Pierre-Simon Laplace

How to use this course

Laplace’s name is attached to an equation, a transform, an operator, a resonance, a demon and a method of asymptotic integration. Each is a fragment of two great programmes. The first was to show that Newton’s law of gravitation, by itself, explains every observed motion in the solar system. The second was to turn the calculus of chance into a method for reasoning from evidence to causes. This course follows both, together with the career of a scientist who served the monarchy, the Revolution, Napoleon and the restored Bourbons in turn.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements in modern notation, derivations and exercises with folded solutions. Lessons 04 to 07 treat celestial mechanics and the physics of attraction, Lessons 08 and 09 probability, and Lesson 10 his contributions to experimental and theoretical physics.

Timeline

Normandy and Caen, 1749–1768

Pierre-Simon Laplace was born on 23 March 1749 in Beaumont-en-Auge, a village in the Pays d’Auge of Normandy. His father, Pierre Laplace, was comfortably off in the cider trade; his mother, Marie-Anne Sochon, came from a prosperous farming family at Tourgéville. Later legends of a childhood in poverty, which Laplace never troubled to correct, are not supported by the records.

From about the age of seven to sixteen he was a day pupil at the school of the Benedictine priory in Beaumont. The family expected a career in the Church. At sixteen, in 1765, he went to the University of Caen to study theology.

Caen

At Caen two teachers changed his direction. Christophe Gadbled and Pierre Le Canu recognised a gift for mathematics, and Laplace found that analysis absorbed him more than theology. He spent two years at Caen and left without a degree. Le Canu gave him a letter of introduction to the most influential mathematician in Paris, and in 1768, aged 19, Laplace set off to use it.

D’Alembert’s protégé, 1768–1773

Jean le Rond d’Alembert was a leading figure of the Académie des Sciences and co-editor of the Encyclopédie. According to the standard account, he gave little weight to the letter of introduction, and Laplace then wrote him a letter of his own on the principles of mechanics. D’Alembert was impressed enough to find him a post as professor of mathematics at the École Militaire, which gave Laplace a salary and time for research.

The output that followed was extraordinary. His first paper was read to the Académie on 28 March 1770, and within three years he had presented work on extrema of functions, difference and differential equations, the theory of chance, and the motions of the planets. He was nevertheless passed over for election in 1771 and 1772, and d’Alembert wrote to Joseph-Louis Lagrange in Berlin asking whether the Berlin Academy might take him. On 31 March 1773 the Paris Académie finally elected him adjoint. He was 24.

Two programmes take shape

Two of the papers of these years set his agenda for life. In the memoir read in 1773 and published in 1776 he attacked the problem of whether the mutual attraction of the planets would slowly change their mean distances from the Sun (Lesson 04). In the 1774 Mémoire sur la probabilité des causes par les événements he formulated, independently of Thomas Bayes, the rule for passing from observed events to the probabilities of their causes (Lesson 08). Laplace saw the two subjects as one: probability was the tool for extracting true laws from imperfect astronomical observations.

The Academy and the Revolution, 1773–1795

Through the 1770s and 1780s Laplace produced a stream of memoirs on celestial mechanics, often in friendly competition with Lagrange, whose results he would sometimes take up, extend and publish quickly. Their exchanges created the secular theory of the planets (Lesson 04). In 1775 and 1776 he developed a dynamical theory of the ocean tides (Lesson 07). In the 1780s he wrote on the attraction of spheroids and the figure of the Earth, where the equation now named after him appears (Lesson 06).

Lavoisier and heat

Between 1780 and 1784 Laplace worked with the chemist Antoine Lavoisier on heat. They built an ice calorimeter, which measures heat by the quantity of ice it melts, and measured specific heats and heats of reaction. Their 1783 Mémoire sur la chaleur showed that a guinea pig produces roughly as much heat as the combustion of the carbon it exhales as carbon dioxide would release, supporting the view that respiration is a slow combustion.

Recognition

In 1784 Laplace became examiner of the Royal Artillery, and in 1785 he examined and passed a 16-year-old cadet named Napoleon Bonaparte. In 1785 and 1786 he solved the long-standing puzzle of the great inequality of Jupiter and Saturn (Lesson 05), and in 1787 he explained the secular acceleration of the Moon. He was promoted to senior rank in the Académie in 1785 and elected to the Royal Society of London in 1789. In 1788 he married Marie-Charlotte de Courty de Romanges, about twenty years his junior. They had a son, Charles-Émile (1789–1874), who followed a military career, and a daughter, Sophie-Suzanne.

