The Life of Marie-Sophie Germain

Caption: Marie-Sophie Germain. Image: Public domain, via Wikimedia Commons.

As many would know– or should know– successful females throughout history have either been scorned, ignored, or referred to briefly from a footnote. As discouraging as that is, many future generations have used that as motivation to make greater accomplishments in areas that traditionally were not open to them. Why, Elizabeth I made England one of the most powerful nations in Europe, Marie Curie discovered Radium and Polonium, and the great Marie-Sophie Germain pioneered elasticity theory.

Wait, who? Elasti-what? Why, everyone knows how Germain was an independent woman who didn’t need a career or formal education to show her what’s what. Obviously. I mean, she only built the foundations for Fermat’s Last Theorem (FLT)– which I’m sure you’re familiar with– and used her own method to show that the modular arithmetic conditions on the exponent n from FLT can lead to particular conditions for x, y, and z for the equation xn + yn = zn. (FLT states that no positive integers x, y, and z can satisfy the previous equation for any integer n greater than 2.)

Fermat’s Last Theorem. Image: Phuong Trinh, via Wikimedia Commons.

Germain had intended on proving many cases of FLT all at once rather than working through each one individually. Of course, this was not a very successful attempt, but her work helped support future generations with their studies on FLT.

Germain’s journey into the world of number theory started with a letter to polynomials and transformation master Adrien-Marie Legendre, when his work Essai sur la théorie des nombres (Essay on Number Theory) was published. In short, they became pen pals and corresponded with each other for many years regarding number theory and eventually elasticity. Legendre was even generous enough to add Germain to a magnificent and grand footnote in his treatise on number theory. And why would he do such a thing? Well, Germain’s unpublished manuscript called Remarque sur l’impossibilité de satisfaire en nombres entiers a l’équation xp + yp = zp, which argued that counterexamples for FLT for p > 5 would need to be numbers nearly 40 digits long, was so brilliant that Legendre used it to prove FLT for p = 5. Afterwards, Germain sent a paper on analysis to the magnificent J. L. Lagrange (one of the big daddies of analysis, number theory, and classical and celestial mechanics) as “Monsieur Antoine-Auguste Le Blanc” and impressed him so much that he became a mentor and supporter of her work, even after Germain confessed to him that she was a woman and it was she who was using the name of a former male student of his to correspond with him. This was also the same with Carl Friedrich Gauss– math guru in number theory, statistics, algebra, matrix theory, differential geometry, optics, analysis, electrostatistics, geodesy, astronomy, and geophysics (phew!). She wrote to him using “Le Blanc” again to discuss number theory, which she studied thoroughly in Gauss’s Disquisitiones Arithmeticae and offered her own work on the theorems listed. Despite her work not having the proper structure that normally would have been apparent from a formal education, Gauss had used Germain’s ideas and proofs for FLT. One of Germain’s unsupported proofs was for the case n = p, with p being a prime number with the form p = 8k + 7. Gauss would provide counterexamples to some of Germain’s proofs as the years went on. Their correspondence would later end due to Gauss no longer expressing any interest in number theory. He moved on to other mathematical fields. If was from there that Germain went on her own tangent.

Caption: Germain’s Elasticity Theorem submitted for the third and final time for the contest. Image: Rational Wiki

Starting in 1809, she began her work on elasticity, specifically with the theory of vibrating elastic surfaces using vibrating metal plates. The Paris Academy of Sciences was having a contest to elaborate E. F. F. Chladni’s study on the subject, and Germain was the only contestant. Well, thanks to the lack of formal education she did not receive the prize due to unsupported work. But the judges thought her results were impressive, so you could say she got the equivalent of a participation sticker. If there’s ever a time to be grateful for a teenage rebellion that resulted in a lifelong pen pal, it was right and there. Lagrange had helped correct Germain’s mistakes, and before long she entered the contest again in 1813. Oh but wait, that whole informal education worked its magic again and she didn’t win again. But this time, she had an honorable mention only because her work had taken an approach that wasn’t derived from physics. And also because there were still several mistakes in her calculations. But hey, third time’s the charm right? The year of 1816 was Germain’s time to shine when she finally won and had her work once again criticized for not completely resulting in what was expected. Ah, c’est la vie. Although her work was not fully supported, it would later become the critical stepping stone for future generations who aimed to significantly improve her work. Now all that was left was to be accepted into society as an educated woman to extend on the topic she had been working on for the past 16 years and become a renowned mathematician for years to come.

Yeah. It’d still take a century or two for that to happen. So why aren’t we teaching our kids about Miss Marie-Sophie Germain or any of her work in schools? It’s only number theory and advanced mathematics. Germain started in 1789 when she was 13 and studied differential calculus so much that her parents found her incurable of her newfound disease. She went through her teenage rebellion in 1794 at the age of 18 when she began to make friends with students at the male-only Ecole Polytechnique and took their lecture notes to study. Did I mention that the majority of Europe at the time did not accept women into colleges? The only exceptions were the wealthy upper class women, only so that they could have more ice breakers when gathering at social functions. Germain, however, was only a middle class maiden who was pushed from exploring her interests. She continued to do what she could outside of the education system. As time went on, Germain continued to be excluded from any sort of research related to mathematics. Any work she did submit to educational institutions were not treated “as a man’s”. Germain’s essays weren’t formally rejected, which meant they actually were rejected but in a very rude way. The logic at the time for that was if the institution sent a letter of rejection to her, they would technically be acknowledging a woman’s work, thus making her work equivalent to a male’s. They didn’t want to portray that to Germain or to anyone, because– you know– that just wasn’t acceptable in the 18th century. This didn’t deter her though. She continued for the rest of her life working on elasticity and math theory, along with philosophy and psychology.

Germain would later die from breast cancer, after submitting a paper to Crelle’s Journal in 1831 explaining elastic surfaces and their curvature. Though this seems anything but the happy ending Germain expected, her life’s work greatly improved the world of math. And not only that, her presence in education during the 18th century demonstrated that the difficulties women have in pursuing math and science were meant to be respected, not ostracized. Gauss became one of the supporters for women’s social justice after he had found out Germain was a woman. He recommended Germain to receive an honorary degree in mathematics before her death, and had exposed the unfair treatment European women faced in education and in social settings.