## George Green, Mathematician. 1793—1841.

BY EDITH M. BECKET.

I wish to acknowledge my indebtedness to the Rev. F. C. Finch, M.A., formerly Vicar of S. Alban's, Nottingham, for drawing my attention to the mathematician of Sneinton, and for information concerning Green which he gave to me ; also to Dr. Piaggio, Professor of Mathematics, University College, Nottingham, for writing the concluding paragraph of this article, and for the loan from his departmental library of a copy of Green's Mathematical Papers.

THERE is one outstanding mystery connected with the education of Nottingham
in the early 19th century of which the complete elucidation will probably
never be reached. As the traveller enters Nottingham by the Midland railway
line from the north-east, he catches a glimpse of a grey old windmill, standing
away on an eminence on his right, a solitary survivor of bygone days, now
almost hidden by rows of modern villas. There, in the days when Sneinton,
remote from the busy life of the town, was a peaceful village of old-world
cottages and little homesteads nestling together on the wind-swept hillside
overlooking the wide vale of Trent, was spent the youth and early manhood
of one of the most original mathematicians of the 19th century, George Green,
who has been described as "standing head and shoulders above all his
companions in and outside the University. George Green was born in 1793,
at Sneinton, where his father, a miller of considerable property, lived at
the mill already mentioned, then the easternmost of three mills situated
on the hill; he also carried on business as a baker in Nottingham. George
worked with his father, until the death of the latter in 1829. It may be
assumed that the younger Green was a man of considerable general education,
for in 1823 he became a member of the Nottingham Subscription Library, already
located in Bromley House.^{1 }In 1828, there was published in Nottingham,
by private subscription, the work which ultimately made Green's name famous
for all time, *An essay on the application of mathematical analysis to
the theories of Electricity and Magnetism, *which Dr. Ferrers referred
to as "perhaps the most important of Green's papers." In this
work the term *potential *was introduced to denote the result obtained
by adding together the masses of all the particles of a system, each divided
by its distance from a given point. The essay, which consists of three parts,
considers in the first the properties of this function, and in the second
and third parts applies them to the theories of magnetism and electricity
respectively.^{2} It is clear from the preface which Green himself
wrote to this work that he had access to the writings of French mathematicians;
for he refers to "two memoirs of singular elegance, relative to the
distribution of electricity on the surface of conducting spheres previously
electrified and put in the presence of each other," presented in 1812
to the French Institute by M. Poisson; further on he refers to other "memoirs" of
the same writer, also to observations made by M. Arago; he acknowledges familiarity
with the investigations of M. Fourier, with work jointly carried out by MM.
Cauchy and Poisson; and he makes mention of the work of the English mathematicians,
Cavendish and Dr. T. Young.^{3} How did Green, the son of a miller
and baker, following his father's calling, living his life on the outskirts
of a provincial town, in the early days of the 19th century when means of
intercommunication between different parts of the country-side were largely
undeveloped, become acquainted with the works of French mathematicians? Such
works were not to be found in Bromley House Library. Dr. Ferrers describes
Green as "an
almost entirely self-taught mathematical genius." But what was the starting
point of his investigations? Who were the friends who were able to appreciate
the power shown in his first Paper, and who assisted in the publication of
it in Nottingham in 1828? Green certainly became connected in some way with
Sir Edward Ffrench Bromhead, whose family seat lay at Thurlby Hall, between
Newark and Lincoln, and who took his degree at Cambridge in 1812. How a friendship
sprang up between the two men remains unexplained. It has been suggested
that Sir E. F. Bromhead lent books to Green,^{4} and he seems to
have encouraged him to go to Cambridge.^{5} There Green entered Caius,
Sir E. F. Bromhead's own College. Green, in his Preface to his Essay published
in 1828, wrote that he had been obliged to obtain the little knowledge which
he possessed at such intervals and by such means as other indispensable avocations,
which offered but few opportunities of mental improvement, afforded.^{6} The
existing Register of the Free Grammar School for Boys dates only from 1809;
it is impossible, therefore, to say whether Green attended the Grammar School
in Stoney Street, or not. It is probable, however, that he did not, for with
the exception of the private pupils of the head master, the pupils of the
school were almost entirely sons of the burgesses of Nottingham,^{7} and Sneinton,
in the days of Green's boyhood, was far outside the borough. After his father's
death in 1829, Green disposed of the business and prepared to go to Cambridge.
In 1832, a Paper entitled *On the Laws of the Equilibrium of Fluids analogous
to the Electric Fluid, *written by Green, was laid before the Cambridge
Philosophical Society by Sir Edward Ffrench Bromhead, and later, another *On
the determination of the attractions of Ellipsoids of variable Densities, *before
the same Society, by the same friend of the author.^{8} A copy of the former
was presented by Green to the Nottingham Subscription Library in 1833.^{9} In
that year he contributed to the Royal Society of Edinburgh a Paper *On
the Vibrations of Pendulums in Fluid Media? *Green, then a man of middle
life, commenced residence in Cambridge, in October, 1833. In 1837 he was
Fourth Wrangler in the mathematical tripos; although only Fourth, Green was
considered the best man of his year, but "want of familiarity with boy's
mathematics prevented him from coming to the top in a time-race."^{10} In
that year he read his Paper *On the Motion of Waves in a variable canal
of small depth and width, *to the Cambridge Philosophical Society. He
continued his studies and did much valuable work on the Theory of Sound and
the Dynamical Theory of Light, producing Papers *On the Reflexion and
Refraction of Sound *and *On the Reflexion and Refraction of Light
at the common surface of two non-crystallized media. *In 1839 he was
elected a Perse Fellow of Caius College, and in the same year he wrote *On
the Propagation of Light in Crystalline Media?*

