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William Fogg Osgood (March 10, 1864-July 22, 1943) was born in Boston, Massachusetts, the son of William and Mary Rogers (Gannett) Osgood. He prepared for college at the Boston Latin School, entered Harvard in 1882, and was graduated with the A.B. degree in 1886, second in his class of 286 members. He remained at Harvard for one year of graduate work in mathematics, receiving the degree of A.M. in 1887, and then went to Germany to continue his mathematical studies. During Osgood's study at Harvard, the great Benjamin Peirce (1809-1880), who had towered like a giant over the entire United States, was no longer there. James Mills Peirce (1834-1906), son of Benjamin, was in the Mathematics Department, and also served later (1890-1895) as Dean of the Graduate School and (1895-1898) as Dean of the Faculty of Arts and Sciences. William Elwood Byerly was also a member of the Department (1876-1913), and is remembered for his excellent teaching and his texts on the Calculus and on Fourier's Series and Spherical Harmonics. Benjamin Osgood Peirce (1854-1914) was a mathematical physicist, noted for his table of integrals and his book on Newtonian Potential Theory. Osgood was influenced by all three of those named they were later his colleagues in the department - and also by Frank Nelson Cole.
Cole graduated from Harvard with the Class of 1882, studied in Leipzig from 1882 to 1885, where he attended lectures on the theory of functions by Felix Klein, and then returned to Harvard for two years, where he also lectured on the theory of functions, following Klein's exposition.
Felix Klein left Leipzig for Göttingen in 1886, and Osgood went to Göttingen in 1887 to study with him. Klein (Ph.D., Göttingen, 1871) had become famous at an early age, especially because of his Erlanger Program, in which he proposed to study and classify geometries (Euclidean, hyperbolic, projective, descriptive, etc.) according to the groups of transformations under which they remain invariant; thus Euclidean geometry is invariant under the group of rigid motions. The group idea was a central unifying concept that dominated research in geometry for many decades. Klein was also interested in the theory of functions, following the great Göttingen tradition, especially in automorphic functions. Later, he took a leading part in organizing the Enzyklopädie der Mathematische Wissenschaften, the object of which was to summarize in one collection all mathematical research up to 1900. Klein also had an abiding interest in elementary mathematics, on the teaching of which he exerted great influence both in Germany and elsewhere.
The mathematical atmosphere in Europe in 1887 was one of great activity. It included a clash of ideals: the use of intuition and arguments borrowed from physical sciences, as represented by Bernhard Riemann (1826–1865) and his school, versus the ideal of strict rigorous proof as represented by Karl Weierstrass (1815–1897), then active in Berlin. Throughout his mathematical career, Osgood chose the best of the two schools, using intuition in its proper place to suggest results and their proofs, but ultimately relying on rigorous logical demonstrations. The influence of Klein on "the arithmetizing of mathematics" remained with Osgood throughout his later life.
Osgood did not receive his Ph.D. from Göttingen. He went to Erlangen for the year 1889-1890, where he wrote a thesis, "Zur Theorie der zum algebraischen Gebilde gehörten Ableschen Functionen." He received the degree there in 1890 and shortly after married Theresa Ruprecht of Göttingen, and then returned to Harvard.
Osgood's thesis was a study of Abelian integrals of the first, second, and third kinds, based on previous work by Klein and Max Noether. He expresses in the thesis his gratitude to Max Noether for help. He seldom mentioned the thesis in later life; on the one occasion that he mentioned it to me, he tossed it off with "Oh, they wrote it for me." Nevertheless, it was part of the theory of functions, to which he devoted so much of his later life.
In 1890, Osgood returned to the Harvard Department of Mathematics and remained for his long period of devotion to science and to Harvard. At about this time, a large number of Americans were returning from graduate school that the French traités d'analyse, also far more rigorous than, say, Forsyth's theory of functions. It was a moment to the care, orderliness, rigor, and didactic skill of its author. When G. Pólya visited Harvard for the first time, I asked him whom he wanted most to meet. He replied "Osgood, the man from whom I learned function theory" even though he knew Osgood only from his book. Osgood generously gives Bôcher part of the credit for the Functionentheorie, for the two men discussed with each other many of the topics contained in it. The book became an absolutely standard work wherever higher mathematics was studied.
