Sophie Germain Dionna Bidny Sophie Germain's Life The term revolutionary can apply to a great array of individuals. Some revolutionaries fight social standards with weapons, some with words, and some with thoughts. During not only her lifetime but well beyond, Sophie Germain was, in more than one sense, a revolutionary. Her life and actions set a new standard for not only mathematics but also for women, emerging from a changing time reflected in the French Revolution itself. Sophie Germain was born on April 1, 1776, the second of three daughters of a middle class family. When Germain was only 13 years old, the French Revolution erupted around her with the fall of the Bastille. Forced to stay indoors, Germain took refuge in her father's library, where she poured over books on mathematics, Latin, and Greek. She read works by Euler, Newton, and many other scholars, and was determined to follow the paths they paved and delve even deeper into the world of mathematics. Germain's parents, however, were unsupportive of her new scholarly interests that were so atypical and socially unacceptable for women of the time, especially those of Germain's age. It is said by her first biographer, Italian mathematician Guglielmo Libri, that Germain was forced to study in the cold library by candlelight at night because of her family's attempts to thwart her budding love of mathematics, language, and philosophy at any other time. Undeterred, it is evident that Germain persevered and did all she could to purse her studies in the subjects that sparked such an interest in her. Her parents, eventually noting her undying zeal, eventually allowed her to continue her education. As Germain diligently studied the works of Bezout, Cousin, and many other scholars in her father's library, the Revolution around her swelled. As a direct result of the Revolution, the Ecole Polytechnique opened successfully in Paris in the year 1794, publishing its first journal one year later. The academy strove to provide an education strong in mathematical and scientific learning to enlighten the middle class and especially to compensate for the death of many engineers and officials during the Revolution. With world-renowned professors as dedicated to research as they were to classroom teaching, the Ecole quickly gained momentum in popularity and prestige. Unfortunately for Germain, however, the Ecole did not admit women as students until much later. Undeterred as always, 18-year old Germain somehow obtained books and lecture notes, which she ever-enthusiastically studied, from students attending the Ecole. Germain especially focused on the works of mathematician Joseph-Louis Lagrange, who taught at the institution. At the Ecole, it was customary for instructors to invite students to write observations or analyses of the works they had been taught by the end of the term. Cleverly, Germain submitted her work on Lagrange's study of mathematical analysis under a pseudonym of a student already attending the Ecole (Antoine-August LeBlanc). Naturally, Lagrange was extremely impressed by Germain's review, and even more impressed and pleasantly surprised when he discovered here true identity after he sought out a meeting with her. It's important to note that at eighteen, Germain had already achieved the educational level of an undergraduate student, enough to impress a top mathematician of the time, simply through her own dedication and self- study. The fact that all of her mathematical prowess came to Germain not only without assistance, but actually during a time when others attempted to suppress her interest, is nowhere short of astounding. Many previous successful female mathematicians started their careers under the mentorship and encouragement of an already established mathematician. Germain, however, had only her incredible passion and insatiable thirst for knowledge as her sole motivator until much later in life. After their initial acquaintance, Lagrange and Germain continued their scholarly correspondence. This corre- spondence gained Germain exposure to much of Paris' notable mathematical circle. She began to collaborate with other renowned thinkers, and engage in new ideas surrounding intricate mathematical, philosophical, and literary concepts. A newfound interest in number theory prompted Germain to contact a prominent mathematician of the time, Legendre, whose book Th´eoriedes Nombres focused on this idea precisely. Their correspondence was so in-depth, that Legendre in fact credited Germain with more developed concepts that he included in a later edition of Th´eorie.Despite now being a more elite member of the Parisian mathematical circle, it is evident that Germain's proactiveness and determination for learning never waned, as she continued to stretch her limits and gain mentorship and insight from anyone with whom she could communicate. Germain similarly began correspondence with Gauss one again under the pseudonym LeBlanc (the real