In honour of International Women’s Day today, I wanted to return to “Inspirational Women in STEM”, a series of blog posts I began writing to celebrate women in STEM history.

The women I’ve written about so far are people who have made significant contributions in their areas in STEM, while battling the gender (and some also racial) discrimination in those fields that is still seen today. They are people we tend not to learn about in school, but whose hard work we benefit from today; they should have a light shone on them, which is something I am attempting to do with the power of my mediocre blogging skills.

Since my educational background is in maths, I thought it would be fitting to write about a mathematician! Someone who pushed forward in her pursuit of education despite these obstacles, and who managed to make significant contributions to number theory and elastic theory despite her lack of a formal education.

### Sophie Germain

Lagrange. Gauss. Legendre. If you have an interest in number theory or calculus, these are names you’ll likely recognise. You might have used Legendre polynomials in your calculus studies, or come across Gauss’ and Lagrange’s many separate contributions in number theory.

It’s possible that you might not have heard of Germain, but hopefully by the end of this post you’ll learn that she was a quiet force, whose contributions in mathematics are significant enough that we still use her ideas today.

Born Marie-Sophie Germain on 1st April 1776 in Paris, France, the middle child of two sisters. When Sophie was 13, the Bastille fell, and so she spent her teenage years housebound during the French Revolution.

*Maths by Candlelight*

It was at home she discovered a passion for maths: in her father’s library she came across J. E. Montucla’s *L’Histoire des Mathématiques*, in which he described the story of Archimedes’ death (Archimedes was killed by a Roman soldier after he declined a meeting with General Marcellus in favour of contemplating a mathematical diagram), a story which intrigued her. She felt that if Archimedes was so captivated by the study of mathematics that he would risk his life for it, then it must be something interesting and therefore something she wanted to learn more about.

And so began her fascination, poring over more of her father’s books on mathematics, reading works by Sir Isaac Newton, Leonhard Euler, Étienne Bézout, and Jacques Antoine-Joseph Cousin, to name a few.

Unfortunately, her parents did not approve of her interest in mathematics as they (as well as society in general at the time) didn’t think it an appropriate interest for a woman. At night, they would deny her a fire and warm clothes in her room, in the hopes that it would stop her studying into the night, but still she persisted, wrapping herself in quilts to study maths by candlelight.

When Germain was 18, an academy called École Polytechnique was founded, a place where male mathematicians and scientists for the country would be trained. Women were not permitted to enrol, so despite her interest and growing knowledge, Sophie was not able to join.

*Mentored by Lagrange*

Still, she was able to obtain the lecture notes for several of the courses, using them to continue her studies; in this way, she was able to learn from many of the prominent mathematicians of the day, notably Joseph-Louis Lagrange. She struck up a correspondence with Lagrange, under the pseudonym of M LeBlanc (a former student of his), and submitted a paper on analysis to him at the end of the term. Impressed with M LeBlanc’s work and wanting to meet the student who had written it, Lagrange was surprised to find that she was a woman, and became her mentor after seeing the potential in her abilities through her paper.

Now with a male mentor to introduce her, Sophie was able to mix with a circle of scientists and mathematicians she never could have before due to her gender and social status, with whom she was able to discuss her ideas and thoughts, rather than learning in isolation as she was used to.

*Correspondence with Gauss*

In 1804, Germain began writing to the German mathematician, Carl Friedrich Gauss, whose famed work in number theory interested her. She sent him some of the results of her own work in number theory, again using her pseudonym of M LeBlanc. They wrote to each other occasionally over the coming years.

In 1807, Germain wrote to Gauss revealing her true identity: “*This leads me to confess that I am not as completely unknown to you as you might believe, but that fearing the ridicule attached to a female scientist, I have previously taken the name of M. LeBlanc in communicating to you those notes that, no doubt, do not deserve the indulgence with which you have responded.*“

To which Gauss responded: “*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.*“

*Exploring Elastic Theory*

In around 1809, the French Academy of Sciences announced a contest to explain the “underlying mathematical law” of German physicist Ernst Chladni’s study on the vibration of elastic surfaces, setting a two year deadline for 1811. Sophie’s was the only entry in the contest, but her lack of formal education seemed to be evident in her anonymous paper and she was not awarded the prize. The contest was extended again, and with input from Lagrange who was able to correct her errors she submitted her paper again two years later, receiving honourable mention this time.

