François Arago

How to use this course

François Arago is less known today than the people whose work he made possible. He saw the bright spot that won the wave theory of light its decisive victory. He reported Ørsted’s discovery to Paris and set Ampère to work. He found the magnetic drag of a spinning copper disc that Faraday later explained by induction. He designed the experiment that decided whether light is slower in water. He also founded the Comptes rendus, launched photography as a public invention and, as a minister in 1848, ended slavery in the French colonies. This course follows the physics in detail and places it in one of the most eventful lives in the history of science.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, derivations and exercises with folded solutions. Arago was above all an experimenter; the lessons give the theory that explains his observations, mostly in the form it took later, and say where his own interpretation differed.

Timeline

Estagel, Perpignan, Polytechnique, 1786–1805

Dominique François Jean Arago was born on 26 February 1786 at Estagel, a Catalan-speaking village in Roussillon, near the Spanish border. His father, François Bonaventure Arago, was mayor of the village and later treasurer of the Mint at Perpignan. François was the eldest of several brothers, most of whom led adventurous lives: Jacques sailed around the world with Louis de Freycinet and became a writer, Étienne a playwright, and Jean a general in the Mexican army.

He was schooled at the municipal college of Perpignan, where he discovered a love of mathematics, and prepared largely on his own for the entrance examination of the École polytechnique. Examined at Toulouse in 1803, he was placed among the top entrants and entered the school late that year.

At Polytechnique he became friends with Siméon Denis Poisson, a young teacher only five years older. Poisson recognized his talent. When Laplace asked for someone to finish the measurement of the Paris meridian, left incomplete by Pierre Méchain’s death in Spain in 1804, Poisson proposed Arago. In 1805 he was given a post as secretary at the Paris Observatory and nominated for the survey. He was nineteen.

The meridian and captivity, 1806–1809

The metre had been defined in 1791 as one ten-millionth of the distance from the North Pole to the equator along the meridian of Paris. Jean-Baptiste Delambre and Pierre Méchain had measured the arc from Dunkirk to Barcelona between 1792 and 1798. The plan now was to carry the triangulation south across the sea to the Balearic Islands, so that the arc would straddle the 45th parallel more symmetrically. Arago and Jean-Baptiste Biot left Paris on 3 September 1806. They spent months on mountaintops in Valencia and the islands, signalling at night with fires and lamps across long stretches of sea. Biot returned to Paris after they had fixed the latitude of Formentera; Arago stayed to complete the measurements.

Three captures

In May 1808 Napoleon placed his brother Joseph on the Spanish throne, and Spain rose against the French. On Mallorca, Arago’s fires on the summit of Mount Galatzó made the islanders suspect a French spy. He surrendered and was imprisoned in the castle of Bellver in June 1808. At the end of July he escaped in a fishing boat to Algiers, which he reached on 3 August. Sailing for Marseille on 16 August, his ship was captured by a Spanish corsair within sight of the French coast, and he was held at Roses and Palamós until late November. Released, he sailed again, but a storm drove the ship to Bougie on the North African coast. He walked back to Algiers, arriving on 25 December, and was again held captive there until the French consul obtained his release.

He sailed at last on 21 June 1809 and landed at Marseille on 2 July, carrying his survey notebooks. In quarantine he received a letter from Alexander von Humboldt, the start of a friendship that lasted more than forty years. His adventures made him famous in Paris before he had published a scientific paper of note.

Academician at twenty-three, 1809–1820

On 18 September 1809, at twenty-three, Arago was elected to the Académie des sciences. In the same year Gaspard Monge chose him to teach analytic geometry at the École polytechnique; he taught there until 1830. He was named an astronomer at the Observatory, where he lived for the rest of his life. In 1811 he married Lucie Carrier-Bescombes; they had sons. From the early 1810s until 1845 he gave public lectures on astronomy at the Observatory, famous for their clarity, which became his posthumous Astronomie populaire.

His research of the decade was in optics. He belonged to the circle of young scientists around Laplace and Claude-Louis Berthollet at Arcueil, where light was understood as a stream of particles. In 1810 he tested that theory with starlight (Lesson 05). In 1811 he discovered the colours of polarized light passing through thin crystal plates and the rotation of the plane of polarization in quartz (Lesson 06). Biot took up both subjects with energy and published extensive particle-based theories of them, and the two former companions became lasting rivals.

