Scientific Justice for George Green

September 17, 2000

The 1828 essay that introduced Green's functions and Green's theorem was most likely written at least in part in the now famous windmill, shown here after its restoration, which was completed in 1993, the bicentenary of George Green's birth.

Robert E. O'Malley, Jr.

Mary Cannell had lots to teach us about George Green, weaving her fascinating tale on a limited warp of specific facts about Green but with a thorough understanding of Victorian life in Nottingham and Cambridge as her very solid weft. Indeed, Cannell, who died earlier this year, confessed that she'd had a "mystic marriage" with GG (1793-1841) for her last twenty years. Her 1993 Athlone Press biography of Green was recommended to me by Lady Bertha Jeffreys, who was the collaborator and wife of the late Sir Harold Jeffreys (the "J" of the WKBJ method, which the authority Frank Olver convincingly argues should more appropriately be named after Liouville and Green).

Most applied mathematicians know of Green's theorem and Green's functions and use his term "potential" without attribution, but have no knowledge of Green and his important essay of 1828. Although Boyce and DiPrima's book on differential equations includes a short biography of Green, the Green's functions books of Bergman and Schiffer, Roach, and Stakgold, for example, include no details about him. In the unusual 1995 book Green, Brown, and Probability, on the other hand, Chung wonders whether the contemporaries Green and Brown could have met.

Much recent scholarship regarding Green is the result of efforts by the George Green Memorial Fund, chaired by Lawrie J. Challis, a prominent low-temperature physicist at the University of Nottingham, who wrote the accompanying story about Mary Cannell, the fund's longtime Honorary Secretary. The fund, for example, pushed for the restoration of Green's mill, organized an elaborate celebration of the bicentennial of Green's birth, and maintains a Web site about Green.

One of Mary's most recent discoveries was Carl Neumann's 1861 definition of the Green's function. "I call the function a 'Green'sche Function' because the importance it currently has in several branches of mathematical physics has first become apparent in Green's analysis," Neumann wrote, referring to Green's essay as reprinted in Crelle's Journal (volumes 39, 44, and 47, 1850-1854). Mary searched painstakingly through Riemann's collected works because of the claim in Kline's Mathematical Thought from Ancient to Modern Times that Riemann had named these kernels Green's functions. Kline documents that Green was more than ten years ahead of Gauss, but only a few years ahead of Ostrogradsky, in publishing his integral theorem.

Most of the credit for the early recognition of Green seems to be due to William Thomson, later Lord Kelvin, who wrote in the Introductory Notices in the journal that much of the previously inaccessible work had been rediscovered since 1828. He also pointed out that the work had been "written during the intervals of a laborious occupation, the author being employed in assisting his father, who was first a baker at Nottingham, and afterwards a miller in the adjacent village of Sneinton." Thomson had first learned of Green's essay from a footnote in an 1832 paper of Robert Murphy.

Many details about Green's work and its impact are included in Sir Edmund Whittaker's A History of the Theories of Aether and Electricity (1910 and 1951) and in P. M. Harman's 1985 collection Wranglers and Physicists (especially J. J. Cross's essay "Integral Theorems in Cambridge Mathematical Physics, 1830-1855"). Green, according to Whittaker, "though inferior to Cauchy as an analyst,
was his superior in physical insight." Amazingly, GG had spent only 18 months in school, and Robert Goodacre's Academy, even if the best in Georgian Nottingham, was certainly no grande �cole! Nonetheless, recognizing his undergraduate education and brief fellowship at Cambridge (both post-1833), Whittaker says that "it is no exaggeration to describe Green as the real founder of that 'Cambridge school' of natural philosophers, of which Kelvin, Stokes, Rayleigh, Clerk Maxwell, Lamb, J. J. Thomson, Larmor, and Love were the most illustrious members in the latter half of the nineteenth century." Cross includes an 1850 letter from Thomson to Stokes that includes the result now generally called Stokes' theorem. It first appeared in print as an 1854 Smith's Prize Examination Paper written by Stokes for Maxwell and Routh. Green's essay acknowledges the earlier work of Poisson as the basis for his theorem, while Stokes, apparently, never corrected the inappropriate popular attribution.

Cannell was preceded in her study of Green by H. G. Green, an unrelated mathematician from Nottingham. His 45-page biography is nearly buried in a 1946 Festschrift volume, Studies and Essays in the History of Science and Learning, edited by Ashley Montagu. H.G. Green documents much personal information about GG, even acknowledging the seven children he fathered between 1824 and 1840 with Jane Smith, daughter of the foreman at the windmill; although GG never married Smith, he did leave his real estate to her and their children. GG seemed to be conveniently celibate at his college (Caius).

H.G. Green's biography includes a 1907 letter from Lord Kelvin to Larmor telling of his getting three copies of Green's essay, "which astonished me." The senior wrangler was given them by his coach William Hopkins in 1845, just after his graduation at the age of 21, as he was about to go to Paris. He had been seeking a copy from Cambridge booksellers for several years. Two days later, he showed Liouville a copy; Sturm visited him late that evening to see the "memoir de Green." Kelvin concludes by reporting that "Green's Essay made a great impression on Sturm and Liouville and others in Paris."

Mary Cannell's biography of George Green describes the structure of 19th-century English society. Despite the scarcity of details about Green, we can learn much about Green and his contemporaries from her extensive study. Indeed, Cannell's persistence and determined scholarship become clear when one compares the American Mathematical Monthly articles of 1995 and 1999 about Green's essay by the historian Grattan-Guinness and Cannell, respectively. Grattan-Guinness suggests that Hopkins may have given Thomson only two copies of the essay, which are now located in the libraries at the University of Nottingham and the University of Keele. Alas, however, the British Society for the History of Mathematics reports "a moste horrible infamy"---the Turner collection of historic mathematics texts was sold secretly by the University of Keele on July 7, 1998. This suggests that an extremely rare copy of Green's essay could turn up at a Sotheby auction soon.

Meanwhile, order your copy of Cannell's updated Green biography from SIAM!

Robert E. O'Malley, Jr., is the editor-in-chief of the SIAM Classics in Applied Mathematics series. He is also the book review editor of SIAM Review.

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