Prime Mystery: The Life and Mathematics of Sophie Germain

Who was Sophie Germain, and why is a mathematical biography particularly timely and interesting, both for the mathematics and the social history?

Sophie Germain (1776–1831) was the first woman we know of who did important original mathematical research. She won the mathematical prize competition of the French Academy of Sciences to explain the fascinating vibrational patterns of elastic membranes. As a girl she was initially actively prevented by her parents from studying mathematics, and as a woman she was denied access by society to higher education and to research and professional environments. But in a true tale stranger than fiction, Sophie Germain nonetheless pursued mathematics successfully through an incredible combination of talent, persistence, and bravado. In an age when Parisian women still had to rely on chaperones for almost every male encounter, Sophie Germain, initially through years of written male impersonation, gained the attention, then mentorship, and eventually professional respect of the greatest mathematicians of the age, including Lagrange, Legendre, Gauss, and Fourier. At one point Gauss even wrote to a friend that “LeBlanc” (Germain’s assumed male pseudonym) was the only person who really understood his landmark book that ushered in modern number theory, Disquisitiones Arithmeticae.

Sophie Germain’s research was in two completely different areas, but little of either was published, since publication avenues were difficult for her. She has long been best known for her work in elasticity theory, and this story has been told in the book Sophie Germain: an Essay in the History of the Theory of Elasticity by L. Bucciarelli and N. Dworksy.

In number theory Sophie Germain has long been known for a single result credited to her by Legendre in a footnote in his own work: a theorem enabling the confirmation of Case I of Fermat’s Last Theorem for many exponents. This was the first published general result towards proving Fermat’s Last Theorem, a historic challenge finally met at the end of 20th century mathematics. For two centuries it was thought that this result of Sophie Germain’s was the extent of her work in number theory. Quite recently, though, hundreds of sheets of Germain’s surviving handwriting on number theory have been studied and analyzed, in a paper by R. Laubenbacher and D. Pengelley in Historia Mathematica, and another independent paper by A. Del Centina in the Archive for History of Exact Sciences. The outcome is a dramatic reassessment of the scope of Germain’s work. We now know that she had a grand plan for proving Fermat’s Last Theorem in its entirety, and that she developed theorems, algorithms, and techniques for carrying out her plan, even though in the end the approach did not succeed.

Against this background we have two recent books by author Dora Musielak. The first, Sophie’s Diary: A Mathematical Novel, published by the MAA, is a delightfully engaging fictional diary of Sophie Germain as a teenager, completely surrounded by the ongoing French Revolution yet focused on teaching herself the mathematics that eventually enabled her to pursue research and gain the attention and respect of great mathematicians. The second book, under review here, Prime Mystery: The Life and Mathematics of Sophie Germain, is a serious mathematical biography in the true sense of both words. Many biographies of mathematicians shy away from any depth of treatment of the mathematics, but this book delves into Germain’s mathematics as well as everything else we know of her life. This is highly appropriate, since so little of Germain’s mathematical work was published by her and until recently precious little was even known through others’ publications.

Prime Mystery painstakingly condenses a comprehensive study of a plethora of original source material to write an in-depth description of what we can know of Germain’s life and work. The book paints a fascinating picture of her interactions, challenges, accomplishments, and frustrations, in large part due to her disfavored status as a woman at the turn of the 19th century. Due to Germain’s lack of stature as a female in her society and in her de facto profession, source material on her life and work is much harder to come by than for a typical male professional. It is a great credit to the author that she has ferreted out so much rich information nonetheless. The interpretations, connections, and conjectures the author makes from this material are well considered, interesting, and often original and stimulating. And the book also raises many fascinating questions either for the reader to consider or for future research.

The scope and depth of the mathematics and the social and political history considered is excellent, making fascinating reading. This includes short connecting biographies near the end, as well as a final discussion of unanswered questions, women and science education, and the legacy of Sophie Germain. All in all, no possible stone is left unturned.

The author has a most engaging writing style, and the story never lacks for interest throughout, despite us actually having little direct evidence about Sophie Germain’s personal life. The author travels all the extant evidentiary connections of her life to weave a truly compelling story of the first female mathematical researcher, far richer than I would have thought possible from what documentation I imagined still existed today.

David Pengelley is professor emeritus at New Mexico State University. His research is in algebraic topology and history of mathematics. He has studied the handwritten manuscripts of Sophie Germain, and published “Voici ce que j’ai trouvé”: Sophie Germain’s grand plan to prove Fermat’s Last Theorem, in Historia Mathematica. David develops the pedagogies of teaching with student projects and with primary historical sources, and created a graduate course on the role of history in teaching mathematics. He has received the MAA’s Haimo teaching award, loves backpacking and wilderness, is active on environmental issues, and has become a fanatical badminton player.