I wrote it as an assignment for a math history class. I could have kept writing. Sophie Germain ROCKS! (and it's written for non-math people to read so if you want to read about a brilliant woman from Revolutionary France here you go!)
Sophie Germain – Revolutionary Mathematician
The streets of Paris were bustling with people. Marie-Madeleine Gruguelin was on an outing to the market with and her two oldest daughters, Marie-Madeleine and Marie-Sophie (just “Sophie” to avoid confusion with all the other Maries in the house) were on an outing to the market. Marie-Madeleine was reticent about taking the girls out, as the air was electric. There had been bread riots occurring with more frequency lately, the King was bankrupt, and the Estates Général had been called for the first time in a hundred and fifty years. Her husband, Ambroise-Francois Germain was asked to serve in the Estates Général, a group composed of what were called the three estates: the clergy, the 1st estate, the nobility, the 2nd estate, and the middle class, which included her husband, the 3rd estate. Everywhere people were writing pamphlets and petitions on their plights to the King. Many were unhappy with what they perceived as a lack of representation in the Estates Général. Marie-Madeleine had to concede that they would always be outvoted due to the structure of the system.
The price of bread had gone up again and Marie-Madeleine feared another bread riot. The women of the working class would instigate these whenever the prices got too high. It didn't affect her family that much as her husband had done well as a silk merchant. There was an assembly outside the Church of Sainte Geneviève; the working women saw her as their own saint. They were there in support of the Estates Général and inadvertently her husband. They were here some days and in front of City Hall other days.
“Maman, why are there so many people here?” Marie-Madeleine asked.
“They're here supporting your father,” her mother replied. “Move along, we've got to get home soon. Your little sister is waiting.”
They made their way down La Rue Clovis back to their apartment. Marie-Madeleine was taking her daughters out less and less these days and refused to take her youngest out at all. There were groups of people amassing on every street, and there were more people begging for food, for work, for money. Food prices were going up, as were taxes. There was so much unrest among the people and many with an overabundance of time on their hands due to the lack of jobs. There were often meetings held at their house. Her husband was among one of the liberal social reformers and he invited many like-minded people over for meetings. Ambroise frequently held meetings in the house with other like-minded individuals. She would continually find Sophie eavesdropping in on the meetings, entranced by what was going on. After a few tries she gave up on trying to keep her away – Sophie would always find a way to sneak back to them. None of her other daughters took any notice of the meetings except to complain that they couldn’t use the front room during these times. Sophie, despite her age, had a keen interest in the tumultuousness happening on the streets of Paris. And she had the opportunity to get the latest news from her father often in these meetings. Ambroise found it amusing that his 13-year-old daughter took such a keen interest in politics. At least this was all she was doing, Marie-Madeleine thought. After all, she could be taking to the streets to find her information and the meetings in the house were much different then those in the street. The air was becoming more desperate and Marie-Madeleine hoped for change to come soon and come peacefully, but she feared the worst. It would only be a matter of months before the Germains would be keeping their daughters inside continually.
In July of 1789 Marie-Madeleine's fears of a violent uprising came to fruition. That afternoon a group of armed people stormed the Bastille, which led to a day of bloody combat and several assassinations. Though they were part of the Third Estate, Marie-Madeleine and Ambroise-Francois felt that it would be best if they kept their children confined to the house until the city calmed down. The children did not react well to this decision. What would they do imprisoned in this house day in and day out with no end in site? Sophie took to her father's immense library to stave off the boredom. She read through everything she could.
It was here that she found a book by J.E. Montucia, History of Mathematics, and in it the story of a man named Archimedes (Osen). Sophie delighted in the odd stories of this man and his inventions, his refusals to take baths, and his obsession with mathematics. But this was not the best part of the story. It was not until she read of Archimedes' demise that she became engrossed with mathematics. The story of Archimedes' death was that as the Romans were capturing his city he was engrossed in a math problem and when engaged by a soldier he ignored that solider, enraging the soldier and prompting him to stab Archimedes. This man gave his life for geometry, for mathematics! This must be one interesting subject indeed. At that moment Sophie put down her book. What could be so interesting about mathematics that someone would give his or her life for it? She sat silently contemplating the subject until she was summoned for dinner. That night she lay in bed thinking of his circles, his diagrams, and his gadgets. Eventually she fell asleep, her mind still racing over the story of Archimedes.
The next day Sophie searched her father's library and found several books on mathematics. Many were older books, but she devoured them voraciously. She found that she would have to learn Greek and Latin in order to study Euler and Newton. Luckily her father had plenty of books on those subjects as well. Every day she spent with her father's books learning as much as she could: Latin, Greek, and about mathematics. She would work through the problems on paper, marveling in their intricate beauty. This went on for some time until one day her parents discovered how she was spending her days. It upset her mother that she was spending her days studying mathematics – this was not something that a young girl did. After dinner one night her upset parents sent her to her room without a book. She sat at her door listening to their conversation.
“This is all she does day and night! How can we keep her from this? It isn't OK for a young girl to study mathematics, especially not this obsessively,” her mother exclaimed.
