Pierre-Simon Laplace

• 1. Introduction

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  Pierre-Simon Laplace stood before Napoleon Bonaparte in the early years of the nineteenth century and handed him a copy of his book on the universe. Napoleon, who enjoyed putting embarrassing questions to learned men, looked up and said: "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 drew himself up and answered bluntly: "Je n’avais pas besoin de cette hypothèse-là." I had no need of that hypothesis.

  That exchange, whether precisely as reported or not, captures something essential about this man. Born on the 23rd of March 1749 in a small Normandy village four miles west of Pont l’Évêque, Laplace spent his life replacing divine intervention with mathematical certainty. He began as the son of a cider merchant and small farmer, attended a school run by a Benedictine priory, and was sent to the University of Caen to study theology. He never became a priest. He became, as Napoleon himself would later call him, a geometer of the first rank.

  What questions does his life raise? How does a boy from rural Normandy build the mathematical architecture of the Solar System? How do you solve a problem Newton himself declared beyond human reach? And what does it mean when the man who explained everything admits, on his deathbed, "Ah! We chase after phantoms."

• 2. The Boy from Normandy

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  At sixteen, Laplace left the school in Beaumont and arrived at the University of Caen in 1765. His father had intended him for the Church. Two mathematics teachers changed that trajectory: Christophe Gadbled and Pierre Le Canu, who awoke in him what contemporaries would later call a phenomenal natural mathematical faculty.

  While still at Caen, Laplace wrote a memoir on integral calculus that opened the first correspondence between him and Joseph-Louis Lagrange in Turin. Lagrange was thirteen years his senior and had recently founded the journal Miscellanea Taurinensia. Laplace's paper appeared in its fourth volume. Before he was twenty, he was in correspondence with one of the leading mathematicians in Europe.

  When he left for Paris, Le Canu gave him a letter of introduction to Jean le Rond d'Alembert, then supreme in scientific circles. According to family tradition, d'Alembert received him poorly and handed him a thick mathematics book, telling him to come back when he had read it. Laplace returned a few days later. D'Alembert refused to believe he had actually read and understood it. He questioned him. He realised it was true. From that point he took Laplace under his care and recommended him for a teaching post at the École Militaire.

  With a secure income and undemanding teaching, Laplace threw himself into original research. For the seventeen years between 1771 and 1787, he produced the bulk of his original work in astronomy. During those same years he examined Napoleon Bonaparte at the École Militaire in Paris when Napoleon graduated in 1785.

• 3. Newton's Unfinished Business

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  Isaac Newton had published his Principia Mathematica in 1687, giving the world the laws of motion and universal gravitation. But Newton himself doubted a purely mathematical solution to the stability of the Solar System. He concluded that periodic divine intervention was probably necessary to keep the planets on course.

  One problem in particular had defeated the best minds of the era. Observers had noticed that Jupiter's orbit appeared to be slowly shrinking while Saturn's was expanding. Leonhard Euler had tried to solve it in 1748. Lagrange tried again in 1763. Neither succeeded.

  Laplace found the key. He recognised that Euler and Lagrange had both made a practical approximation by dropping small terms from their equations. Laplace noticed that even tiny terms, when integrated over long stretches of time, could accumulate into something significant. He pushed his analysis into higher-order terms, including the cubic. Through that more exact calculation, he discovered something crucial: the ratio of Jupiter's mean orbital motion to Saturn's is very nearly equal to the ratio of two small whole numbers. Two periods of Saturn's orbit around the Sun almost equal five of Jupiter's.

  The apparent drift of the two planets was not a catastrophe in progress. It was a long, slow oscillation with a period of nearly 900 years, driven by that near-perfect numerical coincidence. The resulting perturbations reached about 0.8 degrees of arc in orbital longitude for Saturn and about 0.3 degrees for Jupiter. It looked like collapse; it was geometry. Gerald James Whitrow described the achievement as "the most important advance in physical astronomy since Newton." With Laplace's theory in hand, Delambre was at last able to compute reliable astronomical tables for the two planets.

• 4. Celestial Mechanics and the Great Synthesis

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  Laplace set himself the task of writing a work that would "offer a complete solution of the great mechanical problem presented by the Solar System, and bring theory to coincide so closely with observation that empirical equations should no longer find a place in astronomical tables." The result was the five-volume Mécanique céleste, published between 1799 and 1825.

