Translation of S. D. Poisson's Premiere Memoire sur la Distribution de l'Electricite a la surface des Corps conducteurs; (1812)

One of my pet projects over the last couple years has been translating the two parts of Simeon-Denis Poisson's Mémoire Sur la Distribution de l'Électricité à la surface des Corps conducteurs; (read 1812) into English. I didn't dedicate all that much time to it, the full memoires together are only 200 pages or so, so it shouldn't have taken so long, but in any case, 2021 has been a good year of translations for me, so I thought, I might as well put an hour every day into the translation. And now the first memoir is complete! 

Translator's Preface

The story of this work begins, for our purposes, with Charles Augustin Coulomb (1736-1806), the military engineer-turned-natural philosopher, who, beginning in 1785, published a series of seven memoirs on the topics of electricity and magnetism. Coulomb used his recently developed torsion balance to make some of the earliest quantitative measurements of the electrostatic force. He performed these measurements using a fine silver wire, hung fixed at one end to a dial with an angular scale, and at the other end to an arm having at one side a small conductive sphere made of pith (possibly pith-of-elder) and gilded with gold, and at the other a small counterweight. By introducing a second pith sphere, and recording the angular deflection required to bring the sphere on the silver wire back into its initial position, Coulomb could measure the electric force with unprecedented accuracy. 

Today, some doubts exist about the quality of Coulomb's data, one hypothesis being that the numbers he recorded were not the values he measured, but were "idealized" to meet his expectations of the electric force. In fact, only three data points were given (albeit averaged data points from many measurements) in the first memoir, to provide proof that the electric force, like gravity, followed an inverse-squared law. Regardless, Coulomb's work became a fixture of electric science, and his study of the electrical repulsion of spheres lent itself directly to the techniques recently developed for celestial mechanics. 

In fact there was a mathematical physicist working in the same period, the mathematician Pierre-Simon de Laplace (1749-1827), a member of the Academie des Sciences de Paris, who contributed extensively to the application of analysis to celestial mechanics and, by extension, to the burgeoning science of electricity. Laplace was present for the founding of the now-famous Ecole Polytechnique of Paris, which would play a dominant role in physics and mathematics throughout the 19th century. Laplace himself published what may be his greatest work, Mechanique celeste, in four volumes between 1799 and 1805. This work, a massive study in analytical mechanics and something of a follow-up to Newton's Principia, would directly inspire Poisson in his memoirs on electricity in the years that followed. 

In fact, it was in 1801, shortly after the publication of the first two volumes of Mecanique Celeste, that Jean-Baptist Biot published Sur un probleme de physique, relatif a l'electricite (1801). I've translated this early paper, which you can read more about here. In that work, Biot takes the two-fluid theory of electricity as a working hypothesis, and proceeds to derive the forces (assuming an inverse-squared law) present in an electrified revolved ellipsoid at equilibrium. While this paper isn't particularly interesting mathematically (almost every step cites an associated step in a section of Mecanique celeste), it remains perhaps the earliest application of analysis to the study of electricity. 

As discussed in R. W. Home's Poisson's memoirs on electricity: academic politics and a new style physics, BJHS, 1983, vol 16, Biot's paper would serve as the inspiration for Poisson's memoirs, specifically to obtain for Poisson a membership with the First Class of the Institut de France (formerly, and again later, called the Academy of Sciences of France). 

For a more extensive review of how Poisson came to hold this important position, see Home's paper. For now, it will suffice to briefly summarize the important points of that paper.

Simeon Denis Poisson (1781-1840) was a French mathematician, or rather mathematical physicist, whose work covers such diverse physical topics as sound, heat, electricity, and elasticity. Poisson was highly influential in the applications of analysis to problems in physics, a mode of thinking arguably pioneered by the French beginning in the late 18th century. Poisson attended the Ecole polytechnique, having done exceptionally well in the entrance exam, and soon impressed his professors with his mathematical abilities. At the age of 19 he presented a memoir to the First Class of the Institut de France, attracting the attention and support of Laplace himself. 

