Sophie Germain – the Queen of Primes

By Jaime Seltzer

Sophie Germain was born in 1776 in France, meaning that she would come to maturity during the French Revolution. Though she was born to a relatively well-off merchant, she was not a member of the aristocracy. On one hand, this could be a good thing (see: French Revolution)! On the other hand, Sophie’s love of math was not as indulged as it might have been in an upper-class miss, who was expected to know at least enough about mathematics to converse about it with young aristocratic gentlemen. (“Oui, monsieur! That Fermat, he is a tricky one, no? Ahaha!” This was the sort of thing a young woman was advised to say at parties.) There was a whole series of books written on the matter in Sophie’s day entitled Sir Isaac Newton’s Philosophy Explain’d for the Use of Ladies. If you are ever struck by the desire to see Newton’s ideas entirely couched in romantic prose, that is your book, and I can also recommend a skilled mental health professional – before you read it, or after, your choice.

Sophie was wandering in her father’s library one day, confined to house and home because people were rioting in the streets outside, when she found a book on the history of mathematics (no, not that book!). The book described how Archimedes had literally died for the love of math. A Roman solider was questioning him while he was absorbed in some bit of geometric confusion. When Archimedes continued to ignore him, the soldier stabbed him.

If you have ever barred a teenager from doing something in particular, you can probably guess that Sophie persevered regardless, determined and square-jawed until her parents relented.

And that is how one of the greatest mathematicians of all time died, or so the historian romantically put it. (Actually, Archimedes had designed Syracuse’s defenses, so it’s unlikely that invading Romans killed him solely because he was absorbed in a math problem.)

It’s not exactly the average inspiring story for the average young lady, but the tale haunted Sophie. What on earth could be so interesting, so absorbing, that it rendered its devotee completely insensible to any danger? To a girl confined to house and home because of the Revolution, this became an improbably absorbing question. Such power was so mysterious that it seemed almost magical. And so she began to study math.

It was still the 1700s, which should have come with a warning label directed towards clever little girls. Like Mary Somerville, Sophie’s parents immediately began to worry that their daughter’s brain would explode from the strain of all that math and reading and she would end up in the madhouse. Just as Mary’s parents had, they hid the extra candles so she couldn’t study at night, and barred her from the library. Unlike Mary’s parents, they went to the extreme of hiding her clothing so that she couldn’t leave her bed at night to study. That’s some seriously next-level oppression right there, but it was the French Revolution: they’d learned from the best.

Sophie responded the way any red-blooded young intellectual would, by secretly stealing candles and wrapping herself in all the bedclothing in the house. Barring Sophie’s room from any light in the evening also meant barring her from lighting a fire, and the nights were so cold in her room that the ink in her inkwell thickened and froze. However, if you have ever barred a teenager from doing something in particular, you can probably guess that Sophie persevered regardless, determined and square-jawed until her parents relented. Thus, Sophie “spent the years of the Reign of Terror studying differential calculus.”

As seems to be the case for many of these stories, once Sophie’s parents reconciled themselves to her genius, they sang a rather different tune, funding her research and probably embarrassing Sophie in public by telling them that their daughter was a mathematician.

Sophie Germain

Sophie Germain and her Women in Science card.

The Ecole Polytechnique, a school for mathematics and science, opened in Paris in 1794. Sophie was shy and restrained around anyone she did not know well; she was not like some of the other ladies described here, who would have marched straight up to the doors and demanded to be let in. Instead, she pulled a full-on Eponine and assumed the identity of a boy she knew who had been there for a few days and then gone home to, I don’t know, open up a coffee shop or take up knitting. His name was Monsieur Antoine-Auguste Le Blanc, and Sophie attended the college by going to his lecturers and getting his notes and homework ‘for him’ and then turning in her own work for a shockingly long time before anyone caught on.

It was Sophie’s brilliance that was her undoing. Let’s be serious, the real Le Blanc cut and run because he was kind of a crappy student, and Sophie’s work and his was like night and day. One of Sophie’s professors, Joseph-Louis Lagrange, simply had to find out what had turned his student from an indifferent and kind of sloppy mathematician to someone who could solve his homework problems in such ingenious ways. Sophie knew the jig was up, and presented herself as the real author of the work. Perhaps Lagrange had thought he would catch Monsieur Le Blanc out as a cheat and a liar, that someone else had been submitting Le Blanc’s work for money or favors; it’s safe to say he wasn’t surprised. When he discovered that the student was Sophie, and why she had hidden her identity, he was seriously impressed with a side of charmed. Untutored Sophie was his very best student.

That was how Sophie finally found a teacher! This allowed her to gain confidence in her work. It is all very well and good for Sophie to believe that she was talented at math, but without any feedback at all, doubt crept in. With Lagrange, Sophie finally had a sounding-board that let her know that her mathematical ability was not just ‘good enough’, but unique, showing genuine talent – maybe even genius.

Then, Sophie’s first real mathematical challenge arose, which brings us right back to Fermat.