Osgood had previously (1897) written a pamphlet on Infinite Series, in which he set forth much of the theory of series needed in calculus, and his text on calculus dates from 1907. This was also written in a careful, precise style that showed on every page that the author knew profoundly the material he was presenting and its background, both historically and logically. It also showed that Osgood knew the higher developments of mathematics and how to prepare the student for them. The depth of Osgood's interest in teaching calculus is also indicated by his choice of that topic for his address as retiring president of the American Mathematical Society in 1907
Osgood wrote other texts for undergraduates, in 1921 an Analytic Geometry with W. C. Graustein, which was again scholarly and rigorous, and in 1921 a revision of his Calculus, now called Introduction to the Calculus. In 1925 he published his Advanced Calculus, a masterly treatment of a subject that he had long taught and that had long fascinated him. He published a text on Mechanics in 1937, the outgrowth of a course he had frequently given, and containing a number of novel problems from his own experience.
After Osgood's retirement from Harvard in 1933, he spent two years (1934–1936) teaching at the National University of Peking. Two books in English of his lectures were prepared by his students and published there in 1936: Functions of Real Variables and Functions of a Complex Variable. Both books borrowed largely from Functionentheorie.
Osgood did not direct the Ph.D. theses of many students; the theses he did direct were those of C. W. McGraw-Hill, L. D. Ames, E. H. Taylor, and (with C. L. Bouton) G. R. Clements. I asked him in 1917 to direct my own thesis, hopefully on some subject connected with the expansion of analytic functions, such as Borel's method of summation. He threw up his hands, "I know nothing about it."
Osgood's influence throughout the world was very great, through the soundness and depth of his function theory, through the results of his own research, and through his stimulating yet painstaking teaching of both undergraduates and graduate students. He was intentionally raising the scientific level of mathematics in America and elsewhere, and had a great part in this process through his productive work, scholarly textbooks, and excellent classroom teaching.
Osgood's favorite recreations were touring in his motor car and smoking cigars. For the latter, he smoked until a little of the cigar was left, then inserted the small blade of a penknife into the stub so as to have a convenient way to continue.
Osgood was a kindly man, somewhat reserved and formal to outsiders, but warm and tender to those who knew him. He had three children by Mrs. Teresa Ruprecht Osgood: William Ruprecht, Freida Bertha (Mrs. Walter Sitz, now deceased), and Rudolph Ruprecht. His years of retirement were happy ones. He married Mrs. Celeste Phelpes Morse in 1932 and died in 1943. He was buried in Forest Hills Cemetery, Boston.
Cole graduated from Harvard with the Class of 1882, studied in Leipzig from 1882 to 1885, where he attended lectures on the theory of functions by Felix Klein, and then returned to Harvard for two years, where he also lectured on the theory of functions, following Klein's exposition.
Felix Klein left Leipzig for Göttingen in 1886, and Osgood went to Göttingen in 1887 to study with him. Klein (Ph.D., Göttingen, 1871) had become famous at an early age, especially because of his Erlanger Program, in which he proposed to study and classify geometries (Euclidean, hyperbolic, projective, descriptive, etc.) according to the groups of transformations under which they remain invariant; thus Euclidean geometry is invariant under the group of rigid motions. The group idea was a central unifying concept that dominated research in geometry for many decades. Klein was also interested in the theory of functions, following the great Göttingen tradition, especially in automorphic functions. Later, he took a leading part in organizing the Enzyklopädie der Mathematische Wissenschaften, the object of which was to summarize in one collection all mathematical research up to 1900. Klein also had an abiding interest in elementary mathematics, on the teaching of which he exerted great influence both in Germany and elsewhere.
The mathematical atmosphere in Europe in 1887 was one of great activity. It included a clash of ideals: the use of intuition and arguments borrowed from physical sciences, as represented by Bernhard Riemann (1826–1865) and his school, versus the ideal of strict rigorous proof as represented by Karl Weierstrass (1815–1897), then active in Berlin. Throughout his mathematical career, Osgood chose the best of the two schools, using intuition in its proper place to suggest results and their proofs, but ultimately relying on rigorous logical demonstrations. The influence of Klein on "the arithmetizing of mathematics" remained with Osgood throughout his later life.
