## Poincaré on Fuchsian Groups

I wanted to represent these functions by the quotient of two series;
this idea was perfectly conscious and deliberate; the analogy
with elliptic functions guided me. I asked myself what properties
these series must have if they existed, and succeeded without
difficulty in forming the series I have called thetafuchsian.

Just at this time, I left Caen, where I was living, to go on a
geologic excursion under the auspices of the School of Mines.
The indicidents of the travel made me forget my mathematical
work. Having reached Coutances, we entered an omnibus to
go some place or other. At the moment when I put my foot on
the step, the idea came to me, without anything in my former throughts
seeming to have paved the way for it, that the transformations
I had used to define the Fuchsian functions were identical with
those of non-Euclidean geometry. I did not verify the idea;
I should not have had time, as, upon taking my set in the
omnibus, I went on with a conversation already commenced,
but I felt a perfect certainty. On my return to Caen, for
conscience' sake, I verified the result at my leisure.

Then I turned my attention to the study of some arithmetical
questions apparently without much success and without a
suspicion of any connection with my preceding researches.
Disgusted with my failure, I went to spend a few days at the
seaside and thought of something else. One morning, while
walking on the bluff, the idea came to me, with just the same
characteristics of brevity, suddenness and immediate certainty, that
the arithmetric transofrmations of indefinite ternary
quadratic forms were identical with those of
non-Euclidean geometry.

--- From * The Psychology of Invention in the Mathematical Field *,
J. Hadamard, Ch. I