In 1815 Arago met Augustin Fresnel, a young engineer with a theory of diffraction based on waves. Arago became his advocate, his collaborator and his link to the Académie. From 1816 he and Joseph Louis Gay-Lussac edited the Annales de chimie et de physique, one of the leading physics journals of the century. The Royal Society of London elected him a fellow in 1818.

Measuring the Earth

For the general reader. To define the metre as a fraction of the Earth, surveyors had to measure a long arc of a meridian and the latitudes at its ends. Because the Earth is flattened at the poles, a degree of latitude is slightly longer in the north than in the south. Arago’s extension of the arc to the Balearics, and his later pendulum measurements of gravity with Biot, fed into the shape of the Earth and the length of the metre.

Meridian arcs

Model the Earth as an ellipsoid of revolution with equatorial radius \(a\) and eccentricity \(e\), where \(e^2=2f-f^2\) and \(f=(a-b)/a\) is the flattening. Latitude \(\varphi\) is the angle between the local vertical (the plumb line) and the equatorial plane; it is what astronomers measure from star altitudes.

Derivation. Parametrize the meridian ellipse \(x^2/a^2+z^2/b^2=1\) by the normal angle \(\varphi\). The normal is parallel to \((x/a^2,z/b^2)\), so \(\tan\varphi=\frac{a^2z}{b^2x}\). Solving gives \(x=\frac{a\cos\varphi}{W}\) and \(z=\frac{a(1-e^2)\sin\varphi}{W}\) with \(W=\sqrt{1-e^2\sin^2\varphi}\). The arc element satisfies \(ds=M\,d\varphi\), and differentiating, \(\big|\frac{d}{d\varphi}(x,z)\big|=\frac{a(1-e^2)}{W^3}\). Expanding \((1-e^2)\approx1-2f\) and \(W^{-3}\approx1+3f\sin^2\varphi\) gives the first-order form.

Gravity and the pendulum

On a rotating, flattened Earth, gravity increases from equator to pole. Alexis Clairaut’s theorem of 1743 links the increase to the flattening:

A pendulum of length \(L\) beating seconds (half-period one second) satisfies \(\pi\sqrt{L/g}=1\ \mathrm s\), so \(L=g/\pi^2\). Measuring the seconds-pendulum length at many latitudes, as Biot and Arago did along the meridian and as far north as Leith and the Shetland Islands, gives \(g(\varphi)\), and through Clairaut’s theorem an independent estimate of \(f\).

Starlight through a prism, 1810

For the general reader. If light is a hail of particles, starlight should arrive at slightly different speeds depending on whether the Earth is moving toward the star or away from it, and a prism should bend the two kinds of light differently. Arago looked for the effect in 1810 and found nothing. The null result was the first of a line of experiments on light and motion that ended with Einstein’s relativity.

Aberration

The Earth’s orbital speed is \(v\approx30\ \mathrm{km\,s^{-1}}\), so \(v/c\approx10^{-4}\). James Bradley’s aberration of 1729 shifts the apparent direction of a star by up to \(v/c\) radians, about 20.5 arcseconds. In the particle theory, refraction occurs because a medium attracts the particles and speeds them up, and the deviation by a prism depends on the speed of the incident particles. Arago placed a prism in front of his telescope and compared the deviation of starlight at times when the Earth moved toward the star and away from it. The particle theory predicted a change of relative order \(v/c\) in the deviation, which he expected to detect. He saw no difference, and proposed that the eye might respond only to particles of a single speed.

Fresnel wrote his drag hypothesis in a letter to Arago precisely to explain the 1810 result. Hippolyte Fizeau confirmed the coefficient in moving water in 1851. Einstein’s theory gives it without any aether: the speed of light in the medium’s rest frame is \(c/n\), and relativistic velocity addition gives

Colours of polarized light and optical rotation

For the general reader. In 1808 Étienne-Louis Malus found that reflected light is polarized. Three years later Arago sent polarized light through a thin sheet of mica and looked at it through a crystal of Iceland spar. Brilliant colours appeared, changing as he turned the crystal. He also found that quartz cut across its axis turns the direction of polarization, by an amount that depends on colour. These two discoveries became tools for mineralogists, chemists and engineers.

Chromatic polarization

A thin plate of a birefringent crystal such as mica has two perpendicular axes with indices \(n_1,n_2\). Let polarized light make angle \(\psi\) with axis 1, and let the plate have thickness \(d\). The two components acquire the phase difference \(\delta=2\pi(n_1-n_2)d/\lambda\).

