Content uploaded by Gareth A. Jones
Author content
All content in this area was uploaded by Gareth A. Jones on Apr 17, 2020
Content may be subject to copyright.
The Life and Work of Sophie Germain
Natascha Hall, Mary Jones and Gareth Jones
School of Mathematics
University of Southampton
Southampton SO17 1BJ
United Kingdom
J.M.Jones@maths.soton.ac.uk
G.A.Jones@maths.soton.ac.uk
Address for correspondence:
Prof. G. A. Jones
School of Mathematics
University of Southampton
Southampton SO17 1BJ
United Kingdom
Short title: Sophie Germain
Abstract:
We discuss the life of Sophie Germain, her struggle to educate herself and to win acceptance
within the French mathematical community, and her contributions to number theory and
to the theory of elasticity.
1
One of the main difficulties in persuading women to take up careers as mathematicans
is the shortage of role models: students are presented with mathematics as a sequence of
achievements obtained almost exclusively by men, and it takes considerable self-confidence
for a talented female student to imagine herself contributing significantly to the develop-
ment of the subject. For this reason, it is important to draw attention to those women like
Hypatia, Agnesi, Kovalevskaya and Noether whose names have become part of the history
of mathematics, and to show how in many cases they needed to overcome considerable
obstacles in order to become successful mathematicians.
A classic example of this is the life of Sophie Germain who, despite initial family resis-
tance and a society structured so as to make women’s careers almost impossible to sustain,
nevertheless made fundamental contributions in both pure and applied mathematics.
She was born in Paris on 1st April 1776, and through her father, a wealthy silk
merchant and elected deputy, she was exposed at an early age to political and philosophical
discussion. By the time she was 13, the French Revolution was under way, and the streets of
Paris were no place for a young girl. Forced to remain indoors, she immersed herself in her
father’s library, teaching herself Latin and Greek in order to read the older books. One book
which particularly fascinated her was Montucla’s Essais Historiques sur la Math´ematique,
especially the section describing how Archimedes was killed by an invading Roman soldier
because he was too engrossed in a geometric diagram to obey the soldier’s commands.
Clearly, she thought, there must be something special about a subject which could lead to
such a fatal obsession.
At this time, educated women in France were expected to be able to make polite
conversation about philosophical, scientific and mathematical ideas, but they were not
considered capable of understanding these subjects in any depth. This patronising attitude
is illustrated by Algarotti’s Sir Isaac Newton’s Philosophy Explained for the Use of the
Ladies, in which an aristocratic young woman and her tutor discuss the inverse square law
by analogy with love diminishing as mutual separation increases. Sophie Germain’s family
initially discouraged her embarrassingly unladylike enthusiasm for mathematics, and she
was forced to study secretly at night, reading by the light of stolen candles, and wrapped
in blankets against the cold that froze the inkwells. Faced with such determination, her
family relented, and indeed her father supported her financially for the rest of her life.
In 1794 the ´
Ecole Polytechnique opened in Paris. Of course, it did not accept women,
so Sophie Germain took advantage of its otherwise enlightened teaching methods by as-
suming the identity of a former student Antoine-Auguste Le Blanc in order to get copies
of the printed lecture notes and to submit work for assessment. Before long Lagrange,
lecturing on Analysis, noticed with amazement a spectacular improvement in Le Blanc’s
normally undistinguished work, and called the student in for a meeting. Showing a lack
of prejudice which was extraordinary for this period, he was not only pleasantly surprised
to learn her identity, but also became her friend and academic mentor. He introduced her
to specific areas of mathematics, especially number theory, where she read the works of
Fermat, Legendre and later the new genius Gauss. However, despite this encouragement,
both the social conventions of the time and her retiring nature prevented her from enjoying
the full mathematical education she clearly desired. For an ambitious young man from a
poor family, like Poisson, there was Napoleon’s new educational system, and the intellec-
2
tual salons would have been open to a woman from a more aristocratic background, but
for the daughter of a merchant these doors were closed.
One problem which she studied was Fermat’s Last Theorem (FLT), his famous asser-
tion, written without proof in the margin of his copy of Diophantus’s Arithmetica some
time in the 1630s, that there are no positive integer solutions of the equation xn+yn=zn
for integers n > 2. Proving this was one of the great challenges of mathematics. Fermat
having dealt with the case n= 4, it was sufficient to prove it in the cases where nis an
odd prime. Euler proved it (more or less) for n= 3 in 1753 but since there are infinitely
many primes, treating them one at a time was clearly inadequate; a different approach was
needed, one which could deal with sets of exponents rather than individual values.
