He began his undergraduate education in 1933 at the California Institute of Technology. He obtained both the A.B. and M.A. from George Washington University in 1937. In 1941, he received his Ph.D. in history from Columbia University and enlisted in the US Navy as an Ensign; he mustered out in 1946 holding the rank of Lieutenant Commander.
After 1 year at Columbia University as an instructor in history and the history of science, Claggett joined the University of Wisconsin's department of History of Science, eventually becoming Vilas Research Professor, in the Department of History of Science. From 1959 to 1964, he was also director of the University's Institute for Research in the Humanities. After two visiting appointments at the Institute of Advanced Study's School of Historical Studies, 1958-59 and 1963, he was made Professor in 1964, and Professor Emeritus in 1986.
The author of more than a dozen volumes on the history of science and mathematics, he was perhaps best known for his landmark 10 tome, 5 volume work, Archimedes in the Middle Ages, published between 1964 and 1984. Over the remainder of his life, he completed 3 of the planned 4 volumes of Ancient Egyptian Science.
He was honored with the following prizes:
A fellow of the Medieval Academy of America and past president of the History of Science Society, he was a member and former vice president of the American Philosophical Society. He was also a member of the Deutsche Gesellschaft für Geschichte der Medizin, Naturwissenschaft und Technik, and the International Academy of the History of Science, which he served as vice president from 1968 to 1971.