Alexis Clairaut

How to use this course

Alexis Clairaut belongs to the generation that turned Isaac Newton’s Principia into working science. Newton had stated the law of gravitation and drawn some of its consequences with geometric arguments that few could extend. Clairaut, with Leonhard Euler and Jean le Rond d’Alembert, recast those arguments in the calculus of Gottfried Wilhelm Leibniz and pushed them to quantitative tests: the shape of the Earth, the motion of the Moon, the orbit of a comet. Each test could have refuted Newton. Each, in Clairaut’s hands, confirmed him.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, derivations and exercises with folded solutions. Lessons 04 to 06 are pure mathematics from his youth; Lessons 07 to 10 follow his great problems in celestial mechanics and geodesy. Formulas are given in modern notation, which differs from Clairaut’s.

Timeline

A prodigy in Paris, 1713–1731

Alexis Claude Clairaut was born in Paris on 13 May 1713. The date 7 May also appears in reference works; 13 May is the date given in the eulogy by Grandjean de Fouchy, the Academy’s secretary. His father, Jean-Baptiste Clairaut, taught mathematics in Paris and was later elected to the Berlin Academy. His mother, Catherine Petit, bore many children, of whom few survived infancy.

His father taught him at home and used Euclid’s Elements as a reading book. By nine Clairaut had worked through Guisnée’s Application de l’algèbre à la géométrie, and at ten he was studying the calculus in the Marquis de l’Hôpital’s Analyse des infiniment petits (1696), the first textbook of the subject. In 1726, aged thirteen, he read to the Academy of Sciences a paper on four new curves. A younger brother was also precocious: he read a paper to the Academy at fourteen and died in 1732 at sixteen.

Curves of double curvature

By 1729 Clairaut had finished a systematic study of curves in three-dimensional space, then called curves of double curvature because their projections onto two coordinate planes are both curved. He was proposed for the Academy on 4 September 1729. The book, Recherches sur les courbes à double courbure, appeared in 1731, and in that year the king confirmed his election as adjoint mécanicien. He was eighteen, below the statutory minimum age, and he remains the youngest person ever elected.

Maupertuis, Basel and the Newtonians, 1731–1736

The Paris Academy of the early 1730s was Cartesian. Most of its members explained planetary motion by René Descartes’s vortices of subtle matter, and they distrusted Newton’s attraction acting at a distance. A small group of younger academicians, led by Pierre Louis Moreau de Maupertuis, set out to bring Newton’s physics to France. Clairaut joined them almost at once.

In 1733 he published on problems of maxima and minima, an early contribution to the calculus of variations, and on the shortest paths on surfaces of revolution (Lesson 06). In 1734 he and Maupertuis went to Basel to work with Johann Bernoulli, the most powerful analyst in Europe and a teacher of Euler. There Clairaut befriended Samuel König, with whom he corresponded for years. In the same period he studied the differential equations now named after him and their singular solutions (Lesson 05).

The shape of the Earth

The great scientific controversy of the decade concerned the shape of the Earth. Newton and Christiaan Huygens had argued from rotation that the Earth is flattened at the poles. The surveys of the Cassini family along the meridian of Paris suggested the opposite: an Earth elongated toward the poles. The question was settled by measuring the length of a degree of latitude at very different latitudes. The Academy sent one expedition to Peru, near the equator, in 1735, and a second to Lapland, near the Arctic Circle, in 1736.

Lapland, 1736–1737

The Lapland expedition left Dunkirk on 20 April 1736 under Maupertuis. Its members included Clairaut, Charles Étienne Louis Camus, Pierre Charles Le Monnier, the abbé Outhier and the Swedish astronomer Anders Celsius. They worked along the Torne River at the head of the Gulf of Bothnia, triangulating across hilltops in summer and measuring a baseline on the frozen river in winter. Astronomical observations of a star at both ends of the arc gave the difference in latitude.