In 1804 Sophie Germain solved FLT in the case where n=p−1 for some prime p≡7
mod (8). She sent this result to Gauss, who had published his Disquisitiones Arithmeticae
in 1801, but fearing that he would not take a woman mathematician seriously, she again
adopted the pseudonym Le Blanc. Gauss responded favourably, and she continued to
correspond with him, maintaining her disguise until 1806 when, worried that Gauss might
meet the fate of Archimedes at the hands of Napoleon’s army, she used her friendship with
General Pernety to protect him. Learning the identity of his correspondent and guardian
angel, Gauss (like Lagrange, not sharing the prejudices of his time) replied with effusive
compliments for her mathematical ability and for her courage in overcoming the obstacles
which society placed in the path of women.
In trying to prove FLT, one may assume that x, y and zare mutually coprime. When
nis prime this implies that either none or only one of x, y and zis divisible by n, and
these are traditionally called Cases I and II of FLT. This distinction is important, since
the methods involved in the two cases are usually different, with Case I rather easier. In
1808 Germain sent Gauss her proof of Case I of FLT for n= 5, but this time she received
no reply: recently appointed Professor of Astronomy at G¨ottingen, Gauss was losing his
enthusiasm for number theory, and with the loss of his encouragement her interests began
to turn towards applied mathematics.
That year, the physicist Chladni visited Paris and demonstrated what we now call
Chladni figures, symmetric patterns on vibrating plates, revealed by scattering sand on
them. Encouraged by Napoleon, the Institut de France announced a competition, with
a prize of a kilogram of gold for an essay explaining these phenomena. Sophie Germain
began to study elasticity, reading Lagrange’s M´ecanique Analytique and Euler’s work on
the vibrations of elastic rods. In 1811, although hers was the only solution submitted, it
was marred by errors and omissions caused by her lack of access to contemporary teaching
at the ´
Ecole Polytechnique in techniques such as the calculus of variations; no prize was
awarded, and the deadline was extended by two years. Lagrange, one of the judges,
corrected some of her errors, and suggested that small deflections wsatisfy
k∂4w
∂x4+ 2 ∂4w
∂x2∂y2+∂4w
∂y4+∂2w
∂t2= 0,
where xand yare local Euclidean coordinates in the plate, tis time, and kis a constant.
In her second attempt, in 1813, she showed that Lagrange’s equation explained Chladni’s
patterns in some simple cases, but she could not derive it from physical principles, so she
3
was merely given an honourable mention. The deadline was extended again, and in 1816
the judges, Legendre, Laplace and Poisson, finally awarded her the prize. However, she
did not appear at the presentation, feeling that the judges had not taken her work seri-
ously. Certainly Poisson, a younger rival who had published his own paper on elasticity in
1814, resented her new approach which contradicted his molecular theory, and he was very
discouraging in his report on her work. This reluctance to acknowledge her achievements
in this area continued throughout her life and long after her death: for instance, when the
Eiffel Tower was built in 1889, her name was omitted from the list of 72 eminent mathe-
maticians, scientists and engineers displayed on it, despite her important contributions to
our understanding of the properties of metals. She also made important advances in the
differential geometry of surfaces: Euler had argued that the elastic force at any point in a
vibrating rod is proportional to the rod’s curvature at that point; in order to extend this
to 2-dimensional vibrating plates, she needed to introduce an analogous concept of surface
curvature, which she obtained by adding the two principal curvatures at each point, or
equivalently by integrating all the curvatures at that point, to obtain the mean curvature.
Encouraged by Fourier, who was no friend of Poisson’s, she began to take a more
active part in Parisian scientific life. She was the first woman, other than the wife of a
member, to attend lectures at the Academy. During the 1820s she published her results
on elasticity, and also returned to number theory, collaborating with the highly-respected
Legendre.
During this period she proved what is now called Sophie Germain’s Theorem, which
generalises her earlier result on the exponent 5 case of FLT. It states that FLT is true in
Case I for an odd prime exponent nif there is an auxiliary prime psuch that
(a) xn+yn+zn≡0 mod (p) implies xyz ≡0 mod (p), and
(b) nis not an n-th power mod (p).
