**Mary Cartwright was a mathematician whose discoveries would not only aid in the defense of Britain during World War II, but lay the groundwork for a whole new field: chaos theory.**

**Mary Cartwright was born on December 17th, 1900 in Northamptonshire, England, the middle child of five.**

In her early schooling she discovered a love of history and it became her favorite topic, but for one aspect: it required the memorization of seemingly endless lists of facts and dates. In her final year of high school she took to mathematics, where one did not have to spend hours learning facts. When she applied to St. Hugh's College in Oxford in October 1919, it was in the mathematics program.

**University study**

It was a difficult time to apply to university as WWI had just ended and the halls were crowded with men who were back to either continue their studies, or begin them for the first time. Mary was one of only five women in the university who was studying maths. Like many, she could not always get into the lectures due to the crowds, but when that occurred she would obtain notes from others who were there.

After two years of study Mary took the Mathematical Moderations exam, a standard for the maths program. She was deeply disappointed when she did not receive first class as her result, a level of excellence that eluded but a few students that year. She did earn second class, but she interpreted the result as a failure and considered leaving maths to go back to history, a subject she still carried a flame for.

It was a decision she spent much time on but Mary finally decided to stick with maths, as she was enjoying it so thoroughly, and she did not miss the long hours of learning facts while in history class. She would forever remain a student of history to some degree however, as many of her mathematical papers included historical perspectives that made her work more interesting and unique. She would also compose several biographical memoirs that would demonstrate her passion for historical narratives.

Having decided to continue with the program, Mary's path was shaped by some of the advice she took from others, such as V. C. Morton's recommendation that she read up on

*Modern Analysis*by Whitaker and Watson, and consider an evening class given by G.H. Hardy.

The weekly lectures and debates that followed under the guidance of G. H. Hardy were illuminating and inspiring for Mary, and she would graduate from Oxford in 1923 with a first class degree.

Not having the financial certainty she would have wanted to continue in her education, Mary chose not to pursue a degree at that time, and she spent the next four years teaching. She was not happy with the lack of time she had to dedicate to experimental maths, however, nor was the pleased with the stringent guidelines she was to adhere to in her classroom. In 1928 she left teaching and returned to Oxford, where she began to pursue her doctorate under the guidance of G. H. Hardy.

**Unconventional solutions**

In a seminar by Hardy, Mary's excellence in mathematics really began coming to the foreground. One evening after he assigned the class a series of problems that he expected them to solve using one method, Mary startled him by solving it in a completely different fashion. This would lead to some of Mary's work being published that year independently, and also as part of an index to a book by Hardy.

It was during her final examination for her doctorate in 1930 that Mary met J. E. Littlewood, an examiner and professor with whom she would collaborate for many years. Their friendship began when the other examiner opened the interview with a question that was somewhat irrelevant and trifling, and Mary stalled for a moment, flustered - but then Littlewood gave her a wink by way of encouragement, letting her know he was on her side.

In 1930 Mary departed for Cambridge, where she continued working on the topic of her doctoral thesis, the theory of functions. She attended some of Littlewood's seminars, capturing his attention when she solved one of the open problems he always posed. In her solution she used a new and unconventional approach, and the theorem would become known as Cartwright's Theorem.

Mary's theorem was published in 1935 and referred to in Littlewood's book,

*Lectures on the Theory of Functions*. It inspired a great deal of interest and new applications and is considered by many to be her greatest work.

For the remainder of the 1930s Mary taught as a part-time lecturer in maths at Cambridge, and the director of studies in maths as Girton College. In 1938, she began work on a new problem, one that would have a major impact on the direction of her research.

**Appeal for help**

In January 1938, the British Government's Department of Scientific and Industrial Research sent a memorandum to the London Mathematical Society appealing to mathematicians to help them solve a particular problem. While it was not revealed at the time, it was to related to the top-secret development of radar that was progressing with new urgency as war in Europe threatened.

The engineers on the project were having difficulty understanding the erratic behavior of the high-frequency waves, and were searching for "a more complete understanding of the actual behavior of certain assemblages of electrical apparatus." Was there something wrong with their apparatus, or their readings? They were hoping the Mathematical Society could help provide an interpretation.

Mary was intrigued by the memorandum, and brought it to the attention of her colleague, Littlewood, suggesting they combine forces. She knew he had a background in the theory of dynamics, and had worked on the trajectories of anti-aircraft guns during World War I.

**The challenge**

The project involved high power amplifiers which transmitted radio waves, or radar. During World War II it was critical for the amplifiers to respond as they were expected, but the soldiers were thwarted by amplifiers that did not behave consistently. They blamed the manufacturers, but when Mary and Littlewood examined the problem, they came up with a different explanation.

