Leah Leddin
History of Mathematics
Biography
Sophie Germain
(1776-1831)
‘An 18th-century woman who assumed a man’s identity in order to pursue her passion – attempting to prove Fermat’s Last Theorem.’[i]
Many women in mathematics are not given enough credit as they are due for their contributions to the studies just because they are women. The women that are acknowledged work long and hard to be recognized and appreciated for their contributions to the field of mathematics. But not many women can say that they not only overcame the social prejudices but also the turmoil of the revolution to become a highly recognized mathematician. Sophie Germain’s life was just that way, overcome by the chaos of the French Revolution yet still destined for mathematical greatness.
Marie-Sophie Germain was born on April 1, 1776 in Paris, France. She was the middle daughter of three girls, Marie-Madeleine and Angelique-Ambroise. Her father was Ambroise-Francois Germain, a thriving middle class silk merchant who later became a director of the Bank of France. Her mother was Marie-Madeleine. She was known as Sophie to distinguish herself from all the Maries in the family.
At the age of thirteen, her parents kept her isolated in their home due to the danger caused by the French Revolution in Paris. During this time Sophie kept herself occupied by reading her father’s extensive library collection. While going through her father’s books she came across one that gave an account of the death of Archimedes of Syracuse. The story is told that “during the invasion of his city by the Romans Archimedes was so engrossed in the study of a geometric figure in the sand that he failed to respond to the questioning of a Roman soldier. As a result he was speared to death.” [ii] She was so affected by the story that she decided she too must study this subject a person could be so captivated by that it resulted in their own death. She thought mathematics must be an interesting subject.
Sophie pursued her desire to study mathematics. She once again turned to her father’s library, now for her mathematical teaching material. Sophie began teaching herself mathematics. She also taught herself Latin and Greek so she could read certain math books. She read late into the night to study the works of Newton and Euler. Her mother and father felt as though her interest in mathematics was inappropriate for a woman, following the usual belief of the times that women were not suited to do mathematics and they did all they could to discourage her from furthering her studies. This just caused her to study at night while they were sleeping. Her parents even got to the point where they were hiding her books, taking away the fire, her light, and taking her clothes from her once she got into bed to force her to not study at night. All the effort they put into keeping her away from mathematics, she found ways around it. She hid candles and wrapped herself in bedding in order to continue her studies at night. Her parents either got tired of trying or they realized she was determined to study mathematics because they finally gave in to desire, even allowing her to remain at home her entire life and support her financially since she never got married.
In 1794, the École Polytechnique opened in Paris. It was an academy ‘of excellence’ to teach mathematicians and scientists. Though Sophie was not allowed to enroll in the academy, men were only accepted, she was some how able to get the lecture notes from several of the courses being taught and teach herself from the notes and problems. This allowed her to continue her studies in mathematics with the opportunity to learn from some of the most influential mathematicians of the time.
Instead of participating in lectures, she had the practice of sending back ideas and insights to the professors and some times including her work on mathematical problems along with them. When sending these she took the name of a former student of the academy, Monsieur Antoine-August LeBlanc. She some how was able to intercept what was intended for LeBlanc and she continued to submit her insights to the problems under her false name. She took on this male name so she would be taken seriously, as many females of the time did.
Sophie had a particular interest in the lecture notes of Joseph-Louis Lagrange. After studying Lagrange’s lecture on analysis she submitted a paper to Lagrange. ‘LeBlanc’s’ insight into the subject impressed him so much, in part due to the change in skills from this previously average student of his, that he wanted to arrange a meeting with the person that wrote it. This request forced Sophie to reveal that a woman was the actual writer behind the paper. Lagrange looked beyond her gender and saw the brilliance of the work itself. He saw her abilities and became her mentor and friend. Though Sophie never had the formal education she would have liked, she now had a teacher who could help her take her skills to another level and inspire her. He remained a strong support and source of guidance to her for several years. He also introduced her to various French mathematicians and scientists. This was important due to the fact that Sophie was not able to acquire these contacts on her own because of her gender.
With the support of her mentor she grew in confidence, not only solving problems in her studies but also exploring areas of mathematics not many had done. Her early work was in number theory. This was motivated by the 1789 work of Andrien-Marie Legendre’s Théorie des Nombres. Sophie wrote to Legendre about problems suggested in his work. By then Legendre must have known that she was a woman because he was also a professor at the École and Lagrange’s colleague. This started a chain of communication that became a type of partnership. Legendre included some of Sophie’s discoveries in a supplement to the second edition of the Théorie.[iii] Her method of Fermat’s Last Theorem formed the basis for his proof of the theorem for the case n = 5. But how did she get involved in such a famous problem?
Through Sophie’s interest in number theory, she came across Fermat’s Last Theorem; xn + yn = zn has no positive integer solutions for x, y, z when n 2. She worked on this for several years and when she got to the point where she thought she had a major breakthrough she needed the insight of someone who knew the subject very well. In 1804, Sophie began consulting the top number theorist, Carl Friedrich Gauss. Her work on Fermat’s Last Theorem, called Germain’s Theorem, is known as her greatest contribution to mathematics but she was not initially given the credit she deserved for her work. She was still very young, in her 20’s, when she wrote to Gauss and even though she was known in Paris for her work, she did not think she would be taken seriously by Gauss due to her gender.