In 1816, she entered the contest for a third time and won with her paper “Memoir on the Vibrations of Elastic Plates”; judges noted at the time that there were some shortcomings in her explanation, which were not addressed until a few decades later, but it was after winning this contest that Germain continued her work on the theory of elasticity, publishing several more memoirs in the field. Her work in elastic theory would prove to be very important.

Winning this contest led to Sophie being the first woman (who was not a wife of a member) to attend the Academy of Science’ sessions with the help of Jean-Baptiste-Jospesh Fourier, and she was praised by the Institut de France who invited her to attend their sessions, noted to be the “the highest honor that this famous body ever conferred on a woman”.

*Sophie Germain’s Theorem*

In 1815, the French Academy of Sciences offered a price for a proof of Fermat’s Last Theorem, which stirred up Germain’s interest in number theory again. At the time, Pierre de Fermat’s Last Theorem had been proved for cases where prime **p** = 3, **p** = 5, and **p** = 7 since the theorem was proposed around 200 years before.

Her method of proof involved proposing a theorem of her own, commonly called “Sophie Germain’s theorem”:

Let **p** be an odd prime. If there is an auxiliary prime **θ** satisfying the two conditions:

1. x^{p} + y^{p} + z^{p} = 0 mod **θ** implies that x = 0 mod **θ**, or y = 0 mod **θ**, or z = 0 mod **θ**, and

2. x^{p} = p mod **θ** is impossible for any value of x,

then the first case of Fermat’s Last Theorem is true for **p**.

By dividing Fermat’s Last Theorem into two cases, she was able to use her theorem to prove the first case for all odd primes **p** and 2**p** + 1 (known today as a Sophie Germain prime, still in use today in cryptography), and showed proofs for these odd primes **p** < 100 (allegedly **p **< 197 accordingly to Andrea Del Centina), where x^{p} + y^{p} = z^{p} has no integer solutions for which x, y, and z are relatively prime to **p** (i.e. in which none of x, y, and z are divisible by **p**).

In an unpublished manuscript, Germain showed that any counterexamples to Fermat’s theorem for *p* > 5 must be numbers “whose size frightens the imagination”, around 40 digits long. We only know her work on this because of a footnote in Legendre’s treatise on number theory, where he used it to prove Fermat’s Last Theorem for *p* = 5. Germain also proved or nearly proved several results that were originally attributed to Lagrange, or were rediscovered years later.

*Honorary Degree*

In 1829, Sophie learned she had breast cancer and continued to work despite the pain, continuing to have more of her work published until her passing on 27th June 1831.

Before her death, Carl Gauss had recommended to the University of Göttingen she be awarded an honorary degree but that didn’t occur until 1837, six years after her death. Even her death certificate lists her as a property holder, not a mathematician. Gauss stated then that “she [Germain] 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 well have deserved an honorary degree”

* This isn’t phrased in the most flattering way here towards women, and Sophie, note that it is a translation.

*Just a few thoughts*

What I like to take away from Sophie Germain’s life story, is that while everything in it was actively working against her pursuing her interest in maths (the expectations of her gender, her social status, the actions of her parents), she pushed forward with it. She spent cold nights learning maths by candlelight, nurturing her skills, and forming correspondences with noted male mathematicians of her time, who would praise her ideas as “ingenious”.

We’re lucky in a way that her passion for this subject burned so brightly that none could extinguish it, and lucky she found mentorship and guidance from prominent mathematicians who did not hold the gender prejudices of their time against her.

The sad truth is that there were many young girls growing up before, during, and after her time who were met with the same gender bias, whose potential in maths and the sciences were snuffed out before they could even be sparked. It is sad to think of the insights and ideas we may have lost to time because of this. Even to this day, there are young girls around the world who are being denied an education, whose potential in life is being blocked for the sorry excuse that they were born female and have different expectations thrust upon them.

The more I read about Sophie Germain, the more thankful I am that I live in a time and a place where we’re not so pigeon-holed because of our gender, where the doors of equality are slowly opening up. Of course, there’s still a long way to go in the complete eradication of discrimination in STEM (and many other areas), with gender, racial, and social discrimination that is still prevalent today, but progress has been made since then, and still is being made, however slowly. I’m looking forward to a future where opportunities are equal for all, no matter our gender, the colour of our skin, or our social backgrounds.

And on that note, I shall end this lengthy, slightly gushing post with a “Happy International Women’s Day” to everyone reading this!

If you want to read about more inspirational women in STEM, please check out:

- Inspirational Women in Stem | Ada Lovelace
- Inspirational Woman in STEM | Dorothy Vaughan
- 3 Inspirational Women in STEM (Rebecca Cole, Joan Clarke, and Radia Perlman)