It seems reasonable to suppose that throughout his years of residence at Cambridge, Green from time to time visited Sneinton, for when his health gave way he settled there in Notintone Place, in those days still a superior residential quarter, within five minutes' walk of the old mill. There, on May 31st, 1841, he died. A plain, flat stone slab marks the spot where his body was laid to rest in S. Stephen's churchyard. In the forties of last century, Nottingham was the home of few mathematicians, and so little was the importance of Green's work appreciated locally, that beyond a short obituary notice of some fourteen words, the local newspaper made no further reference to him at the time of his death. His fame, however, is no doubt much greater now than in his life, and his ideas have proved much more fruitful as years passed than they probably appeared to his contemporaries. Green's most valuable researches were fated for long to escape the notice not only of foreign, but also of British mathematicians, and "it is a singular fact in the history of science that all his general theorems were rediscovered by Lord Kelvin, Chasles and Sturm, and Gauss."

Of Green's direct descendants, who might presumably be able to throw light
on the mathematician's early history, none are known to be living. While
still a young man, Green fell in love with Miss Jane Smith, but his father
is said to have threatened to disinherit him if he married her. She became,
however, the mother of his seven children. One son, another George Green,
went to St. John's College, Cambridge. The Register of the College gives
12th July, 1829, as the date of his birth, omits his father's name, but gives
his mother's name as Jane Smith, and shows him to have been recommended by
the Rev. W. H. Wyatt, M.A., Vicar of Sneinton. A surviving member of Green's
year describes the mathematician's son in the following terms:—"He
was by far the most impressive in his appearance of all the freshmen of his
year. He had a broad and lofty forehead and a grave and dignified manner.
He must have been about 30 years of age and had been a schoolmaster before
he came up." This George Green was 3rd Senior Optime in 1859. One or
two of his contemporaries recollect him as very reserved, not seeking companionship
and keenly disappointed with his degree, as he hoped to be a wrangler. Apparently
none of his Cambridge acquaintances kept in touch with him or heard what
became of him after he left Cambridge. For a time he seems to have lived
with his sisters at Sneinton.^{11} The name of this George Green
appears on the stone which marks the family grave in S. Stephen's churchyard
at Sneinton; the year 1870 is there given as the date of his death.

Clara,^{12} the youngest child of the mathematician, the last survivor of her
family, died in 1919. Papers, books and mathematical instruments, formerly
the property of her distinguished father, are said to have been in Miss Green's
possession at the time of her death, but they have mysteriously disappeared.
As Miss Green died intestate, the old mill-house and some acres of surrounding
land became the property of the Crown. This was offered for sale and purchased
in November, 1921, by Mr. Oliver W. Hind, who intends to preserve the mill,
the brickwork of which is in remarkably good condition, and to portion out
into gardens the part of the land previously uncultivated.

The extinction of the family of the great mathematician, and the disappearance within the last few years of his books and papers, have finally destroyed the hope of discovering either the sources whence Green drew his early inspiration or the means by which his mathematical genius was brought to light and developed.

The importance of Green's work seems to increase rather than diminish, with
the lapse of time. In Potential, Electricity and Hydrodynamics, Green's Theorem
connecting surface- and volume-integrals, takes a prominent place.^{13} Green's
Lemma (also called Green's Theorem in two dimensions) connecting line- and
surface-integrals is important in the Theory of Functions.^{14 }Recent works
on the Conduction of Heat, Differential Equations and Integral Equations,
introduce Green's Functions.^{15} But in this connection Green's ideas were
neglected until fairly recently, so that they still appear too novel for
their importance to be fully appreciated.

(1) Minute Book, N.S.L.

(2) Geo. Green, Mathematical Papers, Preface, p. 6

(3) Geo. Green, Mathematical Papers, pp. 1—8.

(4) Manuscript Notes of the Rev. F. C. Finch, M.A.. formerly Vicar of St. Alban's,
Sneinton.

(5) Letter of Mr. R. F. Scott, M.A., Master of S. John's College, Cambridge,
Aug. 17th, 1920.

(6) Geo. Green, *op. cit., *p. 8.

(7) S*. *Corner. *The Forester. *July, 1905, p. 67 and p. 68.

(8) N. M. Ferrers, *Preface to Mathematical Papers of the late George Green, *pp.
6, 7.

(9) Minute-book of Nottingham Subscription Library.

(10) Dictionary of
National Biography.

(11) Recollections of an old Sneinton resident.

(12) At one time, governess in the family of the Rev. W. H. Wyatt.

(13) For this and some other work due to Green, such as Green's Equivalent Layer,
Green's Theorem on the discontinuity in field on crossing a boundary with a surface
distribution, Green's method of solving condenser problems, etc.; see Routh's *Analytical
Statics, *Vol. II.

(14) See Goursat's or De la Valtee Poussin's *Cours d'Analyse.*

(15) See Carshaw's *Mathematical Theory of the Conduction of Heat.*