Pierre-Louis Moreau de Maupertuis

How to use this course

Pierre-Louis Moreau de Maupertuis was not a great calculator. His gift was for choosing questions and for organising people and arguments around them. He brought Newtonian gravitation into the Paris Academy, led the expedition that showed the Earth is flattened at the poles, proposed the principle of least action, and gave one of the first quantitative arguments for biological inheritance. His later years were consumed by a priority dispute and by Voltaire’s ridicule, which long overshadowed his work.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements in modern notation, derivations and exercises with folded solutions. Maupertuis’s own statements were often loose; the lessons say clearly where his versions differ from the modern ones.

Timeline

Saint-Malo and the army, 1698–1722

Maupertuis was born in Saint-Malo, the walled Breton port, in 1698 and baptised on 28 September. His family had grown rich through trade and privateering; his father, René Moreau de Maupertuis, sat on the royal Council of Commerce as a representative of Brittany. He was educated at home by tutors before being sent in 1714 to the Collège de la Marche in Paris, where he studied philosophy under a teacher named Le Blond. After two years he returned to Saint-Malo.

He was drawn to mathematics and music, and in Paris and Saint-Malo he took lessons in mathematics from Nicolas Guisnée, a member of the Academy and author of a textbook applying algebra to geometry. His father thought a military career more suitable. In 1718 he joined a cavalry regiment stationed at Lille. Garrison life left time for study, and he read widely in mathematics and natural philosophy. By 1722 he had left the army and settled in Paris.

In the cafés and salons of the Regency he quickly made a name as a clever, witty and ambitious young man. He would keep a soldier’s manner all his life, and some of the vanity that his enemies later mocked.

Academician, London, Basel, 1723–1731

In 1723, aged about 25, Maupertuis was elected adjoint of the Académie des Sciences, the junior rank. His first papers, from 1724, dealt with mechanics and the shape of curves in the analytical style of Pierre Varignon. They were respectable, but they did not yet show the direction he would take.

Discovering Newton

In 1728 he spent some months in London and was elected a Fellow of the Royal Society. There he met Newton’s followers and absorbed the Principia. The Paris Academy of the 1720s was firmly Cartesian: gravity was explained by vortices of subtle matter, and Newton’s attraction at a distance was regarded as a return to occult qualities. Maupertuis came back convinced that the Newtonian theory was the better physics.

Basel

He knew that his mathematics was not strong enough to argue the case. In September 1729 he enrolled at the University of Basel and lived in the house of Johann Bernoulli, the greatest living master of the Leibnizian calculus and himself a Cartesian in physics. He worked with Bernoulli for most of a year, returning to Paris by July 1730 with far better technique and a lifelong friendship with the Bernoulli family. He was promoted within the Academy, and began writing the work that would announce his Newtonianism.

Newton in Paris and the Lapland expedition, 1732–1740

His Discours sur les différentes figures des astres (1732) was the first work by a member of the Paris Academy to defend Newtonian attraction openly. Maupertuis argued cautiously: attraction may be no more mysterious than impulsion, and its mathematical consequences can be tested. He compared the figures that rotating bodies should take under different laws of attraction. The book won the attention of Voltaire and of Émilie du Châtelet, whom Maupertuis taught mathematics and who became a friend.

The shape of the Earth

The figure of the Earth had become a test case. Newton and Christiaan Huygens had predicted an Earth flattened at the poles (oblate). The geodetic surveys of the Cassini family along the meridian of France, finished by Jacques Cassini, suggested the opposite: a degree of latitude seemed to shrink toward the north, implying an Earth elongated at the poles (prolate). The Academy decided to settle the matter by measuring degrees as far apart as possible: one expedition went to Peru in 1735, another, under Maupertuis, to Lapland.

Lapland

The Lapland party left Dunkirk on 2 May 1736: Maupertuis, Alexis Clairaut, Charles Étienne Louis Camus, Pierre Charles Le Monnier, the Abbé Réginald Outhier, and the Swedish astronomer Anders Celsius. They based themselves at Tornio at the head of the Gulf of Bothnia, reached on 19 June 1736. Through the summer they built a chain of triangles up the Torne valley to the hill of Kittisvaara, fighting mosquitoes, forest and rapids. In midwinter they measured a baseline of about 13.5 km on the frozen river and made astronomical observations of latitude at both ends with a zenith sector made by George Graham in London.

