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| Linocut 'Sophie Germain' by Ele Willoughby, 2021 |
French mathematician, physicist and
philosopher Marie-Sophie Germain (1776 – 1831), known as Sophie, taught herself
mathematics using books in her father's library and by corresponding with
leading mathematicians of her day, including Lagrange, Legendre and Gauss,
initially using the pseudonym Monsieur LeBlanc since she knew it was unlikely
her contemporaries would take a woman in math seriously. She later wrote to
Gauss, the leading mathematician of her day that she used the pseudonym
"fearing the ridicule attached to a female scientist." Her work on
elasticity theory won her the grand prize from the Paris Academy of Sciences.
Her trailblazing work on Fermat's Last Theorem set out a strategy and framework
for mathematicians pursuing the problem for hundreds of years. She was denied a
formal mathematics education or professional standing due to her sex, but Gauss
argued she deserved an honourary degree (though it was never granted).
She was born to a bourgeois family; her
father is usually described as a wealthy silk merchant who was elected to the
États-Généraux (later the Constitutional Assembly), as a representative of the
bourgeoisie. By the time she was 13, the Bastille fell, and she remained
indoors for safety during those tumultuous times, her father's library her only
source of entertainment. She was fascinated reading about Archimedes' death at Roman
soldiers' hands during the siege of Siracusa, unable to tear himself away from
mathematics. It piqued her interest and she devoured the mathematics books she
found, even teaching herself Latin and Greek in order to read Sir Isaac Newton
and Leonard Euler's works. Her parents disapproved this sudden interest in a
subject deemed inappropriate for a woman and they denied her warm clothes or a
fire to study at night. She simply wrapped herself in quilts and studied by
candlelight, when it was so cold her ink froze, eventually winning over her
mother's support. Sophie was lucky in that she was not forced to marry and her
family was wealthy enough to support her throughout her life, and though mostly
excluded from formal education and the society of mathematicians, she was able
to pursue her self-study.
The École Polytechnique opened - to men
only - in 1794, but lecture notes were made available on request. Students were
requested to send solutions to faculty and Sophie began submitting notes
to Joseph Louis Lagrange under the name of a former student who had died
young, Monsieur Antoine-Auguste Le Blanc. Lagrange recognized "Le
Blanc's" ability and requested a meeting, so her rouse was up. Luckily,
Lagrange was not opposed to a woman studying mathematics and agreed to mentor
her. In 1798, Adrien-Marie Legendre published Essai sur la théorie des nombres and
Sophie became interested in number theory and began corresponding with
Legendre. Impressed, he included some of her work in the supplement to the
second edition of his book, praising it as très ingénieuse ("very
ingenious"). Then Gauss published his magnum opus Disquisitiones Arithmeticae. She
worked through it for three years before writing him, again as M. Le Blanc, to
discuss his book and tell him about her work on Fermat's Last Theorem, a famous
grand problem of number theory. Unfortunately, she had made a weak assumption
in one of her proofs, and Gauss did not reply to this first letter.
Mathematician Pierre de Fermat famously
scrawled his eponymous last theorem in the margin of a book around 1637,
without supplying any proof, noting simply that the proof was too long to fit.
After 354 years of effort by countless mathematicians, Andrew Wiles was finally
able to prove the theorem correct in 1995. The theorem states that no three
positive integers x, y, and z satisfy the equation xp
+ yp = zp for any integer value of p greater than 2. Sophie was working on this
problem and making real in-roads.
During the Napoleonic wars, France
occupied the German town of Braunschweig, where Gauss lived, and Sophie feared
he might suffer the same fate as Archimedes. She wrote family friend General
Pernety pleading for him to ensure Gauss' safety. Soldiers were dispatched and
found the confused Gauss perfectly safe. Gauss, of course, did not know that
Sophie Germain was none other than M. Le Blanc. She decided to reveal her
identity and he replied,
How can I describe my
astonishment and admiration on seeing my esteemed correspondent
M. Le Blanc metamorphosed into this celebrated person ... when a
woman, because of her sex, our customs and prejudices, encounters infinitely
more obstacles than men in familiarising herself with [number theory's] knotty
problems, yet overcomes these fetters and penetrates that which is most hidden,
she doubtless has the noblest courage, extraordinary talent, and superior
genius.
