by I. Grattan-Guinness

© Oxford University Press 2004 All rights reserved

**Green, George** (1793-1841), mathematical physicist and miller, was born on 14 July 1793, probably at Wheatsheaf Yard, Nottingham, the eldest child and only son of George Green (1758-1829), miller, and his wife, Sarah (1770-1826), *née* Butler. After study for four terms at an 'academy' in Nottingham run by Robert Goodacre, George had to work in his father's bakery from the age of nine, and later at the corn mill which his father constructed at Sneinton, then a village outside Nottingham. By the 1820s he seems to have been the mill manager, but he was teaching himself mathematics and perhaps contemplating research topics; the Nottingham Subscription Library contained a respectable stock of scientific publications, including even some foreign material, and, since the mill was apparently prosperous, Green may have been able also to buy some of this literature. He probably also knew John Toplis, a Cambridge graduate mathematician, then headmaster of Nottingham Free Grammar School, who was unusually sensitive to the superiority of continental mathematics.

Whatever the motivations and means of access, Green made himself familiar with the current state of mathematical analysis and mathematical physics, especially French work, and made a most remarkable contribution in an *Essay on the Mathematical Analysis of Electricity and Magnetism* (1828). At that time these branches of physics were gaining considerable attention, from both experimental and mathematical points of view. In particular, the French mathematician S. D. Poisson (1781-1840) had published papers in 1826 in which he treated a magnetic body 'M' as composed of separate dipoles and analysed mathematically the strength of its attraction to internal and to external monopoles. In the first paper he found a theorem which converted triple integrals over the volume of M to double integrals over its surface. An English translation of a summary of the paper appeared in the *Quarterly Journal of Science* in 1824 and, while the mathematical details were not given, it could have excited Green's curiosity. When the full paper appeared, he must have been inspired by the above theorem, which was novel at the time and of which Poisson himself was not fully appreciative, entitling it merely a 'simplification' of certain formulae, reducing triple integrals to double integrals. Green realized that it had the far more profound consequence of relating properties inside a body to properties on its surface and vice versa. Whatever the motive, in his book he produced a similar theorem (the version now called 'symmetric', using two functions over M). Such theorems are rightly named after him, for he was the first to stress their physical interpretation.

In a feature of the book that was without precedent, Green sought the function which satisfied the basic differential equation representing the physical situation which also took given values on the surface of M and vanished to zero at an infinite distance from M. He called it 'the potential function' (later known as 'Green's function'), and in the rest of his book he sought particular cases for electrostatic and magnetic situations.

Another important result has become known as Green's 'reciprocity theorem', relating the electrostatic charges and the attendant potentials of a collection of conductors in two different states of equilibrium. He also studied the total charge of Leyden jars arranged in series. In a long final section he found expressions for the potentials of magnetic bodies of various kinds, such as solids and shells, and compared his predictions with experimental data published by C. Coulomb and J. B. Biot.

As a piece of research and development, Green's book is of the highest calibre, extraordinary for an autodidact; but his sales and marketing were lamentable. He published it in Nottingham at his own expense with the help of a local subscription list--still a common manner of publication then but hopeless for a book of this kind. The recent revival in mathematics in Britain meant that he might have had a chance with a publisher, and he certainly could have summarized his main result in a paper in, say, the *Quarterly Journal* or the *Philosophical Magazine.* He did none of these things, and probably never thought of them. However, the book did find one informed reader among the subscribers: Edward ff. Bromhead (1789-1855), also a Cambridge graduate and a member of the Analytical Society there which had helped to rejuvenate British mathematics in the 1810s. In correspondence during the 1830s he encouraged Green's formal education and further researches.

In the meantime, in 1829, Green's father had died, leaving him and his sister to inherit the mill and various properties. He was now financially established to pursue his mathematical researches, but one residue of the past remained: about 1824 he had commenced a liaison with a lace-dresser, Jane Smith (1802-1877), and seven children were born to them between 1824 and 1840. They never married, seemingly because of his father's disapproval of the loss of status entailed, and perhaps also because Green had set his sights on a Cambridge career where fellowship would require celibacy.

Bromhead helped Green to secure entry into his own old college, Gonville and Caius, in 1833, as a very mature student. Green graduated four years later, but only as fourth wrangler in the competitive ranking system then in operation. A fellow student, J. J. Sylvester, was ranked second, and colleagues at Caius included the mathematicians Robert Murphy (who knew of his book) and Matthew O'Brien. Green was appointed to a Perse fellowship (worth £10 a year) in 1839, which he held until his death two years later, but he spent some time at the mill, with his family living nearby.

During this Cambridge decade Green produced nine papers on various aspects of mathematical physics. The majority of them appeared with the Cambridge Philosophical Society, of which Bromhead was already a member and which Green himself joined in 1837. A few notable features are noted here, largely in chronological order of publication.