In particular, this proves Case I of FLT if 2n+ 1 is a prime p, since conditions (a) and
(b) now follow easily from the fact that all n-th powers are congruent to ±1 or 0 mod (p);
such primes nare called Sophie Germain primes. Legendre extended this to the cases
where kn + 1 is prime for k= 4,8,10,14 or 16, enabling him to prove Case I of FLT for
all primes n < 197. It is not known whether there are infinitely many Sophie Germain
primes, but there are 26569515 of them less than 1010 . In 1985 Adleman, Heath-Brown
and Fouvry, adding modern analytic methods to those of Germain and Legendre, showed
(non-constructively) that there are infinitely many primes nfor which Case I is true. Of
course, most of this has now been superseded by Wiles’s complete proof of FLT, published
in 1995. Nevertheless, Sophie Germain’s Theorem has an important place in the history
of this problem, as a major step forward in progressing from individual cases to a more
systematic approach.
Sophie Germain did not publish her theorem herself: instead, it appeared, with a
graceful acknowledgement by Legendre, in a paper of his in 1823, reprinted as a supplement
to the second edition of his Th´eorie des Nombres in 1825. (This also contained his proof
of FLT for n= 5, simultaneous with Dirichlet’s.) Perhaps because of this modesty, her
contributions to number theory remained virtually unknown for many years. Indeed,
during the next century and a half, a number of mathematicans published results on
FLT which subsequently turned out to be special cases of her theorem (see Ribenboim’s
4
delightful book [9] for details). In recent years, interest in the solution of FLT has caused
greater awareness of her achievements, but she received little recognition during her life,
apart from the admiration of a few great figures such as Lagrange, Legendre and Gauss.
She died of breast cancer on 27th June 1831, and in 1837, when the University of G¨ottingen
celebrated its centenary by awarding honorary degrees, Gauss deeply regretted that she
was no longer alive to receive one. Dunnington [3, p. 68] quotes his opinion that “She
proved to the world that even a woman can accomplish something worthwhile in the
most rigorous and abstract of the sciences and for that reason would have well deserved
an honorary degree.” Her Oeuvres Philosophiques, including an unfinished philosophical
essay and some of her letters, were published in 1879 [6].
In Paris one can find a street, a hotel and a school named after Sophie Germain.
The courtyard of the latter contains a rather graceful statue of her, used to illustrate a
short biographical article by Dahan Dalm´edico [2], which curiously omits Sophie Germain’s
Theorem. This, together with many other related results, is discussed in Ribenboim’s very
accessible treatment of FLT [9]; for a popular and undemanding introduction to the history
of this problem (ancient and modern) see Singh [10], for a concise account suitable for
students see Laubenbacher and Pengelley [8], and for an encyclopedic account (pre-Wiles)
see Edwards [4]. Bucciarelli and Dworsky [1] discuss Sophie Germain’s contributions to
elasticity, also covered briefly (along with those of Poisson and others of this period) in
Timoshenko’s history of the subject [11]. Gillispie’s biography of Laplace [7] gives an
excellent picture of the French scientific world and its leading personalities during this
period, while Dunnington [3] gives a detailed account of Gauss’s life and work. For up-to-
date information on Sophie Germain primes, see Caldwell’s Prime Pages website,1and for
the 72 savants, visit the Eiffel Tower.2
Principal related characters
Ernst Florens Friedrich Chladni, 1756–1827.
Leonhard Euler, 1707–1783.
Pierre de Fermat, 1601–65.
Jean-Baptiste-Joseph Fourier, 1768–1830.
Carl Friedrich Gauss, 1777–1855.
Joseph-Louis Lagrange, 1736–1813.
Pierre-Simon Laplace, 1749–1827.
Adrien-Marie Legendre, 1752–1833.
Sim´eon-Denis Poisson, 1781–1840.
1http://www.utm.edu/research/primes/lists/top20/SophieGermain.html
2http://www.tour-eiffel.fr/teiffel/fr/documentation/dossiers/page/savants.html
5
References
[1] Bucciarelli, L. L., Dworsky, N.: Sophie Germain: An Essay in the History of Elasticity.