The degree came out about 57,400 toises, several hundred toises longer than the degree near Paris. A longer degree at high latitude means the Earth is flatter there: it is oblate. Maupertuis reported to the Academy on 13 November 1737 and published La Figure de la Terre in 1738. Jacques Cassini and his allies contested the result for several years, but a re-measurement of the French arc by César-François Cassini de Thury and Nicolas-Louis de Lacaille confirmed the flattening by 1740. Voltaire hailed Maupertuis as the man who had flattened the Earth and the Cassinis. He became a celebrity, and was painted in Lapland fur pressing down on a globe.

The figure of a rotating Earth

For the general reader. A spinning ball of fluid bulges at its equator, because the rotation partly offsets gravity there. How large is the bulge? It depends on how the mass is distributed inside. Huygens and Newton gave two different answers in the 1680s and 1690s, and Maupertuis made the question the centre of French science in the 1730s.

Two small parameters

Let the Earth have equatorial radius \(a\), polar radius \(c\), mass \(M\) and angular velocity \(\omega\). Its flattening is \(f=(a-c)/a\). The ratio of centrifugal to gravitational acceleration at the equator is

Both \(f\) and \(m\) are small, and to first order the shape depends only on \(m\) and on the internal structure. The surface of a fluid in equilibrium is a level surface of the combined gravitational and centrifugal potential.

Derivation. The effective potential is \(U=-\frac{GM}{r}-\tfrac12\omega^2r^2\cos^2\varphi\), with \(\varphi\) the latitude. The surface is a level set, so \(U(a,0)=U(c,\tfrac\pi2)\): \(\frac{GM}{a}+\tfrac12\omega^2a^2=\frac{GM}{c}\). Hence \(\frac ac-1=\frac{\omega^2a^3}{2GM}=\frac m2\), and \(f\approx m/2\).

The flattening is larger because the bulge itself attracts: the extra matter at the equator adds to the deformation. Newton’s argument balanced the weights of two fluid columns, one from the centre to the pole and one from the centre to the equator. Colin Maclaurin proved in 1742 that the oblate spheroid is an exact equilibrium figure for any rotation rate below a critical value.

Clairaut proved this after returning from Lapland, in Théorie de la figure de la terre. It linked the shape of the Earth to measurements of gravity by pendulums, which Maupertuis’s party also made in the north. The modern value is \(f\approx1/298.26\), which lies between the two extremes, as Clairaut’s theorem requires for an Earth denser at the centre.

Measuring a degree

For the general reader. To find the shape of the Earth without leaving it, measure how far you must travel north for the stars overhead to shift by one degree. On a sphere this distance is the same everywhere. On a flattened Earth it is longer near the poles. The Lapland expedition did this measurement close to the Arctic Circle, using a chain of triangles over the forest and a baseline measured on the frozen river.

Geodetic latitude and the meridian degree

Latitude, as measured with a zenith sector, is the angle between the local vertical and the equatorial plane. On an ellipsoid with semi-axes \(a\) and \(c\) and eccentricity \(e^2=1-c^2/a^2\approx2f\), the radius of curvature of the meridian at latitude \(\varphi\) is

Derivation. Expand with \(e^2\approx2f\): \((1-e^2\sin^2\varphi)^{-3/2}\approx1+3f\sin^2\varphi\) and \(1-e^2\approx1-2f\). The ratio formula follows by dividing and keeping first-order terms. For \(f>0\) the degree grows toward the pole: this is what the Lapland measurement showed.

Triangulation

Distances over tens of kilometres cannot be measured with rods. The surveyors measured one baseline directly and then only angles. In a triangle with a known side \(b\) and angles \(A,B,C\), the law of sines \(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\) gives the other sides, and each new triangle shares a side with the last. The chain carries the scale from the baseline to the ends of the arc, and the projected north–south distance between the ends gives the length of the arc. The latitude difference comes from the zenith distances \(z_S,z_N\) of the same star crossing the meridian at the two ends: \(\Delta\varphi=z_S-z_N\) for a star north of both zeniths, with corrections for refraction, aberration and precession.

