Sonya Kovalevsky

Philosophenweg ("Philosopher's Way")

A significant part of my sabbatical is to walk - quite literally - where the famous mathematicians of the past walked, and I am certain that Sonya Kovalevsky (1850-1891) walked in the woods above.  This is on a path just off of Philosopher's Way.  It's an area in Heidelberg, just across the river from what's now called "Old Town," that began to be cultivated with vineyards in Roman times.  In the early 19th century  poets would walk through the vineyards, and between 1837 and 1841 the paths that are here today began to be laid out.  Sonya came to the university here in 1869.

Above is the Alte Brucke, or old bridge, which was built in 1788.  She would have been familiar with it and with its gate, a very recognizable landmark in Heidelberg that has been there since the middle ages.

Above is the oldest building in the university - built in 1712.  The university itself was founded in 1386, but Heidelberg has been hit hard by war over the centuries, particularly the Thirty Years' War, so not many of the current buildings date much before that even though "modern Heidelberg" dates back to the fifth century AD.

As was the case for Mary Somerville and Sophie Germain, both of whom I've already posted about, it was nearly impossible for Sonya to become a mathematician because she was a woman.  She was born in Russia where it was certainly impossible for her to study mathematics.  Heidelberg University in Germany also would not allow a woman to take classes, but it would allow a woman to audit classes if the professor gave his permission.

Over her parents' objections Sonya (also known as Sofia Kovalevskaya) entered a marriage of convenience - nominal and platonic (initially) - for the sole purpose of gaining the ability to travel, something single women could not do.  Her husband Vladimir was a student of paleontology and was quite amenable to the plans.  At least three Heidelberg professors were open to having her audit their courses: Hermann von Helmholtz, Gustav Kirchhof, and Robert Bunsen.

The motto of Heidelberg University is "Semper Apertus," or "Always Open" - meaning that the book of learning is always open and that our minds should be always open - open through curiosity, open to new ideas both inside and outside the classroom.  This must have suited Sonya well.  In saying that I'm thinking of a story from her childhood.  it was her uncle Petr that first shared mathematical concepts beyond elementary arithmetic and geometry with her.  Though Uncle Petr was not a trained mathematician, he had a great deal of respect for that branch of learning, and he shared enthusiastically with her.  Of this experience she writes:

"It was from him, for example, that I heard for the first time about squaring the circle, about the asymptote, toward which a curve approaches constantly without ever reaching it  .  .  .  .  The meaning of these concepts I naturally could not yet grasp, but they acted on my imagination, instilling in me a reverence for mathematics as an exalted and mysterious science which opens up to its initiates a new world of wonders  .  .  ."

From the Collection of Mathematical Models and Instruments, Gottingen University

Another experience of mathematics in childhood happened when her family moved to Palibino when she was 11 years old.  Her parents were intent on redecorating the house from top to bottom, but they had not ordered quite enough wallpaper, so her bedroom was papered with lithographed notes on differential and integral calculus from her father's days as a student.  Here too she was "apertus."  She writes:

"I remember how I spent whole hours of my childhood in front of that mysterious wall, trying to make out a single sentence and find the order in which the pages ought to have followed one another.  From long daily contemplation of them, the appearance of many of the formulas burned itself into my memory, and the text itself left a deep imprint in my brain, although at the time I was studying it I could not understand it at all."

Her biographer, Ann Koblitz adds:

"[Sofia] desperately wanted to understand what was written on the wall and consquently was disappointed in her first arithmetic lessons, which did not help her to understand [the] symbols  .  .  .  .  [Sofia] soon convinced herself that in order to get to the point where she could read her wall, she would first have to learn boring things like division and multiplication"

From the Collection of Mathematical Models and Instruments, Gottingen University

Sonya did have some instruction in mathematics from a tutor at home in her younger days, more than a young girl would normally have been exposed to. Her attraction to mathematics became so intense that she started to neglect her other studies.  That's when her father decided to put an end to her mathematical education.  But, as with Mary and Sophie before her, she secretly read about mathematics in her bedroom at night.

Eventually her path did take her to university, and though she could audit courses, she could not earn a degree.  After two years in Heidelberg, she moved to Berlin to study under Karl Weierstrass, but in Berlin she was not even allowed to audit courses!  This ended up working to her advantage, though, as Weierstrass took her under his wing and tutored her privately.  By 1874 she had completed three papers, each one of which Weierstrass considered worthy of a doctorate - yet she could not earn such a thing at the University of Berlin, no matter how worthy her work was, simply because she was a woman.

