Wednesday, November 12, 2008

Sophie Germain: Mathematical Genius

In 1795 a shy young man by the name of Antoine-August Le Blanc enrolled at the Ecole Polytechnique in Paris. Le Blanc's brilliance in a mathematics course caught the attention of the class's supervisor, Joseph-Louis Lagrange, because the student had been notorious for his poor math skills. The student eventually was forced to confess the truth to Lagrange: the real Le Blanc had dropped out and, in fact, he was really a she named Sophie Germain.

Germain was the daughter of a merchant whose interest in mathematics was inspired by the reading a history of Archimides, who, legend has it, was killed because he was so focused on studying a geometric figure he failed to hear the questioning of a Roman soldier. Equally fascinated, she taught herself basic number theory and calculus, often studying late into the night. Her parents tried to deter her from her new-found passion by taking away her candles and source of heat, but Sophie continued her studies despite those hardships. Eventually she received her parents' blessing, and her father ended up supporting her research financially.

Her outing to Lagrange turned out to be a blessing:

Lagrange was astonished and pleased to meet the young woman, and became her mentor and friend. At last Sophie Germain had a teacher who could inspire her, and with whom she could be open about her skills and ambitions.

Germain grew in confidence and she moved from solving problems in her course work to studying unexplored areas of mathematics. Most importantly, she became interested in number theory and inevitably she came to hear of Fermat's Last Theorem. She worked on the problem for several years, eventually reaching the stage where she believed she had made an important breakthrough. She needed to discuss her ideas with a fellow number theorist and decided that she would go straight to the top and consult the greatest number theorist in the world, the German mathematician Carl Friedrich Gauss.

Unsure of how Gauss would respond to a woman, she wrote to him using the Le Blanc pseudonym, and under that name continued a regular mathematical correspondence with him. Her true identity was only revealed when she asked a friend who was a General in Napoleon's army to guarantee Gauss's safety during the French invasion of Prussia. Like Legrange, Gauss turned out to readily accept her true identity, writing:
But how to describe to you my admiration and astonishment at seeing my esteemed correspondent Monsieur Le Blanc metamorphose himself into this illustrious personage 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 not astonished at it: 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.
Eventually Gauss broke off his correspondence, and Germain shifted her own research from number theory to applied mathematics and physics, at which she also excelled.
The occasion was the demonstration by a visitor to Paris, one E. F. F. Chladni, of curious patterns produced on small glass plates covered with sand and played, as though the plates were violins, by using a bow. The sand moved about until it reached the nodes, and the array of patterns resulting from the "playing" of different notes caused great excitement among the Parisian polymaths. It was the first "scientific visualization" of two-dimensional harmonic motion. Napoleon authorized an extraordinary prize for the best mathematical explanation of the phenomenon, and a contest announcement was issued.

Sophie Germain's entry was the only one. While it contained mathematical flaws and was rejected, her approach was correct. All the other possible entrants in the contest were prisoners of the ruling paradigm, consideration of the underlying molecular structure theorized for materials. The mathematical methodologies appropriate to the molecular view could not cope with the problem. But Germain was not so encumbered.

With the help of other mathematicians, she reapplied and eventually won the prize. Her paper "Memoir on the Vibrations of Elastic Plates" laid the foundation of the modern theory of elasticity. The prize helped Germain meet other prominent mathematicians and gave her entrance to sessions at the Academy of Sciences and Institut de France, the only woman so honored.

Her old friend Gauss eventually convinced the University of Gottengen to award her an honorary degree, but sadly she lost her two year battle with breast cancer before she could receive it. She was only 55 at the time of her death. She never married.

Despite the awards and honors she received during her lifetime, she was not completely accepted because of her sex. HJ Mozans noted in his 1913 history Women in Science:
All things considered, she was probably the most profoundly intellectual woman that France has ever produced. And yet, strange as it may seem, when the state official came to make out her death certificate, he designated her as a renti√®re-annuitant [a single woman with no profession]—not as a math√©maticienne. Nor is this all. When the Eiffel Tower was erected, in which the engineers were obliged to give special attention to the elasticity of the materials used, there were inscribed on this lofty structure the names of seventy-two savants. But one will not find in this list the name of that daughter of genius, whose researches contributed so much toward establishing the theory of the elasticity of metals—Sophie Germain. Was she excluded from this list for the same reason she was ineligible for membership in the French Academy—because she was a woman? If such, indeed, was the case, more is the shame for those who were responsible for such ingratitude toward one who had deserved so well of science, and who by her achievements had won an enviable place in the hall of fame.
Today there is a street named after her in Paris, and her statue stands in the courtyard of the Ecole Sophie Germain.

More information about Sophie Germain:
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