Osgood did not receive his Ph.D. from Göttingen. He went to Erlangen for the year 1889-1890, where he wrote a thesis, "Zur Theorie der zum algebraischen Gebilde gehörten Ableschen Functionen." He received the degree there in 1890 and shortly after married Theresa Ruprecht of Göttingen, and then returned to Harvard.
Osgood's thesis was a study of Abelian integrals of the first, second, and third kinds, based on previous work by Klein and Max Noether. He expresses in the thesis his gratitude to Max Noether for help. He seldom mentioned the thesis in later life; on the one occasion that he mentioned it to me, he tossed it off with "Oh, they wrote it for me." Nevertheless, it was part of the theory of functions, to which he devoted so much of his later life.
In 1890, Osgood returned to the Harvard Department of Mathematics and remained for his long period of devotion to science and to Harvard. At about this time, a large number of Americans were returning from graduate school that the French traités d'analyse, also far more rigorous than, say, Forsyth's theory of functions. It was a moment to the care, orderliness, rigor, and didactic skill of its author. When G. Pólya visited Harvard for the first time, I asked him whom he wanted most to meet. He replied "Osgood, the man from whom I learned function theory" even though he knew Osgood only from his book. Osgood generously gives Bôcher part of the credit for the Functionentheorie, for the two men discussed with each other many of the topics contained in it. The book became an absolutely standard work wherever higher mathematics was studied.
Osgood had previously (1897) written a pamphlet on Infinite Series, in which he set forth much of the theory of series needed in calculus, and his text on calculus dates from 1907. This was also written in a careful, precise style that showed on every page that the author knew profoundly the material he was presenting and its background, both historically and logically. It also showed that Osgood knew the higher developments of mathematics and how to prepare the student for them. The depth of Osgood's interest in teaching calculus is also indicated by his choice of that topic for his address as retiring president of the American Mathematical Society in 1907
Osgood wrote other texts for undergraduates, in 1921 an Analytic Geometry with W. C. Graustein, which was again scholarly and rigorous, and in 1921 a revision of his Calculus, now called Introduction to the Calculus. In 1925 he published his Advanced Calculus, a masterly treatment of a subject that he had long taught and that had long fascinated him. He published a text on Mechanics in 1937, the outgrowth of a course he had frequently given, and containing a number of novel problems from his own experience.
After Osgood's retirement from Harvard in 1933, he spent two years (1934–1936) teaching at the National University of Peking. Two books in English of his lectures were prepared by his students and published there in 1936: Functions of Real Variables and Functions of a Complex Variable. Both books borrowed largely from Functionentheorie.
Osgood did not direct the Ph.D. theses of many students; the theses he did direct were those of C. W. McGraw-Hill, L. D. Ames, E. H. Taylor, and (with C. L. Bouton) G. R. Clements. I asked him in 1917 to direct my own thesis, hopefully on some subject connected with the expansion of analytic functions, such as Borel's method of summation. He threw up his hands, "I know nothing about it."
Osgood's influence throughout the world was very great, through the soundness and depth of his function theory, through the results of his own research, and through his stimulating yet painstaking teaching of both undergraduates and graduate students. He was intentionally raising the scientific level of mathematics in America and elsewhere, and had a great part in this process through his productive work, scholarly textbooks, and excellent classroom teaching.
Osgood's favorite recreations were touring in his motor car and smoking cigars. For the latter, he smoked until a little of the cigar was left, then inserted the small blade of a penknife into the stub so as to have a convenient way to continue.
Osgood was a kindly man, somewhat reserved and formal to outsiders, but warm and tender to those who knew him. He had three children by Mrs. Teresa Ruprecht Osgood: William Ruprecht, Freida Bertha (Mrs. Walter Sitz, now deceased), and Rudolph Ruprecht. His years of retirement were happy ones. He married Mrs. Celeste Phelpes Morse in 1932 and died in 1943. He was buried in Forest Hills Cemetery, Boston.