Derivation. Take the incident field \(E_0(\cos\psi,\sin\psi)\) in the plate axes. After the plate, up to a common phase, it is \(E_0(\cos\psi,\ \sin\psi\,e^{i\delta})\). The crossed analyser projects on \((-\sin\psi,\cos\psi)\): amplitude \(E_0\sin\psi\cos\psi\,(e^{i\delta}-1)\), whose square modulus is \(E_0^2\tfrac14\sin^22\psi\cdot4\sin^2(\delta/2)\). The parallel case follows from energy conservation in a lossless plate. Arago called the effect depolarization; the explanation by interference is due to Fresnel and Thomas Young.

Optical rotation

A quartz plate cut perpendicular to its optic axis turns the plane of polarization by an angle proportional to its thickness. Biot found that the rotation varies roughly as \(1/\lambda^2\), and that some liquids such as turpentine and sugar solutions do the same. Fresnel explained the effect in 1822 by resolving linear polarization into left and right circular components that travel at slightly different speeds:

The Fresnel–Arago laws and the bright spot

For the general reader. Between 1816 and 1819 Arago and Fresnel showed that light polarized in perpendicular planes never interferes. This was the key evidence that light vibrates across its direction of travel, as waves on a rope do, and not along it like sound. In 1819 Arago tested a strange prediction of Fresnel’s theory, that the shadow of a small disc has a bright centre, and found the spot. The two results ended the dominance of the particle theory in France.

If the vibration is a transverse vector, laws 1 and 2 are immediate: \(|\mathbf E_1+\mathbf E_2|^2=|\mathbf E_1|^2+|\mathbf E_2|^2+2\,\mathrm{Re}(\mathbf E_1\cdot\mathbf E_2^*)\), and the cross term vanishes when the vectors are perpendicular. Laws 3 and 4 depend on coherence: the two components of unpolarized light have phases that vary randomly relative to each other, so the cross term averages to zero, while components split from one polarized beam keep a fixed relative phase.

The bright spot

Poisson, a judge of the 1818–19 prize on diffraction, deduced from Fresnel’s memoir that the centre of the shadow of a circular disc should be as bright as if the disc were absent. He regarded this as a refutation. Arago, chairing the commission, made the experiment with a disc about 2 mm across and saw the spot. He noted later that Joseph-Nicolas Delisle and Giacomo Maraldi had seen it in the early eighteenth century.

Sketch with Fresnel zones. Seen from an axial point \(P\) at distance \(z\), divide the unobstructed wavefront outside the disc into annular zones whose distances from \(P\) increase by \(\lambda/2\). Successive zones contribute amplitudes \(A_1,A_2,\dots\) of alternating sign whose magnitudes decrease slowly. The sum \(A_1-A_2+A_3-\cdots\) is close to \(A_1/2\), half the first unobstructed zone. For a free wave the same argument starting from the centre gives \(A_1^{\text{free}}/2\). When the disc is small, the first zone outside it differs little from the first free zone, so the intensities agree. For larger discs the spot persists, growing narrower and fainter only slowly as the obliquity of the zones increases.

The polarized sky, comets and the Sun

For the general reader. Arago was the first to use polarization as a tool of astronomy. He noticed that the blue sky is partly polarized, most strongly about ninety degrees from the Sun, and that there are special points where the polarization vanishes. He found polarization in the light of a comet’s tail, showing it shone partly by reflected sunlight. He argued from the absence of polarization in light from the edge of the Sun that its visible surface is gaseous.

Single scattering

A small particle, much smaller than the wavelength, scatters light by oscillating as an induced dipole. A dipole radiates no field along its own axis. Unpolarized sunlight has two independent components perpendicular to its direction; let the scattering angle be \(\Theta\) between the incident and scattered directions. The component perpendicular to the scattering plane radiates fully, the component in the plane radiates with a factor \(\cos^2\Theta\).

Single scattering predicts zero polarization only at \(\Theta=0\) and \(\Theta=180^\circ\), toward the Sun and directly away from it. Observed skies show neutral points displaced from these directions. The Arago point lies about 20° above the antisolar point; Jacques Babinet and David Brewster later found neutral points above and below the Sun. They arise because light scattered more than once adds a component polarized at right angles to the single-scattering pattern.