There was an equation the engineers were using the predict the behavior of the amplifiers, and Mary and Littlewood were able to demonstrate how as the wavelength of the radio wave shortens, the performance of the amplifier become unstable and unpredictable. They identified this as not a failure of the equipment, but a phenomenon that could be expected.

This did not solve the issue entirely for the engineers as they could not eliminate this range of erratic fluctuations, but it did tell them to direct their attention away from blaming the manufacturers, and instead keep the radio waves within a range where they knew it was predictable.

This breakthrough would contribute to the success of radar during wartime, and the defense of Britain against enemy air attacks.

**Chaos Theory**

The duo's study of the predictability of the oscillation of radio waves was not just applicable to radar during wartime, of course. The results would become the foundation for the modern theory of chaos that accounts for the unpredictable behavior of all physical phenomenon, including a pendulum's swing, the flow of a body of water, even the stock market.

For example, when you steadily increase the flow of water through a water-wheel, it will spin faster and faster, in proportion to the amount of water. If the flow is increased too much, however, the wheel will respond in a way that cannot be predicated - it may slow down or speed up, or even change direction.

The importance of accounting for chaotic behavior in our physical environment became apparent in 1961, when Edward Lorenz was running ether simulations through an early computer. When he unwittingly made a very small error in the input when he misplaced a decimal point and it lead to two very different outcomes, he publicized his findings in a lecture famously called "Does the Flap of a Butterfly's Wings in Brazil Set off a Tornado in Texas?"

This was building on the work of Cartwright and Littlewood, who had pinpointed the unpredictability that could result from small changes in their work with radio waves in the late 1930s.

**A modest achievement**

As for Mary, her work went relatively unnoticed at the time, perhaps in part to her unwillingness to boast of her accomplishments. Her work on radio waves was published in the Journal of the London Mathematical Society shortly after the war, but the potential of it was still not fully understood by many.

As stated by Freeman Dyson, a physicist of the day:

Mary's genius extended not only to her understanding of the theory, but in seeing its potential application. According to her peers (who only understood this later), she was the only one to realize the chaos theory was not just the answer to a problem, but a whole new field."When I heard Cartwright lecture in 1942, I remember being delighted with the beauty of her results. I could see the beauty of her work but I could not see its importance. I said to myself, 'This is a lovely piece of work. Too bad it is only a practical wartime problem and not real mathematics.' I did not say, 'This is the birth of a new field of mathematics.' I shared the tastes and prejudices of my contemporaries. Only Cartwright understood the importance of her work as the foundation of chaos theory, and she is not a person who likes to blow her own trumpet."

**After the War**

**In 1947 Mary was the first woman to be elected as a member of the The Royal Society, an exclusive group of scientists in England, and in 1948 she became Headmistress of the Girton College of Cambridge. By all accounts, she was an excellent supervisor who gave encouragement when it was due and would correct her students, but never discourage them or put them down. She was always available for advice and conversation, and dedicated much time to reading and working with her students.**

While her administrative duties prevented her from devoting all her time to mathematics, between the years of 1950 to 1989 she still published several papers in differential equations as well as memorial articles and historical papers. In 1956 she published a book called

*Integral Functions,*based mostly on her research before the war, that was more detailed and precise than any prior work of its kind.

From 1961 to 1963 she was President of the London Mathematical Society, and received its highest honor, the de Morgan Medal, in 1968. In 1969 she gained the title of Dame Commander of the British Empire,

Mary composed close to ninety articles in her lifetime and would forever change the field of modern mathematics, but remained typically modest about her achievements in later years. Again, in the words of colleague Freeman Dyson:

"Cartwright published her discoveries at the end of the war, but nobody paid much attention to her papers and she went on to other things. She became famous as a pure mathematician. Twenty years later, chaos was rediscovered by Ed Lorenz and became one of the most fashionable parts of physics. In recent year I have been calling attention to Cartwright's work. In 1993 I received an indignant letter from Cartwright, scolding me because I gave her more credit than she thought she deserved. I still claim that she is the original discoverer of chaos. She died, full of years and honors, in 1998."Mary died in Cambridge, England, on April 3rd, 1998 at the age of 97. She left strict instruction that there would be no eulogies at her memorial service. It was the end of a lifetime of contribution to mathematics and university administration.

**Sources**

- Mary Lucy Cartwright, 1900-1998. CWP and Regants at the University of California, 1997-2001, cwp.library.ucla.edu. 2016.
- Mary Cartwright (1900-1998), Shawnee McMurran and James Tattersall, Notices of the AMS, ams.org, February 1999
- Headstrong: 52 Women Who Changed Science - and the World. Rachel Swaby, Broadway Books, 2015
- A Point of View: Mary, Queen of Maths. Lisa Jardin, BBC.com, March 8th, 2013.

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