Revolution

In May 1790 Laplace joined the Académie’s commission on weights and measures, which designed the metric system. The Académie was suppressed in August 1793, and during the Terror he was removed from the commission. Lavoisier, his former collaborator, was guillotined in May 1794. After the fall of Robespierre the new institutions of the Republic drew heavily on Laplace. In 1795 he was a founding member of the Bureau des Longitudes and of the Institut national, and he lectured on probability at the short-lived École Normale. Those lectures grew into the Essai philosophique of 1814.

Perturbations and the stability of the solar system

For the general reader. Each planet is pulled mainly by the Sun but also, slightly, by every other planet. Newton feared these small pulls would accumulate until the solar system needed divine correction. Laplace and Lagrange showed that, to the accuracy of their approximations, the sizes of the orbits do not drift, and their shapes and tilts only oscillate within fixed limits.

Osculating elements

For a planet of mass \(m\) around a Sun of mass \(M\), the unperturbed motion is a Kepler ellipse with semi-major axis \(a\), eccentricity \(e\), inclination \(i\), longitude of perihelion \(\varpi\), longitude of node \(\Omega\) and mean longitude \(\lambda=nt+\varepsilon\), where the mean motion satisfies \(n^2a^3=G(M+m)\). With other planets present, the equations of motion become

where the disturbing function \(\mathcal R\) is of the order of the perturbing masses. At each instant the position and velocity define an ellipse, the osculating ellipse, and the elements now vary slowly. The method of variation of constants, developed by Euler, Lagrange and Laplace, gives their rates of change. The simplest is

Secular invariance of the mean distances

The disturbing function is a Fourier series in the angles. Its terms depending on the mean longitudes \(\lambda_j\) oscillate quickly and average out. If no combination \(k_1n_1+k_2n_2\) of mean motions with small integer coefficients nearly vanishes, the long-term evolution is governed by the secular part \(\overline{\mathcal R}\), the average over all mean longitudes.

The proof at first order is the remark above: \(\partial\overline{\mathcal R}/\partial\lambda=0\) because \(\overline{\mathcal R}\) has been averaged over \(\lambda\). The work lies in justifying the averaging and in Poisson’s second-order computation. Laplace drew a broad conclusion: no planet will spiral into the Sun or escape because of mutual gravitation, at least over the times his approximations cover.

The secular system

The eccentricities and perihelia do vary. Set \(z_j=e_j\,e^{\mathrm i\varpi_j}\) for planets \(j=1,\dots,N\). Keeping terms of second degree in the eccentricities in \(\overline{\mathcal R}\), the secular equations become linear with constant coefficients:

where the real matrix \(A\) depends on the masses and semi-major axes through Laplace coefficients (Fourier coefficients of \((1-2\alpha\cos\psi+\alpha^2)^{-s}\), \(\alpha=a_j/a_k\)). Lagrange derived the analogous system for inclinations and nodes in 1774; Laplace did it for eccentricities; Lagrange computed the full solution for the six known planets in 1781–84. The solution is a sum of normal modes,

with real frequencies \(g_k\), the eigenvalues of \(A\). Each perihelion precesses and each eccentricity oscillates between fixed bounds.

The argument depends on all planets orbiting in the same direction (all \(\Lambda_j>0\)) and on the truncations. Urbain Le Verrier found in the 1840s that higher-order terms and small divisors make the inner planets’ secular theory delicate, and Henri Poincaré showed in the 1890s that the perturbation series generally diverge. Numerical integrations by Jacques Laskar from 1989 found that the solar system is chaotic with a Lyapunov time of about 5 million years, and that over 5 billion years Mercury’s eccentricity has a probability of order 1% of growing large enough for collisions. Within the horizon Laplace had in view, his conclusion holds.

The great inequality and resonances

For the general reader. Since the late seventeenth century astronomers had seen Jupiter speeding up and Saturn slowing down. If this continued, Saturn would eventually leave the solar system. Laplace showed that the effect is a slow swing lasting about 900 years, produced because two Jupiter years are almost exactly five Saturn years.

