**Ruth Moufang**'s father, Eduard Moufang (1874-1941), was a chemist who was an expert on brewing and beer, and had written books on that topic. He acted as a scientific consultant to industry. Eduard Moufang was the son of Friedrich Carl Moufang (1848-1885) from Mainz, who was a businessman in Frankfurt am Main, and Elisabeth von Moers from Mainz. Ruth Moufang's mother was Else Fecht, who was the daughter of Alexander Fecht (1848-1913) from Kehl and Ella Scholtz (1847-1921). Ruth was the younger of her parents' two daughters, having an elder sister Erica [5]:-

The two sisters appear to have been encouraged in their intellectual pursuits by their family, as was common among the educated German middle class of that period.

Both girls attended the Realgymnasium in Bad Kreuznach, Ruth beginning her studies there in 1913. Both were taught mathematics by the teacher Wilhelm Schwan who began teaching at the Realgymnasium in Bad Kreuznach in 1918. Schwan wrote a widely used mathematics text and the two Moufang sisters helped him by making the drawings for the diagrams in the book. Both Ruth and Erica Moufang had considerable artistic skills and this was put to good use in their work on Schwan's geometry book. Ruth became fascinated with mathematics through Schwan's inspirational teaching and her sister Erica went on to become an artist. Let us note that Wilhelm Schwan, in addition to the geometry text, edited a volume of mathematics lectures. He became affiliated with the University of Frankfurt and obtained a doctorate in mathematics from the University.

In 1924, Moufang passed her Abitur examination while at the gymnasium in Bad Kreuznach and later that year she began her studies at the Johann Wolfgang Goethe University in Frankfurt am Main. This university had been founded in 1914 and admitted female students from the start. Arthur Schönflies was the first full professor at Frankfurt but he retired before Moufang began her studies, being replaced by Carl Ludwig Siegel. Other leading mathematicians teaching there when Moufang was a student included Ernst Hellinger, Max Dehn, who was appointed to succeed Ludwig Bieberbach after he left in 1921, Otto Szász, and Paul Epstein. She took the examinations to qualify her to teach at a Gymnasium in 1929, and then continued studying at Frankfurt where she undertook research for a doctorate. She was supervised by Max Dehn and obtained a Ph.D. in 1931 for her thesis *Zur Struktur der projektiven Geometrie der Ebene* on projective geometry [4]:-

Moufang's graduate work would be the most significant of her career. In it, she transported the axioms David Hilbert had postulated for the realm of plane geometry into the field of projective geometry.

Following the award of her doctorate, Moufang was awarded a fellowship which let her spend the academic year 1931-32 undertaking research at the University of Rome. She returned to Germany in 1932 and was appointed to a temporary teaching position, a Lehrauftrag (Lectureship), at the University of Königsberg. After teaching for the academic year 1932-33 in Königsberg, she returned to Frankfurt where again she was appointed to a lectureship. The next few years were ones of great difficulty for the mathematics department at Frankfurt after the Nazi party came to power in 1933. Because he was Jewish, Szász was banned from teaching in 1933. For the same reason Dehn and Hellinger were forced to retire in 1935 and Epstein resigned voluntarily to avoid being dismissed. The remarkably productive period for mathematics at Frankfurt from its founding in 1914 ended in the early 1930s with these dismissals. During the years that she was teaching at universities, Moufang was also working on her habilitation thesis. She submitted her thesis to the Johann Wolfgang Goethe University in Frankfurt am Main in 1936.

From 1931 to 1937 she had studied projective planes introducing Moufang planes and non-associative systems called Moufang loops. In [2] Chandler and Magnus describe her contributions to geometry, putting them into context as follows:-

A large part of her work is dedicated to the foundations of geometry. Her most outstanding contribution to this field is a result which adds a third important discovery to two others made previously by Hilbert(1901and1930). Reversing a development going from Euclid to Descartes in which geometry is replaced by algebra as a fundamental discipline of mathematics, Hilbert had shown that a subset of his axioms for plane geometry(essentially the incidence axioms)together with the incidence theorem of Desargues permits the introduction of coordinates on a straight line which are elements of a skew field. If Desargues' theorem is replaced by that of Pappus, the coordinates become elements of a field. Moufang(1933)showed that another incidence theorem, called the theorem of the complete quadrilateral(or of the invariance of the fourth harmonic point), allows one to introduce coordinates which are elements of an alternating division algebra. This and a subsequent paper had the effect of stimulating further research of these algebras and of other nonassociative algebraic structures(Moufang loops). Her work is based both on a powerful geometric intuition and on the development of difficult algebraic techniques. It is supplemented by a sequence of papers on continuum mechanics.

