## An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism.

Nottingham: Printed for the author, by T. Wheelhouse, 1828.

First edition, “excessively rare” (Wheeler Gift), of “one of the major publications in the mathematics of the 19th century” (*Landmark Writings*, p. 403), in which Green revolutionized the study of potential theory as we now understand it (and coined the term ‘potential’ in this context), and introduced ‘Green’s theorem’ and ‘Green’s functions,’ which have proved to be of the greatest importance in all branches of mathematical physics. “Through Thomson, Maxwell, and others, the general mathematical theory of potential developed by an obscure, self-taught miller’s son would lead to the mathematical theories of electricity underlying twentieth-century industry” (*DSB*). “Latimer Clark, a nineteenth-century historian of electricity, described [Green’s] *Essay* of 1828 as ‘one of the most important works ever written on electricity’; and Sir Edmund Whittaker, in his authoritative *History of the Theories of Aether and Electricity*, states that ‘it is no exaggeration to describe George Green as the real founder of that “Cambridge school” of natural philosophers of whom Kelvin, Stokes, Lord Rayleigh and Clerk Maxwell were the most illustrious members in the latter half of the nineteenth century’” (Lawrie Challis in Cannell, p. xxi). Enormous as the influence of Green’s book on nineteenth-century mathematical physics was, “the scale of its application to modern physics is perhaps even more remarkable. It started in 1948 when Julian Schwinger, who had been using Green’s functions in his wartime work on radar, used them for the first time to solve a problem in quantum mechanics. The problem was on quantum electrodynamics – how light interacts with electrons – and Schwinger shared a Nobel Prize with Feynman and Tomonaga for solving it. This pioneering step was followed by a dramatic increase in the use of Green’s functions, particularly in the theory of fundamental particles and nuclei, and in solid-state physics. An assessment of the value of the technique is given in a letter from Robert Schrieffer, who shared the Nobel Prize for the theory of superconductivity with Bardeen and Cooper: “I have, in most of my scientific publications, dealt in one way or another with the techniques of Green’s functions … Not only are Green’s functions of great significance to the theoretical physicist in the solution of physical problems, these functions are directly related to physical observations in the laboratory. Almost every experiment which weakly probes a physical system can be described in terms of the relevant Green’s functions for this observation. Thus the theoretical physicist has a direct link to the experimental results through the work of George Green”” (*ibid*., pp. xxi-xxii). Among nineteenth-century works on mathematics and physics of comparable importance, this is certainly one of the rarest. “Apparently, almost all of the fifty-two subscribers were patrons and friends of Green’s; … the list of subscribers suggests only limited circulation outside Nottingham” (*ibid*.). The only copy located on ABPC/RBH is the Honeyman copy (which lacked the list of subscribers). The present copy is complete, uncut and preserves the original front printed wrapper. The existence of printed wrappers for this book seems to be hitherto unknown to bibliographers, and it is doubtful if many, or any, other copies with the wrappers are extant.

Green’s book was probably inspired by work of the French mathematical physicist Siméon-Denis Poisson (1781-1840), then professor at the *Ecole Polytechnique* in Paris and follower of Laplace and Lagrange. In 1826 he published a paper in which he attempted a mathematical analysis of magnetism (‘Mémoire sur la théorie du magnetisme’, *Mémoires de l’Académie des Sciences *5 (1821-22), pp. 247-338). “Taking a magnetic body *A* to be composed to discrete ‘magnetic elements’ *D*, he set to zero certain surface integrals over *D* expressing internal equilibrium, and wrote down volume integrals to state the components of attraction of *A* to an external point *M* relative to his imposed rectangular coordinate system (*x*, *y*, *z*). The second part of the paper dealt with a ‘simplification of the preceding formulae’; integrating these integrals by parts with respect to (say)* z* led him to convert the volume integral to an integral over the surface *S *of *A* … Adding this formula to its brothers for the *x*- and *y*-directions gave him the first general divergence theorem in mathematics” (Grattan-Guinness, pp. 389). But Poisson failed to realise the significance of his discovery: he regarded it simply as a mathematical trick, replacing triple integrals by double integrals. Green, on the other hand, realised that the importance of Poisson’s results lay in *relating properties inside bodies to properties on their surfaces and vice versa*.