They became friends and Gauss truly
respected her ability, though he was not a reliable correspondent and generally
did not review her work (and she would have really benefited from such
feedback, lacking mentorship in number theory, and having gaps in her knowledge
since she was self-taught).
She became interested in other
problems. German physicist and musician Ernst Chladni had published his experiments
on vibrating plates (following the trailblazing work of Robert Hooke). He used
a violin bow to vibrate a metal plate covered in sand, so that the sand would
concentrate on nodal lines marking divisions between regions that moved in
opposing directions. His drawings of the patterns produced are known as Chladni
figures (like those shown in lavender in my print). Germain was able to attend
his demonstration in Paris. The Paris Academy of Sciences launched a contest to
develop the mathematics explaining the vibration of an elastic surface and
comparing this to experimental data like Chladni's, with a reward of 3,000
francs. Lagrange pointed out that a new branch of analysis would be required
and scared off all would-be contestants with the exception of Sophie and Denis
Poisson. But Poisson was elected to the Academy, thus became a judge, leaving
only Sophie. She began, mentored by Legendre, but her submission was deemed
insufficient, though she provided some ingenious results, which allowed
Lagrange to derive an equation, correct under certain conditions. Lagrange died
within two years, and Sophie lost a mentor. The Academy decided to extend the
contest and Sophie persisted. After initially helping, Legendre withdrew his
support. Sophie submitted another attempt anonymously, but it had several
errors (of the sort she would have been taught to avoid had she been allowed to
study math at a university). She consulted Poisson, and he had access to all
her notes as a judge. He then published his own work on elasticity without
acknowledging any of her work or their conversations on the subject. At that
point in 1816, she published under her own name, "Recherches sur la théorie des surfaces élastiques"
partially so what Poisson had done would be clear, and to point out the errors in his work. They extended the contest
again partially in response to the breach of confidentiality by Poisson and she
persisted with her efforts. She won the gold medal and became the first woman
to win a prize from the Paris Academy of Sciences but did not attend the prize
ceremony. The Academy was not entirely satisfied; she had the correct
differential equation, but having used an incorrect equation by Euler she had
incorrect boundary conditions. Even as a prize winner, she was still denied
entry to academy meetings as a woman for several years until she made friends
with Joseph Fourier, a secretary of the Academy, who got tickets on her behalf.
She published her prize-winning essay in 1821, at her own expense as the
Academy had neglected to do so, complete with her notes on errors she had made.
In 1826, she submitted a revised version of her work; the Academy considered it
trivial but they did not want to reject her as they would a man and
professional colleague. They both denied her access and were patronizing in
their misguided attempt at kindness. She published this essay on the advise of
mathematician Augustin-Louis Cauchy. Her nephew later made sure she had a final
publication on elasticity, publishing "Mémoire sur la courbure des surfaces"
posthumously on her behalf in 1831.
In 1815, the Academy offered an award
for a solution to Fermat's Last Theorem, rekindling her first love of number
theory. She wrote Gauss with her strategy for a general proof and the
significant in-roads towards a proof she had made, but Gauss never replied. She
produced what is now known as Sophie Germain's Theorem. To show that Fermat's
Last Theorem holds, you can divide the powers p into numbers which are not
divisors of x, y or z, or powers p which are a divisor of at least one of x, y
or z. She proposed her theorem:
Let p be an odd prime. If there exists an
auxiliary prime P = 2Np + 1 (N is any positive integer not
divisible by 3) such that:
- if xp + yp + zp ≡ 0
(mod P), then P divides xyz,
and
- p is not a p-th power residue (mod P).
and she used this result to show that
Fermat's Last Theorem holds true for all odd primes p < 100.
