GREEN, George.

I. On the reflexion and refraction of sound. Offprint from: Transactions of the Cambridge Philosophical Society, Vol. VI, Part III. [Bound with:] II. On the motion of waves in a variable canal of small depth & width. Offprint from ibid., Vol. VI, Part III. [And with:] III. On the laws of reflexion and refraction of light at the common surface of two non-crystallized media. Offprint from ibid., Vol. VII, Part I. [And with:] IV. Note on the motion of waves in canals. Offprint from ibid., Vol. VII, Part I. [And with:] V. Supplement to a memoir on the laws of reflexion and refraction of light. Offprint from ibid., Vol. VII, Part I. [And with:] VI. On the propagation of light in crystallized media. Offprint from ibid., Vol. VII, Part II.

Cambridge: Printed at the Pitt Press by John W. Parker, 1838-38-38-39-39-40.

First edition, extremely rare offprints, all inscribed in the author’s hand, of the last six published papers of George Green – he published only nine scientific journal articles in total, in addition to his famous book An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism (1828), “one of the major publications in the mathematics of the 19th century” (Grattan-Guinness, p. 403). “In his Cambridge papers, Green abandoned the topics of electricity and magnetism and made original contributions in hydrodynamics, sound and light” (Cannell, p. xxvii). In paper III, Green is the first to enunciatethe law of conservation of energy – nine years before its appearance in Hermann Helmholtz’s Über die Erhaltung der Kraft. In paper II, he anticipates the WKB approximation method (named after the 20^th century physicists Gregor Wentzel, Hendrik Kramers, and Leon Brillouin) by almost a century. Despite its importance, the Essay initially attracted little attention, and following its publication Green returned to his original occupation as a miller in the provincial town of Nottingham. In 1830, however, “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 to a College Fellowship. During these two years he published six more papers in the Transactions of the Cambridge Philosophical Society [the six papers offered here], of which he became a member” (Cannell, p. xxvii). “Green’s major contribution to mathematical science after that of the Essay of 1828 is to be found in his three papers on light: ‘On the laws of reflexion and refraction of light at the common surface of two non-crystallized media’, ‘Supplement to a memoir on the laws of reflexion and refraction of light’, and ‘On the propagation of light in crystallized media’ [III, V, VI]. These publications were his last and indicate that Green’s intellectual powers in his forties were as strong as ever” (Cannell & Lord, p. 35). Indeed, in paper III Green made a major theoretical advance. “For the purpose of explaining the propagation of transversal vibrations through the luminiferous ether, it becomes necessary to investigate the equations of motion of an elastic solid. It is here that Green for the first time enunciates the principle of the Conservation of Work, which he bases on the assumption of the impossibility of a perpetual motion. This principle he enunciates in the following words: ‘In whatever manner the elements of any material system may act upon each other, if all the internal forces be multiplied by the elements of their respective directions, the total sum for any assigned portion of the mass will always be the exact differential of some function.’ This function, it will be seen, is what is now known under the name of Potential Energy, and the above principle is in fact equivalent to stating that the sum of the Kinetic and Potential Energies of the system is constant” (Ferrers, pp. vii-viii). Several, perhaps all, of these offprints were published before their appearance in the Transactions. For example, offprint VI is dated 1840 on the title page, but Vol. VII, Part II of the Transactions, in which the paper appears, is dated 1842. OCLC lists between one and four copies of each offprint worldwide: I: Senate House, University of London; National Library of Wales; Strasbourg, Turin. II: Senate House, University of London. III. BNF; Edinburgh; Trinity College, Cambridge; Yale. IV. Trinity College, Cambridge. V. Yale. VI. Trinity College, Cambridge; Strasbourg. There are also copies of all the offprints at the University of Nottingham.

Provenance: Each offprint inscribed in Green’s hand, ‘Revd. J J Smith with the authors Complements’. Smith was a fellow and tutor at Gonville and Caius College, Cambridge, where Green himself was a student and later fellow. Smith compiled a catalogue of the manuscripts in the college library, which was published in 1849.

“George Green, born in 1793, was the only son of a Nottingham baker. George Green senior, semi-literate but with a head for business, who 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” (Cannell, pp. xxv-xxvi).

