A Locally Compact REU in the History of Mathematics: Involving Undergraduates in Research
2007: The Year of Euler. That was the year we decided to try an experiment -- doing summer research in the history of mathematics with a group of undergraduate students. The choice of a topic seemed obvious: how could we study anyone but Euler? But so much had been done already in anticipation of the big year; the MAA alone had just published five new books about Euler and his mathematics. It seemed unlikely that we could contribute anything new over the course of one summer. So we shifted our focus slightly. Drawing on our interest in women’s contributions to mathematics, and the fact that we ended up with four young women as our research partners, we chose as our topic female mathematicians in the time of Euler. We planned to look at their educational opportunities, the kinds of mathematics they did, who they were, what their lives were like. If we were particularly fortunate, maybe we would even discover someone new!
Each of us had supervised independent research projects before, including one history project, but we had always worked with one student at a time. But when we heard colleagues describe their Research Experiences for Undergraduates programs, we were envious – of the field trips, the invited speakers, the seminar-like atmosphere, the collaboration, the fun. And so we decided to find four students to work with the two of us. We also decided to invite students with different levels of skill and experience; we did not want to work with four graduating seniors, only to have them leave the College, taking all of our collective knowledge with them. We introduced this scheme of “vertical integration” into the project by inviting students from different classes. In the end our group was composed of two students who had just graduated, one rising senior and one rising sophomore, and the two of us: one full professor who had previously done historical research, and one untenured assistant professor with no previous historical research experience.
The vertical integration gave us a variety of perspectives and experiences to draw upon. Our recent graduates had just completed a senior seminar in the history of mathematics, and could share their knowledge with the younger students. The rising senior was a double major in mathematics and economics, and so brought a different set of interests and knowledge to the group. And our rising sophomore was a potential mathematics major who had just finished calculus.
We pitched our idea for summer research to the appropriate committee, and after some discussion between us and the dean, we were granted Hood College funding for an eight-week research project, set to culminate with a trip to MathFest, the summer meeting of the MAA, in San Jose, California, that August. The students were given dorm rooms for the summer; students and instructors received a stipend.
Our goal was to have the students do enough research so that each could give her own talk at MathFest. For the first several weeks, we worked in the library together, trying to get our bearings and finding appropriate books and articles. We also started finding helpful sources on the Internet; more on this later.
Eventually, our weekly schedule evolved into the following one: Mondays were devoted to the students’ individual research topics. On Tuesdays the students gave presentations on Euler and his works using secondary sources. Wednesdays were our field trip days, and on Thursdays we worked with primary sources from Euler and his contemporaries.
Mondays: Individual research topics
On Mondays, each student gave a report on her previous week’s research and talked about what she would work on in the coming week. We all gave feedback and suggestions; often another student could answer a presenter’s question or direct her to a possible source. Eventually, as MathFest neared, this became our time to practice talks, first just for our little group, then with a few extra audience members (generally other professors, but not always math professors). Once the reports (or practice talks) were completed, we adjourned to the library or the computer lab to work independently. The students could also use this time to request help from the professors or each other.
Tuesdays: By and About Euler
On Tuesdays, the students each gave a presentation on Euler and his work, from sources assigned by the two faculty, including the previously-mentioned MAA books, [1,2,3,4,5], the online Euler Archive, and Ed Sandifer’s online MAA column How Euler Did It. The articles in these sources were generally bite-sized pieces of work from Euler that the author had streamlined and interpreted. Each student could pick any article from her resource, so long as it had not previously been presented. Her job was to read it, try her best to understand it, then present and explain it to the rest of us (who had not read it). If a student did not understand part of her article, then we would all puzzle it out together, generally with the presenter still at the board. This gave us a nice appreciation for the breadth and beauty of Euler’s work. Once all the presentations were finished for the day, the students traded sources and picked articles for the next week. This way, all four students had the opportunity to work with all of our resources.
Wednesdays: Field trips and speakers
Our original plan was to invite speakers to come to campus and make presentations and then spend some time discussing our project with us. But we quickly realized that our potential speakers were in much more interesting places than our college campus in Frederick, Maryland, and it would be more fun, and more helpful, to visit them in their own habitats.
