The Elasticity of Sophie Germain
Marie-Sophie Germain. Source - womwnyoushouldknow.net
Women who want to be a scientist or mathematician today have a much better opportunity to go to university and study towards that ambition, although sadly not in all countries. This was not so back at the end of the 18th century. Sophie Germain is a hero to me not so much because of her work in elasticity and number theory, although that was heroic, but because of her determination when every part of her environment was against her ambition. It’s a wonderful example of the pursuit of knowledge against the odds and even better, for its own sake. Heroes are often lauded for their physical prowess, but overcoming multiple obstacles just to have a better understanding of the world is, in my book, a lot more impressive.
Marie-Sophie Germain was born in 1776 in Paris to a quite wealthy family. She was thirteen when the Bastille fell signalling the start of the French Revolution. Because it was a dangerous time She had to stay indoors for a long period. To pass the time she started reading books from her father’s library. She came across a book called L’Histoire des Mathematiques. This book contained a story about the death of Archimedes. Somehow this story sparked her interest in mathematics and she read every book on the subject in her father’s library. Amazingly for a 13 year old, she taught herself Latin and Greek so that she could read some of the books. She read Isaac Newton’s Principia and many works by Euler, a prolific mathematician. One of the authors, Jacques Antoine-Joseph Cousin even visited her and encouraged her studies.
Joseph Louis Legrange. Source - Wikimedia
Her parents were against her interest in mathematics, agreeing with the popular English notion that ‘brainwork’ was dangerous for girls. In the book written for children ‘Nothing Stopped Sophie’ Cheryl Bardoe recounts how, when night came, they would deny her warm clothes and a fire for her bedroom to try to keep her from studying, but after they left she would take out candles, wrap herself in quilts and do mathematics. In another book ‘Women In Mathematics’ Lynn Osen writes that when her parents found Sophie "asleep at her desk in the morning, the ink frozen in the ink horn and her slate covered with calculations," they realised that their daughter was serious and relented. After some time, her mother even secretly supported her.
When Sophie was 18 the Ecole Polytechnique opened. It is still a prestigious institution today, well known for its engineering degree programmes. Because she was a woman Sophie was not allowed to enrol. Not giving up, she arranged to get the lecture notes and sent her work to a faculty member, Joseph Louis Legrange. Legrange is immortalised as the discoverer of ‘Legrange Points’, locations in local space where the gravitational effects of Earth, Moon and Sun cancel out, allowing objects to remain stationary. This is similar in principle to the places in the oceans, like the Sargasso Sea where four currents conspire to create a stationary area accumulating sargassum seaweed. The same thing is now happening with plastic. Technically though, ocean currents don’t know how to conspire.
To protect her identity She used the name of a former student, LeBlanc, to hide the fact that she was a woman. Lagrange was impressed enough to ask if they could meet. This forced her to disclose her true identity. Lagrange not only didn’t have a problem with her gender, but became her mentor.
Ernst Chladni and the vibrating plate
Chladni, his vibrating plate and some of the patterns produced. Source - Wikimedia
Ernst Chladni’s best known work was on the characteristics of a vibrating plate. He discovered that if a rigid plate is vibrated (he used a violin bow, see left) it flexed up and down is some areas, but not in others. He demonstrated this by sprinkling sand on the plate. When vibrated the sand moved around until it accumulated in the areas of the plate that were not moving. By stroking the bow while placing his finger in different positions on the plate’s edge the sand produced a wide range of patterns. Chladni’s problem was that he could not work out the mathematics that explained how and why these patterns were emerging. It turns out to be a really difficult problem which, according to Legrange, would require a new branch of mathematical analysis. Figuring out the maths was considered important enough for the Paris Academy of Sciences to sponsor a competition "to give the mathematical theory of the vibration of an elastic surface and to compare the theory to experimental evidence." As my son Steve points out in the video left, this behaviour is an example of wave dynamics which threads its way through most of science, so the importance of trying to understand it can’t be overstated. Steve’s video has a really cool animated overlay that shows what’s happening. Incidentally Chladni looks a bit like Paul Whitehouse.
In 1811 Germain submitted a paper or ‘Memoir’, but didn’t win the prize. In fact nobody did. The competition was extended twice in 1813 and 1816. Sophie submitted memoirs each time and won the third time. Although she won under her own name and was the first woman to win an Academy of Science prize, she could not attend the award ceremony or take part in Academy sessions. This was because of the Academy’s tradition of excluding women.
Germain’s work was a major leap forward in understanding wave mechanics. It established a way of working but wasn’t taken as seriously as it should have been because it contained faults in the methodology. The main reason for the errors was that she had to work outside of the academic system and missed out on underlying principles of the scientific method. How much more could she have achieved by being fully included from the start. Other male scientists built on Sophie’s work without acknowledgement and thereby gained the credit.
Left - Pierre de Fermat (1607-1667) Centre - Arien-Marie Legendre (1752-1833) Right - Carl Frederick Gauss (1777-1855) Source - Wikimedia
Fermat’s Last Theorem
In 1637 Pierre de Fermat put forward a mathematical conjecture that was devilishly difficult to prove was true. I don’t propose to describe the conjecture itself because its complicated. I read the excellent book ‘Fermat’s Enigma’ by Simon Singh and although it gave me a headache for about a week, I think I understood it. Thing is, it was a while ago and my memory is crap. As I don’t want another of those headaches I will just link here to a slide show that displays Germain’s partial proof, the one that formed the backbone for all subsequent attempts. Have a look if you’re interested, but have some paracetamol handy. The intriguing thing about it was that Fermat claimed in some margin notes that he had the proof but that there wasn’t room in the margin to set it down. Fermat died without revealing his proof, and since it took another 358 years before it was solved in 1994, there is some doubt that Fermat really had worked it out. Germain’s groundbreaking work was important enough to eventually be given it’s own name ‘Sophie Germain’s Theorem.’ Germain didn’t publish her pioneering work and it only survived as a supplement to ‘Theorie des Nombres’ by Adrien-Marie Legendre, a distinguished mathematician that corresponded with her. Germain’s work was attributed in her lifetime to her mentor Joseph Louis Legrange.
Carl Frederick Gauss
Carl Frederick Gauss has been described as "the greatest mathematician since antiquity" and although they never met Germain corresponded with him using her disguise as leBlanc. When Germain revealed her true identity he replied,
How can I describe my astonishment and admiration on seeing my esteemed correspondent M leBlanc metamorphosed into this celebrated person. . . when a woman, because of her sex, our customs and prejudices, encounters infinitely more obstacles than men in familiarising herself with [number theory's] 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.
Gauss was instrumental in gaining recognition for Germain, but unfortunately just after she had died at 55 of breast cancer. Gauss had recommended her for an honorary doctorate at the University of Gottingen, and she would have met him for the first time if she had lived.
On the one hand this is a dark story showing how the talents of half the human race were suppressed or ignored for most of recorded history. It is impossible to know just how much benefit we all might have obtained had this not been so, but what we might have added to the sum total of human knowledge must be immense. On the other hand it’s inspirational. So much that we have discovered about our world has been the result of a determined personal yearning to understand, and Sophie Germain epitomised that.