**Luther Eisenhart**'s parents were Charles Augustus Eisenhart and Emma Catherine Pfahler who were from long established York families. His father Charles Eisenhart had a surprisingly large number of different occupations as Eisenhart himself described (see [5]):-

Luther was the second of his parents' six sons. He was brought up in a religious home with much of the family's social life being centred around St Paul's Lutheran Church. His mother provided him with such a good foundation to his education that, after beginning primary schooling at the age six and a half, he took only three years to complete the normal course of six years. He attended York High School but took the final year off school to prepare for entry to College undertaking independent study of Latin and Greek.My father, after being a student at the York County Academy, taught in a country school until ... he went to Marshall, Michigan, and worked in a store. At the same time, he was apprentice to a local dentist. Being very expert with his hands, he soon acquired competence in the technique of dentistry. In due time he returned to York, set up a dentist office, and was married. He made sufficient income to meet the expenses of his growing family, but his intellect was too active to be satisfied by dentistry. Electricity appealed to him and he organized the Edison Electric Light Company in the early eighties. The telephone also made an appeal. He experimented with telephones and in the late nineties organized the York Telephone Company.

He was a student at Gettysburg College from September 1892 and won the prize for excellence in his first year and the mathematics prize in his second year. He also excelled at baseball, a sport that the young Eisenhart boys had been passionate about as they grew up. He had to spend the final two years at Gettysburg College studying mathematics with guidance from his professor but with no classes since he was the only student taking the subject. He was awarded his A.B. in 1896.

After teaching in the preparatory school of Gettysburg College for a year, he began graduate study at Johns Hopkins University in October 1897. He wrote [5]:-

His thesis was entitled... Thomas Craig aroused my interest in differential geometry by his lectures and my readings of Darboux's treatises. Toward the close of1900I wrote a thesis in this field on a subject of my own choosing and in June the degree of Doctor of Philosophy was granted.

*Infinitesimal deformation of surfaces*. As Eisenhart indicated in the above quote, this work was heavily influenced by Darboux's treatise on the subject and he received little supervision for his doctorate.

Eisenhart spent most of his career at Princeton where he became an instructor in mathematics taking up his appointment in September 1900. In 1905 he was selected to be a preceptor, a position which had been recently created. In 1908 he married Anna Maria Dandridge Mitchell; they had one son Churchill Eisenhart. Sadly his wife died in 1913 and, five years later he married Katharine Riely Schmidt of York, Pennsylvania; they had two daughters. He was promoted to full professor in 1909 and worked at Princeton until he retired in 1945. He served as Dean of the Faculty from 1925 to 1933 when he became Dean of the Graduate School. After Henry Fine died following a bicycle accident in December 1928, Eisenhart became Head of Mathematics at Princeton and Dod Professor of Mathematics, continuing in these roles until he retired in 1945.

Before looking at his research contributions, let us look at some of the reforms he introduced at Princeton. He proposed the four course plan (instead of the five courses which was the scheme then in place) which involved the undergraduates doing independent reading and research during their final two years. They completed their course by writing an undergraduate thesis. The plan was adopted in 1923 and, as we can see from the details we gave above of Eisenhart's own education, it was very much based on the experiences he had gone through. He believed that:-

He also stated that:-... teaching methods . . . must be designed to encourage independence and self-reliance, to evoke curiosity, and stimulate the imagination and creative impulse.

There was opposition to the four course plan - both from fellow academics and from students. The students sang:-The real test of an educational process is what is becoming of the student as he proceeds with his education - how he is being prepared to continue his education and become an educated man.

However Eisenhart was prepared to fight for what he believed in, and his arguments won the day.Luther Pfahler Eisenhart,

Efficient from the very start;

But he's condemned in the eyes of man

For originating the four-course plan.