The puzzle

Comparing ancient and modern observations, Edmond Halley had inferred an acceleration of Jupiter and a deceleration of Saturn. Leonhard Euler won Paris prizes in 1748 and 1752 for attacking the problem but found an acceleration for both planets, contrary to observation; Lagrange’s work of the 1760s also failed to settle it. In 1773 Laplace found no secular term in the mean motions (Lesson 04) and for a time suspected causes outside Newtonian gravitation. In 1785 Lagrange pointed out that near-commensurable mean motions can make otherwise tiny periodic terms large. Laplace took this up and presented his solution in November 1785.

Small divisors

The disturbing function contains terms of the form \(\mathcal R_{k}\cos(k_1\lambda_J+k_2\lambda_S+\cdots)\). A term with frequency \(\nu=k_1n_J+k_2n_S\) enters \(\dot a\) with amplitude proportional to \(\mathcal R_k\), and so enters \(\lambda\), after two integrations, with amplitude proportional to \(\mathcal R_k/\nu^2\).

The ratio of the effects

Over times short compared with the secular evolution, the energy of the Sun–Jupiter–Saturn system is approximately \(-\tfrac{G M m_J}{2a_J}-\tfrac{GMm_S}{2a_S}\) plus a small interaction term with no long-period part at this order. Conservation gives

so that

When Jupiter speeds up, Saturn must slow down, by a definite ratio. Observations gave about \(-2.3\), close to the value from the masses (Exercise 5.2).

The moons of Jupiter

For the Galilean satellites Io, Europa and Ganymede, Laplace showed that the mean motions and longitudes satisfy, exactly in the long run,

and that this configuration is stable: the angle librates about \(180^\circ\). The relation is now called the Laplace resonance. One consequence is that the three moons can never all be in conjunction.

The Moon’s secular acceleration

Ancient eclipse records showed the Moon’s mean motion increasing slowly. In 1787 Laplace traced this to the slow secular decrease of the Earth’s orbital eccentricity, which changes the average solar perturbation on the Moon. In 1853 John Couch Adams showed that this accounts for only about half the observed effect; the rest is due to tidal friction, which slows the Earth’s rotation.

Potential theory and spherical harmonics

For the general reader. To compute how a flattened planet attracts a moon, or what shape a rotating fluid Earth must take, one needs the gravitational pull of an extended body. Laplace found that the whole problem is governed by a single scalar function satisfying a simple equation, and that its solutions on a sphere can be broken into standard pieces. Both ideas now run through electrostatics, fluid dynamics and quantum mechanics.

The potential and Laplace’s equation

For a body of density \(\rho\), define the potential

Joseph-Louis Lagrange had already used such a function; Laplace made it the central object of the theory of attraction in his memoirs of the 1780s.

The exterior statement follows by differentiating under the integral, since \(\nabla^2(1/|\mathbf x-\mathbf x'|)=0\) for \(\mathbf x\ne\mathbf x'\). Laplace wrote the equation in spherical coordinates, the form useful for planets:

Spherical harmonics

Separating \(V=r^{-(l+1)}Y(\theta,\phi)\) gives the equation for a surface harmonic of degree \(l\):

For axially symmetric \(Y\) the solutions are the Legendre polynomials \(P_l(\cos\theta)\), introduced by Adrien-Marie Legendre in the same years, with generating function

The proof inserts the generating function with \(t=r'/r\) and \(x=\cos\gamma\) into \(1/|\mathbf x-\mathbf x'|=\frac1r(1-2t\cos\gamma+t^2)^{-1/2}\). For an oblate planet the leading correction is the quadrupole term \(-\frac{GMR^2}{r^3}J_2P_2(\cos\theta)\), which makes the orbits of satellites precess. Laplace used such expansions to relate the flattening of the Earth to the variation of gravity with latitude and to the motion of the Moon.

The dynamical theory of tides

For the general reader. Newton explained tides as the ocean taking the shape the Moon’s pull would give it if water could respond instantly. Real oceans cannot keep up with the turning Earth. Laplace wrote the equations for a thin rotating ocean forced by the Moon and Sun, the starting point of all later tidal science and of much of modern ocean and atmosphere dynamics.

