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She’s the Man: The Life of Sophie Germain

In a letter to a friend, Carl Friedrich Gauss—dubbed “the prince of mathematics”—wrote:

I recently had the pleasure of receiving a letter from LeBlanc, a young mathematician in Paris, who made himself enthusiastically familiar with higher mathematics and showed how deeply he penetrated into my ‘Disquisitiones Arithmeticae’.

Unbeknownst to Gauss, this young mathematician, LeBlanc, who had showed a remarkable aptitude towards mathematics, was in fact not a man, but a woman. Her name was Marie-Sophie Germain.

Sophie Germain was born on April 1st, 1776, and died 55 years later on June 27th, 1831, from breast cancer. She lived during a time of great political upheaval: the American Revolution kick-started the same year she was born; in 1789, the French Revolution rocked her home city of Paris; from 1803 to 1815, Napoleon engulfed Europe in widespread and seemless war; and shortly after she died, Parisians rose to overthrow the bourbon monarchy that was installed after Napoleon’s expulsion to Saint Helena. In a sense, she embodied the ideals of the revolution, upending societal expectations, despite the French proclaiming “Liberty, Equality and Fraternity”.

Daughter to a silk merchant, Sophie Germain’s passion towards mathematics began at the start of the French Revolution. Because people back then couldn’t binge-watch Netflix series, Sophie spent her days inside her father’s extensive library. One day, she came across a math history book and read the scene of Archimedes’ death: as the city of Syracuse was being overrun by the Romans, a soldier had burst into Archimedes’ house, only to discover that the old man was engrossed in a mathematical problem. The soldier was so infuriated that the geometer was ignoring his questioning that he slew him right then and there. Sophie was so moved by this account that she had to learn what this Archimedes fellow was studying. She was thirteen.

If Sophie wanted to study mathematics, she needed the basics covered. She taught herself long-division, proportions, fractions, and logarithms. Later, she discovered a math book that dealt with differentiation and integration. She mastered the subject in a quick breath of time. Yet in order to climb the mountain of higher mathematics, whose titans were Newton and Euler, she had to learn Latin and Greek. This she taught herself as well.

While Sophie was toiling away studying, her parents caught wind of the situation. Middle-class families in 18th century Paris saw mathematics as something inappropriate for a girl, so Germain’s parents tried dissuading her from continuing her mathematical journey. When words failed to change her mind, they stripped her room of fire and candlelight; they even took away her clothes, letting her sleep at the mercy of the frosty nights.

Fortunately for this story, Germain’s parents’ efforts were in vain.

One cold winter night, the parents came over to Germain’s room, but there was no sight of her. Instead, they discovered their daughter sleeping at the library over a book, wrapped in quilts, the inkwell dried up. The room was freezing.

That scene convinced Sophie’s parents their daughter’s obsession was “incurable”.  They would financially support their daughter’s investigations.

Sophie continued her private mathematical studies through the head-chopping Reign of Terror. Eventually, as the revolution toned down, the École Polytechnique opened in 1794, when Sophie turned 18. Women could not, however, gain entry to the university. Luckily, Sophie had a few things up her sleeves. For one, lecture notes from the École were published freely to non-students, which meant Sophie could gain notes from J.L Lagrange, one of mathematic’s titans of the 18th century; secondly, in order to gauge her mathematical prowess, she needed Lagrange’s feedback, but since she was barred from entering the École, she used a pseudonym of “LeBlanc”, who was in fact an actual student that dropped out from Lagrange’s course.

At the end of the course, Sophie—under the guise of LeBlanc, of course—sent Lagrange her mathematical observations and insights. The man was deeply impressed by LeBlanc’s ingenuity. How could a student that had no gift in mathematics suddenly experienced a metamorphosis? In order to find out, Lagrange arranged a meeting with LeBlanc. Sophie was afraid Lagrange might rebuke her for hiding her real identity. But to Sophie’s surprise, Lagrange did not mind that she was a girl and decided to be her tutor.