Let us now look at Eisenhart's research contributions. There are two stages in his work although it is all in differential geometry. The first stage continued his doctoral work studying deformations of surfaces. His first book *A Treatise in the Differential Geometry of Curves and Surfaces* , published in 1909, was on this topic and was a development of courses he had given at Princeton for several years. In [2] this book is described as:-

The second stage started after 1921 when Eisenhart, prompted by Einstein's general theory of relativity and the related geometries, studied generalisations of Riemannian geometry. He published... in textbook form, with numerous problems, introducing the student to classical and modern methods. One of the most interesting novelties of the volume was the so-called 'moving trihedrals' for twisted curves as well as surfaces so freely used in writings of Darboux and others. From the first, methods of the theory of functions of a real variable are employed. The work was of great value in introducing the American student to an important field by the most modern method of the time.

*Riemannian Geometry*in 1926 and

*Non-Riemannian Geometry*in 1927. The scene is set for the first of these works in [1]:-

In 1933 Eisenhart publishedRiemann proposed the generalisation of the theory of surfaces as developed by Gauss, to spaces of any order, and introduced certain fundamental ideas in this general theory. Important contributions to it were made by Bianchi, Beltrami, Christoffel, Schur, Voss, and others, and Ricci-Curbastro coordinated and extended the theory with the use of tensor analysis and his absolute calculus. The book gave a presentation of the existing theory of Riemannian geometry after a period of considerable study and development of the subject by Levi-Civita, Eisenhart, and many others.

*Continuous Groups of Transformations*which continues the work of his earlier books looking at Lie's theory using the methods of the tensor calculus and differential geometry. Again quoting [1]:-

After he retired, Eisenhart continued to undertake research. In fact he published 21 papers between 1951 and 1963, for example:The study of continuous groups of transformations inaugurated by Lie resulted in the developments by Engel, Killing, Scheffers, Schur, Cartan, Bianchi and Fubini, a chapter which closed about the turn of the century. The new chapter began about1920with the extended studies of tensor analysis, Riemannian geometry and its generalizations, and the application of the theory of continuous groups to the new physical theories. Eisenhart has thus developed a remarkable body of original material and has notably served his colleagues by frequent surveys of fields in which he had become a specialist.

*Generalized Riemann spaces and general relativity*(1953);

*A unified theory of general relativity of gravitation and electromagnetism*(1956);

*The cosmology problem in general relativity*(1960); and

*The Einstein generalized Riemannian geometry*(1963).

Eisenhart had a long association with the American Mathematical Society being vice president in 1914, and Colloquium lecturer in 1925 when he lectured on non-Riemannian geometry. He edited *The Annals of Mathematics* from 1911 to 1925 and the *Transactions of the American Mathematical Society* from 1917 to 1923, being managing editor in 1920-23. He was President of the American Mathematical Society from 1931 to 1932. He was also honoured by being elected president of the American Association of Colleges (1930), and vice president of the National Academy of Sciences (United States) (1945-49) and of the American Association for the Advancement of Science. He was also elected an officer of the American Philosophical Society serving from 1942 to 1959. In addition he received honorary degrees from Gettysburg College (1921), Columbia University (1931), University of Pennsylvania (1933), Lehigh University (1935), Duke University (1940), Princeton University (1952) and Johns Hopkins University (1953). In 1937 he was named Officer of the Order of the Crown of Belgium.

Lefschetz writes about Eisenhart's character in [1]:-

After his death his faculty colleagues said, in a tribute, that:-He was par excellence a family man and found in his family a great source of happiness and strength. Eisenhart was essentially a most modest man. The intimate atmosphere which surrounded him, its very serenity, was due in large measure to the care and devotion which he received from Mrs Eisenhart. The Dean, as we all called him, did not seem to realize that he was an outstanding leader both in his field and in higher education. For outside his family he had two "loves": differential geometry(as research and study)and education.

and the trustees declared that he had:-... in two centuries of Princeton's history few scholars had done more to shape the future of the University ...

... earned an enduring place in the front rank[of those who had]made Princeton great.

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

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