She published seven papers on this work. These are:* Zur struktur der projectiven Geometrie der Ebene* (1931); *Die Einführung in der ebenen Geometrie mit Hilfe des Satyes von vollstandigen Vierseit* (1931); *Die Schnittpunktssätze des projektiven speziellen Fünfecksnetzes in ihrer Abhängigkeit voneinander *(1932); *Ein Satz über die Schnittpunktsätze des allgeimeinen Fünfecksnetzes* (1932); *Die Desarguesschen Sätze von Rang 10* (1933); *Alternativkörper und der Satz vom vollständigen Vierseit* *D*_{9} (1934); and *Zur Struktur von Alternativkörper* (1934). Moufang published only one paper on group theory, *Einige Untersuchungen fiber geordenete Schiefkörper*, which appeared in print in 1937. In this paper, which was motivated by the two papers of Hilbert on geometry mentioned above (published in 1901 and 1930), she examines the group *M* = *F*/*F*'', the free metabelian group on two generators. She proves that the rational group algebra of this group can be embedded in an ordered division ring. As a consequence it is easy to show that *M* contains a copy of the free semigroup on two generators. Moufang also gives applications of the result to number theory, knot theory and the foundations of geometry.

Moufang submitted her habilitation thesis in the summer of 1936 and habilitated on 9 February 1937, being only the third German woman to habilitate in mathematics. However, the Nazis, to be precise Hitler's minister of education, refused Moufang permission to teach (because she was a woman), so from 1937 she became an industrial mathematician working on elasticity theory. In fact this gives Moufang the unique position of being the first German woman with a doctorate to be employed in industry. She may actually be the first ever such woman anywhere. Bhama Srinivasan writes in [5] that, following her habilitation:-

... the logical course of events would then have been for her to become a Privatdozent. However, in March1938she received a letter from the Minister of Education informing her that the policies of the Third Reich required a professor to be a "leader" of the students in more than just the academic sphere; since the student body was almost exclusively male, they did not think it feasible to appoint women professors. They did not, however, have any objection to her holding a job which involved only research. Since there were no permanent positions in universities which consisted of research alone, Moufang left academic life and joined the Krupps Research Institute in Essen where she remained until1946.

Her first appointment at the Krupps Research Institute was as a research assistant but in 1942 she was appointed as Head of the Department of Applied Mathematics and Mechanics. At this stage in her career Moufang undertook research on applied mathematics topics, in particular on the theory of elasticity. She published *Das plastische Verhalten von Rohren unter statischem Innendruck bei verschwindender Längsdehnung im Bereich endlicher Verformungen* in 1941 which is based on a discussion of a stress-strain law of deformation type. Her next paper *Volumentreue Verzerrungen bei endlichen Formänderungen* (1946) generalised to the finite case the known decomposition in the infinitesimal case of the strain tensor into the strain deviation, giving the change of shape, and a spherical tensor, giving the change of volume. In a 1947 paper, which appeared after she left the Krupps Research Institute, she extended the results of this last paper and gave rational approximations to the irrational intensity of the spherical tensor. A final paper by Moufang on this topic, *Strenge Berechung der Eigenspannungen, die in plastisch aufgeweiteten Hohlzylindern nach der Entlastung zurückbleiben*, was published in 1948 and reviewed by William Prager:-

The paper contains an analysis of the residual stresses in a thick-walled tube which has been plastically deformed by interior pressure. This analysis is based on the strain-hardening law of Schmid and on the assumption of plane strain.

After World War II ended, various German universities wanted to recruit staff who had not been members of a Nazi organisation. In particular the Johann Wolfgang Goethe University in Frankfurt am Main looked to recruit such people and Moufang, having habilitated nine years earlier, was awarded the right to teach there as a docent in 1946. She became an extraordinary professor at Frankfurt in 1951 and a full professor in 1957, but she published nothing further after the applied mathematics papers which resulted from her research at the Krupps Research Institute except for an obituary of Max Dehn in 1954 (written in collaboration with Wilhelm Magnus). We note that Moufang holds a unique position here as the first German woman professor of mathematics [5]:-

Although she did not publish much in later years she had many Ph.D. students. One can only speculate on what her mathematical output might have been had she not been forced to spend ten years in industry at a productive stage in her career.

Moufang retired in 1970. She was honoured in 1965 with a special part of volume 87 of the *Mathematische Zeitschrift* devoted to papers published to celebrate her 60th birthday. In 2006 a street in Frankfurt was named in her honour and in 2010 the University of Frankfurt set up the Ruth Moufang Fund to support students and scientists from the university.

Let us end our biography by quoting Bhama Srinivasan [5]:-

Moufang had many intellectual interests besides mathematics. She was very modest about her work; when I contacted her about the possibility of writing this article she replied that she did not feel her work was important enough.

**Article by:** *J J O'Connor* and *E F Robertson*

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