Heidelberg University Library

Heidelberg University Library

Heidelberg University Library

Amazingly, though she had never take a course or any exams there, it was arranged that she could earn her doctorate through Gottingen University in 1874.  Her degree was granted summa cum laude.  Yet despite this and despite letters of strong recommendation from Weierstrass (a very well-respected mathematician) she could not obtain an academic position.  Eventually her friend Gosta Mittag-Leffler was able to get permission granted for her to lecture in Stockholm (1884); he had to raise personal funds in order to support her, as she was not paid by the university or government.  Five years after she began lecturing he was able to secure a tenured position for her.

Throughout her life Sonya had connections with circles of learned people.  Both she and her sister Anna were gifted with literary talents, and her sister had a short story published in a magazine edited by Dostoevski when she was still a teen.  This began for them a close friendship with an elite group of European intellectuals in Moscow.

There is so much more I could write about Sonya - more on her family life - details of her mathematical work - her tragic, early death, but at this point I should probably just recommend the biography by Ann Koblitz: A Convergence of Lives: Sofia Kovalevskaia - Scientist, Writer, Revolutionary (Lives of Women in Science).  Below are pictures of the graves of some of her personal connections in London - the writer George Eliot (Mary Ann Evans) and also Karl Marx, both buried in Highgate Cemetery, London.

This picture is included in order to show my students I was really here checking out all of this history!

Sophie Germain

She's buried here at Pere Lachaise Cemetery, Paris.

She died here, in her home at 13 Rue de Savoie, Paris

But Sophie Germain (1776-1831) should be here - listed among the 72 French scientists, engineers and mathematicians recognized for their contributions.  In particular she should be here as her work on the elasticity of metals was crucial in the building of the Eiffel Tower.

As I mentioned in my post about Ada Byron Lovelace, prior to the 20th century women were discouraged - a better word might be disallowed - from going into mathematics.  The only exceptions that I am aware of are Ada Lovelace (1815-1852) and Hypatia (c. 360-415) - and things didn't really end well for either one of them.  Ada's work was forgotten until Alan Turing rediscovered it in the 1940s, and Hypatia was murdered by a mob.

Sophie Germain was born in Paris 13 years before the beginning of the French Revolution.  During her youth it was necessary for her to remain indoors due to the bedlam in the streets relating to the revolution, so she spent a great deal of time in her father's library.  It was here that she came across the story of the death of Archimedes in a book on the history of mathematics by J. E. Montucla.  As the story goes, Archimedes was working on a geometrical problem when his hometown of Syracuse was invaded by the Romans.  Archimedes was well-known and was credited with devising inventions that had held off the Roman invasion.  Soldiers were given orders to bring Archimedes back alive, but Archimedes incensed the soldier who came for him by saying, when that soldier's shadow obscuered his drawing in the sand, "Do not disturb my circles."  The soldier ran him through with a sword.

Sophie felt that if mathematics was THAT engaging then it was a topic that she needed to learn.

Sophie's parents were concerned about her studying mathematics and did their best to put a stop to it.  When they didn't let her do mathematics, she began to work on math in her bedroom at night.  When her parents realized what she was doing they took away her candles and her clothing so that it would be too cold and dark for her to do mathematics.  Sophie snuck candles into her room and wrapped up in her blankets for warmth.  Her parents finally relented when they found her asleep at her desk one morning with the ink frozen in the ink-horn.  It is said that after some time her mother secretly supported her.  Sophie never married, and her father supported her financially throughout her life.

ASIDE - As one small example of the bedlam going on in the streets of Paris, the row of kings on the facade of Notre Dame Cathedral was beheaded by the mob, thinking they represented kings of France, when actually they represented biblical kings of Judah.  Thankfully it was later restored.

West Facade Notre Dame

When Sophie was 18, the now-famous Paris university Ecole Polytechnique was opened, but, of course, as a woman, she could not attend.  Here too she found a way.  She used the identity of a young man who was no longer attending, Monsieur Antoine-August Le Blanc.  Under this pseudonym she was able to receive lecture notes and to submit work.  The person she began submitting her work to is one of those whose name can be found on the Eiffel Tower, Joseph-Louis Lagrange.

Sophie's intellect stood out, so much so that the great mathematician Lagrange took notice and asked for a meeting.  Thus Sophie had to give up her secret identity.  Thankfully, Lagrange did not mind that she was a woman, and he became her mentor, visited her home and also provided moral support.

Germain later became interested in number theory after reading work of Legendre, and she initiated a correspondence with him about number theory and elasticity.  He included some of her work in a supplement to one of his books.  Her interest in number theory was heightened when she read Carl Friedrich Gauss's famous work Disqusitiones Arithmeticae.  She also began a correspondence with him using her old pseudonym Monsieur Le Blanc.  With Gauss as well the secret was finally revealed - this time by way of her protecting him during a time of war.  It was about 1807, and the French were occupying the area near Braunschwieg where Gauss lived.  Sophie was concerned that Gauss might meet the same fate as Archimedes, so she wrote to a general in the army who was a family friend asking him to ensure Gauss's safety.  The general went to meet Gauss to make sure that he was safe, which was certainly a welcome gesture, but Gauss was puzzled when the general said he owed the visit to Sophie Germain - whom Gauss had never heard of!

