The Coutances Illumination

In the summer of 1880, Henri Poincaré joined a geological excursion organized by the École des Mines, spending his days examining rock formations near the small Norman town of Coutances and his evenings in polite conversation about stratigraphy. Mathematics was consciously absent. He had spent the previous fifteen days in intense and fruitless work on a class of functions he would later call Fuchsian functions; every approach had failed; he had set the problem aside not strategically but from genuine exhaustion. Then, as he reported with the precision of a man who understood the diagnostic significance of what had happened, “the ideas came to me, without anything in my former thoughts seeming to have paved the way for them.” His foot touched the step of the omnibus. The Fuchsian transformations were identical to those of non-Euclidean geometry. The recognition was instantaneous, complete, and accompanied by an absolute