After a preface, dated March 1828, and some pages of ‘Introductory observations’, the *Essay* is divided into three Parts: ‘General preliminary results’ and then applications to electricity and to magnetism. The most important mathematical innovations are contained in the ‘General preliminaries’. First was the explicit specification of ‘the potential function,’ as he called it (p. 9), and now named after him: “It only remains therefore to find a function which satisfies the partial differential equation, becomes equal to [a given function] when [the point]* p* is upon the surface *A, *vanishes when *p* is at an infinite distance from *A, *and is besides such that none of its differential coefficients shall be infinite when the point *p* is exterior to *A*”. Second was Green’s type of divergence theorem which, while similar in mathematical form to Poisson’s, was understood at a far deeper level as physics. ‘Green’s functions’ were found for various cases with the help of his theorem; their symmetrical form (pp. 37-39) launched what have become known as ‘reciprocity relations’.

In the second part of the book Green gave a variety of applications and examples of his theorems in electrical situations. The first one was the ‘Leyden phial’, both on its own and with several deployed ‘in cascade’. A variant upon Poisson was to consider two electrified spheres joined by ‘an infinitely fine wire’, when some simple forms for potential functions were found; a harder case was taken with a ‘very thin spherical shell, in which there is a small circular orifice’. Green then analyzed phenomena involving ‘long metallic wires, insulated and suspended in the atmosphere’, for example, when joining two spheres. In all these cases Green presumed electrical conduction to be perfect; he then allowed for imperfection, which he likened to the effect of friction in mechanics. His main example concerned the production of magnetism within a rotating body.

Green passed on to magnetism proper for the last Part of his book. He first determined potential for ‘a very small body’, perhaps like the monopole which had served a major role in Poisson’s analysis. He then imitated Poisson in studying the ‘magnetic state’ of any body, using his eponymous theorem to convert the potential function to a surface integral. His last case dealt with magnetism in ‘cylindric wires’, where differential operators and complex variables led him to a closed form for density from which he could calculate values to set against the experimental data found by C.A. Coulomb as recorded in J.B. Biot’s *Traité de physique *(1816).

The *Essay* was privately published by subscription in a small provincial town, and its appearance went almost unnoticed. Even Green himself mentioned it only twice in his subsequent writings, and gave no indication of its importance. After his death in 1841 it might well have been forgotten. Fortunately, however, Green had given a few copies of his book to the Cambridge coach William Hopkins, who gave two copies to his student William Thomson (1824–1907), later Lord Kelvin, in 1845. The young man immediately recognized the book’s importance. Shortly afterwards Thomson travelled to Paris, where he met and befriended Joseph Liouville. Thomson presented a copy of the *Essay* to Liouville, which, he wrote, ‘causes a great sensation here, Chasles and Sturm find their own results and demonstrations in it.’ Thomson also arranged for its republication, not in Liouville’s *Journal de Mathématiques*, but in August Crelle’s *Journal für die reine und angewandte Mathematik*, where the three Parts appeared in 1850, 1852 and 1854.

“When Green’s book became well known it was clear that he had brought in a new phase the branch of mathematics that became known after him as ‘potential theory’. Prior to the book various figures, especially Laplace and Lagrange and a few special results due to C.F. Gauss, had produced results usually concerning equipotential surfaces, extending those for potentials at points on principal axes of a body as established by Isaac Newton. Poisson had gone somewhat further, and more than he realised, with his ‘simplifying’ theorem. Now after Green’s book the subject possessed new ways to handle many kinds of phenomena in mechanics and physics involving continuous bodies” (*Landmark Writings*, p. 410).

