I thought, before this past week, that I appreciated quite well the important but often unacknowledged role that women have played in the history of science and mathematics. It turns out that I've hardly scratched the surface of their contributions, which go back even further than I imagined. Perhaps even more fascinating is the fact that a number of truly great male researchers realized the brilliance of these women, even if the bulk of the academic community did not. As a supplement to my Ada Lovelace day post, I thought I'd present a little more musing on the role of women in science from the point of view of some of these researchers.
A truly jaw-dropping bit of correspondence centers around a woman whose work I knew little about before this week, Sophie Germain (1776-1831), who was born into a well-to-do Parisian family .
Sophie Germain at age 14
The French Revolution had begun in earnest when Germain was a teenager and, forced to stay inside, she turned to books on mathematics in her father's library for entertainment. It is said that she was fascinated with the story of the death of Archimedes (at the end of the siege in which he supposedly used his "death ray"), and to further her interest in mathematics she taught herself Latin to study the work of Newton and Euler. Though she was barred as a woman from attending the École Polytechnique when it reopened in 1794, she obtained the freely-available lecture notes and smuggled her homework papers in for consideration under the pseudonym "M. LeBlanc". LeBlanc's work drew the attention of the mathematician Joseph Louis Lagrange, who quickly learned the true identity, and gender, of his correspondent, and was so impressed that he became her friend, counselor and mentor.
Germain would go on to make significant contributions to the mathematical theory of elasticity and to number theory, providing ideas and making inroads in solving the infamous Fermat's last theorem that would still be used over a hundred years later.
The correspondence of Germain's I would like to highlight seems to have been sparked by the publication of Carl Friedrich Gauss' book on number theory, Disquisitiones Arithmeticae, in 1801. (If you're not familiar with Gauss' name, let me just state that it seems like half of the theorems and functions of mathematical physics are named after him -- it is simply impossible to learn physics without learning and using Gauss' discoveries.) Sophie worked through Gauss' theorems for three years and finally wrote to him directly under the familiar pseudonym "M. LeBlanc" in 1804 to discuss his results and present her own work.
Their correspondence continued for several years. Around 1806-1807, the French had occupied Gauss' home of Braunschweig, and Sophie had become concerned for his safety. To quote from ,
On 27 November a French officer named Chantel entered a room where Gauss was with his wife of one year. His general, Pernety had been asked by one Demoiselle Sophie Germain to ask after his health. Gauss was perplexed for he knew only one woman in Paris, the wife of an astronomer. Only three months later did Gauss hear from Germain that she was his correspondent Leblanc. Bolyai wrote teasingly to Gauss when he heard of this. "You wrote to me once of a Sophie in Paris; if I were your wife I would not be too pleased. Write me more of her." On 20 February 1807 Germain wrote to put matters straight. "[I have been led] to confess that I am not as completely unknown to you as you might believe, but fearing the ridicule attached to a female scientist, I have previously taken the name of Leblanc in communicating to you those notes that, no doubt, do not deserve the indulgence with which you have responded.... I include with my letter a note intended to show you that I have maintained an appetite for analysis that the reading of your work has inspired...."
Gauss' reply is, to me, mind-blowing and an amazing validation of the talents of his female correspondent. He wrote,
. . 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 familiarizing 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. ... The scientific notes with which your letters are so richly filled have given me a thousand pleasures. I have studied them with attention. ...
If you ever encounter anyone who claims that women cannot be mathematicians, and could historically have been if they wanted to, point them at Gauss' letter! He shows that he recognized both her brilliance and the unique obstacles she had to face as a woman mathematician.
It seems that the correspondence between Gauss and Germain lasted only a few letters beyond her "coming out", though the reasons for this are unclear. I have read that Gauss simply lost interest in number theory and his correspondence became less important to him, though other reasons seem just as likely. His letters ceased in 1809, which corresponds directly with the death of Gauss' first wife, followed soon after by the death of one of his sons. Wikipedia states that Gauss plunged into a depression from which he never fully recovered, and it is not too great a leap to think that he had other things on his mind other than his mathematics and letter writing. Nevertheless, his respect for Sophie was significant; he recommended to the University of Göttingen that she be awarded an honorary doctorate, but she died in 1831 before such a degree could be awarded. In 1837, Gauss lamented the missed opportunity :
She proved to the world that even a woman can accomplish something worthwhile in the most rigorous and abstract of the sciences and for that reason would well have deserved an honorary degree.
For those interested in reading more about Germain, an early account can be found in the 1894 issue of The Century Illustrated Monthly Magazine.