“In 1833, when he was forty, Green decided he would go to Cambridge. ‘Several kind and respected friends were anxious that he should adopt an University education several years before that circumstance actually took place’, wrote William Tomlin in 1845. Green’s father had died in 1829 leaving him free and with sufficient resources to sell the milling business and devote his time to mathematical study. He was now sufficiently affluent to aspire to the status of gentleman and academic, financing himself for the six years he was now to spend in Cambridge. In October 1833 he entered Bromhead’s own college of Gonville and Caius and was soon recognized for his mathematical ability. ‘He stood head and shoulders above all others in the University’, wrote a contemporary student, Harvey Goodwin, and was expected to head the final examination list as Senior Wrangler. In the event, in January1837, he became Fourth Wrangler (the colourful algebraist James Joseph Sylvester was Second). He was now forty-five, older even than the Tutors and Fellows in the College, and the contemporary of such eminent Cambridge figures as Whewell, Peacock, Airy and the great Cambridge coach, William Hopkins. Now a graduate with a first class degree Green stayed on at Caius hoping for a Fellowship. With examinations behind him he returned to his own researches and published six papers in the next two years [the six papers offered here]” (Cannell & Lord, pp. 33-34).

Two of these papers deal with water waves in canals. “The initial paper treats the motion of ‘small’ water waves in a thin, shallow rectangular canal of width β and depth γ. He obtains the relevant equation governing φ₀, the leading term of the disturbance, by a standard ‘expand and ignore the second order terms’ approach, but then allows that β, γ ‘be functions of x [the distance along the canal] which vary slowly [compared with the wave], so that … β = ψ(wx) where w is a very small quantity. ‘This time ignoring w² terms, in an approach which anticipates the WKB method by almost a century, he is able to complete his analysis showing that the height of the wave depends on β^-1/2γ^-1/4, on its length λ, on γ^1/2, and that its speed is (gγ)^1/2 [where g is the acceleration due to gravity]. He is spurred to a follow-up note on reading of ‘cases of propagation of what Mr Russell denominates the ‘Great Primary Wave’’, the earliest report of experimental work on what are now called solitons. He derives the modifications of the wave speed formula to (gγ/2)^1/2 for a triangular canal and to (gλ/2π)^1/2 for deep sea waves; just as Laplace’s adiabatic argument had corrected Newton’s isothermal formula for the speed of sound in air, so the latter formula corrects Newton's (gλ)^1/2/π for the speed of water waves. Green also points out for the first time that the particles in a deep sea wave execute circular motion” (ibid., pp. 46-47).

In ‘On the reflexion and refraction of sound,’ “Green investigates the phenomena that occur when waves travelling through one fluid encounter a different fluid, for example, sound waves passing from air to water. In order to expose an inadvertent, but manifest, error in Poisson's work, he deliberately concentrates on a special case: a plane wave and two media separated by an infinite plane. Simply by analysing the wave equation in fluids together with mathematically appropriate boundary conditions at the interface, he is able faithfully to reproduce the laws of reflection and refraction, deduce the intensities of reflected and refracted rays, and to explain fully the phenomenon of total internal reflection with, for elastic media, the accompanying change of phase which occurs. He notes that, in the latter case, his formulae tantalizingly agree with experimental observations on light polarized perpendicular to the plane of reflection.

“Light posed one of the most pressing problems to the physicists of the time. The experimental work of Young, Fresnel and others led to the supposition that light consisted of a transverse wave motion and the luminiferous ether was posited as that elastic (necessarily) solid medium which transmitted these vibrations: Maxwell’s electromagnetic theory was still twenty years away! In the end, all attempts to model the ether as a conventional substance had problems in reconciling theory to observation – Green's was no exception and thus controversial – but this revolutionary approach to obtaining the equations governing the vibrations of an elastic solid (remembered today in the ‘Cauchy-Green tensor’) has earned him a prominent place in the history of elasticity: in Love’s estimation [A treatise on the mathematical theory of elasticity, 1906], ‘The revolution which Green effected … is comparable in importance with that produced by Navier’s discovery of the general equations [of motion of fluids]’” (ibid., pp. 47-48).

“Light posed one of the most pressing problems to the physicists of the time. The experimental work of Young, Fresnel and others led to the supposition that light consisted of a transverse wave motion and the luminiferous ether was posited as that elastic (necessarily) solid medium which transmitted these vibrations: Maxwell’s electromagnetic theory was still twenty years away! In the end, all attempts to model the ether as a conventional substance had problems in reconciling theory to observation – Green's was no exception and thus controversial – but his revolutionary approach to obtaining the equations governing the vibrations of an elastic solid (remembered today in the 'Cauchy-Green tensor') has earned him a prominent place in the history of elasticity: in Love's estimation, ‘The revolution which Green effected … is comparable in importance with that produced by Navier's discovery of the general equations’ …

“It is convenient to consider his three papers on light [III, V, VI] together. Rather than assume some ad hoc physical hypothesis about the structure of the ether, he proceeds more abstractly:

‘The principle selected as the basis of the reasoning contained in the following paper is this: In whatever way the elements of any material system may act upon each other, if all the internal forces exerted be multiplied by the elements of their respective distances, the total sum for any assigned portion of the mass will always be the exact differential of some function.’ Although he does not phrase this in energy terms, this amounts to assuming the existence of a potential energy function φ for the ‘elastic energy’ stored in the medium for which the conservation of energy holds – the first time that this principle had been used. It is hard to gauge precisely how Green viewed this principle but he does seem to have regarded it as more than a mere mathematical convenience; rather tantalisingly, the desirable mathematical property that φ be an exact differential coincides with the physically desirable property that φ be a conservative field!