The degree measured in Lapland was markedly longer than the degree near Paris. A longer degree means a flatter meridian, so the Earth is flattened at the poles, as Newton had said. The expedition returned to Paris on 20 August 1737. Voltaire celebrated Maupertuis as the man who had flattened the Earth and the Cassinis. The Royal Society elected Clairaut a Fellow on 27 October 1737, and Louis XV later granted him a pension for his work on the Earth’s figure. In 1738 he became a full member of the Academy.

The measurements raised a deeper question. Given a rotating, self-gravitating body, what shape should it take, and how does its shape relate to gravity at its surface? Clairaut’s first attempts, published in the Philosophical Transactions during the expedition, were incomplete. The full answer was his book of 1743 (Lessons 07 and 08).

Space curves, mixed partials and exact differentials

For the general reader. Before Clairaut, the calculus was mostly used for curves in a plane. His first book treated curves in space, and his later papers treated functions of several variables. He was among the first to state when an expression like \(M\,dx+N\,dy\) is the differential of a single function, a question that sits at the root of thermodynamics, potential theory and the hydrostatics of Lesson 07.

Curves of double curvature

Clairaut represented a space curve by two equations \(y=\phi(x)\), \(z=\psi(x)\), its projections onto the \(xy\)- and \(xz\)-planes. The tangent direction is \((1,\phi',\psi')\) and the arc length is \(\int\sqrt{1+\phi'^2+\psi'^2}\,dx\). More generally a curve is the intersection of two surfaces \(F=0\), \(G=0\), and its tangent is orthogonal to both normals \(\nabla F\) and \(\nabla G\).

Clairaut and Euler used this identity around 1740; the hypotheses were made precise by Hermann Schwarz in 1873. Its main use for Clairaut was the following criterion, which appears in his memoirs of 1739 and 1740 on the integration of first-order equations.

Necessity is the equality of mixed partials. For sufficiency put \(\Phi(x,y)=\int_{x_0}^xM(s,y_0)\,ds+\int_{y_0}^yN(x,t)\,dt\); then \(\Phi_y=N\) and \(\Phi_x=M(x,y_0)+\int_{y_0}^yN_x(x,t)\,dt=M(x,y_0)+\int_{y_0}^yM_y(x,t)\,dt=M(x,y)\). When the criterion fails, one looks for an integrating factor \(\mu\) such that \(\mu M\,dx+\mu N\,dy\) is exact. Clairaut showed such factors exist for first-order equations, a fact anticipated by Nicolas Fatio de Duillier and Johann Bernoulli.

Clairaut’s equation and singular solutions

For the general reader. Most differential equations have a family of solutions depending on a constant. Clairaut found a simple equation whose solutions are a family of straight lines, and which has one more solution: the curve that all the lines touch. That extra curve cannot be obtained from the family by any choice of constant. It was the first clear example of a singular solution.

Derivation

Differentiate \(y=xp+f(p)\) with respect to \(x\): \(p=p+xp'+f'(p)p'\), so

Either \(p'=0\), so \(p=c\) and \(y=cx+f(c)\); or \(x=-f'(p)\), which with the original equation gives the parametric curve. The envelope of \(F(x,y,c)=y-cx-f(c)=0\) is found by eliminating \(c\) from \(F=0\) and \(\partial F/\partial c=-x-f'(c)=0\), which is the same pair of equations. At each point of the singular curve the solution is not unique: a line of the family and the envelope pass through it with the same slope.

The singular solution is the graph of the Legendre transform. If \(g(x)=\sup_p\big(xp+f(p)\big)\) is attained at an interior point, then \(x=-f'(p)\) there and \(g(x)\) is the singular \(y\). The duality between a curve and its family of tangent lines, implicit in Clairaut’s equation, is the same one used in Hamiltonian mechanics and thermodynamics.