The tide-generating potential

Let the Moon, of mass \(M_m\), be at distance \(d\) from the Earth’s centre. At a point on the Earth’s surface (radius \(a\)) at angular distance \(\psi\) from the sub-lunar point, the Moon’s potential minus the uniform part that accelerates the whole Earth is, to leading order in \(a/d\),

In Newton’s equilibrium theory the sea surface rises by \(\zeta_{\rm eq}=\Phi_{\rm tide}/g\), producing two bulges, one under the Moon and one opposite.

Laplace’s tidal equations (1775–76)

Treat the ocean as a thin layer of depth \(D(\lambda,\varphi)\) on a sphere rotating with angular velocity \(\Omega\), with longitude \(\lambda\), latitude \(\varphi\), eastward and northward velocities \(u,v\) and surface elevation \(\zeta\). Linearizing and neglecting vertical accelerations,

The terms in \(2\Omega\sin\varphi\) are the Coriolis terms; Laplace included them half a century before Gaspard-Gustave Coriolis analysed the effect in general. The equations describe forced oscillations of a rotating fluid, and their free solutions are the Kelvin, Poincaré and Rossby waves of geophysical fluid dynamics.

Species

Expanding \(P_2(\cos\psi)\) in the latitude and hour angle of the Moon splits the forcing into three species: long-period (independent of the Earth’s rotation), diurnal (once per day) and semidiurnal (twice per day). Each species has its own spatial pattern and responds separately. Laplace showed that for an ocean of uniform depth covering the globe, the diurnal species produces no rise and fall of the surface, He tested his theory against long series of tide records from Brest. He also introduced the practice of analysing tidal records as sums of harmonic constituents, each forced at an astronomical frequency.

Inverse probability

For the general reader. Ordinary probability goes from causes to effects: given a fair coin, how likely are ten heads? Laplace asked the reverse question: given the effects we observe, how probable is each possible cause? His answer, found independently of the English minister Thomas Bayes, is the foundation of what is now called Bayesian statistics.

The binomial case

An urn contains white and black tickets in an unknown proportion \(p\), taken uniform on \([0,1]\) a priori. In \(n\) draws with replacement we see \(s\) white. Then the posterior density is

using Euler’s beta integral \(\int_0^1q^s(1-q)^{n-s}dq=\frac{s!(n-s)!}{(n+1)!}\).

In the Essai philosophique Laplace illustrated the rule with the sunrise: if the sun has risen every day for 5000 years, the odds that it rises tomorrow are 1,826,214 to 1. He added that anyone who knows the regularity of nature would be far more confident, a reminder that the uniform prior expresses ignorance, not knowledge.

Data

Laplace applied these methods to the excess of male over female births in Paris and London, asking whether the observed excess could be due to chance. He showed that the posterior probability that \(p\le\tfrac12\) was vanishingly small, a problem that forced him to develop the asymptotic methods of Lesson 09. He also estimated the population of France from a sample of parishes, using the ratio of population to annual births, and gave an estimate of the error: an early survey with a stated margin of uncertainty.

Asymptotics, the central limit theorem, least squares

For the general reader. Probability with large numbers of observations involves enormous products and integrals that cannot be computed exactly. Laplace found how to approximate them. The method showed him that sums of many small independent errors tend to follow the bell curve, whatever the individual errors look like. That fact, the central limit theorem, is why the normal distribution is everywhere in science.

Near \(x_0\), \(f(x)\approx f(x_0)-\tfrac12|f''(x_0)|(x-x_0)^2\), and the integral is dominated by a Gaussian whose integral Laplace evaluated, \(\int_{-\infty}^{\infty}e^{-t^2}dt=\sqrt\pi\). Applied to the posterior of Lesson 08 with \(f(p)=\frac sn\log p+\frac{n-s}{n}\log(1-p)\), it shows that the posterior is approximately normal with mean \(\hat p=s/n\) and variance \(\hat p(1-\hat p)/n\).

The central limit theorem

Abraham de Moivre had found the normal approximation to the binomial with \(p=\tfrac12\) in 1733. Laplace extended it to general \(p\) and, in a memoir of 1810, to sums of independent errors with an arbitrary common distribution. His tool was the function \(\phi(t)=E[e^{\mathrm itX}]\), now called the characteristic function.

