May 27, 1902 - August 29, 1989

Lulu Hofmann Bechtolsheim was one of only 110 U.S. women to have earned the Ph.D. in mathematics before 1930 and she was one of the 228 women profiled in Judy Green's and Jeanne LaDuke's 2009 history, *Pioneering Women in American Mathematics: The Pre-1940 PhD's.* Born May 27 (=3^{3}, as she was fond of pointing out), 1902, in New York City to German parents, Lulu Hofmann grew up in Frankfurt, Germany, with her older sister Emy, also born in New York, and her younger sister Ilse, born in Frankfurt. Her father, Otto Hofmann, was a banker and her mother, Clara Olshauson Hofmann, a homemaker from a "long line of academicians," according to Lulu's niece, Marie-Gabriele Lindenborn, who also reported that Otto and Clara Hofmann encouraged their daughters to pursue higher education and careers. All three girls graduated from the Realgymnasium der Schillerschule in Frankfurt, earning the Abitur, which was equivalent to a U.S. high school diploma plus one or two years of university work. Lulu later recalled, "I loved school, all learning, in particular, mathematics and languages." After graduating from the Gymnasium in 1922, she studied at the University of Freiburg for one year (1922-23) and then at the University of Zürich from 1923 to 1926. She received the Ph.D. in Mathematics from the University of Zürich in 1927, with a dissertation titled, "Über einige spezielle Strahlenkongruenzen, die mit analytischen Funktionen zusammenhängen" ("On some special congruences of rays connected with analytic functions").

While in Zürich, Hofmann also audited courses at the Eidgenössiche Technische Hochschule (Swiss Federal Institute of Technology), known as ETH, where she would have participated in the mathematics seminar offered every semester by mathematics chair Hermann Weyl (1885-1955) and mathematics professor George Polya (1887-1985). She audited courses with Weyl and most likely with Polya as well; Weyl taught a wide variety of courses in algebra, analysis, geometry, mathematical physics, and philosophy of mathematics during this time. Hofmann worked so closely with Weyl during her years in Zürich and afterwards that both her stepson, Claus von Bechtolsheim, and her niece, Lindenborn, were certain that Weyl had been her *Doktorvater* (literally, doctoral father) or Ph.D. advisor. In fact, Hofmann's thesis advisor was Eugenio Giuseppe Togliatti of the University of Zürich, but she would remain in contact with both Weyl and Polya in the years to come. She also met Albert Einstein and John von Neumann at ETH.

Hofmann moved to the United States soon after completing her Ph.D. and, in April of 1927, gave a talk based on her Ph.D. dissertation at a meeting of the American Mathematical Society (AMS) in Chicago. In 1927-28, she was an assistant in mathematics at Columbia University in New York City and, in 1928-29, a lecturer in mathematics there. By this time, Columbia was admitting women to most, but not all, of its graduate programs, and some of these programs had female instructors. In 1929, Columbia appointed its first female assistant professor (in chemistry), in 1937 its first female tenured associate professor (in anthropology), and in 1940 its first female full professor (in English). Columbia would not admit women to its undergraduate college until 1983. In 1929, Hofmann became a mathematics instructor at Barnard College, Columbia's "sister" liberal arts college for women and next-door neighbor, where she would remain through the end of the 1936-37 academic year. During the summer of 1937, she was an instructor at Hunter College, another women's college in New York City, and from 1937 to 1940 she was an instructor at the brand-new co-educational Queens College in nearby Flushing, New York.

While she was at Columbia, Hofmann served as assistant to mathematics professor Edward Kasner (1878-1955), who had been a leading American geometer throughout his career. Today, Kasner is best known for co-authoring with James Newman the book *Mathematics and the Imagination* (1940), which popularized the terms "googol" (10^{100}) and "googolplex" (10^{googol}). Hofmann published "Synthetic proof of Professor Kasner's pentagon theorem," a companion piece to an article by Kasner, in the *American Mathematical Monthly* in 1928 [JSTOR]. Hofmann and Kasner published the paper "Homographic circles or clocks" in the *Bulletin of the AMS* in 1928 [Article], and Hofmann published "On a certain metric aspect of plane projective transformations," also in the *Bulletin*, in 1929 [Article]. She pursued research in geometry at least through 1931, giving two talks per year on her work at AMS meetings in New York City during each of the years 1928, 1930, and 1931. Green and LaDuke report that in 1937 Bechtolsheim was writing a new geometry text, based on courses she had offered at Barnard, but, as far as we know, this book was never published. This may have been the same analytic geometry text she was working on in January of 1945, again, never published as far as we know. (For a list of Hofmann's presentations and publications, see Green's and LaDuke's *Supplementary Material for Pioneering Women in American Mathematics*.)