Sophie Germain is not the only woman to draw the attention of brilliant researchers. I have previously blogged about some correspondence between the brilliant experimental researcher Michael Faraday and a "lady of the highest talent", who proposed "to become his disciple and go through with him all his own experiments." In an 1844 letter, Faraday declined, but in the most flattering and cordial way one could imagine:
Dear Lady ___, __ Your letter ought to have been answered before, but there are two circumstances which have caused delay — its high character and my want of health; for since I returned from a very forced journey to Durham, I have been under the doctor’s hand. I am quickly recovering, and now have the difficult pleasure of writing to you. I need not say how much I value your letter — you can feel that; and even if it were possible that you did not, no words of mine would convey the consciousness to you: the thanks which I owe you can only properly be acknowledged by an open and sincere reply, and the absence of all conventional phrase. I wonder that, with your high object, and with views, determinations, and hopes consistent with it — all of which are justified by the mind and powers which you possess, which latter are not known to yourself only, but, as I say in perfect simplicity, are now made fully manifest to others. I wonder that you should think as I believe you do of me. But whilst I wonder, and at the same time feel fully conscious of my true position amongst those that think and know how unworthy I am of such estimation, I still receive it with gratitude from you, as much for the deep kindness as for that proportion of the praise which I may perhaps think myself entitled to, and which is the more valuable because of the worthiness of the giver.
That with your deep devotion to your object you will attain it, I do not doubt. Not that I think your aspirations will not grow with your increasing state of knowledge, and even faster than it; but you must be continually passing from the known to the unknown, and the brightness of that which will become known, as compared to the dulness, or rather obscurity, which now surrounds it, will be, and is worthy to be, your expected reward. And, though I may not live to see you attain even what your mind now desires, yet it will be a continually recurring thought in my imaginings, that if you have life given you you will do so.
That I should rejoice to aid you in your purpose you cannot doubt, but nature is against you. You have all the confidence of unbaulked health and youth both in body and mind; I am a labourer of many years’ standing, made daily to feel my wearing out. You, with increasing acquisition of knowledge, enlarge your views and intentions; I, though I may gain from day to day some little maturity of thought, feel the decay of powers, and am constrained to a continual process of lessening my intentions and contracting my pursuits. Many a fair discovery stands before me in thought which I once intended, and even now desire, to work out; but I lose all hope respecting them when I turn my thoughts to that one which is in hand, and see how slowly, for want of time and physical power, it advances, and how likely it is to be not only a barrier between me and the many beyond in intellectual view, but even the last upon the list of those practically wrought out. Understand me in this: I am not saying that my mind is wearing out, but those physico-mental faculties by which the mind and body are kept in conjunction and work together, and especially the memory, fail me, and hence a limitation of all that I was once able to perform into a much smaller extent than heretofore. It is this which has had a great effect in moulding portions of my later life; has tended to withdraw me from the communion and pursuits of men of science, my contemporaries; has lessened the number of points of investigation (that might at some time have become discoveries) which I now pursue, and which, in conjunction with its effects, makes me say, most unwillingly, that I dare not undertake what you propose — to go with you through even my own experiments. You do not know, and should not now but that I have no concealment on this point from you, how often I have to go to my medical friend to speak of giddiness and aching of the head, &c., and how often he has to bid me cease from restless thoughts and mental occupation and retire to the sea-side to inaction.
If I were with you, I could talk for hours of your letter and its contents, though it would do my head no good, for it is a most fertile source of thoughts to my mind; and whether we might differ upon this or that point or not, I am sure we should not disagree. I should be glad to think that high mental powers insured something like a high moral sense, but have often been grieved to see the contrary, as also, on the other hand, my spirit has been cheered by observing in some lowly and uninstructed creature such a healthful and honourable and dignified mind as made one in love with human nature. When that which is good mentally and morally meet in one being, that that being is more fitted to work out and manifest the glory of God in the creation, I fully admit.
You speak of religion, and here you will be sadly disappointed in me. You will perhaps remember that I guess, and not very far aside, your tendency in this respect. Your confidence in me claims in return mine to you, which indeed I have no hesitation to give on fitting occasions, but these I think are very few, for in my mind religious conversation is generally in vain. There is no philosophy in my religion. I am of a very small and despised sect of Christians, known, if known at all, as Sandemanians, and our hope is founded on the faith that is in Christ. But though the natural works of God can never by any possibility come in contradiction with the higher things that belong to our future existence, and must with everything concerning Him ever glorify Him, still I do not think it at all necessary to tie the study of the natural sciences and religion together, and, in my intercourse with my fellow creatures, that which is religious and that which is philosophical have ever been two distinct things.
And now, my dear Lady, I must conclude until I see you in town; being indeed your true and faithful servant,
Lady in question was evidently none other than Ada Lovelace (1815-1852) herself!
