Six-minute summary: Sophie Germain

Hello readers! Today we’re taking a look at the life of Sophie Germain, a French mathematician. I wouldn’t be surprised if you’d never heard of her, and by the end of this summary, I hope you’ll agree that this is a terrible shame.

Childhood

Sophie Germain was born in Paris in 1776, with the full name Marie-Sophie Germain. She just went by “Sophie” – probably because half the women in Paris at the time were called Marie-Something (indeed, her sister and mother were both called Marie-Madeleine). Sophie Germain had an interest in mathematics from an early age, after reading about Archimedes in L’Histoire des Mathématiques. Apparently, she was intrigued by its story of his death, where he was so engrossed in his maths that a Roman soldier killed him. To her, this seemed like a worthy pursuit – to be so invested in a discipline that even a sword-wielding soldier couldn’t distract you.

Unfortunately, Sophie’s parents took a dim view of their daughter studying maths – and so did she, as she was forced to study secretly at night. Her parents took away her warm clothes to try and stop her, but it didn’t work; she even taught herself Latin and Greek so that she could read more complicated works by Euler and Newton.

Education (or lack thereof)

The École Polytechnique opened in Paris in 1794, when Sophie Germain was eighteen – the perfect age to enrol, had she been a man. Luckily for her, the school ran a home-learning scheme, and she found a way to bend the rules. She enrolled under a male pseudonym, and the lecturers sent notes out to her, and she sent written work back. Needless to say, her abilities were noticed by a certain Joseph-Louis Lagrange, and he requested a meeting. On discovering that she was a woman, he agreed to keep mentoring her (although I notice that she isn’t recorded as a “notable student” on his Wikipedia page).

Unfortunately, Sophie Germain was never allowed to attend university properly, and so her education was patchy at best. As a young woman, it was deemed inappropriate for her to meet with her male teachers, let alone other students, and so the transfer of ideas occurred through letters, which were unreliable and slow.

Sophie Germain became interested in number theory in 1798, after reading the work of Legendre. She sent him several letters, and even though she was only 22, and hadn’t received a formal education, he was very impressed with her work, calling it “très ingénieuse”. He included her work in the second edition of his book, Théorie des Nombres, and he credited Germain in helping him.

Correspondence with Gauss

Sophie Germain also corresponded with Gauss, another very famous contemporary. She used her male pseudonym again, frightened that he would ignore her otherwise, and the two of them exchanged letters discussing number theory. He found out who she really was under the strangest of circumstances. During the Napoleonic Wars, the French army occupied the German town where Gauss lived, and upon hearing about this, Sophie Germain contacted a French general (a family friend) and asked him to ensure that Gauss was safe. The head of the battalion was sent to check on him, and Gauss was very bemused that some random French woman was asking after him.

Sophie Germain revealed her identity to him in a letter three months after the incident. He didn’t mind that she was a woman – indeed, he told her that he was filled with “astonishment and admiration”, and he spoke very highly of her in letters to his male colleagues too. He acknowledged that “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” and that she had “the noblest courage, extraordinary talent, and superior genius”.

Work on elasticity

In 1808, Ernst Chladni visited the French Academy of Sciences and demonstrated his magic vibrating plates to Napoleon. For those unfamiliar, Chladni had a rectangular plate, covered with sand, and he then ran a violin bow against the edge, causing the sand to move into beautiful, symmetric patterns. Napoleon was so enthralled that he set a prize for the best mathematical explanation of the phenomenon.

Most mathematicians didn’t bother to try and explain the Chladni figures. Many were put off when Lagrange said that it would require a whole new type of analysis. As such, there were only two entrants to Napoleon’s competition: Sophie Germain, and Poisson – yet another one of her famous male contemporaries. Poisson was five years younger than her, but before the deadline for the competition, he was elected to the academy, which meant that he became a judge for the prize, not a contestant. As such, Sophie Germain was the only one who submitted a solution.

Legendre aided Germain in her work by supplying her with equations, references and current research, which would have been inaccessible to her otherwise. However, even though she was the only entrant, she did not win. The judges didn’t think she answered the question fully – although Lagrange derived equations from her work which he said could be used under special assumptions.

Undeterred, Germain tried to win the prize a second time, although Legendre stopped helping her. She showed that Lagrange was right – that her equations could be used under certain conditions. Still, the judges were not convinced that she had addressed the problem fully.

Even then, she was determined to solve it, and this time she reached out to Poisson for help. They exchanged many letters, but in 1814 he published his own work on elasticity. He didn’t credit her, despite their correspondence, and despite having read all of her unpublished work that had been submitted for the competition. In 1816, Germain submitted her third and final attempt under her own name, and this time, the judges were convinced. She became the first woman to win a prize from the French Academy of Sciences.