Fermat was a famous mathematician who had provided a tantalizing question to his students. Could it ever be true that xn + yn = zn, or must such a mathematical formula always be false? The formula and question are known as Fermat’s Last Theorem because it was a question he posed to his students… right before he died. The devious old mathematician never answered his own question, and did not leave any notes on the matter. However, Fermat specifically stated that a proof existed, implying he’d worked it the challenge successfully himself. The world may never know.

Germain thought that a math problem that no one in the world had ever been able to solve sounded like fun, so she set her magnificent brain to the task. Sophie’s approach was, rather than to try and plug values in (which would literally take forever) she would group values together so that she could disprove huge sets of numbers at a time.

She ended up creating a formula for a specific kind of prime number, in which the prime number is called ‘p’, so long as 2p + 1 is also prime. For example, 2 was this sort of prime – because 2(2) + 1 is five, which is prime as well. By proving that using these sorts of values did not work for the formula, Sophie was able to eliminate a huge number of possible solutions to Fermat’s Theorem.

Now Sophie thought she had something of value, she decided to consult Carl Friedrich Gauss (if you’ve heard the word ‘Gaussian’ used in math… yes, that Gauss.) She begged him for his advice to see if her approach really did make sense. But for reasons of temerity or even habit, Miss Germain signed her name Monsieur Antoine-Auguste Le Blanc.

Gauss was quite impressed with the work of the young ‘man’ who had written him so tentatively, and gamely tried various primes out to test Germain’s theory. Then, in 1806, Napoleon invaded Germany. Sophie was desperate to save the life of one of the only men who had encouraged her in her studies. One of her good friends was a general under Napoleon, and she begged him to take especial care of Gauss. Of course, the General informed the mathematician immediately that his ‘friend’, Sophie Germain, had given him special instruction to ensure Gauss’s safety.

Well, who was Gauss to look a gift horse in the mouth? Yet he had to admit he had never met a ‘Sophie Germain’.

The General responded that of course he knew Sophie; everyone who knew Sophie knew that Sophie knew him; why, she’d sent him a mathematical proof not so long ago and found great comfort in how he had supported her efforts.

All became clear for the second time in Sophie’s young life, and she was forced to write a scarlet-faced letter of apology for deceiving the world’s greatest mathematician.

Instead of the rejection Sophie had feared, Gauss’s support increased tenfold. “…the enchanting charms of this sublime science reveal only to those who have the courage to go deeply into it. But when a person of the sex which, according to our customs and prejudices, must encounter infinitely more difficulties than men to familiarize herself with these thorny researches, succeeds nevertheless in surmounting these obstacles and penetrating the most obscure parts of them, then without doubt she must have the noblest courage, quite extraordinary talents, and superior genius.”

Presumably, when Sophie Germain read this, she clung to it and wept, because that is the totally appropriate response from anyone, man or woman, who has the world’s foremost mathematician tell them that they are brave and clever and amazing. Unfortunately, Gauss would soon turn from abstract mathematics to other work, leaving Sophie out in the cold, mentor-wise.

Luckily, at about this time a new challenge loomed: the Institut of France was offering a prize to anyone who could create a formula that would describe the behavior of elastic (flexible) surfaces and describe how they had used empirical evidence to obtain it. Sophie eventually won the prize, though it was not without trial and tribulation. The prize date was extended twice because no amateur mathematicians but Sophie wanted to tackle it! The level of difficulty was such, and Sophie’s education so spotty that she kept making mistakes that her better-educated judges were able to catch. Still, no one else seemed capable of even approaching the problem, and so the judges were forced to award the prize of one kilogram of gold to Sophie. Germain was well aware of the ‘forced’ part of the equation, and did not deign to show up at her own awards ceremony. She knew she had been slighted by the three judges, who refused to participate in conversations about the work with her, and ignored her in public.

Over the course of her lifetime, Sophie wrote seminal works in mathematics, physics, psychology and philosophy; she discovered Germain Primes, and there is a Sophie Germain Prize in mathematics awarded by the Institut de France every year since 2003. Her accomplishments are sometimes downplayed because her lack of a formal education was a significant hindrance to her, leaving gaps and illogic in some of her most revolutionary work.

One wonders what she might have accomplished if she had been able to attend the Ecole Polytechnique and earn a degree in mathematics as Sophie Germain.

References
Maisel, M. and Smart, L. (n.d.) Sophie Germain. In The San Diego Supercomputing Center Presents: Women In Science, A Selection of 16 Significant Contributors. Retrieved from http://www.sdsc.edu/ScienceWomen/germain.html
O’Connor, J. J., & Robertson, E. F. (1996, December). Marie-Sophie Germain. In School of Mathematics and Statistics: University of St. Andrew’s. Retrieved from http://www-history.mcs.st-andrews.ac.uk/history/Biographies/Germain.htm
Singh, S. (1997, October 28). Math’s Hidden Woman. In Nova. Retrieved from http://www.pbs.org/wgbh.nova/physics/sophie-germain.html
Swift, A. (1995, April). Sophie Germain. In Biographies of Women Mathematicians: Agnes Scott College. Retrieved from http://www.agnesscott.edu/lriddle/women/germain.htm

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