Lovelace is credited with writing the first computer program, designed for Charles Babbage's mechanical difference engine. She had finished her work on Babbage's difference engine the year before and threw herself wholeheartedly at trying to convince Faraday to take her on as an apprentice! Faraday seems to have been somewhat shocked by the correspondence, an excerpt of which may be read here, and politely declined the offer while still praising her intellect and her scientific potential. His correspondence with her seems to have continued until her untimely young death.
Another woman whose brilliant mathematical work I am currently exploring is Sofia Kovalevskaya (1850-1891), a Russian mathematician who, like Lovelace and Germain, was not shy in pursuing her scientific ambitions.
Over Kovalevskaya's career (another one which was unfortunately too short), she made significant contributions to analysis, differential equations and mechanics. In fact, I describe the Cauchy-Kovalevskaya theorem in my upcoming textbook, a theorem which describes the uniqueness of solutions of certain classes of differential equations.
Though Russia was going through an intellectual Renaissance at the time and intellectuals widely believed in the equality of the sexes , women were barred from academia by the tsarist government which saw such integration as a threat. The result is that many women sought education in Western Europe. Written permission was needed from a father or husband to study abroad, so Sofia found an accomplice, Vladimir Kovalevsky, to join her in a "fictitious marriage" and the couple moved to Heidelburg, Germany.
In Heidelburg she would attend the University of Heidelburg, and sought out the brilliant mathematician Karl Weierstrass as her sponsor at university. Quoting from ,
Weierstrass had numerous students, was busy with administrative responsibilities, and was rumored to be opposed to the education of women. But Sofia could be persistent, especially in connection with her mathematical career. She resolved to approach Weierstrass at his home.
The great mathematician later told Anna Carlotta Leffler that at their first meeting he had no idea of Kovalevskaia's age or appearance, because she wore an unbecoming bonnet which hid her face and was suitable to a much older woman. When he heard her request to study with him, he was astounded. But Weierstrass was too understanding a man to turn anyone away without a hearing. He gave Sofia a series of problems and told her to come back when/if she could solve them.
In a week Kovalevskaia returned with solutions in hand. Weierstrass sat her down and had her wait while he reviewed her answers. To his surprise, not only was everything correct, but the solutions were original and demonstrated Sofia's complete command of her field. Her paper showed him that she had "the gift of intuitive genius to a degree he had seldom found among even his older and more developed students," he later told Anna Carlotta Leffler.
After investigating her credentials with other faculty, Weierstrass approached the faculty senate to support Sofia's admission to the University. Even with the support of additional faculty, the senate as a whole voted to reject her. Weierstrass was not so easily discouraged, however:
Although the senate ruled against Kovalevskaia, Weierstrass could not conceive of abandoning someone of such outstanding mathematical gifts as Sofia. Despite his busy schedule, he proposed private lessons. He would recapitulate his lectures for her, and expound upon his own research. Kovalevskaia agreed eagerly. For the next four years she visited him every Sunday that she was in Berlin, and he in turn came to her one other day each week at the apartment she shared with Iulia Lermontova [another student].
The friendship and collaboration lasted through Sofia's life, even through its rather turbulent ups and downs. In 1874 she was able to earn her doctorate at the University of Göttingen, the first woman in Europe to hold that degree. By 1884, with the help of Weierstrass' student Gösta Mittag-Leffler, she was able to secure a professorship at Stockholm University, the first woman in Northern Europe to earn such a distinction. Mittag-Leffler had himself been wowed by both the charm and intellect of Sofia :
More than anything else in Petersburg what I found most interesting was getting to know Kovalevskaia. ... As a woman, she is fascinating. She is beautiful and when she speaks, her face lights up with such an expression of feminine kindness and highest intelligence, that it is simply dazzling. Her manner is simple and natural, without the slightest trace of pedantry or pretension. She is in all respects a complete 'woman of the world'. As a scholar she is characterized by her unusual clarity and precision of expression. ... I understand fully why Weierstrass considers her the most gifted of his students.
I had three points to make in writing this post, which the examples presented hopefully get across. First, the early women in mathematics faced extraordinary obstacles in pursuing their interests, and showed amazing determination and boldness in overcoming them. Second, though these woman were clearly prohibited by society and individuals from studying science, there were in fact a surprising number of male academics who were far ahead of their time in recognizing the abilities of women to do so. Third, these women did not just "get by" in science, but excelled, and were recognized by some of the greatest minds of their time as being brilliant.
I've started myself a new category of "women in science" for the blog; I'll hopefully be coming back and discussing the historic contributions of women to various disciplines in future posts, including more details about the women discussed here.
 N. Mackinnon, "Sophie Germain: or was Gauss a feminist?" The Mathematical Gazette, vol. 14 (1990), 346-351.
 A.H. Koblitz, A Convergence of Lives: Sofia Kovalevskaia (Birkhauser, Boston, 1988).