The numbers

Jean Picard’s degree between Paris and Amiens (1669–70) was 57,060 toises; the Lapland degree came out at about 57,400 toises. With the later corrected values used by historians, a degree at latitude 66°10′ in Lapland was 57,395 toises and a degree at 0°30′ in Peru was 56,768 toises. The modern value at the Lapland latitude is about 57,210 toises, so the Lapland degree was too long by roughly 200 toises, exaggerating the flattening. The direction of the effect was nevertheless right, and decisive.

Least action and refraction (1744)

For the general reader. When light passes from air into water it bends. Pierre de Fermat had explained this in the 1660s by saying light takes the path of least time. Maupertuis disliked the idea that light should “know” about time, and proposed instead that nature spends the least of a quantity he called action. Strangely, his rule gives the same law of refraction from the opposite assumption about the speed of light.

The 1744 memoir

In 1744 Maupertuis read to the Paris Academy his Accord de différentes lois de la nature qui avaient jusqu’ici paru incompatibles. He defined the action along a path as the sum of the products of each distance travelled and the speed along it, and asserted that light follows the path for which the total action is least. He took the Newtonian view that light is a stream of particles moving faster in a denser medium.

Derivation. Put \(P=(0,h_1)\), \(Q=(d,-h_2)\), and let the ray cross the interface \(y=0\) at \((x,0)\). Then \(A(x)=v_1\sqrt{x^2+h_1^2}+v_2\sqrt{(d-x)^2+h_2^2}\) and

Setting \(A'(x)=0\) gives the law. \(A\) is a sum of convex functions, so the critical point is the unique minimum.

Two theories, one law

Both principles give Snell’s law \(\sin\theta_1/\sin\theta_2=\text{const}\). They disagree about what the constant means. For light entering water, \(\theta_2<\theta_1\). Maupertuis’s version then needs \(v_2>v_1\), faster in water, as in Newton’s corpuscular theory; Fermat’s needs \(v_2<v_1\), as in Huygens’s wave theory. Léon Foucault’s measurement of 1850 showed light is slower in water. In modern terms Maupertuis’s “speed” is better read as the momentum of the light corpuscle, and his principle is the mechanical counterpart of Fermat’s optical one (Lesson 08).

Least action in collisions and statics (1746)

For the general reader. Two years later, as president of the Berlin Academy, Maupertuis raised his principle into a universal law: whenever any change occurs in nature, the quantity of action used is the least possible. He derived the rules of collision and of the lever from it, and claimed that such economy was evidence of a wise Creator.

The rule of 1746

In Les lois du mouvement et du repos déduites d’un principe métaphysique, read to the Berlin Academy in 1746, Maupertuis defined the quantity of action as mass times speed times distance, \(mvs\). For a change of velocity occurring over a fixed time, distance is proportional to speed, so the action of a change from speed \(u\) to speed \(w\) is proportional to \(m(u-w)^2\). He asserted that the change that actually occurs makes the total of these quantities a minimum.

Derivation for the elastic case. Write \(\beta=\alpha+a-b\) and minimise \(F(\alpha)=A(a-\alpha)^2+B(2b-a-\alpha)^2\). Then \(F'(\alpha)=-2A(a-\alpha)-2B(2b-a-\alpha)=0\) gives \(\alpha(A+B)=Aa+2Bb-Ba\), as stated. Note that the minimisation again reproduces conservation of momentum, \(A\alpha+B\beta=Aa+Bb\).

Assessment

The derivations reach correct results, but the choice of what to minimise is adjusted to each case: Maupertuis needed the extra condition on relative velocity for elastic bodies, and the lever argument rests on a quantity chosen to give the right answer. The principle as he stated it was too vague to be a law of mechanics, and he gave it a theological reading. Euler had already given the precise version for a particle in 1744 (Lesson 08), and it was that version, not Maupertuis’s, that entered physics.

From Maupertuis to Euler, Lagrange and Jacobi

For the general reader. Maupertuis named the principle; others made it exact. Within a century it had become one of the foundations of physics: the laws of motion can be written as the statement that a single number, the action, is stationary along the true path. The same idea guides the path integrals of quantum mechanics.

