We all know about Green's theorem, and applied mathematicians probably know about Green's functions, but details of Green's life may have eluded many of us. A perusal of thirteen calculus or advanced calculus books on my shelves found nine stating Green's theorem without any mention of the person behind it, while the other four gave his first name and dates (but no other information). History of mathematics books in most cases do not go much further: Burton and Eves have just one page reference in the index for Green, and Boyer and Katz are only slightly more generous. If we rely on such sources, we are left to imagine perhaps a conventional British gentleman making his way through the usual educational steps, then embarking on a career of teaching and research.
In fact, George Green was the son of a prosperous miller (George Green Senior) at Sneinton, at the time a village about one mile from Nottingham. A notable near contemporary of Green's was (General) William Booth of Salvation Army fame, born in Nottingham in 1829. As a youth Green had only four terms of formal education (not uncommon for the time), and at the age of nine went to work at the bakery that was part of his father's milling operation. Green continued mathematical and scientific study on his own, possibly with the aid of a tutor. He became acquainted with Jane Smith, daughter of the mill manager, and fathered seven illegitimate children by her from 1824 to 1840. In 1828 Green published privately, through the Nottingham Subscription Library, his first and most important paper: "An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism." 51 individuals "subscribed" to this publication, but few of them could have understood any of it. One of these, however, Sir Edward Ffrench Bromhead, had contributed an article on calculus to the Encyclopedia Brittanica, and recognized the worth of the Essay. Green had enough money on the occasion of his father's death in 1829 to sell the business and pursue mathematics, shortly producing another paper, this one on fluid equilibrium. Bromhead and Green were in communication, and eventually met, and Bromhead helped Green have the second and a subsequent paper published by the Cambridge Philosophical Society. In 1833, we find Green, at the age of forty, arriving at Cambridge to begin undergraduate studies at Gonville and Caius College. He completed his B. A. by 1838, and became a Fellow of the College, but ill health forced him to retreat to Nottingham, where he died in 1841. The cause of his death was listed as influenza, but biographer Mary Cannell suspects "miller's disease," a condition caused by the flour and dust present in a mill, and analogous to lung disease afflicting coal miners. Other accounts wonder if excessive alcohol consumption might have contributed to his demise. Green's total output consisted of ten papers on mathematical analysis of electricity and magnetism, and of wave motion of fluids, sound, and light. The all-important Essay of 1828 was little known until rediscovered in 1845 by William Thomson (later Lord Kelvin), who arranged to have it republished in three parts in Crelle's Journal (1850, 1852, and 1854).
The book under review is the second edition of a work published by Althone Press in 1993, Green's bicentenary year. Features new to this edition are: inclusion in the final chapter of an account of the Green bicentenary celebration that was held in Nottingham, Cambridge, and London; an expanded list of scientific references; and a new appendix containing the bicentenary talks given at Nottingham University by Julian Schwinger and Freeman Dyson. After Cannell's prefaces to each edition, there is a brief Foreword by Lawrie Challis (Nottingham physics professor) that includes Edmund Whittaker's claim that Green should be described as the real founder of the "Cambridge school" of natural philosophers (e.g., Kelvin, Stokes, Rayleigh, and Maxwell), and the observation that not only were Green's theorem and functions important tools in classical mechanics, but were revived with Schwinger's 1948 work on quantum electrodynamics that led to his 1965 Nobel prize (shared with Feyman and Tomonaga — later in the book Sylvan Schwebe is quoted as suggesting Dyson should also have had a share of the prize). Green's functions later also proved useful in analyzing superconductivity, and Dyson ends his talk saying he would not be surprised if Green's functions were eventually to be reincarnated in superstring theory. An Introduction by Challis outlines what is known of Green's life, and is followed by "Mary Cannell: In Memoriam" — she died in April of 2000 before the second edition had been published.
The bulk of the book is Cannell's 179-page biography of Green. It is meticulously researched, as 44 pages of Notes attest, but it also includes a certain amount of speculation, since our record about Green's life is far from complete: there are no portraits of Green, and much correspondence is missing. Among several possible scenarios that the author constructs from scanty evidence is that after Green's brief elementary education at Goodacre's Academy, his father employed as mathematics tutor for his son one John Toplis, headmaster of the local grammar school. Toplis was a Cambridge graduate who had translated the first book of Laplace's Mécanique Céleste, and who in 1819 returned to Cambridge as a Fellow and Dean of Queens' College. He was instrumental in introducing continental — especially French — mathematics to a tradition-bound Cambridge (and Royal Society), so that fluxion notation was finally superseded by that of Leibniz, and the works of Laplace, Lagrange, Legendre, Poisson and others sparked a revival of learning. Further speculation is that Green was aware of these French analysts through the efforts of Toplis, and was therefore mathematically up to speed when he arrived at Cambridge in 1833, by which time (page 41) "...continental analysis was the accepted form of mathematics."
Mention of Toplis and French mathematics leads Cannell to report extensively on the educational and scientific situation in France, and there are similar detours throughout the narrative. Some of them are fascinating, such as the vivid description of windmill workings and conditions (pages 6-7), but the excursions into the founding and facilities of the Nottingham Subscription Library, various branches of Green's family (or of his common-law wife's family), the restoration of Green's mill, etc., may try the patience of some readers. Mary Cannell's background was in French language and literature and in school administration, and she understandably keeps the mathematical details at arm's length. She certainly acquired, though, a solid knowledge of who the important mathematicians and scientists of the time were: in addition to the names already mentioned, there is material on Babbage, Cayley, Coulomb, Gregory, Hamilton, Jacobi, Stokes, and many others. She reminds us that Boole and Faraday, like Green, were self-taught, points out that Sylvester left Cambridge in 1837 because "...as a Jew he was denied a degree on religious grounds" and that DeMorgan "...left Cambridge for London, since as a Unitarian he was debarred from a Fellowship." [In fairness, she also tells us that both Cambridge and Oxford removed all such religious restrictions in 1883.] There is a wealth of detail in the biography, and those mainly interested in the mathematician and the mathematics may need to employ some skimming or speed-reading techniques.
Following the biography, there are several appendices. Appendix I ("The Mathematics of George Green" by M. C. Thornley) is an analysis of the 1828 Essay, quoting and commenting on several passages. A second appendix simply lists Green's ten published papers, and Appendix III contains two very brief accounts of Green's life by two contemporaries, Willam Tomlin (a brother-in-law) and Bromhead. The next two appendices give the Green, Butler, Smith, and Tomlin family trees, and time charts of mathematicians and scientists mentioned in the text. The two-part Appendix VI (new to the second edition) is an outstanding addition to the volume. It consists of Julian Schwinger's wittily titled "The Greening of Quantum Field Theory: George and I" and "Homage to George Green: How Physics Looked in the Nineteen-Forties" by Freeman Dyson. It is here that we learn how the nineteenth-century Green's functions took on new life and meaning in the second half of the twentieth century. The book concludes with the Notes on chapters of the biography (and one page of Notes on Thornley's contribution), four pages of biographical references on Green, five pages of scientific references, and a list of four websites (one discussing the Moon's George Green crater).
This is an interesting, if somewhat uneven, package of a book that is an obvious choice to be ordered for any college or university library, and will be a welcome addition to the personal libraries of those interested in the recent history of mathematical physics.
David Graves (email@example.com) is Associate Professor of Mathematics at Elmira College, where he is active as a pianist, and has taught courses in cryptology, opera, and history of astronomy as well as the usual run of mathematics courses.