George Green (14 July 1793 – 31 May 1841) was a British mathematical physicist born in Sneinton, Nottinghamshire, now part of the city of Nottingham. Despite receiving little recognition during his lifetime, he is best remembered for his Essay on electricity and magnetism published in 1828. In this essay he introduced an early version of Green's theorem in vector calculus, the notion of potential functions as currently used in physics, and a method of solving differential equations called Green's functions. This paper formed the foundation for the work of later scientists such as William Thomson (later Lord Kelvin), among many others. His work on potentials ran parallel to that of Carl Friedrich Gauss.
As a mathematician, he was almost entirely self-taught, receiving only about one year of formal schooling as a child, between the ages of eight and nine. Green then worked his father's bakery from a young age, and later at the mill his father constructed in 1807, now known as Green's Windmill. Whilst working as a miller, he became a member of Nottingham Subscription Library through which he was able to access contemporary mathematical knowledge.
Green formed a relationship with Jane Smith, the daughter of the mill manager William Smith. Although the couple never married, they had seven children together and she later adopted Green's surname.
After the death of his father in 1829, he was left a substantial inheritance, allowing him to give up milling to pursue mathematics. In 1833, Green began a formal education at the University of Cambridge and was elected a fellow at Caius College after his graduation. Throughout his time at the university, he wrote six papers on hydrodynamics, acoustics, and optics.
In 1840, he fell ill and returned to Nottingham where he died of influenza the following year, on 31 May 1841. After his death, his Essay was rediscovered by William Thomson (later Lord Kelvin) who promoted Green's work and used it in his own research on electricity and magnetism. Today, Green's functions remain a standard tool for solving differential equations in modern physics, including in quantum electrodynamics (QED) and condensed-matter physics.
Early life Green was born and lived for most of his life in the English town of Sneinton, Nottinghamshire, now part of the city of Nottingham which then had a population of around 30,000. He was baptized on 14 July 1793. His father, also named George, was a baker who had built and owned a brick windmill used to grind grain.
Robert Goodacre's Academy During this era, most children received little schooling beyond being taught to read and write at Sunday schools. Green was fortunate in that his father was able and willing to have him privately educated. He entered the academy of Robert Goodacre, a man known for his enthusiasm towards the sciences and education. The academy's "philosophical instruments" included an "electrical machine" and an orrery. However, Green stayed for only 18 months before returning to work at his father's bakery.
Move from Nottingham to Sneinton In 1773, George's father moved to Nottingham, which at the time had a reputation for being a pleasant town with open spaces and wide roads. By 1831, however, the population had increased nearly five times, in part due to the budding Industrial Revolution, and the city became known as one of the worst slums in England. There were frequent riots by starving workers, often associated with special hostility towards bakers and millers on the suspicion that they were hiding grain to drive up food prices.
For these reasons, in 1807, George Green senior bought a plot of land in Sneinton. On this plot of land he built a "brick wind corn mill", now referred to as Green's Windmill. It was technologically impressive for its time, but required nearly twenty-four-hour maintenance, which was to become Green's burden for the next twenty years.
Adult life
Miller
Just as with baking, Green found the responsibilities of operating the mill annoying and tedious. Grain from the fields was arriving continuously at the mill's doorstep, and the sails of the windmill had to be constantly adjusted to the wind speed, both to prevent damage in high winds, and to maximise rotational speed in low winds. The millstones that would continuously grind against each other, could wear down or cause a fire if they ran out of grain to grind. Every month the stones, which weighed over a ton, would have to be replaced or repaired.
Family life In 1823, Green formed a relationship with Jane Smith, the daughter of William Smith, hired by Green Senior as mill manager. Although Green and Jane Smith never married, Jane eventually became known as Jane Green and the couple had seven children together; all but the first had Green as a baptismal name. The youngest child was born 13 months before Green's death. Green provided for his (so-called) common-law wife and children in his will.
Nottingham Subscription Library When Green was thirty, he became a member of the Nottingham Subscription Library, then the intellectual heart of the town. It still exists today as the Bromley House Library. This library was likely the main source of Green's advanced mathematical knowledge, giving him access to the Philosophical Transactions of the Royal Society, among other scientific journals. While foreign journals were not available, the Transactions did mention their contents, which allowed Green to obtain reprints directly from their authors.
Formal education and academic career
By the time Green's father died in 1829, the senior Green had become one of the gentry due to his considerable accumulated wealth and land owned, roughly half of which he left to his son and the other half to his daughter. The young Green, now thirty-six years old, consequently was able to use this wealth to abandon his miller duties and pursue mathematical studies.
