The beheading of the great French chemist Antoine-Laurent de Lavoisier was swift, but its effects were long lasting. Lavoisier’s head was not the only one to roll during the Reign of Terror of 1792-1794, led by Maximilien de Robespierre and his fellow Jacobins. The city of Lyon was particularly contentious, as it was a stronghold of the competing Girondists who controlled the city even after Robespierre had secured power in Paris and executed the Parisian Girondists en masse. Caught in the middle of this power struggle was a well-meaning local magistrate named Jean-Jacques Ampère.
Ten years earlier, Ampère moved his family to his country estate so he could concentrate on homeschooling his children. He filled his house with books and encouraged his children to grow up with a love of ideas. As an example of his great teaching, his son, André-Marie, wrote a treatise of mathematical proofs and submitted them to the local academy at age 13. While this work was not new to mathematics, it did teach André-Marie how to produce original ideas and articulate them using mathematics and the written word. In this fashion, André-Marie developed a keen mind and strong academic skills.
The Reign of Terror brought this to a complete halt, when a kangaroo court of Jacobin nationalists sentenced and executed his father. Thus, at the age of 17, André-Marie Ampère suffered his first bout of depression, which would punctuate the rest of his mostly unhappy personal life.
Ampère started teaching high school mathematics in 1795, fell in love in 1796, and fathered a son in 1800. In 1802 he got a college teaching job in another town, and spent time there alone concentrating on his research. His hard work paid off, as he secured a college teaching position in Lyon in 1803, where he could be with his family and close friends. Unfortunately, his wife had been ill for some time, and died. Ampère was soon offered a teaching position at the prestigious École Polytechnique in Paris.
Meanwhile, during the Reign of Terror, the Royal Academy of Sciences was abolished, and most of its members quietly stayed out of the spotlight doing their work. After Robespierre went to the guillotine himself, things greatly improved. The new leader of the Academy, Lazare Carnot , was an engineer and great military tactician, who had studied at Coulomb’s alma mater and published a paper on kinetic energy. In 1795 the new National Institute of Sciences and Arts was formed, and divided into three classes for the following subjects: (1) the natural sciences, (2) the social sciences and (3) the arts. Thus, the First Class of the Institute succeeded the old Royal Academy retaining the same members, except for Lavoisier of course. As the change was largely in name, we will refer to it simply as the Academy.
Pierre-Simon Laplace took the primary leadership role in the mathematical sciences of the First Class, where he dictated the research agenda over the next two decades. Laplace’s clear and forceful leadership, with an emphasis on careful experimental techniques, led to great advances in those areas of physics where Laplace’s ideas were correct. Thus, Laplace ran the first truly modern research laboratory with exacting standards for mathematical rigor, experimental precision, and the empirical testing of theory.
Laplace also promoted the astronomical view of nature, which supposed that molecules act on each other from a distance, in much the same way that gravity acts from a distance between astronomical objects. The Laplacians made a distinction between ordinary ponderable matter, and the imponderable fluids of heat, light, electricity and magnetism. These fluids were thought to comprise of mutually repulsive particles, so they would spread out, and then interact with ordinary ponderable matter. The primary goal of early nineteenth century physics was to carefully characterize these forces and fluids experimentally, or as Laplace put it:
All terrestrial phenomena depend on forces of this kind, just as celestial phenomena depend on universal gravitation. It seems to me that the study of these forces should now be the chief goal of mathematical philosophy. I even believe that it would be useful to introduce such a study in proofs in mechanics, laying aside abstract considerations of flexible or inflexible lines without mass and of perfectly hard bodies. A number of trials have shown me that by coming closer to nature in this way one could make these proofs no less simple and far more lucid than by the methods used hitherto.
One problem with Laplacian physics was that its followers did not allow for dissent. So long as a young scientist agreed with Laplace in general, his work was considered worthy and he was given more opportunities to advance his career. As it turns out, the Laplacian paradigm was correct for electricity, but incorrect for heat, light, and magnetism.