Three months later Sophie admitted her identity to Gauss.  Here is his response:

"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"

Sophie is one of many mathematicians who worked on Fermat's Last Theorem, and as a result of the specifics of that work a particular type of prime number bears her name.  A Sophie Germain prime is a prime number p where 2p+1 is also prime.

In spite of her achievements she never received a degree and even on her death certificate she is not named a mathematician or scientist but only as a property holder.  When the matter of honorary degrees came up as a topic at Gottingen it was six years after her death, and Gauss (the most prominent mathematician in the world at that time) lamented that she who had proved to the world that even a woman could contribute to that most rigorous and abstract of sciences, mathematics, could not receive the degree.

Here's to you Sophie!  Thinking of you and the intellectual heights you achieved from the height of the tower on which your name should appear!

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Post Script for Personal Blog:

This is a TREASURE HUNT!  If  I thought finding Colin Maclaurin's grave in Greyfriar's Cemetery in Edinburgh was like looking for a needle in a haystack - how much more so Sophie Germain's grave in Pere Lachaise Cemetery?!  That cemetery is a city unto itself!  While there I was berating myself for not having done more research before leaving in the morning so that I could find her grave, but when I returned to the hotel and looked up her grave I found conflicting information about what "boulevard" she was near and what section she was in and what other grave she was near - so I'm not sure that the research would have helped.  (I have also wanted to find the grave of Joseph Fourier who is also buried here, but Sophie is more of a focus for me.)

Here are some imagines in which I tried to capture how this is boulevard upon boulevard, grave upon grave:

I also got lost trying to find the place Sophie had lived at the end of her life.  The streets off of the main boulevards are positively labyrinthine!  I got lost on my way there AND on my way back from there!  Her house is not far from the Seine and not far from Notre Dame - but just tucked away in these little streets on the left bank (rive gauche) of the Seine.

I guess it's good that I like labyrinths - have taken pictures of quite a few on my travels.  Here is one I saw today just beside the Seine.  I guess I'll have to be accept the fact that much of what I'm doing involves solving mazes!  Much of what I'm looking for is not on any maps!

 I saw this on the way - name of a restaurant nearby:

 And I can't believe I didn't post this as part of the main blog!  This is the plaque right outside the home - top right of the doorway:

Lady Ada Augusta Byron Lovelace

Babbage's Difference Engine

Ada, Countess of Lovelace (1815-1852) was born Augusta Byron, daughter of the poet George Gordon, Lord Byron and his wife Anne Isabelle Milbanke.  They were married in January of 1815 but separated in January 1816 when Ada (as she was nick-named by her father) was one month old.  Four months later he left the country and never saw her again.  Despite this, and his reputation as a womanizer (one woman with whom he had a publicly well-known affair called him "mad, bad and dangerous to know"), Ada requested to be buried next to her father in the family vault at the Church of St. Mary Magdalene, Hucknall, England.

A poet of Byron's stature would normally have been buried in Poet's Corner at Westminster Abbey, and his remains were sent there, but the Abbey refused to bury him there for the reason of "questionable morality."  (The Abbey finally put in a memorial stone to him in Poet's Corner in 1969; he died in 1824.)  Following are more pictures of Lady Ada and Lord Byron's resting place at Hucknall, which seems happy to have them there.  (Personal note - I had to post a ton of pictures, as this is one of two locations for which I drove a car in England!  This and Newton's home were not really accessible by public transportation, but both were close to Lincoln, England, so I made that my home base and drove here from there.)

 The light you see shining up out of the grave is for a wreath that was given at the time of his death that has been well preserved and proudly displayed. (That light isn't shining all the time; they turned it on for me to show off the wreath!)

There's also an opening that allows for viewing into the vault.

Along with the grave, there are also wall plaques and a stone in the floor.  The stone in the floor was sent by the King of Greece.  Lord Byron had died in Greece - having given money to the Greeks for their war of independence from the Ottoman Turks.  His money went to refit the Greek fleet.  Byron had planned to be part of the attack on the Turkish-held fortress of Lepanto, but he fell ill and died of his illness.  The Greeks saw him as a hero, and the marble slab in the floor of the church in Hucknall was given by the King of Greece.

Though Ada never knew her father, she was fascinated by him and came to call herself a "poetical scientist."