Weyl had visited Columbia University in 1926 and in 1928-29 accepted a prestigious professorship at Princeton University. He was offered the position permanently but chose to accept it for just one year, returning to Zürich afterwards. In 1930, he took over David Hilbert's chair at Göttingen, only to resign it in 1933, after the Nazis gained control of the German government, and to return to Princeton permanently, this time to the newly formed Institute for Advanced Study. Lindenborn reported that Hofmann helped Weyl learn English when he moved to America in 1933; she may well have helped him with English in 1928-29 and during shorter visits to the U.S. before 1933, too. Weyl indicated in the Preface to his book, *The Open World: Three Lectures on the Metaphysical Implications of Science* (1932), which recorded lectures he had given at Yale University during the spring of 1931, that Hofmann's translations included the contents of this book, as well as other manuscripts, writing:

The lectures were originally written out in German. I do not want to omit acknowledging my indebtedness to my friend, Dr. Lulu Hofmann of Columbia University, New York, for the devoted assistance which she has rendered me in the translation of my manuscripts into English on this as well as on similar previous occasions.

Whether it was due to Hofmann's excellent tutoring or to Weyl's strong interest in and talent for languages, Weyl learned English quickly and, indeed, soon was writing his papers and books in English and having them translated into German! Hofmann herself would translate his famous book, *Symmetry*, published in 1952, from English to German. The German edition, *Symmetrie*, was published by Birkhäuser in 1955, and Weyl, in his Foreword, wrote:

It is important to the author to add a word of thanks to this foreword ... for the dedicated work of the translator, a long-time friend. Years ago when I still struggled with the English language, she frequently assisted me in rendering lectures written down in German into English. It is thus an extraordinary experience for me to see her now use her mastery of both languages in the other direction!

Sometime before 1936, Hofmann met Baron Wilhelm Alfred von Bechtolsheim on board a ship en route from Germany to New York City. Much like herself, he was a tall, dignified German with a great love of learning. He had been born in Germany in 1881, had served in the Imperial German Navy from 1899 to 1921, rising to the rank of Fregattenkapitän (Frigate Captain), and had been a prisoner of war in Japan for six years during World War I (from 1914 to 1919). At the time of their meeting, von Bechtolsheim was recently divorced and was working for the German movie company, Ufa-film, in New York City. He and Hofmann were married in 1936 and would remain in New York until 1940 or later.

The World War II years of 1939 to 1945 could not have been easy ones for the Bechtolsheims. Lulu Bechtolsheim was without an academic position from the end of the 1940 academic year to the end of 1943, and, at some point during this time period, she and Wilhelm moved from New York to California. Polya moved to the United States in 1940, and he accepted a permanent position at Stanford University in 1943. It probably was Polya who obtained for Bechtolsheim a position as a mathematics instructor at Stanford during the 1943-44 winter quarter and again during the 1947 summer term, this time as an assistant professor.

In the meantime, Bechtolsheim had accepted a position as an assistant professor at the University of Redlands in the city of Redlands, 70 miles east of Los Angeles in the Southern California citrus belt. She took up her new post in September of 1944, was promoted to associate professor in 1950 and to full professor in 1956, and retired in June of 1961. A small, private, co-educational university affiliated with the American Baptist Church, the University of Redlands had had just four long-term mathematics professors since its founding in 1907, and two of them had been women. The first, Emma Kirtland Whiton, who had earned a masters degree from the University of California in 1916, was professor of mathematics from 1918 to 1924. The second, Mary Newton Keith, Dean of Women from 1922 to 1941, also was assistant professor of mathematics during her tenure at Redlands. She had majored in mathematics at Wellesley College and usually taught four lower division mathematics courses each year. After Bechtolsheim, the next women hired as tenure-track professors of mathematics at the University of Redlands were two of the authors of this article, Mary Scherer in 1980 and Janet Beery in 1989.