Typically for a mathematical physicist of that time, Green exploited analogies between types of phenomena and theories. His first paper (1835) investigated 'the equilibrium of fluid analogous to the electric fluid', expressing potentials in terms of Legendre functions without any specific interpretation. A paper of the same year on 'attractions of ellipsoids of variable densities' was more novel, for he formed the equation by optimizing the value of a volume integral by a principle to which the name of his German contemporary J. P. G. Dirichlet has become attached, and which became a major technique in 'potential theory' (as the subject came to be called). He solved the equation by a special method now known as 'WKB', the initials of the surnames of three independent rediscoverers in 1926.

In an analysis of 'the vibration of pendulums in fluid media' (1836) Green treated the effect of the surrounding air on the motion of the pendulum. Taking it to be ellipsoidal in shape so as to draw upon work by P. S. Laplace, he investigated its motion in the directions of its three principal axes; then he could state its motion as a combination of these three, in a clever use of superposition of special solutions (another well-known technique of the time).

A paper of 1838 dealt with 'the reflection and refraction of sound', giving the first detailed study of total internal reflection. Reflection and refraction were properties more closely associated with light, which Green examined in two papers. Following A. L. Cauchy and A. J. Fresnel, he assumed that the ether was an elastic solid and that optical phenomena were caused by activity within it. In 1838 he analysed them at the intersection of two non-crystalline media (that is, media with no special properties); he worked out the conservation of potential energy when the media were placed under strain. Four years later he applied similar methods to the propagation of light in general in crystalline media, setting up two different models based upon certain assumptions about the stress properties of the ether and the types of vibration possible within it; in one model he obtained a version of one of Cauchy's models.

Green died on 31 May 1841 at Jane Smith's house, 3 Notintone Place, Sneinton, and was buried on 4 June in St Stephen's churchyard, Sneinton, near the mill. Green's papers entered in the general mass of writing on mathematical physics, though they did not gain all the attention that they deserved, but his book remained unknown; two rather passing mentions in those papers had not given his public much chance. However, while at Cambridge he had given some copies to the mathematical coach William Hopkins (1793-1866), who in turn gave one in 1845 to a bright young student, William Thomson (1824-1907). At last Green had his first active reader; Thomson told his friends in both Britain and France, and arranged for the book to be reprinted in three parts in a leading German mathematical journal between 1850 and 1854. He also profited from its contents, for example to invent his 'method of images' to calculate potentials in various contexts.

Thanks to this publicity and use of its ideas, Green's importance was recognized in the later development of potential theory; his theorem and functions were central tools, and his name was attached to both. An edition of his works appeared in 1879: it included the book, which itself was reprinted and translated into German late in the century, and the edition was reprinted in 1903, in Paris. His ideas, especially the theorem and the function, were to receive fresh boosts later in the twentieth century, in some parts of engineering and above all the development of quantum mechanics (the contributions of 'W', 'K', and 'B' belonged to this movement).

However, scholarly work on Green came too late to save the manuscripts which he seemed to have left but which were destroyed after the death of his last child in 1919. Thus there is no access to his other thoughts, and not even a known likeness. Only in the mid-1970s was a major effort launched, centred on the reconstruction of the mill, which had fallen into desuetude; it was opened in 1985 both as a working mill and as a science centre for educational purposes. This county recognition was enhanced to the national level in 1993, the bicentenary of Green's birth, when a plaque bearing a depiction of the mill was unveiled in the section for scientists of Westminster Abbey, near to the tomb of Isaac Newton and next to plaques of Thomson (as Lord Kelvin), James Clerk Maxwell, and Michael Faraday. A fine biography by Mary Cannell also appeared in that year, making Green appreciable as a historical figure at last.

I. GRATTAN-GUINNESS

**Sources **

H. G. Green, 'A biography of George Green', *Studies and essays in the history of science and learning in honor of George Sarton,* ed. A. Montagu (1946), 545-94

R. M. Bowley and others, *George Green, miller, Sneinton,* 2nd edn (1980)

D. M. Cannell, *George Green: miller and mathematician, 1793-1841* (1988)

D. M. Cannell, *George Green: mathematician and physicist, 1793-1841* (1993); 2nd edn (2000)

I. Todhunter, *A history of the mathematical theories of attraction and the figure of the earth,* 2 vols. (1873)

P. M. Harman, ed., *Wranglers and physicists: studies on Cambridge physics in the nineteenth century* (1985)

I. Grattan-Guinness, 'Why did George Green write his essay of 1828 on electricity and magnetism?', *American Mathematical Monthly,* 102 (1995), 387-96

G. J. Whitrow, 'George Green, 1793-1841', *Armali dell' Istituto di Storia della Scienza di Firenze,* 9 (1984), 45-68

G. Green, *Mathematical papers,* ed. N. Ferrers (1871)

E. T. Whittaker, *History of the theories of aether and electricity* (1951)

D. M. Cannell, 'George Green: an enigmatic mathematician', *American Mathematical Monthly,* 106 (1999), 136-51

**Archives **

U. Nott. L., George Green Library, archives

**Wealth at death **

various houses and monies: Green, 'Biography'

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