Portrait of Lord Byron - Charlotte Bronte Bicentennial Celebration - National Portrait Gallery - London

Ada's mother, on the other hand, did everything she could to make sure that Ada did not grow up to be a crazy poet like her father.  Anne Isabelle (Annabella, as she was known) made sure that Ada's instruction was focused on mathematics and science in order to make sure she grew up with a disciplined and balanced mind.  This is in strong contrast to other female mathematicians prior to the twentieth century, all of whom I know had a tremendous struggle to gain mathematical education due to societal or familial concerns that studying mathematics would be too much for a woman's mind to handle and would perhaps drive her mad.

One of these other female mathematicians who had had to struggle in order to study mathematics and whose family feared she would go mad (her father said they would have her in a strait-jacket if she didn't stop studying mathematics) was Mary Fairfax Somerville of whom I posted earlier.  She was 35 years Ada's senior and became one of Ada's math tutors.  Another of Ada's math tutors was Augustus de Morgan, but though he was her tutor, even he felt that mathematics was too much for the mind of a woman in general.  He wrote to her mother that women should avoid doing hard mathematics - "the reason is obvious - the very great tension of mind which mathematics requires is beyond the strength of a woman's physical power of application."  In the same letter he stated, however, that Ada unquestionably had as much power as would require all the strength of a man's constitution.

Mary Somerville and Ada developed a close friendship, as did their families.  Mary mentored Ada, and it was Mary who introduced Ada to Charles Babbage, he who developed ideas for early computing devices - the Difference Engine and the Analytical Engine - and is considered by some to be a father of modern computing.  He is described as a pre-eminent polymath among the many polymaths of his day - involved in mathematics, inventing, philosophy and mechanical engineering.

The pictures below and the picture at the very top of this post are of the uncompleted Difference Engine on display at the National Science Museum, London.

Babbage was not the only well-known personage with whom Ada was in contact. Among other acquaintances were Michael Faraday and Charles Dickens. Ada's life was one of privilege, and at one point she nearly became a lady-in-waiting to Queen Victoria.

I had opportunity as part of these studies to read the correspondence between the Byron/Lovelace and Somerville/Grieg families, and it was amazing to see this relationship unfold.  Early-on Ada's mother writes to Mary Somerville of her appreciation for the affect she has on balancing Ada's mind.  Later Ada herself writes to Mary and sweetly asks her if she will be her chaperone to go to parties put on by Babbage so that she could interact with him and learn more about his computing machines.  Ada threw herself into understanding the Analytical Engine, and saw, even more than Babbage did, the full extent of what it could do.  She saw that the hardware was only half of the story - and that the computer needed software if it were to be able to calculate any type of equation (Babbage's vision having been mostly about number-crunching and creating tables of logarithms).

Ada Byron Lovelace - from a display in St. Mary Magdalene Church, Hucknall

Display at St. Mary Magdalene Church, Hucknall

From a display about Ada in St. Mary Magdalene Church Hucknall

In her twenties Ada said to her mother that she wanted to compensate for her father's misguided genius, expressing that if he had transmitted any portion of his genius to her that she wanted to use it to bring out great truths and principles.  She is indisputably the world's first published computer programmer.  She saw the possibility of interchanging numbers and symbols, and one example of her vision is that of an engine that might compose elaborate, scientific music.  After her death her works seems to have been forgotten for a while, but when Alan Turing built his machine The Bombe to help break the German Enigma Code during WWII he came across Ada's work, and it is thanks to him that her work has come fully to light.  A new programming language developed in 1986 and used by UK air traffic control is called ADA, which is very fitting since Ada had invented a mechanical bird when she was young; now Ada has finally gotten to fly!

There's so much I wanted to share about Ada that I fear this post is very disjoint!  I think that any commentary about her life needs a book, or maybe an epic poem, rather than a blog post!  But this is the best I can do for now.

Lord Byron's memorial outside St. Mary Magdalene Church, Hucknall, England

Ada died at age 36, the same age at which her father had died - he of fever, she of cancer.  Because of the closeness of their families, Mary Somerville's son Woronzow Grieg wrote to Ada's doctor asking him to send on any news.  I know this from the reading I did of the correspondence in the Bodleian, and when I read the letter from the doctor I was in tears.  The doctor let him know that it was cancer and that there was no hope.  His despair was evident as he ended his letter by saying that while a doctor is necessary in such circumstances, the best a doctor can ever hope for is to RESTORE health, but in this case the best he could hope for was euthanasia - in other words a "good death."

On her father's memorial outside the Hucknall church is a  quote from his work Childe Harold's Pilgrimage, "But there is that within me  .  .  .  that shall breathe when I expire."  I'd certainly say that is true of Ada!