Euler’s version

In an appendix to his Methodus inveniendi lineas curvas (1744), Leonhard Euler showed that a particle moving under central forces follows a path for which \(\int v\,ds\) is a minimum, among paths with the same endpoints, provided the speed is determined by the conservation of energy. Euler gave Maupertuis the credit for the general principle, and defended him later in the König affair. Joseph-Louis Lagrange extended the result in 1760 to systems of particles.

Jacobi’s form

For one particle of mass \(m\), \(2T\,dt=mv\,ds\) and \(v=\sqrt{2(E-V)/m}\), so

Carl Gustav Jacob Jacobi observed in the 1830s that time has disappeared: the trajectories are the geodesics of the metric \(d\sigma^2=2m(E-V)\,ds^2\). The same equation describes light rays in a medium of refractive index \(n(\mathbf x)\propto\sqrt{E-V(\mathbf x)}\). Maupertuis’s corpuscular optics of 1744 is exactly this analogy, with light faster where the index is larger.

Hamilton’s principle

William Rowan Hamilton in 1834–35 gave a different principle: with both endpoints fixed in time, the actual motion makes \(S=\int(T-V)\,dt\) stationary. Hamilton’s principle varies paths with fixed duration; Maupertuis’s fixes the energy instead. The two are related by a Legendre transform, \(S=S_0-Et\), and they lead to the same equations of motion. In quantum mechanics the phase of the wave function is \(S/\hbar\), and Louis de Broglie took the parallel between Maupertuis’s and Fermat’s principles as the starting point of wave mechanics.

Heredity and probability

For the general reader. In the 1740s most naturalists believed that each living thing was already formed in miniature in the egg or the sperm. Maupertuis rejected this. Children resemble both parents, he argued, so both must contribute material. To show that traits are inherited, and not repeated by chance, he followed an unusual trait through a family over several generations and calculated the odds.

Against preformation

The Dissertation physique à l’occasion du nègre blanc (1744), prompted by an albino child shown in Paris, and its expansion Vénus physique (1745), published anonymously, argued that preformation could not explain resemblance to both parents or the appearance of hybrids such as mules. Maupertuis proposed that particles from every part of each parent’s body meet and combine, guided by something like chemical affinity, to form the embryo. Errors in the arrangement of particles could produce new varieties, and he speculated that new species might arise in this way. Historians have read these passages as early anticipations of particulate inheritance and of variation as a source of new forms, though Maupertuis gave no mechanism of selection.

The Ruhe family

In Système de la nature (1751) and his Lettres (1752) he described polydactyly, extra fingers and toes, in the family of Jacob Ruhe, a Berlin surgeon. The trait appeared in several generations and was passed on through both mothers and fathers. Maupertuis estimated that polydactyly occurs in about one person in 20,000, and argued that its repeated appearance in one line by chance was so improbable that it must be inherited.

With \(q=\tfrac12\), as for an autosomal dominant trait, the ratio is \(10^8\). Maupertuis did not have the concept of a likelihood ratio, but his argument was one of the first uses of probability to test a biological hypothesis.

Cosmology and utility

In the Essai de cosmologie (1750) he criticised arguments for God drawn from the intricate details of animals, and argued that the evidence should come from general laws, above all least action. He also proposed to measure happiness, treating a pleasure or pain as the product of its intensity and its duration, an early version of the calculus later associated with utilitarianism.

Laws of attraction and the figures of the stars

For the general reader. Maupertuis’s case for Newton in 1732 was cautious. He did not claim that gravity must fall off as the inverse square of distance; he asked what the world would look like under other laws, and argued that the inverse square is singled out by its consequences. This lesson works through the mathematics behind that argument.

Spheres that attract like points

Newton showed in the Principia (Book I, Section 12) that a uniform spherical shell attracts an outside particle as if its mass were at the centre when the force varies as \(r^{-2}\), and also when it varies as \(r\). These are the only power laws with this property, up to adding the two together. Maupertuis made this one of his arguments: only under the inverse square (among the decreasing laws) do the planets and the Sun act on each other as points, which is why Kepler’s laws hold so simply.