LeBlanc having died by now) after reading on of Gauss' books. Germain included her work on Number Theory and her preliminary work on Fermats Last Theorem. Gauss, naturally, was impressed by the thoroughness of Monsieur LeBlanc's thought processes. When Gauss learned the true identity of is intelligent young correspondent, he was surprised but full of encouragement and praise, as is evident in a statement he made after discovering the true Germain: But how can I describe my astonishment and admiration on seeing my esteemed correspondent Monsieur LeBlanc metamorphosed into this celebrated person, yielding a copy so brilliant it is hard to believe? The taste for the abstract sciences in general and, above all, for the mysteries of numbers, is very rare: this is not surprising, since the charms of this sublime science in all their beauty reveal themselves only to those who have the courage to fathom them. But when a woman, because of her sex, our customs and prejudices, encounters 2 infinitely more obstacles than men, in familiarizing herself with their knotty problems, yet overcomes these fetters and penetrates that which is most hidden, she doubtless has the most noble courage, extraordinary talent, and superior genius. Nothing could prove to me in a more flattering and less equivocal way that the attractions of that science, which have added so much joy to my life, are not chimerical, than the favor with which you have honored it. The scientific notes with which your letters are so richly filled have given me a thousand pleasures. I have studied them with attention and I admire the ease with which you penetrate all branches of arithmetic, and the wisdom with which you generalize and perfect. At this point, Sophie Germain embarked on a number of notable mathematical journeys (the details of which will be covered in a later section). Among these was a major portion of a proof of the vibrational and elastic waveforms observed on a surface by German physicist Ernst Chaladni in 1808. By demonstrating nearly to completion the underlying mathematics of the demonstrated phenomenon, Germain eventually won a contest by the French Academy of Sciences to develop such a proof. This award was highly influential in Germain's life, as it propelled her to the ranks of a higher circle of mathematicians. After winning the award, Germain continued to improve her work and became ultimately very influential in the field of elasticity. In addition to her work in elacticiy, Germain dedicated extensive work towards developing a proof of Fermat's Last Theorem. While Germain never completed the proof, Germain is credited for developing the first proof that generalizes the theorem and makes it applicable for a range of exponents. In addition, manuscripts of Germain that were lost and later recovered show that her work extended far past this portion of the proof and delved deep;y into a method for proving the theorem in its entirety. Germain continued to rise as a prominent mathematician of her time. She was the first single woman to attend the Academy of Sciences session after an invitation by the Insitute of France, and was conintually praised by the Institute and her mathematical peers. By now, Gauss had convinced the University of Gottingen to award Germain with an honorary degree. However, before she could receive the honorary degree, Germain died to brest cancer on June 27th, 1831. She was 55 years old. While premature, her death did nothing to minimize the revolutionary life she had led. Germain persevered through political upheaval, social prejudice, and lack of formal education to achieve more than seems possible. She left an extremely significant imprint on the face of mathematics, and her works and memory continue to influence many to this day. 3 Sophie Germain's Mathematical Works FERMAT'S LAST THEOREM: As has been mentioned, one of Sophie Germain's most prominent achievements was her work with Fermat's Last Theorem. The origin of the theorem stems back to a note scrawled by Pierre de Fermat in the margin of a book in the 1630s: It is impossible to separate a cube into two cubes, or a biquadrate into two biquadrates, or in general any power higher than the second into two powers of like degree; I have discovered a truly remarkable proof which this margin is too small to contain The proof that went unwritten in the margin was, for all we know, never written down at all by Fermat, but rather continued to dog at the minds of mathematicians for centuries. In mathematical notation, the theorem Fermat developed is as follows: xn + yn = zn has no positive integer solutions for x, y, z when n > 2 While this theorem wasn't fully proved until Andrew Wile's 1995 publication, Sophia Germain played an essential role in first creating an overarching plan for how to tackle a majority of the proof. Previously, Fermat had proven by his method of infinite decent that for a right triangle with rational sides, the area cannot be a square.