“He then assumes that φ can be expanded as a power series in the six linear and angular strain components and observes that, since at equilibrium the degree one terms vanish, the leading terms are those twenty-one terms of (total) degree two. All twenty-one may be needed for a ‘crystallized’ (= non-isotropic) medium, but any assumptions about spatial symmetry lead to a reduction, as far as two for a non-crystallized (= totally isotropic) medium. He does not stop to dwell on their physical meaning, but proceeds to analyse refraction and reflection phenomena in a similar manner to that he employed for sound; unfortunately, though, his conclusions do not match with observation as well as they might have done” (ibid., pp. 47-49).

“Green almost wholly shook off the molecular treatment, and worked out all that was to be worked out for the wave theory of light, by the dynamics of continuous matter. Indeed, I do not know that it is possible to add substantially to what Green has done in this subject. Substantial additions are scarcely to be made to a thing perfect and circumscribed as Green's work is, on the explanation of the propagation of light, of the refraction and the reflection of light at the bounding surface of two different mediums, and of the propagation of light through crystals, by a strict mathematical treatment founded on the consideration of homogeneous elastic matter Green's treatment is really complete in this respect, and there is nothing essential to be added to it” (Sir William Thomson, Baltimore Lectures on Molecular Dynamics and the Wave Theory of Light (1904), pp. 5-6).

“This succession of works secured Green’s election in October 1839 as Perse Fellow of Caius College, although he apparently made no significant contribution to academic life. It is reported that he set the problem papers for two college examinations but never lectured. In May 1840 he was in Nottingham; and it is doubtful that Green ever returned to Cambridge. His will, dated 28 July 1840, was probably written at Nottingham and confirms that he was in poor health, but no details of his illness are given. A codicil, added four months before his death, is his last known action. A locally published obituary, referring to Bromhead’s support, concluded that ‘had his life been prolonged, he might have stood eminently high as a mathematician’” (DSB).

“In any attempt to evaluate the corpus of Green’s published work, the Essay stands alone. It is a work of striking originality and he hit a rich vein. The later papers are arguably more mainstream and reflect more closely the specific concerns of the age, but Green’s hallmark is clear on them as well – a reluctance to make unwarranted physical assumptions (preferring instead more general mathematical hypotheses) and an eye for the incisive special case. Thus, one of his champions, Stokes, enthused [Lord Rayleigh, Scientific Papers, Vol. V, p. 176]:

‘Indeed Mr Green’s memoirs are very remarkable, both for the elegance and rigour of the analysis, and for the ease with which he arrives at most important results. This arises in a great measure from his divesting the problems he considers of all unnecessary generality: where generality is really of importance, he does not shrink from it.’

“And a century later, [Sir Edmund] Whittaker [A History of the Theories of Aether and Electricity (1989), p. 153] reflected enticingly:

‘Green undoubtedly received his own early inspiration from [the great French analysts] … but in the clearness of insight and conciseness of exposition he far excelled his masters, and the slight volume of his collected papers has to this day a charm wanting in their more voluminous writings’” (Cannell & Lord, pp. 49-50)

Honeyman 1546 (I-III, V, and two earlier offprints). Cannell, George Green, Mathematician and Physicist 1793-1841. The Background to His Life and Work, Second edition, 2001. Cannell & Lord, ‘George Green, Mathematician and Physicist 1793-1841,’ The Mathematical Gazette 77 (1993), pp. 26-51. Ferrers (ed.), Mathematical papers of the late George Green, 1871. Grattan-Guinness, ‘George Green, An Essay on the Mathematical Analysis of Electricity and Magnetism [sic]’, Chapter 30 in Landmark Writings in Western Mathematics 1640–1940, 2005.

Six vols. bound in one, 4to (271 x 214 mm). I. pp. [2], [403]-413, [1, blank]. II. pp. [2], [3]-8 (journal pagination 457-462). III. pp. [2], [1]-24. IV. pp. [2], [3]-11, [1, blank] (journal pagination 87-95). V. pp. [2], [3]-10 (journal pagination 115-120). VI. pp. [2], [1]-20 (journal pagination 121-140).

Item #6355

Price: $18,000.00