“We'll just have to watch what she reads. Perhaps we shall no longer give her unfettered access to the library during the day that way we can monitor what she reads,” Ambroise-Francois replied. “There are many other places where she can sit and read since that is how she likes to fill her days. We do not have to keep such a close watch on the other children. When I am not here, if we limit her access to the library, it should not be too hard for you to make sure that she is not reading mathematics books.” The Germains were in agreement: they wouldn't let Sophie into the library on her own during the day and they would then be able to make sure that she did not continue her study of mathematics.
Sophie softly closed her door and tiptoed away from the door. She sat down and thought about ways in which she could sneak the books away from the library without her parents noticing. Surely they would keep close watch on all the books and would notice if they went missing. In that case hiding them in her room was not an option. Sophie decided that she would wait until after everyone went to bed and then go into the library and continue her studies. * Why her parents wouldn't understand her love of mathematics confused her, after all Archimedes gave his life for it, this was obviously a very important subject.
Sophie spent many nights in the library. Then one night, while she was consumed with a geometry problem, her father saw a light coming from the library. He opened the door and discovered his daughter poring over one the books that they had forbidden her to read. He immediately sent her back to her room and removed all of the candles from her room. And so it began. Sophie's parents tried everything: stealing her candles, her clothes, even cutting off the heat to her room at night. No matter what she always had a stash of hidden candles and quilts that she would wrap herself in and continued her studies. Despite all their efforts they continued to find Sophie curled up next to her books morning after morning. At this point the Germains realized that Sophie's love of learning mathematics could not be willed away or forced out of her and they allowed her free reign of the library once again. She was allowed to study mathematics freely during the day and with all her clothes.
In 1794 the École Polytechnique opened in Paris. It's mission statement was to “train mathematicians and scientists for the country”(Perl 64). The school did not admit women, but as with her earlier studies, she didn't let this obstacle stop her from learning even though she could not physically sit in on the classes. She was able to obtain the lecture notes from students in the classes and would send comments to the professors, which would at times include original notes on mathematics problems, but unlike other students she had to use a pseudonym to disguise her femaleness. And so Sophie became M. le Blanc.
During this time she became especially interested in the work of one professor, Joseph-Louis Lagrange. Lagrange was so impressed by her work that he insisted upon meeting the student who had produced it. * Upon discovering that it was a woman who had created it he was surprised, but not put off by it. He praised her for her analysis and would continue to support her and her work, becoming a mentor and a friend. Such encouragement from such a prominent mathematician energized Sophie and gave her more confidence in her work as a mathematician. With this newfound confidence Sophie moved from solving problems in her course work and into studying unexplored areas of mathematics. It was at this point that she became aware of Fermat's Last Theorem**. Fermat's Last Theorem continues to puzzle mathematicians to this day. It states that there are no solutions to equations of the form x^n+y^n=z^n where n ≥ 3.
After many years of work she believed that she had made a breakthrough in Fermat's Last Theorem. She felt that at this time she would need to talk to a fellow number theorist. After reading Gauss's Disquisitiones arithmeticae, a work on the theories of cyclometry, having to do with measuring of circles, and arithmetical forms she became obsessed with the works of Gauss. Using her alternate identity, M. le Blanc, she then sent Gauss the results some of her of work in number theory in 1804. Gauss was impressed by her findings and this began a correspondence between the two. She approached the theorem in a way that no other mathematicians had, possibly owing to the self taught nature of her education. She used a far more general approach then any of the other mathematicians of the day and this intrigued Gauss, whose contempt for Fermat's Last Theorem was well known.
In her letters she showed that she was not trying to prove that there was no solution for each equation one at a time, but instead she approached the Theorem holistically and tried to make a general statement that would cover all the equations. She focused on the equations in which n was a prime number. A prime number is a number such that itself and the number one can only divide it without leaving a remainder. For example 7 is a prime number because only 1 and 7 divide 7 without leaving a remainder whereas 9 is not since 1, 3, and 9 all divide 9 without remainders. Sophie was only interested in a particular brand of prime numbers, p, which were such that 2p+1 was also a prime number. For example 3 is a Germain prime because 2*3+1=7 and 7 is also prime. But 7 is not a Germain prime because 2*7+1=15 and 15 can be divided by 1, 3, 5, and 15 without remainders. She used these numbers to show that it was highly unlikely that any answers existed for the equation x^n+y^n=z^n, when n = p, with p being a Germain prime (as described above)**.