“Not only Green's insights and results were used by his successors; his own work, especially the *Essay*, were made available *four times *in the last thirty years of the 19^{th} century to an extent surpassing all other literature of his own time. The edition of his works by N. Ferrers appeared in London in 1871, and was reprinted in facsimile in 1903 in (of all places) Paris. The *Essay* itself was also reprinted in facsimile, in 1890 in Berlin, in a series of classic reprints of science; five years later it appeared in an annotated German translation by A. van Oettingen and A. Wangerin, in Wilhelm Ostwald's famous booklet series of editions of major scientific works. Green's successors in the classical phase not only absorbed his contributions into their own heritage; they wanted to read the words of the master himself. Their modern successors have maintained the tradition; for Ferrers’s edition appeared again in 1970, and the *Essay* itself in 1993, in the university of his home town Nottingham, as part of their bicentennial celebrations of their remarkable citizen” (Grattan-Guiness, pp. 394-5).

“George Green, born in 1793, was the only son of a Nottingham baker. George Green senior, semi-literate but with a head for business, prospered sufficiently to build a new windmill at Sneinton, a village just outside the town boundary, and grind his own corn for the bakery. Some years later he built a substantial family house next to the mill, where George Green and his parents went to live. George Green was a reluctant miller, but he had no choice in the matter until his father died in 1829, and he found himself sufficiently affluent to be able to dispose of the milling business and devote himself to mathematics. This had been his passion since his youth — so much so that his father had sent him at the age of eight to the town’s leading academy, run by Robert Goodacre, an enthusiast for ‘the mathematics’ and for the popular science of the day. Young George stayed only four terms, by which time he had learnt all his masters could teach him, so he was set to work in his father’s bakery. From then until 1823, over twenty years later, there is little information on Green, and none on his personal development.

“In that year, George Green joined the Nottingham Subscription Library. There he had access to books and also to journals and periodicals. Equally important was the fact that membership of the Library eased the thirty-year-old miller out of his intellectual isolation and the company of manual workers, of farmers and shopkeepers, millers and stableman, into a different stratum of society: that of gentlemen of leisure, of professional men such as clergymen and doctors, and well-to-do businessmen. For them the Library, situated in Bromley House in the centre of Nottingham, functioned as a gentlemen’s club, where politics and public affairs were debated and current topics of social and scientific interest were discussed.

“Association with these people and the use of the Library facilities were beneficial for Green. In five years he had written his first work, ‘An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism’. This was published in 1828 by private subscription and more than half the fifty-odd subscribers were members of Bromley House. The Essay attracted scant attention, however, and despondently, Green returned to his milling, until in 1830 he came into contact with Sir Edward Bromhead, a Lincolnshire landowner and mathematics graduate of Gonville and Caius College Cambridge. Bromhead recognized the originality of Green’s work and encouraged him to resume his mathematical studies. As a result Green wrote three papers which Bromhead sponsored for publication in the Transactions of the Cambridge Philosophical Society and the Royal Society of Edinburgh. Finally, in 1833, Green, at the age of forty, enrolled as an undergraduate in Bromhead’s own College*, *Caius. Green took the Mathematical Tripos in 1837, emerging as fourth wrangler, and two years later he was elected into a College Fellowship. During these two years he published six more papers in the Transactions of the Cambridge Philosophical Society, of which he became a member. Ill health compelled him to leave Cambridge some two terms after his election. He returned to Nottingham, where he died in May 1841 at the age of forty-seven. In his early twenties, Green formed a relationship with the daughter of his father’s mill manager, Jane Smith. She bore him seven children, the last just thirteen months before his death. They never married, though Jane and the children were known by the name of Green” (Cannell, pp. xxv-xxvii).

4to (280 x 221 mm), pp. viii, [2], 72. Disbound, uncut, preserving the fragile original front printed wrapper. Custom half calf clamshell box with gilt spine lettering.

Item #5637

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Price:
$88,500.00
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