Geodesics on surfaces of revolution

For the general reader. On a sphere the shortest paths are great circles. On a flattened Earth they are more complicated, and a navigator or surveyor needs to know them. In 1733 Clairaut found a conserved quantity that governs every shortest path on any surface of revolution. It is the geometric form of the conservation of angular momentum.

Let \(S\) be obtained by rotating a curve about the \(z\)-axis, parametrized by \(\mathbf x(s,\theta)=(r(s)\cos\theta,\ r(s)\sin\theta,\ z(s))\) with \(s\) the arc length of the profile curve. The metric is \(ds^2+r(s)^2\,d\theta^2\).

Proof

Geodesics are critical points of the energy \(\int\big(\dot s^2+r(s)^2\dot\theta^2\big)\,dt\). The Lagrangian does not depend on \(\theta\), so the conjugate momentum \(r^2\dot\theta\) is conserved (compare Noether’s theorem). For a unit-speed curve the component of the velocity along the parallel is \(r\dot\theta=\sin\psi\), so \(r^2\dot\theta=r\sin\psi\).

Consequences: meridians (\(\psi=0\)) are geodesics; a parallel is a geodesic exactly where \(r'(s)=0\); and a geodesic with Clairaut constant \(c>0\) stays in the region \(r\ge c\), turning back where \(r=c\) with \(\psi=\pi/2\).

The figure of the Earth: equilibrium and flattening

For the general reader. A spinning ball of fluid bulges at its equator. How much it bulges depends on how fast it spins and on how its mass is arranged inside. Clairaut put this problem on a sound mathematical footing in 1743, starting from a clean principle of fluid equilibrium and ending with formulas that could be checked against the Lapland and Peru measurements.

Equilibrium

Clairaut imagined thin channels of fluid running through the body. The fluid is in equilibrium only if the forces produce no net flow around any closed channel. In modern terms:

This is the exactness criterion of Lesson 04 used in physics. For the rotating Earth, \(U\) is the sum of the gravitational potential and the centrifugal potential \(\tfrac12\omega^2(x^2+y^2)\). Clairaut’s principle introduced the potential, though the name came later, with George Green in 1828.

Degrees of latitude

For an ellipse of equatorial radius \(a\), flattening \(f=(a-b)/a\) and eccentricity \(e^2=f(2-f)\), the radius of curvature of the meridian at geodetic latitude \(\varphi\) is

A degree of latitude has length \(M(\varphi)\,\pi/180\), so it grows from equator to pole in proportion to \(3f\sin^2\varphi\). Comparing degrees at two latitudes gives \(f\).

For the Earth \(m\approx1/288\), so \(f\approx1/230\), Newton’s value. Huygens, who assumed all attraction directed toward the centre, found \(f=\tfrac12m\approx1/578\). The true value, about \(1/298\), lies between. Clairaut showed why: a real Earth is denser toward the centre (Lesson 08).

Clairaut’s theorem on gravity

For the general reader. A pendulum clock runs slower near the equator, because gravity is weaker there. Jean Richer had noticed this in Cayenne in 1672. Clairaut found a simple formula linking the variation of gravity with latitude to the Earth’s flattening and its rotation. Remarkably, the formula does not depend on how mass is distributed inside, so pendulum measurements at the surface reveal the shape of the planet.

Clairaut proved this for a body made of homogeneous spheroidal layers of different densities. George Stokes showed in 1849 that it holds for any internal density distribution, provided the outer surface is a level surface of spheroidal shape.

The internal layers

Let \(\varepsilon(a)\) be the flattening of the level surface of mean radius \(a\), \(\rho(a)\) the density there, and \(\bar\rho(a)\) the mean density inside it. Clairaut derived the second-order equation

with \(\varepsilon\) regular at the centre and the surface condition \(R\,\varepsilon'(R)+2\varepsilon(R)=\tfrac52m\). Since density does not increase outward, \(\rho\le\bar\rho\), and one shows that \(\varepsilon\) increases outward and that \(\tfrac12m\le f\le\tfrac54m\). The two bounds are Huygens’s and Newton’s values. Later, Pierre-Simon Laplace, Adrien-Marie Legendre and Harold Jeffreys used this equation to constrain the density inside the Earth.