  The first two volumes, published in 1799, covered methods for calculating planetary motions, determining the figures of planets, and resolving tidal problems. The third and fourth volumes, published in 1802 and 1805, applied those methods and included several astronomical tables. The fifth volume, published in 1825, was mainly historical, but appended Laplace's latest research.

  Jean-Baptiste Biot, who helped Laplace revise the work for the press, noted a telling habit. Laplace was frequently unable to reconstruct the full chain of reasoning that led to his conclusions. When he was satisfied that a result was correct, he was content to write: "Il est aisé à voir que..." It is easy to see that. The phrase became notorious as a signal for something true but hard to prove.

  The work had a complicated relationship with credit. Many results were drawn from other writers with little or no acknowledgement. Historians have described the Mécanique céleste as the organised result of a century of work by multiple hands, presented by Laplace as if his alone. The companion work, Exposition du système du monde, published in 1796, gave a general account without the mathematics and became so celebrated that it secured Laplace's admission to the French Academy, the body of forty that governs French literary life.

  In that same Exposition, Laplace developed the nebular hypothesis for the origin of the Solar System. The idea had first been sketched by Emanuel Swedenborg and expanded by Immanuel Kant in 1755. Laplace's version held that the Solar System evolved from a rotating mass of incandescent gas. As it cooled and contracted, successive rings broke from its outer edge. Those rings cooled and condensed into planets. The Sun was the remaining central core. From this Laplace predicted that the more distant planets would be older than those closer to the Sun. That hypothesis remains the most widely accepted model for the origin of planetary systems.

• 5. The Ocean, the Earth, and Light

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  In 1775, Laplace developed the dynamic theory of tides, a mathematical account of how the oceans actually respond to tidal forces. Earlier work by Newton described the forces themselves; Daniel Bernoulli described how a stationary ocean would react to them in theory. Neither captured what real tides do.

  Laplace's theory incorporated friction, resonance, and the natural periods of ocean basins. It predicted the large amphidromic systems found in the world's oceans and explained observed tidal ranges. The equilibrium theory, which ignored the Earth's rotation and the presence of continents, calculated a maximum tide height of less than half a meter. Laplace's dynamic theory explains why tides in some locations reach up to fifteen meters. Satellite observations, including data from the CHAMP satellite matched against TOPEX models, have since confirmed the theory's accuracy. Tides worldwide can now be measured to within a few centimeters.

  In the years 1784 to 1787, Laplace published papers that completely determined the gravitational attraction of a spheroid on a particle outside it. This work introduced spherical harmonics into analysis, functions that are now essential to mapping the sky and solving Laplace's equation in spherical coordinates. The work also developed the concept of gravitational potential, transforming a vector problem into a scalar one, which is computationally far simpler to handle. Alexis Clairaut had first suggested the idea in 1743, and there are claims that Lagrange had used it earlier; Laplace himself described Clairaut's work as belonging to "the class of the most beautiful mathematical productions."

  Laplace also supported Newton's corpuscle theory of light and worked with Étienne-Louis Malus to show that Huygens's principle of double refraction could be recovered from the principle of least action applied to light particles. When Augustin-Jean Fresnel presented a wave theory of diffraction to a commission of the French Academy in 1815, Laplace was one of the commission members. They ultimately awarded the prize to Fresnel.

  In 1816, Laplace became the first to identify the reason Newton's calculation of the speed of sound in air came out too low. Newton had not accounted for the adiabatic compression of air that produces a local rise in temperature and pressure. Laplace's correction gave a more accurate measurement.

• 6. Probability and the Demon

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  From his earliest published work in 1771, Laplace had been thinking about probability. He understood it not as gambling mathematics but as the proper instrument for reasoning under uncertainty. As one formulation in the source puts it: "Laplace took probability as an instrument for repairing defects in knowledge."

  His Théorie analytique des probabilités, published in 1812, laid down many fundamental results in statistics. In two papers from 1810 and 1811, he developed the characteristic function as a tool for large-sample theory and proved the first general central limit theorem. He then showed that the central limit theorem provided a Bayesian justification for the method of least squares, which Legendre had published in 1805 and Gauss had approached from a different angle in 1809.