Proof via the potential. The potential of the shell is \(\Phi(r)=-G\int dM/s\). A ring at polar angle \(\theta\) has mass \(\tfrac M2\sin\theta\,d\theta\) and distance \(s^2=r^2+R^2-2rR\cos\theta\), so \(s\,ds=rR\sin\theta\,d\theta\). Then

which is \(-GM/r\) for \(r>R\) and the constant \(-GM/R\) for \(r<R\). Differentiating gives the force.

Stable circular orbits

Proof. With angular momentum \(L\), the radial equation is \(m\ddot r=L^2/(mr^3)-k\,r^{-n}\). Circular orbits satisfy \(L^2/(mr_0^3)=k\,r_0^{-n}\). Linearise with \(r=r_0+\delta\): \(m\ddot\delta=\big[-3L^2/(mr_0^4)+nk\,r_0^{-n-1}\big]\delta=-(3-n)\frac{L^2}{mr_0^4}\,\delta\). Since \(\Omega^2=L^2/(m^2r_0^4)\) is the square of the orbital angular velocity, \(\ddot\delta=-(3-n)\Omega^2\delta\). For \(n=2\) the radial and orbital frequencies are equal, so the orbit closes after one turn: the ellipse of Kepler, with fixed apsides.

Flattened stars

In the Discours Maupertuis applied the theory of rotating fluid bodies to the heavens. A star spinning fast enough should be strongly flattened, and he suggested that some faint nebulous patches seen in telescopes might be such flattened stars seen edge-on, and that changes in the brightness of some stars might come from their rotation. He also discussed how a body passing close to Saturn might leave a ring. The details proved wrong, but the idea of explaining the shapes of celestial objects by gravity and rotation became a central theme of astronomy.

President in Berlin, 1740–1753

Frederick II became king of Prussia in 1740 and wanted a great academy. He invited Maupertuis, the most famous French scientist of the moment, to direct it. Maupertuis went to Berlin, followed the king on campaign in Silesia, and in April 1741 was captured by Austrian soldiers after the battle of Mollwitz. He was robbed, then treated with courtesy in Vienna and released, and was back in Paris by June. He was elected director of the Paris Academy for 1742 and was received into the Académie française on 27 June 1743.

He returned to Berlin in 1745 and on 25 August married Eleonore von Borck, from a Prussian noble family. In 1746 he became president of the reorganised Royal Prussian Academy of Sciences and Letters, with near-absolute authority over appointments, prizes and publication. Euler, already in Berlin, directed the mathematics class and was his strongest support. Maupertuis set prize questions, recruited foreign members, and used French as the Academy’s language. He continued to publish: the Essai de cosmologie (1750), Système de la nature (1751), and in 1752 a collected Œuvres with Lettres and the Lettre sur le progrès des sciences, a programme of research that proposed, among other things, experiments on hybridisation and a city where only Latin was spoken.

His administrative style was autocratic, and his health, troubled by lung disease, was declining. These two facts help explain how badly the next crisis went.

König, Akakia, death and legacy, 1751–1759

The König affair

In March 1751 Samuel König, a Swiss mathematician whom Maupertuis had brought into the Berlin Academy, published a paper criticising the principle of least action. He quoted a letter of Gottfried Wilhelm Leibniz to Jakob Hermann, apparently stating the principle decades earlier. König said the original had belonged to Samuel Henzi, a Swiss executed in Bern in 1749, and could not produce it. Maupertuis had the Academy investigate. In 1752, with Euler leading the case, the Academy declared the letter unauthentic, in effect accusing König of forgery. König resigned his membership. Most scholars now think the letter was probably a genuine copy, or at least not a forgery by König, and the Academy’s judgement has generally been seen as a misuse of authority.

Voltaire

Voltaire, then at Frederick’s court and now hostile to Maupertuis, took König’s side. In late 1752 he published the Diatribe du docteur Akakia, a satire mocking Maupertuis’s character and the more speculative proposals of the Lettres. Frederick had copies burned, and Voltaire left Prussia in 1753. The satire was widely read and fixed an absurd image of Maupertuis for generations. Voltaire had already ridiculed the Lapland voyage in Micromégas.

Last years

Ill and humiliated, Maupertuis left for France in 1753, returned to Berlin in 1754, and left for good in 1756. His membership of the Paris Academy, cancelled when he took office in Berlin, was renewed with a pension. In 1757 he set out for Italy but stayed seven months in Bordeaux, and in October 1758 he reached Basel, where he lived with the Bernoulli family. He died there on 27 July 1759, before his wife could arrive from Berlin, and was buried at Dornach, near Basel.

