | first1=Pierre Simon Marquis De | publisher=Cosimo }}, translated from the French 6th ed. (1840)
** {{Internet Archive|philosophicaless00lapliala|A Philosophical Essay on Probabilities (1902)}}
* {{cite book |title=Philosophical Essay on Probabilities |others=Translated by Andrew I. Dale |series=Sources in the History of Mathematics and Physical Sciences |volume=13 |isbn=978-1-4612-8689-9 |date=1995 |publisher=Springer |doi=10.1007/978-1-4612-4184-3|last1=Dale |first1=Andrew I. |last2=Laplace |first2=Pierre-Simon |hdl=2027/coo1.ark:/13960/t3126f008 |url=http://archive.org/details/philosophicaless00lapliala }}, translated from the French 5th ed. (1825)

==See also==
* [[History of the metre]]
* [[Laplace–Bayes estimator]]
* [[Ratio estimator]]
* [[Seconds pendulum]]
* [[List of things named after Pierre-Simon Laplace]]
* [[Pascal's wager]]

== References ==
=== Citations ===
{{reflist}}

=== General sources ===
{{refbegin}}
* {{cite journal | author=Andoyer, H. | title = L'œuvre scientifique de Laplace | url=https://archive.org/details/loeuvrescientifi00andouoft | journal=Paris | year = 1922 | publisher = Paris Payot | bibcode = 1922osdl.book.....A | language = fr }}
* {{cite journal | author=Bigourdan, G. |year=1931 | title=La jeunesse de P.-S. Laplace | language = fr | journal=La Science Moderne | volume=9 | pages=377–384 }}
* {{cite book | title = The Society of Arcueil: A View of French Science at the Time of Napoleon&nbsp;I | isbn=978-0-435-54201-6 | author=Crosland, M. | year = 1967 | location = Cambridge, MA | publisher=Harvard University Press }}
* – (2006) "A Science Empire in Napoleonic France", ''History of Science'', vol. 44, pp.&nbsp;29–48
* {{cite journal | author=Dale, A. I. |year=1982 | title = Bayes or Laplace? An examination of the origin and early applications of Bayes' theorem | journal=Archive for History of Exact Sciences | volume=27 | pages=23–47 | doi = 10.1007/BF00348352 |s2cid=116147039 |url=https://link.springer.com/article/10.1007/BF00348352}}
* David, F. N. (1965) "Some notes on Laplace", in [[Jerzy Neyman|Neyman, J.]] & LeCam, L. M. (eds) ''Bernoulli, Bayes and Laplace'', Berlin, pp.&nbsp;30–44.
* {{cite journal | author=Deakin, M. A. B. |year=1981 | title=The development of the Laplace transform | journal=Archive for History of Exact Sciences | volume=25 | pages=343–390 | doi=10.1007/BF01395660 | issue=4 |s2cid=117913073 }}
* {{cite journal | last=Deakin | first=Michael A. B. | title=The development of the Laplace Transform, 1737-1937 II. Poincaré to Doetsch, 1880-1937 | journal=Archive for History of Exact Sciences | publisher=Springer Science and Business Media LLC | volume=26 | issue=4 | year=1982 | issn=0003-9519 | doi=10.1007/bf00418754 | pages=351–381| s2cid=123071842 }}</ref>
* {{cite journal | author=Dhombres, J. |year=1989 | title = La théorie de la capillarité selon Laplace: mathématisation superficielle ou étendue | language = fr | journal = Revue d'Histoire des Sciences et de Leurs Applications | volume=62 | pages=43–70 | doi = 10.3406/rhs.1989.4134 }}
* {{cite journal | author1=Duveen, D. | author2=Hahn, R. | name-list-style=amp |year=1957 | title = Laplace's succession to Bézout's post of Examinateur des élèves de l'artillerie | journal=Isis | volume=48 | pages=416–427 | doi=10.1086/348608 | issue=4 | s2cid=143451316 }}
* {{cite journal | author=Finn, B. S. |year=1964 | title=Laplace and the speed of sound | journal=Isis | volume=55 | pages=7–19 | doi=10.1086/349791 |s2cid=20127770 }}
* {{cite journal | author=Fourier, J. B. J. | author-link=Joseph Fourier | title=Éloge historique de M. le Marquis de Laplace | journal=Mémoires de l'Académie Royale des Sciences | volume=10 | pages = lxxxi–cii |year=1829 | language = fr |url = http://www.academie-sciences.fr/activite/archive/dossiers/Fourier/Fourier_pdf/Mem1829_p81_102.pdf | archive-url=https://web.archive.org/web/20130724030237/http://www.academie-sciences.fr/activite/archive/dossiers/Fourier/Fourier_pdf/Mem1829_p81_102.pdf | archive-date=24 July 2013 }}, delivered 15 June 1829, published in 1831.
* {{cite journal | title = Probability and politics: Laplace, Condorcet, and Turgot | author=Gillispie, C. C. | journal = Proceedings of the American Philosophical Society | volume=116 |year=1972 | pages=1–20 | issue=1 }}
* {{cite book | last=Gillispie | first=Charles | title=Pierre-Simon Laplace, 1749–1827 : a life in exact science | publisher=Princeton University Press | publication-place=Princeton, N.J | year=1997 | isbn=0-691-01185-0 | oclc=36656386 | page=}}