In 1799 Napoleon Bonaparte took power in a coup d’état, and soon appointed Laplace as his minister of the interior. While Laplace was forceful and decisive for a scientist, he was not for a Napoleonic executive—so Bonaparte promoted him. As a member of the senate, Laplace received a handsome salary, a great deal of influence over scientific policy, and time for research. It was the perfect position for an ambitious mid-career scientist such as Laplace. The chemist Claude Louis Berthollet and the elderly Joseph-Louis Lagrange also received senatorial posts.
In 1806, Laplace bought an estate next door to Berthollet. Each summer, Laplace, Berthollet, and their German friend Alexander von Humboldt, would invite a small number of the most promising young scientists to spend their summers in a idyllic environment of intellectual interchange. The group called themselves the Society of Arcueil, as they were located in the small town of Arcueil about five kilometers south of central Paris. Society of Arcueil participants who contributed to electrodynamics included: Dominique François Jean Arago, Jean-Baptiste Biot, Joseph Louis Gay-Lussac, Étienne-Louis Malus, and Siméon Denis Poisson.
On July 21 of 1820, Hans Christian Ørsted published a short Latin paper summarizing his discovery that a current carrying wire deflects a compass needle in a circular pattern around the wire. But it was not until late summer that, while visiting Geneva, Arago learned of the discovery. As the news was received with disbelief when Arago reported it on the first Monday in September, he experimentally demonstrated it the following Monday. This sparked a race for an explanation, primarily between Biot and Ampère.
Ampère got off to a running start with a demonstration that a compass points in the tangential direction surrounding the wire. And at the last Monday in September, Ampère demonstrated that coils of current carrying wires behave like permanent magnets. Following on the heels of Ampère’s success, Arago wrapped an iron needle with wire, and discovered that the poles of this new electromagnet reverse when the current reverses.
By the meeting of October 2, Ampère presented a draft paper describing the electrodynamic forces between current-carrying wires, and in the next meeting he demonstrated these forces using an apparatus. By the very next meeting, he had demonstrated the force on a current-carrying wire due to the Earth’s magnetic field.
After all of these demonstrations, and discussions, Biot presented his analysis of the same problem on the last Monday in October. Biot, and his protégé Félix Savart, had made very careful measurements of the torque on a magnetized needle as a function of the distance from the wire, which they reported as:
Biot and Savart were led to the following result which rigorously expresses the action experienced by a molecule of astral or boreal magnetism located at an arbitrary distance from an extended, very fine, cylindrical wire magnetized by a voltaic current. Draw a perpendicular to the axis of the wire from the point where the is molecule resides: the force which influences the molecule is perpendicular to the line and to the axis of the wire. Its intensity is inversely proportional to the simple distance.
In November, Ampère demonstrated that the magnetic field adds with the number of current carrying wires in the same direction, and had outlined the main points of his eventual force law.
On the last Monday before Christmas, Biot and Savart presented measurements with bent wires, with the goal of finding the force between infinitesimal magnetic fluid elements. Now they found that the force is proportional to the distance squared between their supposed poles of magnetic fluid, just like Coulomb’s law for electricity.
Thus, by the end of the calendar year of 1820, Biot and Savart had characterized empirically the torque on a magnetic needle, but interpreted their result using the existing Laplacian paradigm which supposed that magnetic forces were caused by forces between the southern (astral) or northern (boreal) molecules which make up the magnetic fluid. Despite a revolutionary experiment that completely contradicted the Laplacian paradigm, Biot over-interpreted the data to conform with Laplacian two-fluid theory of magnetism.
It was the provincial André-Marie Ampère, a person with little formal education, who first correctly explained the magnetic forces between current-carrying wires. While Ampère’s explanation was clearly correct, the tenacious Biot would not concede defeat. So, Ampère and Biot published rival papers, and were given joint credit for the discovery of the force law between wires.
In general, the Society of Arcueil members supported Biot, with the notable exception of François Arago. The careers of Arago and Biot were launched together by Laplace, who gave them the task of measuring the meridian arc through Paris, as the meter was defined as one ten millionth (10-7) of the distance from the north pole to the equator through Paris. As it turns out, after taking some measurements near the Spanish border, Biot returned to Paris. Arago stayed to finish the work, was captured, and became a prisoner of war. Eventually he escaped and through a series of great adventures returned to France, while somehow keeping all of the data safe and intact.