The Moon’s apogee and the three-body problem

For the general reader. The Moon’s elliptical orbit slowly turns, so that its farthest point from the Earth goes once around the sky in about nine years. The Sun’s pull causes this. But when Clairaut, Euler and d’Alembert each calculated the effect from Newton’s law, all three found only half the observed rate. For eighteen months it looked as if the law of gravitation itself was wrong. Clairaut then found the missing half in terms everyone had neglected.

Precession from a perturbing force

For a nearly circular orbit of radius \(r\) under a central attraction \(F(r)\) per unit mass, the angular frequency \(\Omega\) and the frequency \(\kappa\) of small radial oscillations satisfy

For \(F=\mu/r^2\), \(\kappa=\Omega\): the ellipse closes. If \(\kappa<\Omega\), the apsides advance by \(2\pi(\Omega/\kappa-1)\) per revolution. Averaged over a month, the Sun’s tidal force on the Moon has a mean outward radial component \(\tfrac12n'^2r\), where \(n'\) is the Sun’s mean motion.

The crisis and its resolution

On 15 November 1747 Clairaut told the Academy that the discrepancy required adding a small term, varying as \(1/r^4\), to Newton’s inverse-square law. D’Alembert had deposited similar results shortly before, and Euler had written to Clairaut in September supporting the same conclusion. Georges-Louis Leclerc de Buffon objected on metaphysical grounds that a law of nature should be simple.

Clairaut went back to his equations. He had written the orbit in a form \(1/r=\frac1p\big(1-e\cos(c\theta)\big)\) plus perturbation terms, and had dropped products of small quantities that looked negligible. In spring 1748 he found that the transverse component of the solar force, combined with the variation of the orbit, contributes a second-order term almost as large as the first. On 17 May 1749 he announced that the inverse-square law accounted for nearly the whole motion. In modern form the series is

whose terms are about \(0.0042,\ 0.0029,\ 0.0010\). The series converges slowly; George William Hill and others computed many more terms in the nineteenth century. Euler, now convinced, praised the result warmly, and Clairaut’s essay won the prize of the St Petersburg Academy. He published it as Théorie de la lune (1752) and produced lunar tables in 1754, important for finding longitude at sea.

The return of Halley’s comet

For the general reader. Edmond Halley had identified the comets of 1531, 1607 and 1682 as one object and predicted its return around 1758. But the comet passes close to Jupiter and Saturn, whose pulls stretch or shrink its orbit. Clairaut, with Joseph Jérôme Lalande and Nicole-Reine Lepaute, spent months computing these effects by hand, and predicted the date of its closest approach to the Sun. The comet arrived within his margin of error. It was the first object other than a planet shown to orbit the Sun, and a public triumph for Newton’s theory.

The computation

Clairaut did not integrate the full motion. He took the known orbit, estimated how the attraction of Jupiter and Saturn had altered the comet’s orbital elements along each of the two previous returns, and compared the two intervals 1607–1682 and 1682–1759. The integrals over the planets’ positions were evaluated numerically, point by point. Lalande and Lepaute carried out much of the arithmetic. On 14 November 1758 Clairaut announced perihelion for mid-April 1759, with an uncertainty of about a month due to neglected effects and the unknown masses of Saturn and other bodies.

Johann Georg Palitzsch, a farmer and amateur astronomer in Saxony, saw the comet on 25 December 1758. Perihelion came on 13 March 1759, about a month earlier than predicted. Clairaut’s calculation had found that Jupiter and Saturn would delay the return by about 618 days compared with an unperturbed orbit, and the observed return confirmed the bulk of that delay. D’Alembert and others attacked the precision of the prediction, and a public quarrel followed.