Lagrange spread the word of Sophie’s talents to other mathematicians. Many came to visit to make correspondence. Most of the time the visits were productive and splendid, while some bad-mouthed Sophie. Nevertheless, this opportunity opened Sophie to the wide mathematical world, although she spent most of her life with her family in Paris.

The next chapter in Sophie’s journey occurred between 1798 and 1801. Adrien-Marie Legendre and Carl Friedrich Gauss wrote their most important respective works in number theory: Essai sur la Théorie des Nombres and Disquisitiones Arithmeticae. Sophie first wrote to Legendre, keeping her identity a secret, exchanging discoveries that she made while reading his work. Her discoveries were published in Legendre’s second edition of Théorie.

However, Germain’s correspondence with Gauss was probably the high point of her (unofficial) career.

After devouring Disquisitiones Arithmeticae, Sophie sent Gauss her observations and insights in 1804. The results astounded Gauss. Their back-and-forth correspondence dealt with many topics, including Fermat’s Last Theorem, a conjecture that was stated a century earlier by the French mathematician Pierre de Fermat.

Basically, Fermat’s Last Theorem states that, when given three positive numbers, say, x, y, and z, you can’t express them as xn+yn=zn when n is greater than 2. Remember that for n=1 we have x+y=z, which means any positive number can be written as the sum of two other positive numbers; for n=2, we have good old Pythagoras. However, mathematicians around Sofie’s time proved that for n=3 and n=4 the formula is incorrect.

Sophie’s real identity came to be known to Gauss in 1806. That year the French were occupying the city of Braunschweig in Germany, where Gauss was living. Fearing that he might suffer the same fate as Archimedes, Sophie contacted the general overseeing the occupation. One can imagine Gauss’ perplexed face when the general told him that he owed his life to a Parisian woman. What Parisian woman?

Eventually, Sophie revealed her real identity to Gauss. She explained that LeBlanc and Sophie were the same person. Thankfully, Gauss’ respect for her increased tenfold and continued collaborating with her until he became professor of astronomy at the University of Gottingen in 1808.

With Gauss focused on more applied mathematics, Sophie tried her luck elsewhere. That would come in 1809 when a scientist produced music and visible patterns in sand on a flat glass plate by striking a violin bow at its edge. This captivated the Paris intellectual community so much that Napoleon Bonaparte offered a prize to whoever developed a mathematical description. The deadline was set for two years by the French Academy of Sciences

Sophie was the only person who submitted an entry. However, because she lacked a formal mathematical education, she needed feedback from Legendre. Her ideas were on the right track, but the judges, who among them was Lagrange, discovered her answer was riddled with errors.

Two attempts later, Sophie won the prize. Although the judges saw certain shortcomings to her memoir, Sophie continued refining her theory for the remaining years of her life.

The prize from the Academy opened her to the crème de la crème of the greatest mathematicians of the era. She was the first woman to attend lectures at the academy and was a collaborator for many male mathematicians. It was here her proofs and mathematical techniques reached new heights; she published remarkable theorems that advanced the study of Fermat’s Last Theorem.for

And now our story has reached its ending. In 1829, Sophie Germain was suffering from breast cancer. She fought the disease until she died on June 27, 1831, in Paris. 

When we think of mathematicians, our minds flicker towards figures like Pythagoras, Newton, Euler, and Gauss, to the detriment of lesser known mathematicians such as Sofie Germain. Besides being brilliant, Sofie has a captivating and sympathetic life story that is better movie material than most mathematicians alive and dead, not to mention that she made a name for herself by proving a theorem related to Fermat’s Last Theorem, a huge leap on a century’s old mathematical problem that took until the 1990s to be solved

Regardless of her legacy, Sofie Germain is an example of persistence in the face of adversity. A timeless lesson for everyone.

Emilio H. Mejill Ortiz is a fifth-year UPRM student pursuing a mayor in pure mathematics. His two dreams after graduating is to one day publish a novel and to drive from Miami to Seattle. He loves reading, learning about history, and strives to master koine greek.
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