  His Essai philosophique sur les probabilités, published in 1814, set out a mathematical system of inductive reasoning. Among its results was the rule of succession: a formula for estimating the probability that the next trial in a series will succeed, based only on the count of previous successes and total trials. Laplace illustrated it by calculating the probability that the sun will rise tomorrow, given that it has never failed to do so. He was fully aware the calculation seemed absurd. He immediately followed it by noting that for anyone who understands the physical laws governing days and seasons, the probability is far higher than the formula alone would suggest.

  That same year, 1814, Laplace published what may have been the first scientific articulation of causal determinism. He imagined an intellect that knew all forces in nature and all positions of every particle in the universe. For such an intellect, he wrote, "nothing would be uncertain and the future just like the past would be the present to it." This hypothetical entity came to be called Laplace's demon, a term Laplace himself never used. He referred simply to "une intelligence." The concept had philosophical predecessors: Boscovich had proposed a similar form of scientific determinism in his 1758 Theoria philosophiae naturalis, and Maupertuis had touched on the idea as early as 1756.

• 7. Napoleon, Politics, and Late Life

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  In November 1799, immediately after seizing power in the coup of 18 Brumaire, Napoleon appointed Laplace Minister of the Interior. The appointment lasted six weeks. Napoleon later wrote in his Mémoires de Sainte Hélène that 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." The historian Grattan-Guinness argues these remarks were tendentious, and that Laplace was only ever intended as a short-term figurehead while Napoleon consolidated power.

  Laplace had been careful throughout his earlier career to stay clear of politics. He withdrew from Paris during the most violent phase of the Revolution. After his brief ministerial tenure, he was raised to the Senate. He became a Count of the Empire in 1806 and was named a marquis in 1817 after the Bourbon Restoration, having shifted his allegiance as Napoleon's hold on power collapsed.

  His biographer Roger Hahn disputes the portrait of Laplace as a mere opportunist. The Laplaces' only daughter, Sophie-Suzanne, died in childbirth in September 1813. Their son Émile was on the eastern front with Napoleon. The family had watched the Russian campaign with serious misgivings. Napoleon's reported response to the news of Sophie's death captured something of his relationship with Laplace by that point. When Laplace mentioned he had lost his daughter, Napoleon replied: "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."

  Laplace died in Paris on the 5th of March 1827, the same day Alessandro Volta died. His physician François Magendie removed his brain, which was reportedly smaller than average, and it was later displayed in a roving anatomical museum in Britain. Laplace was buried at Père Lachaise in Paris but in 1888 his remains were moved to Saint Julien de Mailloc in Normandy and reinterred on the family estate, on a hill overlooking the village where the records of his early life had once been held. Those records had been burned in 1925 when the family château was destroyed.

• 8. The Last Words

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  Laplace's religious views resisted easy classification throughout his life. Raised Catholic, he appears in his adult writings to have held something closer to deism. Some contemporaries thought him an atheist. Napoleon, on Saint Helena, told General Gaspard Gourgaud that Laplace had told him he was an atheist. The chemist Jean-Baptiste Dumas, who knew Laplace in the 1820s, wrote that he "provided materialists with their specious arguments, without sharing their convictions." On the 17th of June 1809, Laplace wrote to his son: "Je prie Dieu qu'il veille sur tes jours." I pray that God watches over your days.

  In manuscripts preserved in a black envelope in the library of the Académie des sciences, published for the first time by Roger Hahn, Laplace mounted a deist critique of Christianity. He wrote that the first and most infallible of principles is "to reject miraculous facts as untrue." He ridiculed the doctrine of transubstantiation and argued that no sovereign lawgiver of the universe would suspend the laws he had established and seems to have maintained invariably.

  In old age, he discussed Christianity regularly with the Swiss astronomer Jean-Frédéric-Théodore Maurice. He told Maurice that "Christianity is quite a beautiful thing" and praised its civilising influence. When the mathematician Poisson complimented the dying Laplace on his brilliant discoveries, Laplace fixed him with a pensive look and replied: "Ah! We chase after phantoms." The astronomer François Arago gave Laplace's eulogy before the French Academy in 1827, the same year the exchange about God was reported. Arago told the astronomer Hervé Faye that Laplace had tried before his death to suppress that story, asking Arago to have it deleted from a forthcoming biographical collection. It was neither deleted nor explained. His name is now among the seventy-two inscribed on the Eiffel Tower.