Sketch in Laplace’s style. Take \(\mu=0\). The characteristic function of \(S_n/(\sigma\sqrt n)\) is \(\phi(t/\sigma\sqrt n)^n\). Since \(\phi(s)=1-\tfrac12\sigma^2s^2+o(s^2)\),

the characteristic function of the standard normal. Laplace then inverted the Fourier transform. A rigorous proof needed the continuity theorem for characteristic functions; Aleksandr Lyapunov gave one under his moment condition in 1901, and Jarl Lindeberg in 1920–22 under a weaker condition.

Least squares

Adrien-Marie Legendre published the method of least squares in 1805, and Carl Friedrich Gauss justified it in 1809 by assuming normally distributed errors. In 1810–11 Laplace gave a justification that needs no such assumption: when each estimate combines many observations, the central limit theorem makes linear estimates approximately normal, and among them least squares minimizes the expected error. Gauss later proved, in 1821–23, the exact finite-sample optimality now known as the Gauss–Markov theorem. Laplace’s generating functions, integrals of the form \(\int e^{-sx}\varphi(x)\,dx\) and characteristic functions all appear in the Théorie analytique; the Laplace transform of modern engineering is named in his honour.

Heat, sound, capillarity and the origin of the planets

For the general reader. Laplace hoped to explain all of physics by short-range forces between particles, just as gravity explained the heavens. The programme failed as a whole, but along the way he fixed Newton’s wrong value for the speed of sound, gave the law of pressure in curved liquid surfaces, and proposed a theory of how the solar system formed that still shapes modern ideas.

The speed of sound

Newton’s Principia gave the speed of sound as \(c=\sqrt{p/\rho}\), which assumes the air’s temperature stays constant during compression. His value was too small by about a sixth. Laplace argued in 1816 that the compressions are too fast for heat to flow, so they are adiabatic: \(p\rho^{-\gamma}\) is constant, with \(\gamma=c_p/c_v\).

Capillarity

In the supplements to Book X of the Mécanique céleste (1806–07) Laplace derived, from short-range attractions between molecules, the pressure jump across a curved liquid surface. Thomas Young had reached the same law by a different argument in 1805.

Heat

The Lavoisier–Laplace ice calorimeter surrounds a sample with ice at 0 °C, itself shielded by an outer jacket of ice; the heat released equals the latent heat of fusion times the mass of meltwater collected.

The nebular hypothesis and dark stars

In the Exposition du système du monde (1796) Laplace proposed that the solar system formed from a hot, slowly rotating nebula that contracted and spun faster, shedding rings that condensed into planets. Immanuel Kant had sketched a related idea in 1755. The hypothesis explained why the planets orbit in nearly the same plane and in the same direction as the Sun rotates. Modern theories of protoplanetary disks descend from it. In the same book Laplace noted that a star with the density of the Earth and 250 times the Sun’s diameter would, under Newton’s corpuscular theory of light, not let its light escape. John Michell had made a similar argument in 1783.

Minister, senator, marquis, 1795–1827

After the coup of 18 Brumaire in November 1799, Napoleon Bonaparte, now First Consul, made his former examiner Minister of the Interior. Laplace held the post from 12 November to 25 December 1799 and was replaced by Napoleon’s brother Lucien. Napoleon later wrote that Laplace brought the spirit of the infinitely small into administration. As compensation he was appointed to the Senate, becoming its vice-president and, in September 1803, its chancellor. He received the Legion of Honour and was made a count of the Empire.

The Mécanique céleste

The Traité de mécanique céleste gathered a century of work on gravitation into five volumes: volumes 1 and 2 in 1799, volume 3 in 1802, volume 4 in 1805 and volume 5 in 1825. Jean-Baptiste Biot helped see it through the press. The style was compressed, and the phrase “it is easy to see” often covered pages of calculation. Nathaniel Bowditch’s annotated English translation (1829–39) and Mary Somerville’s Mechanism of the Heavens (1831) made it accessible to English readers. A famous story has Napoleon remarking that the book never mentions God and Laplace replying that he had no need of that hypothesis. The only eyewitness record of a conversation between them on the heavens, in William Herschel’s diary for August 1802, does not contain the phrase, and historians treat the exchange as at best garbled.