Had it not been for Napoleon's soldiers marching on Germany and Sophie fearing for Gauss's safety we may never have known that she was responsible for this breakthrough. We might know these numbers as le Blanc primes. Though Sophie and Gauss had been corresponding for 2 years by the time Napoleon invaded Germany, Gauss had no idea that M. le Blanc was actually Sophie Germain. Just as Archimedes' death inspired Sophie to begin her studies, so too did it weigh heavily on her mind as she thought of Gauss in Germany and the French army marching through. She worried that Gauss may suffer the same fate that Archimedes did and contacted a General friend of hers, Joseph-Marie Pernety, and asked him to guarantee Gauss's safety (Osen 85). True to his word General Pernety sent an emissary to Gauss's home to make sure that he was safe while they were in nearby Breslau, but confusion arose at the mention of Sophie's name. He did not understand why this woman was making inquiries about his safety – he had been corresponding with M le Blanc! It was after this that Sophie was forced to reveal her identity to Gauss. Upon learning of her true gender Gauss's respect and admiration for Sophie grew. For a woman to study mathematics at this time, and at this level required a high level of determination. He writes:
“But how to describe to you my admiration and astonishment at seeing my
esteemed correspondent M. le Blanc metamorphose himself into this illustrious
personage [Sophie Germain] who gives such a brilliant example of what I would
find it difficult to believe. A taste for the abstract sciences in general and above all
the mysteries of numbers is excessively rare; one is astonished at it; the
enchanting charms of this sublime science reveal themselves only to those who
have the courage to go deeply into it…Indeed nothing could prove to me in so
flattering and less equivocal manner that the attractions of this science, which has
enriched my life with so many joys, are not chimerical, as the predilection with
which you have honored it.” (Bell 1937, pg 262)(Osen, 86-87)
Sophie and Gauss would continue to correspond throughout her lifetime, though they unfortunately were never able to meet face to face.
Though Sophie's work with Number Theory and Germain primes would prove to be her most famous work, she became consumed with the work of Ernst Chladni. Chladni was a physicist who was interested in the mathematics of the vibrations of elastic surfaces in two dimensions. Chladni observed that figures would form when a bow was dragged across the edge of an elastic surface covered in fine powder. Lagrange declared that the current form of mathematical thinking would not be enough and that a revolutionary new form of analysis would be the only way that this problem would be solved. So in 1808, by order of Napoleon, the French Academy of Sciences issued a challenge for all mathematicians and scientists:
“Formulate a mathematical theory of elastic surfaces and indicate just how it agrees with empirical evidence. (Perl)”
Sophie, despite being wholly unschooled and up against highly trained mathematicians found the problem challenging and took to the task of trying to solve it with the passion that marked all her endeavors. In 1811 she submitted her first attempt in an anonymous memoir. Her mentor, Lagrange, was part of the commission that evaluated the submissions and had written that her method of jumping from a line to a surface did not seem complete or accurate. Her work was rejected, but that did not stop Sophie from working on the problem. In 1813 another competition was held. This time Sophie's memoir won her an honorable mention. As you should know by now, this was not enough for Sophie. She tried again in 1816 and this time her work Memoir on the Vibrations of Elastic Plates earned her the prize. But her win did not come without immense criticism. She had won, but the commission had not been completely satisfied with her entry, and Sophie admitted that her connection between the theory and the observation was not rigorous. Many mathematicians for this criticized her, but one may wonder if that is because they themselves could not come up with something better.
Despite this criticism she had an impact on those around her. One of the judges, M.H. Navier wrote on her paper “it is a work of which few men are able to read and which only one woman was able to write.”(Perl) Her win also thrust her into the world that she belonged to. She was welcomed into circles with other noted mathematicians of the time, and able to talk with her peers openly and not have to hide behind letters and the fictitious M le Blanc. She was not one of the most noted mathematicians in the world, and her work would finally be known as her own. She was now able to attend sessions at the Institute de France with other prominent mathematicians, and was the first woman who was able to do so.
Sophie continued to publish papers on elasticity, including one on the nature and extent of elastic surfaces. She also continued to try and perfect her theory that won her the prize in 1816. One of her papers drew heavily from the works of her mentor, Gauss, though it was widely criticized for not thoroughly understanding the potential of Gauss's theories.
In 1829 Sophie developed breast cancer. Though this became her biggest battle she continued her work in mathematics until her death at the age of 55. At Gauss's urging Sophie was granted an honorary PhD from the University of Gottengen (Osen 89), but unfortunately died before she was able to receive the award. Though she made great strides in elasticity and was one of the leading minds of her time, when the Eiffel Tower was erected and the engineers inscribed the names of those who contributed to the study of elasticity of materials of the 72 people listed her name is conspicuously absent (Osen 89). So too in her death certificate was her membership to the French Academy of Sciences overlooked as she was designated a rentière-annuitant (a single woman with no profession) and not as a mathematician.
Her work as a woman who defied the expectations and restrictions of her time is one of the most lasting parts of her legacy. Sophie Germain was not only one of the most brilliant minds of her time, and she excelled almost only through her own devices. Without any formal training and guided only somewhat through her correspondence with leading mathematicians she snubbed societal conventions and changed mathematics in lasting ways. The child of revolution, she gave us a revolution all her own and that was by following her passion and not letting the mores of the day or the obstacles that she faced dissuade her from following her heart. And mathematics is a richer field for that.
Bibliography
Osen, Lynn M. Women in Mathematics. Cambridge: The Massachusetts Institute of Technology, 1974
Perl, Teri. Math Equals: Biographies of Women Mathematicians + Related Activities. Menlo Park: Addison-Wesley Publishing, 1978.
*http://www.sdsc.edu/ScienceWomen/germain.html
**http://www.pbs.org/wgbh/nova/proof/germain.html
then G is abelian.
and 
we can cancel and therefore 
which shows that G is abelian.
