Archimedes leapt from his bath in joy and ran through the streets of Syracuse, crying ‘Eureka!’ as he went. He had solved the problem that had been worrying him for days: how to measure the volume of an irregularly-shaped object, such as a golden (or not-so-golden) crown. – Friedrich von Kekulé, dozing by the fire, had a dream suggesting that the structure of the troublesome benzene molecule might be a ring. A whole new branch of science (aromatic chemistry) was founded as a result. The mathematician Jacques Hadamard, more than once, found a long-sought solution ‘at the very moment of sudden awakening’. And Henri Poincaré, as he was boarding a bus to set out on a geological expedition, suddenly glimpsed a fundamental mathematical property of a class of functions he had recently discovered and which had preoccupied him for days.