When the heroic Arago sided with the awkward Ampère over the popular Biot, it finally made other members of the Academy more comfortable voicing their true thoughts on scientific ideas. Laplacian hegemony was beginning to end, as now the two interpretations of magnetism could be freely debated. This opened the door for Fourier and Augustin-Jean Fresnel, whose ideas regarding heat and light were now taken seriously, but this new openness also meant that the rivalry between Biot and Ampère grew more intense over the next few years.
Poisson developed a theory of magnetism in the Laplacian mold, which explained the new electromagnetic phenomena in terms of the imponderable (i.e. massless) fluid of magnetic poles. This theory was consistent with the measurements of both Biot and Ampère, and at the same time preserved Laplacian magnetism. While Poisson’s theory was mathematically elegant, it was physically wrong.
Ampère, for his part, obtained the services of the brilliant young scientist Félix Savary (not to be confused with Biot’s young collaborator Félix Savart) whose work propelled the Ampèrian theory of magnetic forces further, until Ampère eventually wrote his complete treatise fully documenting the forces between wires. Ampère and his collaborators also promoted the idea that permanent magnets contain small circular currents within molecules.
In the hands of James Clerk Maxwell, Ampère’s force law became the new basis for defining the magnetic field. In this chapter we will develop Ampère’s idea of equilibrium forces between wires, but do so within the context of Maxwell’s field theory and the mathematics of vector calculus.
We begin with a presentation of a fundamental law that is mistakenly attributed to Ampère: the circulation of a magnetic field around a closed loop is directly proportional to the total current that flows through any surface bounded by that loop. Maxwell first stated this theorem in a letter to William Thomson in 1854. The local, or differential, version of this law is that the curl of a magnetic field is proportional to the current density. Maxwell presented this local version in differential form in his 1855 paper, “On Faraday’s lines of force.”
 James R. Hofmann, André-Marie Ampère, (Oxford: Blackwell, 1995), which is one of the Blackwell science biographies.
 P.S. Laplace, Traité de mécanique céleste, 5 vols, (Paris 1799-1825), 5, 99. Translation and discussion of Laplacian physics from the book chapter by Robert Fox, “The Rise and Fall of Laplacian Physics,” in Historical Studies in the Physical Sciences 4, edited by R. McCormmach, (Princeton: Princeton University Press, 1974).
 James R. Hofmann, André-Marie Ampère, Blackwell science biographies, (Oxford: Blackwell, 1995), 232.
 See the book chapter by Robert Fox, “The Rise and Fall of Laplacian Physics,” in Historical Studies in the Physical Sciences 4, edited by R. McCormmach, (Princeton: Princeton University Press, 1974), 89-136.
 The Scientific Letters and Papers of James Clerk Maxwell, ed. P. M. Harman. 2 vols. (Cambridge: Cambridge University Press, 1990-95), 1: 256-7. This is discussed in some detail in O. Darrigol, Electrodynamics from Ampère to Einstein (Oxford: Oxford University Press, 2000), 139-42. Maxwell’s letter of November 13, 1854 is also reprinted in full in “The origins of Clerk Maxwell’s Electric ideas, as described in familiar letters to William Thomson,” 32 Proc. Cambridge Philos. (1936), 701- 705 (see especially p. 702). Maxwell states his mathematical result in words rather than symbols, and he uses terms we would consider arcane. For a clear explanation of the evolution of Maxwell’s ideas, beginning with his letter to Thomson, see M. Norton-Wise, “The Mutual Embrace of Electricity and Magnetism,” Science 203 (30 March 1979), 1310-18. Norton-Wise does a good job of translating Maxwell’s original terminology into modern language.
 J.C. Maxwell, “On Faraday’s Lines of Force,” Transactions of the Cambridge Philosophical Society, 10, no. 1, (1856), 27-83. See the equations on p. 56.