Discovery of Fuchsian Functions
Creative discoveries do not always take place by thinking through the problem in a systematic, step-by-step manner. Often the innovative solution requires a period of gestation or incubation, when the creative person abandons the problem and gets involved in other unrelated activities. The discovery of Fuchsian Functions by the mathematician Henri Poincare provides one such instance of incubation, ending in sudden illuminations which came to him with "brevity, suddenness and immediate certainty".
Poincare was working on certain mathematical transformations which defined the Fuchsian Functions, when he had to leave his work midway. On account of a geological expedition organised by the school of mines, he decided to take a vacation, and left Caen where he was living at that time. The excursion made him forget his mathematical work. One day during the trip, the group boarded a bus in Countance to go to some place. The insight came to Poincare as he was boarding the bus:
"At the moment I put my foot on the steps the idea came to me, without anything in my former thought seeming to have paved the way for it, that the transformation I had used to define the Fuchsian functions were identical with those of non-Euclidian geometry."
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