If the life of any 20th century mathematician can be said to be a history of mathematics in his time, it is that of David Hilbert. To the enchanted young mathematicians and physicists who flocked to study with him in Goettingen before and between the World Wars, he seemed mathematics personified, the very air around him''scientifically electric.'' His remarkably prescient proposal in 1900 of twenty-three problems for the coming century set the course of much subsequent mathematics and remains a feat that no scientist in any field has been able to duplicate. When he died, Nature remarked that there was scarcely a mathematician in the world whose work did not derive from that of Hilbert.

Constance Reid's classic biography is a moving, nontechnical account of the passionate scientific life of this man-from the early days in Koenigsberg, when his revolutionary work was dismissed as ''theology,'' to the golden years in Goettingen before Hitler came to power and within a few months destroyed the entire Hilbert school.

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