William Osgood entered Harvard College in 1882, graduating in 1886. At first he studied classics but Benjamin Osgood Peirce, a mathematical physicist, and F N Cole persuaded him to study mathematics. Osgood undertook graduate work at Harvard for a year.
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 did this and also undertook graduate work at Erlangen in 1889. His thesis was based on work of Klein and Max Noether on Abelian integrals. After being awarded a doctorate by Erlangen in 1890, he returned to Harvard where he taught until he retired.
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. His Lehrbuch der Funktionentheorie (1907) became a classic.
After he retired from Harvard, Osgood taught for two years in Peking.
He is important in bringing the latest ideas of mathematical research to the USA. A kindly but reserved man, he liked to travel by car, smoking cigars and playing tennis and golf.