Bechtolsheim joined Professor Orrin Wilson Albert in the Mathematics Department. She also taught an astronomy course in the Physics Department nearly every spring semester and occasionally taught French, German, and Italian courses. Albert, who had joined the University of Redlands faculty in 1923, retired in 1954 and was replaced in 1956 by Judson "Sandy" Sanderson, who would serve until 1988. The two mathematics professors' teaching loads were heavy. The University of Redlands *Catalog* shows Bechtolsheim teaching 11 *different* courses, including astronomy, during the 1948-49 academic year, plus directed studies and advising of senior research projects. Albert had a similar schedule, as did Sanderson in the late 1950s. Most mathematics and science majors took Mathematical Analysis and Analytic Geometry their first year and Calculus their second year of college, and the two mathematics professors taught these courses, along with all of the other courses for mathematics majors and most if not all of the courses for non-majors. They also advised the research projects they required of every senior mathematics major and hosted the seniors' final presentations of their projects in their homes. Mathematics major Sue (Blackwell) Hurlbut (Redlands '59) remembered that at least one examination in her College Geometry course also took place in the Bechtolsheims' home, with her professor serving delicious homemade German pastries afterwards.

Former University of Redlands students reported that Bechtolsheim's classes consisted primarily of lectures by the professor, with ideas outlined and illustrated on the blackboard, but also that students themselves spent a significant amount of time at the blackboard explaining their solutions to problems to the rest of the class. Recalled philosophy major Howard Hurlbut (Redlands '59), who began his studies as a science major:

The expectation was that ... you'd be able to explain it as well as do it. ... You knew that you had to be in class and be prepared. You didn't want her to be disappointed in you. She was not mean – rather gentle, really. Her discipline was the discipline of her field, not personal animosity. ... [She was] arealprofessor!

Sue Hurlbut agreed: "She had high expectations; she really wanted students to enjoy the classes in a serious way. ... You didn't dare slouch, talk, or chew gum." Physics major Dick Carlson (Redlands '57) remembered, "Students really respected her. They saw her as very smart, very tough, with very high standards, but fair. She gave the best possible class – 100% plus." Sanderson remembered that his colleague was "very willing to help students, extremely gracious and polite, and stern but fair." He described himself and Bechtolsheim as "eclectic mathematicians," which he considered to be fortunate considering the wide variety of courses they taught. Bechtolsheim herself later wrote, "I was not a researcher, only a teacher, but all the teaching I did with great enjoyment and enthusiasm through all the 30 years."

Bechtolsheim had enjoyed translating Hermann Weyl's *Symmetry* into German, and her retirement in June of 1961 gave her the opportunity to complete several more translation projects. Her first project, which she had begun during a 1959-60 sabbatical, was to translate her old friend George Polya's famous two-volume *Mathematics and Plausible Reasoning* from English into German. Birkhäuser published German versions of *Volume I, Induction and Analogy in Mathematics*, in 1962, and *Volume II, Patterns of Plausible Inference*, in 1963. In 1965, Bechtolsheim's translations from Portuguese to English of Leopoldo Nachbin's *The Haar Integral* and *Topology and Order* were published by Van Nostrand. She then translated two more of Polya's books from English to German, again for Birkhäuser, his two-volume *Mathematical Discovery: On understanding, learning, and teaching problem solving*, with the first volume published in 1966 and the second in 1967. Her last published mathematical translation was another for Birkhäuser: she translated Bruno de Finetti's Italian *Il saper vedere in matematica* (*Knowing How to See in Mathematics*) to German. Her translation was published as *Die Kunst des Sehens in der Mathematik* (*The Art of Seeing in Mathematics*) in 1974. (For a list of Bechtolsheim's presentations and publications, including her translations, see Green's and LaDuke's *Supplementary Material for Pioneering Women in American Mathematics*.)

Wilhelm Bechtolsheim, who had been a successful pharmaceutical salesman in California, died in 1968. From 1970 onward, Lulu Bechtolsheim's main interest was a religious philosophy called anthroposophy. Her friend, University of Redlands German professor Barbara Pflanz, remembered, "She told me once that she woke up one day and realized that she did not have a *Weltanschauung* (philosophy of life). And that is when, I have the impression, she ... embraced the anthroposophic philosophy." The religion attracted Bechtolsheim intellectually but she also eventually embraced its spiritual and emotional components. She attended services at the Christian Community Church in West Los Angeles; taught acolytes in the Redlands area; and translated religious materials, most likely from German to English.

Bechtolsheim died August 29, 1989, after surgery to remove an abdominal obstruction. Following a three-day vigil at the Christian Community Church and funeral services there on September 1, 1989, she was interred next to her husband at Hillside Memorial Park in Redlands.