Boston Dordrecht: Reidel 1980
[2] Dahan Dalm´edico, A.: Sophie Germain. Scientific American. December 1991, 76–81
[3] Dunnington, G. W.: Carl Friedrich Gauss: Titan of Science. New York: Hafner 1955
[4] Edwards, H. M.: Fermat’s Last Theorem. New York: Springer 1977
[5] Germain, S.: Recherches sur la Th´eorie des Surfaces ´
Elastiques. Paris 1821
[6] Germain, S.: Œuvres Philosophiques (ed. Stupuy). Paris 1879
[7] Gillispie, C. C.: Pierre-Simon Laplace. Princeton: Princeton University Press 1997
[8] R. Laubenbacher, R., Pengelley, D.: Mathematical Expeditions: Chronicles by the
Explorers. New York: Springer 1998
[9] Ribenboim, R.: Fermat’s Last Theorem for Amateurs. New York: Springer 1979
[10] Singh, S.: Fermat’s Last Theorem. London: Fourth Estate 1997
[11] Timoshenko, S. P.: History of Strength of Materials. New York: McGraw-Hill 1953;
reprinted Dover 1983
6
Chronology
1776: born 1st April, Paris; parents Ambroise-Fran¸cois and Marie-Madelaine.
1789: reads about Archimedes’s death in Syracuse in Montucla’s History of Mathe-
matics.
1789–95: reads Bezout, Newton, Euler, etc. in father’s library.
1795: ´
Ecole Polytechnique founded; gets Lagrange’s analysis lecture-notes; submits
work to Lagrange under name Antoine-August Le Blanc.
1798: Legendre publishes Th´eories des Nombres.
1801: Gauss publishes Disquisitiones Arithmeticae.
1804: writes to Gauss (as Le Blanc), solving FLT for n=p−1, prime p≡7 mod (8);
Gauss responds favourably; further correspondence (12 letters).
1806: asks army commander Pernety to protect Gauss, who discovers her identity;
Gauss delighted.
1808: writes to Gauss about FLT n= 5, shows x, y or zdivisible by 5 (i.e. proves
case I); no reply, Gauss now professor of astronomy at G¨ottingen.
1808: Chladni demonstrates vibrating plates in Paris.
1809: Academy of Science offer gold medal for explaining vibrations, deadline 1811.
1811: only contestant in competition; work incomplete, no award, but Lagrange cor-
rected and developed her work (4th-order PDE); competition extended by two years
1813: submitted new entry, showing Lagrange’s PDE sometimes worked; honourable
mention, no medal.
1814: Poisson’s paper on elasticity.
1815: Germain’s essay on elasticity submitted, judges Legendre, Laplace, Poisson
award prize; misses ceremony.
1820s: collaboration with Legendre on number theory.
1825: Dirichlet and Legendre prove FLT for n= 5.
1827: 2nd ed. of Legendre’s Th´eorie des Nombres includes Germain’s work on FLT,
proving case I for primes n < 100; “very ingenious, quite simple, and of an almost absolute
generality”; Legendre extends to n < 197.
1830: Gauss recommends honorary degree, G¨ottingen???
1831: dies of breast cancer, 27 June, 1831, Paris, aged 55.
1832: Lam´e proves FLT for n= 14.
1839: Lam´e proves FLT for n= 7.
1847: Kummer proves FLT for regular primes.
1993: Wiles announces proof of FLT.
1995: Wiles publishes proof of FLT.
7
Related figures
Ernst Chladni.
Pierre de Fermat, 1601–65.
Jean-Baptiste-Joseph Fourier, 1768–1830.
Carl Friedrich Gauss, 1777–1855.
Ernst Eduard Kummer, 1810–1893.
Joseph-Louis Lagrange, 1736–1813.
Gabriel Lam´e, 1795–1870.
Adrien-Marie Legendre, 1752–1833.
Joseph-Marie Pernety.
Sim´eon-Denis Poisson, 1781–1840.
Fermat’s Last Theorem
FLT. There are no positive integer solutions of an+bn=cnfor integers n≥3.
Fermat did n= 4, so sufficient to do odd primes n. Can also assume a, b, c are coprime.
Case I. nis prime, and does not divide a, b or c.
Case II. nis prime, and divides one of a, b or c.
Theorem. If nis prime and 2n+ 1 is prime, then FLT is true in case I for exponent n.
Primes nsuch that 2n+1 is prime are Germain primes, e.g. n= 3,5,11 but not 7,13.
This generalises to:
Sophie Germain’s Theorem. If nis an odd prime, and there is a prime psuch that
(i) xn+yn≡znmod (p)implies x, y or z≡0mod (p),
(ii) xn≡nmod (p)has no solutions,
then FLT is true in case I for exponent n.
For example, n= 7, p = 29. Germain used this to prove case I for odd prime n < 100.
8