To increase his chances of obtaining membership with the First Class, Poisson directed his studies towards those that would win him regard and nomination to particular seats. After several unsuccessful attempts, with many members pulling their weight around to win Poisson a seat, the sudden illness and death of Etienne Malus (1775-1811) provided just the opportunity needed. A competition was announced by the Academy in 1812, to "determine by calculation [...] the manner in which electricity is distributed at the surface of electric bodies". It is believed that the competition was set specifically with Poisson in mind, and that he had already completed much of the analysis required. His major competitor being a qualitative work in Latin by Francois Joseph Gardini, but this work was far from comparable to Poisson's. 

Poisson's memoirs would go on to influence further works, including those by George Green (1793-1841), which is how I myself learned of them. 

In translating this work, I've made a point to follow the punctuation and word order rather more closely than is typical of translation, but I felt it served well to "let Poisson talk for himself". In some places, I have chosen to keep words literal, though they could be better translated with a little freedom of interpretation. An example is les differences partielles, which I have translated literally, as "partial differences", although today we would call these mathematical devices "partial derivatives." 

Along side some of the introduced vocabulary I have included footnotes with the original French word, but I believe the choices in translation will be sufficiently exact for most. I will continue to revise the first several pages, translated before I had much of a "feeling" for Poisson's style, and which were written perhaps too hastily. 

Fortunately for the reader unfamiliar in electricity at this point in history, Poisson's account is mostly self-contained. It serves as an interesting look at how one of the electrical theories (the two-fluid theory of Franklin) is conceptualized. Poisson also explains the difference between a conductor and an insulator, and remarks that the electric fluid residing within a conductor must be held on that body by the pressure from the insulating dry air around it. In all, it is an interesting account, with top-notch analysis being applied to rather complicated topics in mathematical physics. 

Speaking of the mathematics, I've been working (April 2022) on annotations for the text, explaining what Poisson is doing from both a modern and historical perspective. The work is known for the introduction of the concept of potential to electrostatics, which Poisson lifts directly from Mechanique Celesete, see section (1) on page 7. It also makes frequent use of spherical and surface harmonics, and by extension Legendre polynomials, which had previously been studied by Legendre and Laplace during work on the gravitational potential. 

A concise summary of this Memoire (as well as a relevant work by Legendre on the attraction of spheroids) can be found e.g. in Annals of Philosophy 1813 vol. I 2nd edition pp. 150-156, available from the Biodiversity Heritage Library

I hope this translation of Poisson's work finds itself useful for someone out there, I believe it still does provide great value, enough that I am happy to have spent the time to translate it. The translation of the second memoir will have to wait now, as I may soon forfeit a large amount of my free "translation" time. 

Sam Gallagher, 2021

Update, 14 Dec. 2021: Fixed wording in (1) to properly describe the spherical coordinates.

Update, 9 April 2022: Fixed typos and formulas in the few sections, updated translator's preface













Comments

  1. Thanks so much for this great translation. How can I talk you into translating Poisson’s “Second mémoire sur la théorie du magnétisme”? Here is a link to it: https://babel.hathitrust.org/cgi/pt?id=inu.30000112182427&view=1up&seq=1

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    1. I can look into it. I still have to translate the second memoire on electricity here, which could take a while depending on how much time I can put towards it! But I'll put that on the list, it looks shorter so maybe I'll do it before the second memoire on electricity.

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    2. Thanks very much, Of course I suggest translating this one first. I will eagerly await the day.

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  2. Thank you so much! I have been looking for an English translation of Possion's memoirs on electricity for several months now. Is this the first translation of the work? I couldn't find any other.

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    Replies
    1. You're welcome, I always hope that something I translate will help someone. I looked for another translation for a while before I sat down to do this one. I'm doubtful one exists, although sometimes translations appear in more obscure journals under different names, and it can be difficult to know for sure. In any case, I've never found one, so I did it.

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