Legacy

• Geodesy: the Lapland measurement settled the oblate figure of the Earth and helped establish Newtonian physics in France.
• Mechanics: the principle of least action, made precise by Euler, Lagrange, Hamilton and Jacobi, is a cornerstone of classical and quantum physics.
• Biology: an early particulate theory of heredity, a family pedigree studied with probability, and speculation on the origin of varieties.
• Institutions: eight years at the head of the Berlin Academy in its most brilliant period.
• Honours: Fellow of the Royal Society (1728), member of the Académie française (1743), and member of most European academies of his time; a lunar crater bears his name.

Connections

People linked to Pierre-Louis Moreau de Maupertuis across this collection. A name without a link has no page here yet. See Pierre-Louis Moreau de Maupertuis in the network of all 337 lives →

Teachers and advisors

• Johann Bernoulli 1729-1730
• Nicolas Guisnée

Students

• Émilie du Châtelet 1733-1735

Influenced by

• Isaac Newton
• Pierre de Fermat

Influenced

• Carl Gustav Jacob Jacobi
• Joseph-Louis Lagrange
• William Rowan Hamilton

Colleagues and collaborators

• Alexis Clairaut 1736-1737
• Jean le Rond d'Alembert
• Leonhard Euler 1746-1756
• Anders Celsius 1736-1737

Rivals

• Giovanni Domenico Cassini
• Samuel König 1751-1752
• Voltaire

Friends

• Daniel Bernoulli
• Émilie du Châtelet

Also mentioned in this life

• Archimedes
• Louis de Broglie
• Léon Foucault
• Christiaan Huygens
• Johannes Kepler
• Adrien-Marie Legendre
• Gottfried Wilhelm Leibniz
• Colin Maclaurin

Fields

• geodesy 12 lives
• mechanics 27 lives
• natural philosophy 12 lives

On CoursesHub

• Classical Mechanics

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Pierre Louis Moreau de Maupertuis, MacTutor History of Mathematics.
2. B. Glass, Maupertuis, Pierre Louis Moreau de, Dictionary of Scientific Biography.
3. Pierre Louis Maupertuis, Wikipedia.
4. French Geodesic Mission to Lapland, Wikipedia.
5. M. Terrall, The Man Who Flattened the Earth: Maupertuis and the Sciences in the Enlightenment, University of Chicago Press, 2002.
6. P. Brunet, Maupertuis, 2 vols., Blanchard, Paris, 1929.

Science

1. P.-L. M. de Maupertuis, Discours sur les différentes figures des astres, Paris, 1732.
2. P.-L. M. de Maupertuis, La Figure de la Terre, déterminée par les observations de Messieurs de Maupertuis, Clairaut, Camus, Le Monnier et de M. l’Abbé Outhier, accompagnés de M. Celsius, Paris, 1738.
3. P.-L. M. de Maupertuis, Accord de différentes lois de la nature qui avaient jusqu’ici paru incompatibles, Mém. Acad. Roy. Sci. Paris (1744).
4. P.-L. M. de Maupertuis, Les lois du mouvement et du repos déduites d’un principe métaphysique, Hist. Acad. Roy. Sci. Berlin (1746).
5. P.-L. M. de Maupertuis, Vénus physique, 1745; Système de la nature, 1751; Œuvres, 4 vols., Lyon, 1756.
6. A. C. Clairaut, Théorie de la figure de la terre, tirée des principes de l’hydrostatique, Paris, 1743.
7. L. Euler, Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes, Lausanne and Geneva, 1744, Additamentum II.
8. B. Glass, Maupertuis, pioneer of genetics and evolution, in Forerunners of Darwin 1745–1859, Johns Hopkins, 1959.
9. Maupertuis provides evidence of hereditary particles, EBSCO Research Starters.
10. H. Goldstein, C. Poole and J. Safko, Classical Mechanics, 3rd ed., Addison-Wesley, 2002, §8.6 (the principle of least action).
11. C. Lanczos, The Variational Principles of Mechanics, 4th ed., Dover, 1970.