She began writing him, under her male name, concerning his work of 1801, Disquisitiones Arithmeticae. He was quite impressed with the work he was receiving and many times told of this to colleagues. Sophie’s work on Fermat’s Last Theorem was more general than other work Gauss had seen. She looked at the broader spectrum of the problem, looking at several equations rather than trying to figure out just one equation. ‘In her letter to Gauss she outlined a calculation which focused on those equations in which n is equal to a particular type of prime number.
Prime numbers are those numbers which have no divisors. For example, 11 is a prime number because 11 has no divisors, i.e. nothing will divide into 11 without leaving a remainder (except for 11 and 1). On the other hand, 12 is not a prime number because several numbers will divide into 12, i.e., 2, 3, 4, and 6. Germain was interested in those prime numbers p such that 2p + 1 is also a prime number. Germain’s list of primes, now known as the Sophie Germain prime, includes 5, because 11 (2 × 5 +1) is also prime, but it does not include 13, because 27 (2 × 13 + 1) is not prime.
For values of n equal to these Germain primes, she could show that there were probably no solutions to the equation: xn + yn = zn. By “probably” Germain meant that it was unlikely that any solutions existed, because if there was a solution, then either x, y, or z would be a multiple of n. this put a very tight restriction on any solutions. Her colleagues examined her list of primes one by one, trying to prove that x, y, or z could not be a multiple of n, therefore showing that for that particular value of n there could be no solutions.’[iv] This proof was significant in the study of mathematics due to the fact that it restricted the possible solutions to Fermat’s Last Theorem. Before her, the only proofs had been for specific cases of n. For prime exponents less than 100, she showed there could be no solutions relatively prime to the exponent. The first few Sophie Germain primes are. 2, 3, 5, 11, 23, 29, 41, 53, 83, 89, 113, and 131.[v] One of her other well known mathematical identities was ‘commonly known as Sophie Germain’s Identity, which states that for any two numbers x and y:’[vi]
x4 + 4y4 = (x2 + 2y2 + 2xy)(x2 + 2y2 – 2xy)
Though not much is mentioned about it when reading about Sophie, we use this today in factorizing certain equations.
It was not until Sophie feared that Gauss’ life may be in danger that Gauss learned of her true identity. Due to the French occupation of Gauss’ hometown of Braunschweig, Sophie asked French commander General Joseph-Marie Pernety, a friend of her family, to ensure Gauss’ safety.[vii] Even though Pernety was not a mathematician or scientist, he knew of the greatest mathematician and took particular care of Gauss. When hearing that Sophie Germain made the request of his safety Gauss was confused due to the fact that he did not know of this name. Sophie wrote another letter to him when she heard this.
“In describing the honourable mission I charged him with, M. Pernety informed me that he made my name known to you. 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.”[viii]
When hearing that the brilliant person he was in touch with was a woman he was taken back and he gave her even more respect as a mathematician.
“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 a superior genius.”[ix]
Her interaction with Gauss inspired much of her work in number theory. But in 1808 the letters stopped coming from Gauss. He had received the position of professor of astronomy at the University of Göttingen and his interest in number theory also stopped. Now that Sophie was without the guidance of Gauss she was in search of something new. She turned to the work of a physicist.
Around this time the French Academy of Sciences was offering a prize for a contest to explain the mathematical account of German physicist Ernst F.F. Chladni’s study of patterns produced by vibration. “Formulate a mathematical theory of elastic surfaces and indicate just how it agrees with the empirical evidence.”[x] Sophie accepted the challenge and spent the next two years trying to derive a theory of elasticity. In 1811 she entered an anonymous paper into the contest. Her paper was the only one submitted. Errors were found and it was evident that she had no formal education, but that did not stop her from moving forward in the subject. Lagrange, being one of the judge’s of the contest, was able to make the corrections in Sophie’s paper. But she was not awarded the prize. The contest was extended two years later and again she was the only one who entered it. Her paper this time was better showing the corrections she missed that last time but she was still not able to fulfill the requirements from the physical principles. Again, she was not awarded the prize but she was given an honorable mention. And again the contest was extended.
Sophie submitted a paper titled “Memoir on the Vibrations of Elastic Plates” and was finally awarded the prize of a medal of one kilogram of gold in 1816. Her paper laid the foundations for the modern theory of elasticity and was published privately in 1821, though she received word that there were still some insufficiencies in her work. ‘What should have been Sophie’s greatest achievement and source of pride, however, turned into a bittersweet victory. She received a laconic response from Siméon Denis Poisson, one of the judges and chief rival on the theory of elasticity, who wrote that her analysis still contained deficiencies and lacked mathematical rigor.’[xi]
Even though her work had earned her the prize, she did not go to the award ceremony. She felt that the judges and the community didn't really respect her work. “Although it was Germain who first attempted to solve a difficult problem, when others of more training, ability, and contact built upon her work, and elasticity became an important scientific topic, she was closed out. Women were simply not taken seriously.”[xii] But she did continue to study in the field of the theory of elasticity and published several additional works. Her work in the theory of elasticity is important today as it is used in the construction of skyscrapers and the applied mathematics associated with them.
Sophie met Jean-Baotiste-Joseph Fourier, well known mathematician of heat conduction and Secretary to the mathematical section of the Académie des Sciences, they became friends and he had quite a few goods words to say about her. As a result of her work on the theory of elasticity and Fermat’s Last Theorem, Sophie received honors from the Institut de France and was invited to attend lectures there. She was even allowed to attend meetings at the Académie des Sciences, with the help of her good friend Joseph Fourier. She was the first woman not related by marriage to have the privilege to be able to do so.