Textbooks, salons and quarrels

Clairaut was a public figure in Enlightenment Paris. He never married, enjoyed society and moved easily in the salons. He was close to Émilie du Châtelet and Voltaire, and he guided du Châtelet’s French translation of Newton’s Principia, checking her commentary and adding his own results. The translation appeared in part in 1756, after her death in 1749, and remained the standard French version.

He also wrote for beginners. His Élémens de géométrie (1741) taught geometry by leading the reader to rediscover results from practical problems of measuring land, instead of from axioms. A companion Élémens d’algèbre followed in the same spirit. Both went through many editions into the nineteenth century.

His relations with d’Alembert, cordial in the early 1740s, became hostile after the lunar crisis of 1747. Both worked on the same problems, the three-body problem and the figure of the Earth, and each accused the other of errors and borrowings. D’Alembert favoured general analytic methods; Clairaut favoured results that could be compared with observation. The quarrel reached the public after the comet’s return in 1759.

Last years and legacy

In 1756 Clairaut took part in measuring a geodetic baseline at Villejuif, south of Paris. In 1761 and 1762 he worked on achromatic lenses combining two kinds of glass; the work was unfinished at his death. He was a member of the Royal Society of London and the academies of Berlin, St Petersburg, Bologna and Uppsala. He died in Paris on 17 May 1765, aged 52, after a short illness.

Legacy

• Geodesy and geophysics: Clairaut’s theorem and his equation for the internal flattening remain the starting point of the theory of the Earth’s figure; the gravity formula used in geodesy today is a refinement of his.
• Celestial mechanics: his resolution of the lunar apsides was the decisive test of the inverse-square law in the three-body problem, and his comet calculation was the first quantitative prediction of planetary perturbations confirmed by observation. Lagrange and Laplace built on both.
• Mathematics: Clairaut’s equation, Clairaut’s relation for geodesics, the exactness criterion and the potential function are standard. His treatment of space curves opened differential geometry in three dimensions, later developed by Euler, Gaspard Monge and Gauss.
• Commemoration: the lunar crater Clairaut is named after him.

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Alexis Claude Clairaut, MacTutor History of Mathematics.
2. Alexis Clairaut, Wikipedia.
3. Académie des sciences, notice on Clairaut (J.-C. Pecker).
4. O. Courcelle, Chronologie de la vie de Clairaut (online chronology, 2007 onward).
5. J.-P. Grandjean de Fouchy, Éloge de M. Clairaut, Histoire de l’Académie royale des sciences (1765).
6. J. Itard, Clairaut, Alexis-Claude, Dictionary of Scientific Biography, vol. 3.

Mathematics and physics

1. A. C. Clairaut, Recherches sur les courbes à double courbure, Paris, 1731.
2. A. C. Clairaut, Théorie de la figure de la Terre, tirée des principes de l’hydrostatique, Paris, 1743.
3. A. C. Clairaut, Théorie de la lune, déduite du seul principe de l’attraction, St Petersburg, 1752.
4. A. C. Clairaut, Théorie du mouvement des comètes, Paris, 1760.
5. Clairaut’s theorem (gravity) and Halley’s Comet, Wikipedia.
6. H. Jeffreys, The Earth, Cambridge University Press, chapter on the figure of the Earth (Clairaut’s equation).
7. C. Wilson, The problem of perturbation analytically treated: Euler, Clairaut, d’Alembert, in R. Taton and C. Wilson (eds.), Planetary Astronomy from the Renaissance to the Rise of Astrophysics, Part B, Cambridge, 1995.
8. M. C. Gutzwiller, Moon–Earth–Sun: the oldest three-body problem, Reviews of Modern Physics 70 (1998) 589–639.
9. M. do Carmo, Differential Geometry of Curves and Surfaces, section on Clairaut’s relation.