Her method was later used to show it holds true for all p < 1700.
Her theorem is known from a footnote in Legendre's treatise on number theory,
where he used it to prove Fermat's Last Theorem for p = 5. The
text in my print is from one of her on unpublished manuscripts:
Remarque sur
l'impossibilité de satisfaire en nombres entiers a l'équation xp + yp = zp. L'impossibilité de cette équation serait hors de doute si
on pouvais démontrer la théorème suivant: Pour toute autre valeur de p que p =
2, il y a toujours un infinité de nombres premiers de la forme Np + 1 pour
lequels on ne peut trouver deux residus 1ièmes puissances dont la différence soit l'unité.
(My gloss: "Remarks on the
impossibility of any whole numbers satisfying xp + yp = zp
. The impossibility of this equation can be shown to be without doubt if we can
demonstrate the following theorem: For all p > 2, there are an infinite
series of primes of the form Np + 1 for which we cannot find two residues of
the first power separated by 1"). She goes on to note that if there's any
numbers which do satisfy Fermat's equation for p > 5 it must be numbers
"whose size frightens the imagination", around 40 digits
long.
She also pursued philosophy and
psychology on her own. Her nephew had two of her works published posthumously: Pensées diverses,
a history of science and math with her commentary and Considérations générales sur l'état des sciences et des
lettres, aux différentes époques de leur culture in
which she argued there no difference between the sciences and the humanities
and gained the praise of philosopher August Comte.
She continued working despite pain
after her diagnosis of breast cancer in 1829. She died in 1831, listed only as
a property owner, not a mathematician on her death certificate. She is now
recognized for her brilliance and originality, but her progress was often sadly
hampered by the lack of instruction and the way her peers treated her as a
novelty, and avoided proper constructive criticism. She wrote, “These facts are
my domain and it is to me alone that they remain hidden. That’s the privilege
of the ladies: they get compliments and no real benefits.” Several scholars
argue that a contemporary man with similar skills and interest would have had
his abilities embraced and talents nurtured. Sophie Germain was able to achieve
what she did through both her tremendous talent and extraordinary persistence. Along
with her theorem, subsequent discoveries in number theory have been named in
her honour, as well as a street in Paris and the Sophie Germain Prize in
mathematics offered by the same Academy which had snubbed her.
References
Sophie Germain,
Wikipedia, accessed March 2021
Sophie
Germain’s Theorem, Wikipedia, accessed March 2021
Ernst Chladni,
Wikipedia, accessed March 2021
Ernst Chladni, Entdeckungen
über die Theorie des Klanges, 1787, via Chladni Figures (1787) on Public
Domain Review
Cristina
P. Tanzi, Sophie
Germain's Early Contribution to the Elasticity Theory, MRS Bulletin , Volume
24 , Issue
11 , November 1999 , pp. 70 - 71 DOI: https://doi.org/10.1557/S0883769400053549
Alexanderson, G.. “About
the cover: Sophie Germain and a problem in number theory.” Bulletin of the
American Mathematical Society 49 (2012): 327-331.
Richard Baguley, Sophie
Germain: The Mathematics Of Elasticity, Hackaday.com, March 20, 2018
Evelyn Lamb, Thank
You, Sophie, and I'm Sorry, Scientific American Blog, April 1, 2017.
Maria Popova, How the
French Mathematician Sophie Germain Paved the Way for Women in Science and
Endeavored to Save Gauss’s Life, Brainpickings, org, February, 2017.
Reinhard Laubenbacher and David Pengelley,
“Voici ce que j’ai trouvé:” Sophie Germain’s
grand plan to prove Fermat’s Last Theorem, Historia Mathematica Volume 37, Issue 4,
November 2010, Pages 641-692
