[James Mills Peirce's lectures] stood as the Old over and against the New, and of the latter Cole was the apostle. The students felt that he had seen a great light. Nearly all the members of the Department attended his lectures. It was the beginning of a new era in graduate education at Harvard ...Cole had attended Klein's lectures at Harvard in 1885-87 on function theory and he persuaded Osgood to go to Göttingen in 1887 and study with Klein. Osgood was awarded a Parker Fellowship and set off for Germany in the autumn of 1887 having already an excellent command of the German language and a deep respect for German learning. After working with Klein for two years, Osgood wondered if he might not benefit by spending some time at another German University before the end of his three year fellowship. He wrote to Harry Tyler, an American student of mathematics who was studying at Erlangen, asking for his advice. Tyler replied :-
I think in the first place that it is much better for you or anyone else who has three years abroad not to spend the whole time in Göttingen unless for special reasons of great importance. I do not think the incidental loss comparable with the gain in broader knowledge of mathematics and mathematicians. So I wouldn't stay at Göttingen even if it were somewhat better than any other university ...Tyler also described to Osgood the two mathematicians at Erlangen who might act as his supervisor, namely Max Noether and Paul Gordan:-
Both men are so peculiar and so irreconcilable that ... [personal relations] must be cultivated with some tact ... As far as I know [Max Noether] like [Gordan] confines himself to pure mathematics ... and both I think run to depth rather than breadth, as compared with Klein. If they have so much in common, that's about all. [Gordan] is outspoken, irascible, exasperating, violent. [Max Noether] is taciturn, serious, equable, patient.Osgood took Tyler's advice and undertook graduate work with Max Noether at Erlangen in 1889. He did this despite having met Theresa Anna Amalie Elise Ruprecht in Göttingen and the two already were thinking of marriage. Theresa's family owned the local publishing firm of Vandenhoeck & Ruprecht. His thesis, written while at Erlangen, was based on work suggested by Klein on abelian functions and this fitted particularly well with Max Noether's interests at that time. After being awarded a doctorate by Erlangen in 1890 for his thesis Zur Theorie der zum algebraischen Gebilde ym = R(x) gehörigen Ableschen Functionen Ⓣ, he married Theresa Ruprecht and they returned to Harvard. Walsh writes  or :-
At about this time a large number of Americans were returning from graduate work in Germany with the ambition to raise the scientific level of mathematics in this country. There was no spirit of research at Harvard then, except what Osgood himself brought, but a year later Maxime Bôcher joined him there, also a student greatly influenced by Felix Klein, and a man of mathematical background and ideals similar to those of Osgood. They were very close friends both personally and in scientific work until Bôcher's death in 1918.At Harvard Osgood was an instructor in mathematics from 1890 to 1893 when he was promoted to assistant professor. He held this post until 1903 when he was promoted to full professor, being named Perkin Professor of Mathematics in 1913 on the death of the holder of the chair William Elwood Byerly who had taught Osgood as an undergraduate and had then been his colleague for 23 years. He must have been a slightly odd character, however :-
It is safe to say that no American mathematician during this period (and certainly none since) bore the mark of German academic training as visibly and as permanently as Osgood. ... He even sympathized with the German cause during World War I. Later in life, Norbert Wiener thought he cut a rather ridiculous figure, self-consciously emulating even the mannerisms of a German professor.Osgood's main work was on the convergence of sequences of continuous functions, solutions of differential equations, the calculus of variations and space filling curves. Some of his early work is described in  or :-
In 1897 he published a deep investigation into the subject of uniform convergence of sequences of real continuous functions ... In 1898 Osgood published an important paper on the solutions of the differential equation dy/dx = f (x, y) satisfying the prescribed initial conditions y(a) = b. ... Osgood showed that if f (x, y) is merely continuous there exists at least one solution ... In 1900 Osgood established, by methods due to H Poincaré, the Riemann mapping theorem, namely that an arbitrary simply connected region of the plane with at least two boundary points, can be mapped uniformly and conformally onto the interior of a circle. ... This theorem remains as Osgood's outstanding single result.Some papers over the next few years included: Sufficient conditions in the calculus of variations (1900), On a fundamental property of a minimum in the calculus of variations and the proof of a theorem of Weierstrass's (1901), A Jordan curve of positive area (1903), On Cantor's theorem concerning the coefficients of a convergent trigonometric series, with generalizations (1909), On the gyroscope (1922), and On normal forms of differential equations (1925).
His Lehrbuch der Funktionentheorie (Volume I, 1907) became a classic. The complete work occupied three volumes: Volume I, Volume II, Part I, and Volume II, Part II. The second volume (Volume II, Part I) appeared in 1924 and the final volume (Volume II, Part II) was first published in 1932. Other classic texts included Introduction to Infinite Series (1897), A First Course in the Differential and Integral Calculus (1909), Topics in the theory of functions of several complex variables published by the American Mathematical Society in 1914, Plane and Solid Analytic Geometry (with W C Graustein, 1921), Advanced Calculus (1925), and Mechanics (1937).
Osgood's marriage to Teresa produced three children, William Ruprecht, Frieda Bertha Ruprecht (who married Walter Silz), and Rudolf Ruprecht. However the marriage ended in divorce and Osgood married Celeste Phelps Morse, the former wife of Marston Morse, in August 1932. Morse, also a professor of mathematics at Harvard, had divorced Celeste in 1930. When Celeste married Osgood, who was 28 years older than Morse and 68 years old at the time of the marriage, not only did Morse receive a great shock, but there was a scandal which led to Osgood retiring. After he retired from Harvard in 1933, Osgood taught for two years at the National University of Peking. At this time he had two texts published based on the lectures he gave in Peking: Functions of real variables (1936), and Functions of a complex variable (1936).
Osgood is important in bringing the latest ideas of mathematical research to the United States. He received many honours for these contributions. He was elected to the National Academy of Sciences (United States) in 1904 and had the honour of being president of the American Mathematical Society during 1905-06. He was twice American Mathematical Society Colloquium Lecturer, first in 1898 and again in 1913.
A kindly but reserved man, he liked to travel by car, playing tennis and golf, and smoking cigars :-
... he smoked until little of the cigar was left, then inserted the small blade of a penknife in the stub so as to have a convenient way to continue.After returning from China he lived in Belmont, Massachusetts, and after his death there he was buried in Forest Hills Cemetery, Boston.
Article by: J J O'Connor and E F Robertson