It happened that William Dunham was giving a lecture on Euler as part of a public lecture series at the Johns Hopkins University Applied Physics Lab in Baltimore, MD, early in June. And so we used that event to kick off our summer project. After the talk we chatted with Prof. Dunham about our plans, and in fact we kept in touch with him throughout the summer for information and advice. Networking like this was invaluable to us in the course of the summer, and we never could predict who would be able to give us what information, so we learned to talk to everybody about everything. In many ways these trips were the most important part of our summer, as we discovered a great many helpful sources and information on them.
At the APL: Laura Printz, Lindsey Nagy, Bill Dunham, Chelsea Sprankle, Melissa Barrick
Our second field trip was to the MAA’s Carriage House Conference Center in Washington, D.C. There we began our day with a tour of MAA headquarters and met some of the staff. Then we gathered in a conference room and met with mathematics historian Victor Katz, who presented an overview of the process of doing research in the history of mathematics and gave us lots of good ideas for topics and resources.
Participants at the MAA: Kimber Tysdal, Lindsey Nagy, Laura Printz, Melissa Barrick, Chelsea Sprankle, Betty Mayfield
A variety of field trips followed. One week we went to the National Museum of American History of the Smithsonian to meet with mathematics curator Peggy Kidwell. This trip was especially exciting as the museum was closed to the public for renovation, and so we had to get special permission to enter and walk by the spookily dark exhibits on our way to her office. Ms. Kidwell showed us artifacts related to mathematics that were made or used during the era we were researching, and talked to us about the importance of family businesses in the manufacture of scientific instruments during that time. We also talked to her about eighteenth-century women’s access to education, both in Europe and America, and she was able to tell us a little about the books they would have used. She also helped us expand the boundaries of our research by looking at mathematics that was written for women, and not just by women, during Euler’s time.
At the NMAH: Peggy Kidwell is third from left
Ronald Calinger from Catholic University was just completing a Dibner Fellowship at the Smithsonian that summer, working on his own book about Euler, and one week we visited him at the Cullman Library of the Smithsonian’s Museum of Natural History. He gave a nice talk about Euler, and we talked to him about women who may have been influenced by him. He also arranged for us to see some beautiful and rare old mathematics books from the library’s collection. We returned later to get a second look at a special edition of Euler’s Letters to a German Princess.
At the Cullman Library: Ron Calinger is fourth from left
But the trip we took most often was to the Library of Congress; we traveled and worked there virtually every week. We all got researcher’s cards and took the regular tour of the Library. We attended research orientation sessions both in the Humanities and Social Sciences Division and in the Science, Technology, and Business Division. We spent some time in the Rare Books Room, looking at first editions of Maria Agnesi’s and Emilie du Châtelet’s work. Everyone we met was phenomenally helpful and nice; in fact they were thrilled to be helping us and excited that there were students doing research. They helped us find print and online resources; they called other libraries and offices for information. The Head Librarian of the Science Library, Constance Carter, was especially accommodating. She and her staff set us up in a conference room and brought us books and helped us locate resources.
At the Science Library: Constance Carter is in the middle
Thursdays: Math in the original
On Thursdays one of the faculty members on the team would present something a little more challenging and lead a discussion on it. We looked at some of Euler’s papers and books and worked through the mathematics. We also used this day to read other primary sources, such as Maria Agnesi’s Analytical Institutions, and compared her presentation of calculus to that of Euler. One week we all spent this time translating Euler from the Latin (or trying to)!
Each student eventually chose a specific topic to study. She spent a lot of time searching for and reading information, studying and developing research questions, and then presenting her work to the rest of the group. And each student did give a talk in the undergraduate paper sessions at MathFest. We were so proud of them. Links to their annotated PowerPoint presentations are provided below.
Laura Printz: Emilie du Châtelet and Maria Agnesi as early feminists.
Melissa Barrick: Euler’s Letters to a German Princess: how did Euler teach mathematics to a young woman?
Chelsea Sprankle: Maria Agnesi’s Analytical Institutions: How did she teach calculus to young people? How was her book different from Euler’s? Did she use Newton’s or Leibniz’s notation, and was it changed in the English translation?
Lindsey Nagy: Rediscovering Laura Bassi, a physicist and mathematician from Bologna who was very famous in her time.
What Did We Learn?
- The history of mathematics is an appropriate, accessible area in which undergraduate students can do meaningful research.
- Working together in a group, structuring a project like a traditional REU, is rewarding and fun.