* [[Ivor Grattan-Guinness|Grattan-Guinness, I.]], 2005, "'Exposition du système du monde' and 'Traité de mécanique céleste'" in his ''Landmark Writings in Western Mathematics''. Elsevier: 242–57.
* [[John Gribbin|Gribbin, John]].  ''The Scientists: A History of Science Told Through the Lives of Its Greatest Inventors''.  New York, Random House, 2002.  p.&nbsp;299.
* {{cite journal | author=Hahn, R. |year=1955 | title=Laplace's religious views | journal=Archives Internationales d'Histoire des Sciences | volume=8 | pages=38–40 }}
* – (1981) "Laplace and the Vanishing Role of God in the Physical Universe", in Woolf, Henry, ed., ''The Analytic Spirit: Essays in the History of Science''. Ithaca, NY: Cornell University Press. {{ISBN|0-8014-1350-8}}.
* {{cite book | last=Hahn | first=Roger | title=Calendar of the correspondence of Pierre Simon Laplace | publisher=Office for History of Science and Technology, University of California, Berkeley | publication-place=Berkeley | year=1982 | isbn=978-0-918102-07-2 | oclc=8877709}}
* {{cite book | last=Hahn | first=Roger | title=The new calendar of the correspondence of Pierre Simon Laplace | publisher=Office for History of Science and Technology, University of California at Berkeley | publication-place=Berkeley, CA | year=1994 | isbn=978-0-918102-20-1 | oclc=31967034}}
* {{cite book | last=Hahn | first=Roger | title=Pierre Simon Laplace, 1749-1827 : a determined scientist | publisher=Harvard University Press | publication-place=Cambridge, Mass | year=2005 | isbn=978-0-674-01892-1 | oclc=58457459 | language=it}}
* {{cite book | author=Israel, Werner | contribution=Dark stars: the evolution of an idea | editor2-last=Israel | editor2-first=Werner | editor1-last=Hawking | editor1-first=Stephen W. | title=300 Years of Gravitation | publisher=Cambridge University Press | year=1987 | pages=199–276 }}
* {{MacTutor Biography |id = Laplace }} (1999)
* {{cite journal | author = Nikulin, M.|year=1992 | title= A remark on the converse of Laplace's theorem| journal= Journal of Soviet Mathematics | volume=59 |issue=4 | pages=976–979 | doi=10.1007/bf01099128|s2cid=121149198 }}
* [[W. W. Rouse Ball|Rouse Ball, W.W.]] [1908] (2003) "Pierre Simon Laplace (1749–1827)", in ''A Short Account of the History of Mathematics'', 4th ed., Dover, {{ISBN|0-486-20630-0}} Also [https://www.gutenberg.org/ebooks/31246 available at Project Gutenberg].
* {{cite journal | last=Stigler | first=Stephen M. | title=Studies in the History of Probability and Statistics. XXXIV Napoleonic Statistics: The Work of Laplace | journal=Biometrika | publisher=JSTOR | volume=62 | issue=2 | year=1975 | pages=503–517 | issn=0006-3444 | doi=10.2307/2335393 | jstor=2335393 }}
* {{cite journal | last=Stigler | first=Stephen M. | title=Laplace's Early Work: Chronology and Citations | journal=Isis | publisher=University of Chicago Press | volume=69 | issue=2 | year=1978 | issn=0021-1753 | doi=10.1086/352006 | pages=234–254| bibcode=1978Isis...69..234S | s2cid=143831269 }}
* {{citation | last = Whitrow | first = Gerald James | author-link = Gerald James Whitrow | year = 2001 | chapter = Laplace, Pierre-Simon, marquis de | title = Encyclopædia Britannica | edition = Deluxe CDROM }}
* {{cite journal | author=Whittaker, E. T. | year=1949a | author-link=E. T. Whittaker | title=Laplace | journal=Mathematical Gazette | volume=33 | doi=10.2307/3608408 | jstor=3608408 | issue=303 | pages=1–12 | s2cid=250442315 }}
* {{cite journal | doi=10.2307/2306273 | title=Laplace | jstor=2306273 | journal=American Mathematical Monthly | volume=56 | year=1949b | pages=369–372 | issue=6 | last1=Whittaker | first1=Edmund }}
* {{cite journal | title=The Great Inequality of Jupiter and Saturn: from Kepler to Laplace | author=Wilson, C. | journal=Archive for History of Exact Sciences | volume=33 | issue=1–3 | pages=15–290 |year=1985 | doi=10.1007/BF00328048 | bibcode=1985AHES...33...15W | s2cid=121751666 }}
* {{cite book | author=Young, T. | author-link=Thomas Young (scientist) | title = Elementary Illustrations of the Celestial Mechanics of Laplace: Part the First, Comprehending the First Book | year=1821 | location=London, England | publisher=John Murray | url = https://archive.org/details/elementaryillus00laplgoog | quote=laplace. |via = [[Internet Archive]] }}
{{refend}}