Arcueil

Around 1806 Laplace bought a house at Arcueil, next to the chemist Claude-Louis Berthollet. The Society of Arcueil that met there gave young scientists such as Siméon Denis Poisson, Jean-Baptiste Biot, François Arago, Étienne-Louis Malus and Joseph Louis Gay-Lussac access to two powerful patrons. Laplace’s programme of explaining light, heat and capillarity by short-range forces between particles shaped their research, and the Institut’s prize questions. When Augustin Fresnel’s wave theory of diffraction won the prize of 1819, it marked the decline of that programme.

Restoration

In 1813 his daughter Sophie-Suzanne, married to the marquis de Portes, died in childbirth; her child survived. In April 1814 Laplace voted in the Senate for Napoleon’s deposition and offered his services to the Bourbons. Louis XVIII made him a peer of France, and in 1817 a marquis. He was elected to the Académie française in 1816 and became the first president of the Société de géographie in 1821. His political flexibility drew contempt from contemporaries, but his scientific standing was unchallenged. He died in Paris on 5 March 1827, aged 77.

Determinism and legacy

The Essai philosophique sur les probabilités (1814) opens with a statement of causal determinism that became the classic formulation of the Newtonian world view. Laplace imagined an intelligence that knew, at one instant, all the forces of nature and the positions of all the bodies that compose it, and could analyse these data. For such an intelligence, he wrote, nothing would be uncertain, and the future like the past would be present before its eyes. Later writers named it Laplace’s demon; he did not use the word. For Laplace the point was the opposite of fatalism: since we are not such an intelligence, our knowledge is partial, and probability is the calculus of reasoning under partial knowledge.

His remains were buried at Père Lachaise and moved in 1888 to Saint-Julien-de-Mailloc in Normandy. His name is one of the 72 inscribed on the Eiffel Tower.

Legacy

• Celestial mechanics: perturbation theory, the secular theory with Lagrange, resonances and the invariable plane. The questions he answered to first order led through Le Verrier and Poincaré to the KAM theorem and to the modern discovery of chaos in the solar system.
• Analysis and physics: Laplace’s equation and the Laplace operator, spherical harmonics, the potential, and the Laplace transform. Potential theory became the model for electrostatics, and through Green, Gauss and Dirichlet for much of nineteenth-century analysis.
• Geophysics: the tidal equations are still the basis of ocean tide models, and their rotating shallow-water form underlies dynamical meteorology.
• Probability and statistics: inverse probability, the central limit theorem, characteristic functions, Laplace’s method and asymptotic normality of estimators. After a century of neglect his inverse-probability approach returned as Bayesian statistics.
• Students and circle: Poisson, Biot and Arago carried his methods into the next generation; Fourier, Cauchy and Fresnel built alternatives.

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Pierre-Simon Laplace, MacTutor History of Mathematics.
2. Pierre-Simon Laplace, Wikipedia; Pierre-Simon de Laplace, Wikipédia (French).
3. C. C. Gillispie, with R. Fox and I. Grattan-Guinness, Pierre-Simon Laplace, 1749–1827: A Life in Exact Science, Princeton University Press, 1997.
4. R. Hahn, Pierre Simon Laplace, 1749–1827: A Determined Scientist, Harvard University Press, 2005.
5. M. Crosland, The Society of Arcueil: A View of French Science at the Time of Napoleon I, Heinemann, 1967.

Science

1. P.-S. Laplace, Traité de mécanique céleste, 5 vols., Paris, 1799–1825; English translation with commentary by N. Bowditch, Boston, 1829–39.
2. P.-S. Laplace, Exposition du système du monde, Paris, 1796.
3. P.-S. Laplace, Théorie analytique des probabilités, Paris, 1812; Essai philosophique sur les probabilités, Paris, 1814.
4. J. Laskar, Is the solar system stable?, arXiv:1209.5996.
5. C. D. Murray and S. F. Dermott, Solar System Dynamics, Cambridge University Press, 1999 (secular theory and resonances).
6. S. M. Stigler, The History of Statistics: The Measurement of Uncertainty before 1900, Harvard University Press, 1986.
7. A. Hald, A History of Mathematical Statistics from 1750 to 1930, Wiley, 1998.
8. D. E. Cartwright, Tides: A Scientific History, Cambridge University Press, 1999.
9. Central limit theorem, Wikipedia (history section).