George Green

The Reluctant Miller's Tale: A Major Mathematical Mystery

Review of
George Green: Mathematician and Physicist, 1793-1841: The Background to his Life and Work.
By D.M (Doris Mary) Cannell
Athlone Press, Atlantic Highlands, NJ, 1993, 265 pages, $70.00

The American Mathematical Monthly frequently runs a "picture puzzle", in which the reader is asked to identify a picture of a famous mathematician. There are no known likenesses of the person I have in mind, so I will post my puzzle verbally: What mathematician, on the defeat of a piece of legislation in 1831, took up a musket to defend his property against a rioting mob?

Very few professionals --- indeed, very few people of any sort --- know much about this person, although every undergraduate who has taken a course in multivariable calculus is familiar with one of his major contributions: Green's theorem (Ostrogradsky's theorem in Russia). Those who have studied differential equations may remember the Green's function of a linear differential equation. Some may even have studies Green's tensors. Other than his work, though, he left behind few traces.

The scientific life of George Green was very unusual even for the day. He was born in 1793 in Nottingham, England. His father, George Green senior, was a hard-working baker who, although only semiliterate, had a good head for business. Green senior build a windmill in Sneinton, a nearby village, to grind grain for his bakery; the business prospered.

At the age of eight, young George was enrolled in a local academy run by a certain Robert Goodacre, a mathematics enthusiast. As the child already had a passion for the subject, this was a fortunate conjunction, or perhaps it was the result of a deliberate decision on the part of George senior. Young Green stayed there only for four terms, having learned all that the masters could teach him.

At the age of nine, he was taken into his father's bakery and milling business, where he worked for 27 years. Although a reluctant miller, he carried on with it until 1823, when his father died. Green later leased the mill and sold the business, and with the addition of the rents from his father's Nottingham property, found himself sufficiently wealthy to support himself as a mathematician.

The riot alluded to earlier occurred as follows:

In 1831 ... angered by the defeat of the Reform Bill, a mob of angry townspeople saw the mill as an immediate target ... The mathematician vigorously defended his property by firing on the crowd, and his young daughter Jane helped him to reload his musket.

We don't know too much about Green's personal life. A dozen or so letters and a few statements by contemporaries have been uncovered. When he was in his early twenties, he formed a relationship with Jane Smith, the daughter of his father's mill manager. He never married her, and she bore him seven illegitimate children. Why didn't he marry Jane, although he provided for her and their children very nicely? The reason is unknown, but conjecture has it that his father considered Jane too lower class.

In 1828, in his mid-thirties, Green published An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism. Published privately and by subscription, this was his magnum opus. About a hundred copies were printed; 51 subscribers had the privilege of buying the monograph at seven shillings sixpence. From a scientific-book collector's point of view, the monograph is very rare. (The rare book library at Brown University has only a copy of a facsimile edition produced in 1889.)

In 1833, having written four major papers (three of which were published in the Transactions of the Cambridge Philosophical Society and the other in the Transactions of the Royal Society of Edinburgh) Green enrolled at the age of 40 as an undergraduate in Gonville and Caius College, Cambridge University! In 1837, at the age of 44, he took the Mathematical Tripos and ended up as Fourth Wrangler (first-class honors, nonetheless; James Joseph Sylvester, aged 23, who came in second, was the only one of the top three who went on to achieve mathematical fame.) Two years later, Green was elected to a college fellowship at Caius (pronounced "Keys"). During those two years he produced six more papers, all published in the Transactions of the Cambridge Philosophical Society.

In the course of her book, Doris Cannell asks the critical question: "If even Cambridge did not teach continental analysis and use the works of Laplace, Lacroix, Poisson, and the rest, how did a provincial miller and untutored mathematician in Nottingham come across them and use them to such advantage?"

How did he learn what he learned? How did he study and plug himself into the latest in French mathematics? By working in some remote corner of his busy, noisy, dusty mill? Why not? Don't graduate students today do their work to canned music at full blast? How did he go beyond the existing knowledge to create what he created and thus bring to a close what Sir Edmund Whittaker called the darkest century in the history of science at Cambridge University? This is the mystery alluded to in the title of this review, and it is a mystery that remains unsolved --- so little is known that biographers can only conjecture. Far more is known about another mathematical autodidact who comes to mind: Srinivasa Ramanujan, who was at Cambridge almost a century later.

in 1823, at the age of 30, Green was admitted to the Nottingham Subscription Library. Located in Bromley House, it was a center for a variety of cultural activities, including scientific and philosophical speculation and debate. The library of Bromley House (for which records are available) did not have Green's major reference: the works, in French, of Biot, Lacroix, Poisson, and Coulomb.

Green was probably familiar with the Reverend John Toplis, a headmaster of the Nottingham Free School and an enthusiast of French mathematics, whose translation of Laplace's Mécanique céleste was published in 1814. Perhaps Toplis, who lived not far from Green's bakery, had been Green's mentor. Perhaps Green picked up references to the French works from Toplis or his translation.

There is information on individuals (among them Sir Edward Ffrench Bromhead) who supported Green psychologically and provided entrée into the intellectual world for "the son of a miller who has had only a common education in the town." And we do not know much more than this.

Green's major work was ignored for years. Mathematical England was not ready for it. By 1845, when it was recognized by William Thomson (Lord Kelvin), English mathematical science, which had been rigidly locked into Newtonian notations and formulations, had rejoined the Continent.

Green died in 1841. The cause of death was given as influenza, although it has been suggested that he had "miller's disease", a condition something like silicosis, caused by the flour and the dust-laden atmosphere of a mill. Felix Klein, writing in Germany 50 years after the fact, asserted that Green had died of alcoholism.

He died a wealthy "gentleman", a technical term in 19th-century England. His direct descendants are numerous. There is now a considerable George Green Memorial Fund, which, among other things, has restored the windmill to grinding condition and has established an adjacent museum of science of considerable popularity. There is a plaque to him in Westminster Abbey. In 1993, the bicentenary of Green's birth was celebrated in Nottingham, and at Gonville and Caius. The restored mill was on display, along with a number of Nobelists who, in their work in quantum mechanics, have made use of Green's function. A few numerical analysis, especially those who have promoted the finite element method, might also have been invited, for their work also carries the mark of Green.

This is the first substantial biography of Green, a man who was remarkable in his own day. Unique? Perhaps not. Boole, Faraday, and Whewell came from humbler circumstances. The author, in the absence of the vast quantities of material that allow the 700-page biographies so common today, has put together all that is now known. She has amplified it with an informative picture of science at Cambridge in the early 1800s --- about which much is known. She has also included a nice mathematical appendix explaining the work of George Green; the author of the appendix, Michael C. Cramley writes in conclusion: "He was a self-taught genius who provided one of the bridges between the earlier generations of Laplace and Lagrange, and the later generations of Thomson and Maxwell." There are many references to today's literature, on quantum theory, for example, where Green's ideas are employed.

Even with enough material for a 700-page biography, the final creative leap from the study of the masters to that which goes beyond, whether in the case of Green or any other mathematician, has been described only unsatisfactorily. The psychology of mathematical invention, despite the work of Jacques Hadamard and later writers, remains an enigma. The George Greens of the world cannot be programmed.

Philip J. Davis, professor emeritus of applied mathematics at Brown University, is an independent writer, scholar, and lecturer. He lives in Providence, Rhode Island, and can be reached at AM188000@brownvm.brown.edu