Lulu Hofmann Bechtolsheim was a gifted and serious scholar with increasingly wide-ranging interests throughout her career and her life. As a woman who had earned a Ph.D. in mathematics in 1927 and who taught mathematics in a variety of colleges and universities from 1927 to 1961, she was a pioneer, but she did not seem to see herself as such. Her friend, University of Redlands English professor Frederick "Fritz" Bromberger, said that Lulu and her husband Wilhelm were the kindest people he and his wife Corrine ever knew, and that the Bechtolsheims became like parents to them. Corrine Bromberger added, "This is a personality who should not be forgotten. She was a feather in the cap of the [University of Redlands]."

Janet Beery is Professor of Mathematics, Barbara Pflanz Professor Emerita of German, and Mary Scherer Professor Emerita of Mathematics at the University of Redlands in Redlands, CA.

- Bechtolsheim, Lulu, University of Redlands Public Relations Data Sheet, 1958.
- Bechtolsheim, Lulu, University of Redlands Sabbatical Project Proposal, 1959.
- Bechtolsheim, Lulu, Autobiographical Statement in Eulogy for Lulu Bechtolsheim, 1989 (see van Schilfgaarde, Paul, 1989).
- Columbia University website, "Columbia 250: Women at Columbia," last edited March 2004, http://c250.columbia.edu/c250_events/symposia/history_women_timeline.html
- Frei, Günther,
*Hermann Weyl und die Mathematik an der ETH Zürich 1913-1930*, Birkhäuser, Basel, Switzerland, 1992. - Green, Judy and Jeanne LaDuke,
*Pioneering Women in American Mathematics: The Pre-1940 PhD's*, American Mathematical Society, Providence, RI, 2009. - Green, Judy and Jeanne LaDuke,
*Supplementary Material for Pioneering Women in American Mathematics: The Pre-1940 PhD's*, American Mathematical Society, Providence, RI, 2009, www.ams.org/bookpages/hmath-34 - Kasner, Edward, "A Projective Theorem on the Plane Pentagon,"
*American Mathematical Monthly*, Aug.-Sept. 1928, 352-356. - Kasner, Edward and James Newman,
*Mathematics and the Imagination*, Simon and Schuster, New York, 1940. - O'Connor, J.J. and E.F. Robertson, "George Polya,"
*MacTutor History of Mathematics Archive*, St. Andrews University, Scotland, http://www-history.mcs.st-and.ac.uk/Biographies/Polya.html - O'Connor, J.J. and E.F. Robertson, "Hermann Weyl," MacTutor History of Mathematics Archive, St. Andrews University, Scotland, http://www-history.mcs.st-and.ac.uk/Biographies/Weyl.html
*Redlands Daily Facts*(newspaper):

"Physics Instructor Appointed by U.R.," January 13, 1945

"Dr. Bechtolsheim Becomes U of R Professor," December 15, 1956

"Bechtolsheim dies here at age of 87," November 5, 1968

"Lulu Bechtolsheim" (obituary), August 30, 1989- Schmidt, Hans-Joachim,
*Die Verteidiger von Tsingtau und ihre Gefangenschaft in Japan (1914 bis 1920) Historisch-biographisches Projekt*, "Kurzbiographien: Bechtolsheim, Freiherr von, Wilhelm," posted 2002-2010, http://www.tsingtau.info/index.html?namen/b.htm - University of Redlands
*Catalog*, 1909-1989. - University of Redlands
*La Letra*(yearbook), 1944-1961 - van Schilfgaarde, Paul, Eulogy for Lulu Bechtolsheim, 1989.
- Weyl, Hermann,
*Symmetry*, Princeton University Press, NJ, 1952. - Weyl, Hermann,
*Symmetrie*, Birkhäuser, Basel, Switzerland, 1955. - Weyl, Hermann,
*The Open World: Three Lectures on the Metaphysical Implications of Science*, Yale University Press, New Haven, CT, 1932; re-printed by Ox Bow Press, Woodbridge, CT, 1989. - Zitarelli, David, "The Other American Mathematical Congress" (lecture), AMS-MAA Special Session on History of Mathematics, Joint Mathematics Meetings, New Orleans, LA, 2006.
- Interviews:

Claus von Bechtolsheim (2007)

Corrine Bromberger (2001)

Frederick "Fritz" Bromberger (2001)

Richard Carlson (2001)

Howard Hurlbut (2001)

Susan Blackwell Hurlbut (2001)

Marie-Gabriele Lindenborn (1997)

Judson "Sandy" Sanderson (2001)

Photo Credit: Photograph used with permission of the University of Redlands