==External links==
{{commons category}}
{{Wikiquote}}
{{Wikisource author}}
* {{cite web | url=http://scienceworld.wolfram.com/biography/Laplace.html | title=Laplace, Pierre (1749–1827) | work=Eric Weisstein's World of Scientific Biography | publisher=[[Wolfram Research]] | access-date=24 August 2007 }}
* "[http://www-history.mcs.st-andrews.ac.uk/Biographies/Laplace.html Pierre-Simon Laplace]" in the [[MacTutor History of Mathematics archive]].
* {{cite web | title=Bowditch's English translation of Laplace's preface | work=Mécanique Céleste | publisher=The MacTutor History of Mathematics archive | access-date=4 September 2007 | url=http://www-history.mcs.st-andrews.ac.uk/history/Extras/Laplace_mechanique_celeste.html }}
* [http://www.oac.cdlib.org/findaid/ark:/13030/kt8q2nf3g7/ Guide to the Pierre Simon Laplace Papers] at [[The Bancroft Library]]
* {{MathGenealogy |id=108295 }}
* [http://www.cs.xu.edu/math/Sources/Laplace/index.html English translation] {{Webarchive|url=https://web.archive.org/web/20121227004253/http://www.cs.xu.edu/math/Sources/Laplace/index.html |date=27 December 2012 }} of a large part of Laplace's work in probability and statistics, provided by [http://www.cs.xu.edu/math/Sources/index.html Richard Pulskamp] {{Webarchive|url=https://web.archive.org/web/20121029123640/http://www.cs.xu.edu/math/Sources/index.html |date=29 October 2012 }}
* [https://web.archive.org/web/20130513142304/http://portail.mathdoc.fr/cgi-bin/oetoc?id=OE_LAPLACE__7 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 ''[https://www.bibnum.education.fr/physique/mecanique/sur-le-mouvement-d-un-corps-qui-tombe-d-une-grande-hauteur BibNum] {{Webarchive|url=https://web.archive.org/web/20150402143731/https://www.bibnum.education.fr/physique/mecanique/sur-le-mouvement-d-un-corps-qui-tombe-d-une-grande-hauteur |date=2 April 2015 }}'' (English).

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| before= [[Nicolas Marie Quinette]]
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