However, although Germain had derived the correct differential equation, her work had been littered with numerical errors, and she could only publish her work privately, using her own money, as the academy refused to review it. She was well aware that her work contained errors, writing “I had made serious mistakes; it only took a simple glance to see them; one could therefore have condemned the piece without taking the trouble to read it. Fortunately, one of the commissioners, M. de Lagrange, noticed the hypothesis; he deduced from it the equation which I should have found myself, if I had conformed to the rules of calculation.”

I find this quote rather sobering. Sophie Germain was aware of her deficiencies, and seems saddened and maybe even ashamed by them. If she had only received the same tuition as her male peers, she could have solved this problem so much faster. Worse still, she is indebted to the people who refused to teach her properly; she sees it as “fortunate” that a superior noticed that she was onto something, and gave her some feedback. The others simply dismissed her work out of prejudice, or for the simple errors that they could have helped her avoid.

Work on Fermat’s Last Theorem

Sophie Germain’s most famous work is on Fermat’s Last Theorem. For those of you unfamiliar, remember Pythagoras’ Theorem from school: a² + b² = c², where a, b, and c are the sides of a triangle. A brilliant mathematician named Fermat had stated that aⁿ + bⁿ = cⁿ cannot be true for any value of n greater than two. However, he only scribbled this in a margin, without any proof. Most tantalisingly of all, he claimed to have proved it, but that the proof wouldn’t fit in the margin. After he died, many mathematicians tried to derive this proof for themselves, and it was finally solved in 1994 by Andrew Wiles. However, it was Sophie Germain who made the earliest breakthrough.

Fermat had shown that the only case where the equation might be true is if n was a prime number. When Sophie Germain approached the problem, there were only proofs for the prime numbers 3, 5 and 7, but she came up with a new proof which worked for a whole class of prime numbers, now called Sophie Germain primes.

Death and legacy

Sophie Germain died relatively young, at the age of 55. Gauss had tried to get her an honorary degree from the University of Göttingen, but she passed away before she could receive it.

In recent years, Sophie Germain’s achievements have started to be recognised and remembered. For example, a “Sophie Germain prime” is any prime number p where 2p+1 is also prime. There has also been an attempt for the mean curvature of a surface to be called the Germain curvature, as she was the one to demonstrate its significance for elasticity. Sadly, this does not seem to have caught on.

What we can learn from Sophie Germain

There is no doubt that Sophie Germain was an incredibly talented individual. As Gauss said, for her to make any headway in the field of mathematics, she had to overcome an incredible number of obstacles, from her parents denying her the warmth she needed to study, to the university denying her access to education, and to her male peers denying her the credit she deserved for her work. Even after winning her prize, she was not allowed to come to any sessions of the Academy of Sciences, because the only women they invited were the wives of its members. She only managed to attend in 1822, by the time she was 46, because she befriended Joseph Fourier (another very famous name to add to the list), who managed to get her tickets. The Academy refused to show her any professional respect; when she tried to publish her prize-winning work in 1825, they didn’t even bother to review it. By consistently ignoring her work, and failing to provide her any feedback, they made it incredibly difficult for her to fix her errors.

Sophie Germain never married, and she was lucky in that her family was wealthy enough to support her throughout her adult life. I’m intrigued as to how she spent her time, ostracised from the rest of the scientific community. She spent years working on problems, but I can’t determine if she treated maths as a hobby, or whether she invested every waking hour into it. In my head, I imagine that somebody so determined might also have been obsessive, in which case mathematics would have been her life. As admirable as this is, it is also rather sad to think that she could have achieved so much more had she only received the same education as her male peers. The fact that she was aware of her deficiencies, imposed through societal restrictions rather than any lack of effort or ability on her part, is particularly heartbreaking.

Sophie Germain’s story also raises questions about how we tackle ingrained, accepted social injustices. As much as Gauss and Legendre may have appreciated her talents, and admired her for succeeding against the odds, they didn’t expend much effort in improving her situation. Of course, it’s unfair to lay blame on individuals for perpetuating an unjust societal norm; we all perceive and interact with our world inside the mental boundaries instilled in us by others. It was probably unthinkable to them that a woman could be allowed to attend university, just as it’s unthinkable to us that an elephant could be allowed to attend university – even if we accept that elephants are clever in their own way.

However, much of the way that Sophie Germain was treated is simply deplorable, and I’ll happily throw accusations at the scientists who refused to review her work or who stole her ideas (difficult to prove anything, but Poisson’s timing was highly suspicious). It brings home just how impossible her situation must have felt, when most of the people around her couldn’t see the need for change, or were benefitting from the status quo.

In summary…

Sophie Germain is a mathematician you may never have heard of, despite her working alongside some of the most famous names in mathematics. She made incredible progress in the fields of number theory and elasticity, but she could never reach her full potential due to the limitations imposed on her by society at the time. We can learn a lot from her experience, because as far as we have come since then, many of the same behaviours persist today. We might find it easy to look back and decry the way she was treated, but we should remember that societal norms can render us blind or accepting of injustices – and to keep making progress, we must force ourselves to confront them.

(Apologies: even for fast readers, this was longer than six minutes! I’ll have to choose someone less interesting next time.)