Comets and the Sun

In 1819 Arago examined the tail of the great comet of that year with his polariscope and found it polarized, so part of its light is reflected sunlight. He also observed that light from the edge of the Sun’s disc is not polarized. Light emitted at a grazing angle from an incandescent solid or liquid surface is polarized, while light from an incandescent gas is not. He concluded that the luminous surface of the Sun is gaseous, a conclusion confirmed in substance by later solar physics.

The electromagnet and magnetism of rotation

For the general reader. In September 1820 Arago brought news of Ørsted’s experiment to Paris. Within days he showed that a current-carrying wire picks up iron filings like a magnet, and with Ampère he magnetized steel needles by placing them inside a coil, the first magnetization of iron by an electric current. In 1824 he found that a spinning copper disc drags a magnet round with it, although copper is not magnetic. Faraday solved this puzzle in 1831: the moving copper carries induced currents, the eddy currents now used in brakes and induction motors.

Magnetizing with a coil

Arago reported in late September 1820 that a copper wire carrying current attracts iron filings, which cling around it and drop off when the current stops. Ampère suggested winding the wire into a helix; inside, a steel needle was permanently magnetized. The field inside a long coil of \(n\) turns per metre is \(\mu_0nI\) (see the Ampère course). Soft iron placed in the coil becomes magnetized, and its magnetization adds a field much larger than that of the current alone; William Sturgeon’s horseshoe electromagnet of 1825 exploited this.

Magnetism of rotation

On 22 November 1824 Arago reported that a magnetic needle swinging above a copper plate comes to rest far sooner than one swinging in the open. Inside a thick copper ring a needle made about a quarter as many swings, before its amplitude fell to a fixed value, as it did inside a wooden ring. In 1825 he showed that a copper disc spun beneath a suspended needle drags the needle round. Charles Babbage and John Herschel reversed the experiment, spinning a magnet beneath a pivoted disc, and found that radial slits cut in the disc weaken the effect sharply.

Derivation. In the frame of the sheet, its charges feel the field \(\mathbf v\times\mathbf B\), of size \(vB\), along the sheet perpendicular to \(\mathbf v\). Ohm’s law gives the current density \(J=\sigma vB\) in the field region. The force density \(\mathbf J\times\mathbf B\) has size \(\sigma vB^2\), directed against \(\mathbf v\) (Lenz’s law). Multiply by the volume \(tA\). Charges piling up at the edges of the field region create an electric field opposing \(\mathbf v\times\mathbf B\), so the true force is smaller by a geometric factor of order one. Radial slits block the return paths, so charge accumulates and the currents, and the force, drop: the effect Babbage and Herschel measured.

The speed of light in water

For the general reader. The particle theory says light speeds up when it enters water; the wave theory says it slows down. In 1838 Arago designed an experiment to decide between them, using a mirror spinning so fast that it turns measurably while light travels a few metres. His eyesight failed before he could do it. In 1850 Léon Foucault, and separately Hippolyte Fizeau with Louis Breguet, carried it out: light is slower in water, as waves require.

Two theories of refraction

In the particle theory, a surface force accelerates light particles perpendicular to the interface, so the tangential velocity is conserved: \(u_1\sin\theta_1=u_2\sin\theta_2\). Snell’s law \(\sin\theta_1=n\sin\theta_2\) then requires \(u_2=nu_1\): faster in the denser medium. In the wave theory, Huygens’s construction with equal phases along the interface gives \(\sin\theta_1/u_1=\sin\theta_2/u_2\), so \(u_2=u_1/n\): slower.

Derivation. The round trip takes \(\Delta t=2D/u\), during which the mirror turns by \(\omega\Delta t\). Rotating a mirror by \(\beta\) rotates a reflected beam by \(2\beta\). Arago’s plan sends two beams from the same turning mirror, one through air and one through a tube of water, and compares their displacements side by side. The method needs no measurement of \(\omega\) to decide the question, only which image is displaced more.

Secretary, deputy, patron, 1830–1847

In 1830 Arago’s career changed direction. On 7 June the Académie elected him perpetual secretary for the mathematical sciences, succeeding Joseph Fourier, and he gave up his teaching at Polytechnique. After the July Revolution he entered politics as a republican, sitting from 1831 as deputy for the Pyrénées-Orientales. His wife died in 1829, leaving him with their sons; Emmanuel became a lawyer and politician, and Alfred a painter and inspector of fine arts.