A pivotal figure in Green's mathematical education was Reverend John Toplis (–1857), who graduated in mathematics from Cambridge as 11th Wrangler in 1802. He then served as headmaster of the Nottingham Free Grammar School 1806–1819, and lived in the same neighbourhood as Green and his family. Toplis was an advocate of the continental school of mathematics, and fluent in French, having translated Pierre-Simon Laplace's celebrated work on celestial mechanics (Traité de mécanique céleste). The possibility that Toplis played a role in Green's mathematical education would resolve several long-standing questions about the sources of Green's mathematical knowledge. In the Preface to his 1814 translation of Laplace, Toplis recommended the works of French mathematicians Joseph-Louis Lagrange, Adrien-Marie Legendre, and Sylvestre François Lacroix. These and other Continental mathematicians extended the works of Sir Isaac Newton on mechanics, written in terms of synthetic geometry, using analytical methods. Toplis—and later Green—employed the notation of Leibniz rather than Newton for the differentials.
Green likely learned much from the 1816 exposition of Jean-Baptiste Biot on the electrical experiments of Charles-Augustin de Coulomb and from the Transactions of the Cambridge Philosophical Society.
1828 essay In 1828, Green published An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism, the 72-page work for which he is most famous today. It was the custom at that time to publish one's scientific paper via the journals of either the Royal Society or the Cambridge Philosophical Society. However, Green chose to publish his first work at his own expense, because he thought it would be presumptuous for a man of his station, with no formal education in mathematics, to submit the paper to an established journal. When Green published his Essay, it was sold on a subscription basis to 51 people, many of whom were members of the Nottingham Subscription Library, though few are thought to have understood its mathematical content. These people were local doctors, clergymen, schoolmasters, and lace and hosiery manufacturers.
While Green had hoped that publishing his Essay would bring recognition from mainstream academics in Great Britain and beyond, there was initially no response, with one exception. The wealthy landowner and mathematician Edward Bromhead bought a copy and responded immediately, encouraging Green to do further work in mathematics. As a wealthy public figure who maintained contacts with academia, Bromhead was in a position to be Green's benefactor. However, not believing the offer was sincere given the difference in social class, Green did not contact Bromhead for twenty months.
Green's decision to include "Mathematical Analysis" in the title of his Essay was an acknowledgement of the influence of Continental mathematics. He began by referring to a 1771 paper by Henry Cavendish on electricity, where he noted and rectified an unsatisfactory claim, then moved on to the memoirs of Siméon Denis Poisson. The notion of the potential was first developed by Laplace in his research on gravitation and celestial mechanics and subsequently applied in electromagnetism by Poisson. In his Essay, Green introduced the term potential and applied this concept masterfully, establishing some key results. One of these was Green's theorem, which was related to Gauss' theorem, relating the properties of a function on the boundary of a domain with the interior. (Also see the Fundamental Theorem of Calculus.) In his 1839 memoir, Karl Friedrich Gauss independently introduced the same terminology. Green also cited Book 3 of Laplace's Mecanique celeste for its use of the Legendre polynomials and Jean-Baptiste Joseph Fourier's Theorie analytique de la chaleur (Analytical Theory of Heat). Whereas Poisson had only determined the charge distribution on the surface of a conductor with simple geometry, Green showed how to do so for arbitrary cases. The key tools he employed here were Green's identities, which he proved by integrating by parts. Today, these are typically obtained by applying Gauss' theorem. Green explained how to obtain the general solution in terms of the system's response to a unit input, that is, Green's function.
This seminal paper was initially ignored during Green's lifetime but was later reprinted many times and translated into German. After this paper, Green turned his attention to other topics, namely, hydrodynamics, wave motion, and optics, as advised by Bromhead, who was of the opinion that electricity and magnetism were not of interest to British mathematicians at the time.
Cambridge Members of the Nottingham Subscription Library who knew Green repeatedly insisted that he obtain a proper university education. After they met, Green let Bromhead know of his dream of attending the University of Cambridge. With Bromhead's help, he fulfilled that wish, and matriculated at Cambridge in 1833, as an undergraduate at Gonville and Caius College, the same one Bromhead went to. He was particularly insecure about his lack of fluency in Greek and Latin, which were prerequisites, but it turned out not to be as hard for him to learn these as he had envisaged, as the degree of mastery required was not as high as he had expected. He won the first-year mathematical prize. After sitting in the Mathematical Tripos in January 1837, he became 4th Wrangler (the 4th highest scoring student in his graduating class, coming after James Joseph Sylvester who scored 2nd) and graduated with a BA in 1838. Another source of concern for Green was the existence of Jane and the children, as the university required students to remain celibate. But this was apparently not a problem because he and Jane were not formally married.