- People love helping students. Librarians, historians, mathematicians, curators are all happy to interrupt their ‘important work’ and give help and advice to undergraduates who show an interest in their field.
- The Internet is a surprising resource for the history of mathematics. Google and other resources for digitized books make rare texts accessible to everyone. For example, all of John Colson’s 1801 translation of Maria Agnesi’s Analytical Institutions is available, as is Henry Hunter’s translation of Euler’s Letters to a German Princess.
- Our students learned to navigate the Library of Congress, a big research library that is very different from the library at our small college. But they also learned to navigate the D.C. Metro system. Because of this project, they were eligible for some travel funds and were able to participate in a mathematics conference. (One of them flew in an airplane for the first time.) They met lots of fascinating people. The experience was educational in many ways, some unexpected and intangible.
- Unusual things will happen to you! For instance, one of our summer adventures involved receiving an encouraging letter from Supreme Court Justice Ruth Bader Ginsburg.
You can do this! Pursuing research in the history of mathematics with undergraduates is very possible and very rewarding. Tailor your project to your resources and your interests. You may not be close to the Library of Congress, but you are probably close to some major research library. Find out about private libraries and collections in your area. Use the Internet; learn about Google Books. Come up with questions you want to answer, and start investigating. You never know where it might lead you.
For Further Reading
The five books about Euler published by the MAA in 2007:
- Edward Sandifer, The Early Mathematics of Leonhard Euler. This volume describes Euler’s early mathematical works: the 50 mathematical articles he wrote before he left St. Petersburg in 1741 to join the Academy of Frederick the Great in Berlin.
- William Dunham, editor, The Genius of Euler: Reflections on his Life and Work. Many of these papers first appeared in the 1983 issue of Mathematics Magazine (vol. 56, no. 5) devoted to Euler.
- Edward Sandifer, How Euler Did It. A collection of 40 monthly columns that appeared on MAA Online.
- N. N. Bogolyubov, G. K. Mikhailov, and A. P. Yushkevich, editors, Euler and Modern Science. An English translation of a collection of papers in Russian, from a conference held in Moscow and St. Petersburg in 1983, 200 years after Euler's death.
- Robert E. Bradley, Lawrence A. D’Antonio and C. Edward Sandifer, editors, Euler at 300: An Appreciation. A collection of papers delivered at conferences in North America in the years leading up to the tercentenary of Euler’s birth.
And one more…
- William Dunham, Euler: The Master of Us All. An introductory biographical sketch, followed by chapters describing Euler’s contributions to eight different topics—number theory, logarithms, infinite series, analytic number theory, complex variables, algebra, geometry, and combinatorics. (MAA, 1999).
About women and mathematics in the time of Euler:
- Vera Lee, The Reign of Women in Eighteenth-Century France. (Schenkman Publishing Company, 1975.) "In the Age of Enlightenment, education did precious little to illuminate the minds of French women." Voltaire, Rousseau, and the role of women in society.
- Some of the works of Laura Bassi, from the Biblioteca Comunale dell'Archiginnasio, Bologna and the Archivio di Stato, Bologna. Bassi's philosophical theses; paintings representing her graduation ceremony, the public discussion of her theses and her first lecture; poems dedicated to Bassi's graduation and the silver medal cast to celebrate the event; and other manuscripts by Bassi.
- James E. Force and Sarah Hutton, editors, Newton and Newtonianism. (Kluwer Academic Publishers, 2004). See especially Essay 9: "Women, Science, and Newtonianism: Emilie du Châtelet versus Francesco Algarotti."
- The Biographical Dictionary of Women in Science: Pioneering Lives From Ancient Times to the Mid-20th Century. Even though Laura Bassi is still not listed in the Dictionary of Scientific Biography, she – and the other women mentioned in this article – have entries in this volume.
Two recent print biographies:
- Massimo Mazzotti, The World of Maria Gaetana Agnesi, Mathematician of God. (Johns Hopkins University Press, 2007). From the publisher’s description: “Using newly discovered archival documents, Massimo Mazzotti reconstructs the wide spectrum of Agnesi's social experience and examines her relationships to various traditions—religious, political, social, and mathematical.”
The authors are grateful for the support of the Hood College Summer Research Institute, and to the many librarians, historians, and mathematicians who freely gave of their time and expertise to aid this project.