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Does any one know actual references for the discovery of Gauss' Law (a corollary of the Divergence Theorem)?

The entry in Wikipedia for Divergence Theorem says it was discovered by

 Lagrange 1762, Gauss 1813, Green, 1825

while the entry in Wikipedia for Gauss' law says it was discovered by

 Gauss in 1835, 

but not published until 1867.

In the case of the divergence theorem, only dates are given, no references. I couldn't find anything in the collected works of Lagrange around 1762 that seemed to be close to any form of the divergence theorem.

In the case of Gauss' law, a reference is given to the book by Balone, "A Word on Paper: Studies on the Second Scientific Revolution", but no page reference is given; the only index entry for Gauss is a brief mention of his name in connection with some one else; and perusing through the subject matter makes it seem very unlikely that anything about Gauss' law is in this book.

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If Gauss's law was first published in 1867, it must be among the posthumous papers in his Werke. –  Franz Lemmermeyer Jun 17 '11 at 18:24
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See also "A history of the divergence theorem" at sciencedirect.com/science/article/pii/0315086078902124 –  Franz Lemmermeyer Jun 17 '11 at 18:34
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Dear Michael Spivak, you could give a look at this paper of Viktor Katz on the subject, I hope it is useful to you.ingelec.uns.edu.ar/asnl/Materiales/Cap03Extras/Stokes-Katz.pdf –  Giuseppe Tortorella Jun 17 '11 at 19:14
    
The eighth entry in its bibliography is to the paper of 1813 by Gauss. –  Giuseppe Tortorella Jun 17 '11 at 19:24
    
I've added the "citation needed" template in appropriate places in the Wikipedia article corresponding to facts in the paragraph mentioned by Michael Spivak above. –  Michael Hardy Jun 17 '11 at 23:07
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1 Answer 1

I found the following excerpt on this web site: http://jeff560.tripod.com/d.html. It gives references to specific papers of both Lagrange and Gauss. The inverse square law for the electric force is usually associated with Coulomb, but was apparently first inferred by Priestley on the basis of Franklin's observation that there is no electric field inside a hollow conductor and analogy to the known analogous property for gravity.

The history of the theorem is bewildering with many re-discoveries.

O. D. Kellogg Foundations of Potential Theory (1929, p. 38) has the following note on the result “known as the Divergence Theorem, or as Gauss’ Theorem or Green’s Theorem”:

A similar reduction of triple integrals to double integrals was employed by Lagrange: Nouvelles recherches sur la nature et la propagation du son Miscellanea Taurinensis, t. II, 1760-61; Oeuvres t. I, p. 263. The double integrals are given in more definite form by Gauss Theoria attractionis corporum sphaeroidicorum ellipticorum homogeneorum methodo novo tractate, Commentationes societas scientiarum Gottingensis recentiores, Vol III, 1813, Werke Bd. V pp. 5-7. A systematic use of integral identities equivalent to the divergence theorem was made by George Green in his Essay on the Mathematical Theory of Electricity and Magnetism; Nottingham, 1828 [Green Papers, pp. 1-115]. Kline (pp. 789-90) writes that Mikhail Ostrogradski obtained the theorem when solving the partial differential equation of heat. He published the result in 1831 in Mem. Ac. Sci. St. Peters., 6, (1831) p. 39. J. C. Maxwell had made the same attribution in the 2nd edition of the Treatise on Electricity and Magnetism (1881). See also the Encyclopaedia of Mathematics entry Ostrogradski formula

For Gauss’s theorem Hermann Rothe “Systeme Geometrischen Analyse, Erster Teil” Encyklopädie der mathematischen Wissenschaften mit Einschluss ihrer Anwendungen Volume: 3, T.1, H.2 p. 1345 refers to Gauss’s Allgemeine Lehrsätze in Beziehung auf die im verkehrten Verhältnisse des Quadrats der Entfernung wirkenden Anziehungs- und Abstossung-Kräfte 1839 in Werke Bd. V (especially pp. 226-8.)

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