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An Essay on the Psychology of Invention in the Mathematical Field – Jacques Hadamard

Henri Poincare’s observations throw a resplendant light on relations between the conscious and the subconscious, between the logical and the fortuitous which lie at the base of the problem…..

Poincare’s example is taken from one of his greatest discoveries, the first which has consecrated his glory, the theory of fuchsian groups and fuchsian functions.  In the first place, I must take Poincare’s own precaution and state that we shall have to use technical terms without the reader’s needing to understand them.

‘I shall say, for example’ he says ‘that I have found the demonstration of such a theorem under such circumstances.  This theorem will have a barbarous name, unfamiliar to many, but that is unimportant; what is of interest for the psychologist is not the theorem but the circumstances’.

So we are going to speak of fuchsian functions.

At first, Poincare attacked the subject vainly for a fortnight, attempting to prove there could not be any such functions; an idea which was going to prove to be a false one.

Indeed during a fortnight of sleeplessness and under conditions to which we shall come back, he builds up one first class of functions.  Then he wishes to find an expression for them.

‘I wanted to represent these functions by the quotient of the 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 theta fuchsian.

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 incidents of the travel made me forget my mathematical work.  Having reached Coutances, we entered an omnibus to go to 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 thoughts 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 Euclidian geometry.  I did not verify the idea; I should not have had time, as upon taking my seat 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'.