As secretary he made the Académie public. He opened its sessions to journalists, and in 1835 he launched the weekly Comptes rendus, which printed short notes within days and changed the pace of scientific publication. His historical éloges of deceased members, among them Fresnel, Ampère, Fourier, James Watt and Thomas Young, remain readable biographies. At the Observatory he took charge of observations in 1834 and became director after Alexis Bouvard’s death in 1843.

He used his influence for science and inventors. In January 1839 he announced Louis Daguerre’s photographic process to the Académie. He persuaded the Chamber to buy the rights and grant Daguerre a pension, so that the process could be given to the world, and on 19 August 1839 he described it in detail at a joint session of the Academies of Sciences and Fine Arts. He supported railways and the electric telegraph and secured public money for the works of Laplace and Fermat. He encouraged the young astronomer Urbain Le Verrier to study the irregularities of Uranus. When Johann Galle found Neptune on 23 September 1846 close to Le Verrier’s predicted position, Arago defended Le Verrier’s priority fiercely against claims for John Couch Adams, and proposed naming the planet after him.

1848, last years, legacy

When the July Monarchy fell in February 1848, Arago joined the Provisional Government on 24 February as minister of the Navy and the Colonies, and from April also of War. At the Navy he abolished flogging and improved sailors’ rations. He appointed the abolitionist Victor Schœlcher as under-secretary for the colonies, and the government’s decree of 27 April 1848 abolished slavery in all French colonies. On 10 May the new National Assembly elected him to the five-member Executive Commission, which he chaired. It resigned on 24 June during the insurrection of the June Days.

His sight was failing, and he never performed the rotating-mirror experiment himself. After Louis-Napoléon’s coup of December 1851, Arago refused the oath of allegiance required of public servants. The government chose to exempt him and kept him at the Observatory. He attended the Académie for the last time in August 1853, visited his family in Roussillon, and died in Paris on 2 October 1853, aged 67. He was buried in Père Lachaise. His Œuvres complètes, edited by Jean-Augustin Barral, appeared in seventeen volumes from 1854 to 1862.

Legacy

• Optics: Arago discovered chromatic polarization, optical rotation in quartz and sky polarization, made the polariscope an instrument of astronomy, and supplied the experimental support that established Fresnel’s wave theory.
• Electromagnetism: he brought Ørsted’s discovery to Paris, with Ampère first magnetized iron and steel with electric currents, and found the eddy-current effects that led Faraday to induction.
• Institutions: the Comptes rendus, public sessions of the Académie and state support for invention, including photography.
• Honours: Copley Medal (1825), Rumford Medal (1850), the Prussian Pour le Mérite (1842). The Académie des sciences created the Arago Medal in 1893. His name is on the Eiffel Tower, and bronze medallions bearing his name, Jan Dibbets’s Hommage à Arago, trace the Paris meridian through the city.

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, François Arago, MacTutor History of Mathematics.
2. François Arago, Wikipedia.
3. D. W. Strickland, François Arago (February 26, 1786 – October 2, 1853), MacTutor.
4. F. Arago, Histoire de ma jeunesse, in Œuvres complètes, vol. 1, 1854 (his own account of the meridian expedition).
5. J. Lequeux, François Arago, un savant généreux, EDP Sciences / Observatoire de Paris, 2008; English edition Springer, 2016.
6. M. Daumas, Arago, la jeunesse de la science, Gallimard, 1943.

Physics

1. F. Arago and A. Fresnel, Mémoire sur l’action que les rayons de lumière polarisés exercent les uns sur les autres, Annales de chimie et de physique 10 (1819).
2. F. Arago, Expériences relatives à l’aimantation du fer et de l’acier par l’action du courant voltaïque, Annales de chimie et de physique 15 (1820).
3. Arago spot and Arago’s rotations, Wikipedia.
4. W. Whewell, History of the Inductive Sciences, 3rd ed., 1858, vol. 2, on the polarization of light.
5. K. Alder, The Measure of All Things: The Seven-Year Odyssey That Transformed the World, Free Press, 2002.
6. J. D. Jackson, Classical Electrodynamics, 3rd ed., Wiley, 1999 (eddy currents, scattering).
7. M. Born and E. Wolf, Principles of Optics, Cambridge, 7th ed., 1999 (polarization, Fresnel zones).