Andre Ampere - his name is still known in the scientific world today thanks to the use of it as a current unit - got a new insight on april , 27th, 1802. Seven years earlier he had found the solution to a problem, which he knew was correct "without being able to prove it. The matter often returned to my mind and I had sought twenty times unsuccessfully for this solution. For some days I had carried the idea about with me continually. At last, I don't know how, I found it, together with a large number of curious and new considerations concerning the theory of probability."

 Ampere realised that what he had found would be a good method to obtain a chair of mathematics in a college, when it would be published, and so it proved.

 Ampere's expererience is one of the many given by famous scientists and inventors, how their discoveries caame to them not by working hard in their laboratory, but by a sudden stroke of enlightenment, or intuitive insight.

Another case of these sudden insights is the mathematician Karl Gauss - his name became a unit of the intensity of a magnetic field. The solution to a problem on which he was working had puzzled him for at least four years, when it suddenly came to him, 'not by painful effort, but, so to speak, by the grace of God, as a sudden flash of light, the enigma was solved'.

The professor for astronomy at the Dublin University, Sir William Rowan Hamilton, had been puzzled by a mathematical problem for longer - fifteen years. In the autumn of 1843 he was walking with his wife along the Grand Canal in Dublin, when ' an electric current seemed to close, and a spark flashed forth, the herald ( as I foresaw immediately) of many long years to come, of definitely directed thought and work, by myself, if spared, and at all events on the part of others, if I should ever be allowed to live long enough directly to communicate the discovery. Nor could I resist the impulse - unphilosophical as it may have been - to cut with a knife on the stone of Brougham Bridge, as we passed it, the fundamental formula.'
His discoveries of quaternions may not be famous any more today, but when in 1865 the National Academy of Sciences was founded in the United States, Rowan Hamilton was the first to be invited as an associate.

Henri Poincaré described, how he was baffled by one particular problem.
'One day, going along the street, the solution of the difficulty, which had stopped me, suddenly appeared to me. I did not try to go deep into it immediately, and only after my military service did I again tak up the question. I had all the elements, and had only to arrange them and put the together. So I wrote out my final memoir at a single stroke and without difficulty.' There were many more experiences, and what was most interesting, was the 'appearance of sudden illumination', 'following long unconscious prior work'.

Henri Poincare, one of the most distinguished mathemticians of his time, admitted, 'I must confess, I am absolutely incapable of doing an addition sum without a mistake.'
When he was very young, he had striven to prove 'that there could not be any functions like those since called Fuchsian functions'. 'I was then very ignorant; every day I seated myself at my work table, stayed an hour or two, tried a great number of combinations, and reached no results. One morning, contrary to my custom, I drank black coffee and could not sleep. Ideas rose in cowards. I felt them collide until pairs interlocked, so to speak, making a stable combination. By the next morning I had established the existince of a class of Fuchsian functions.'
He then had to write out the results, 'which took but a few hours'.

For Poincare it was even not essential to have his conscious mind on the problem.
'At the moment when I put my foot on the step of the bus, 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 functionswere identical wuth those of non-Euclidean geometry. I did not verify the idea; I should not have had time, as, upon taking my seat in the bus, I went on with a conversation already commented, but I felt a perfect certainty. On my return (to Caen), for cience's sake, I verified the result at my leisure.

Jaques Hadamar's interest had initially aroused by an occasion, when he solved a problem during sleep.
He had not actually had a dream, providing the solution, but 'on being very abruptly awakened by an external noise, a solution long searched for appeared to me at once without the slightest instant of reflection on my part. The fact was remarcable enough to have struck me unforgettable - and in quite different direction from that which I had previously tried to follow.'
Intrigued by Poincare, he began to investigate to find if other mathematicians and scientists had had similar experiences.
They had Farady among them. He was, like Poincare, feeble at mathematics.
It was to the highest degree astonishing, what a large number of general theorems, the mathematical deduction of which requires the highest powers of mathematical analysis, Faraday had found by a kind of intuition. With the security of instinct, without the help of a single mathematical formula.
He 'saw' the ansers to his questions in his inner eye. It was a vision of tubes, which rose up before him like things, that was to guide him to his invention of the dynamo and the electric motor.

 Creativity among mathematicians appeared often to depend upon the ability to use imagery in this way.
The most remarkable report comes from Friedrich von Kekulé. When he stayed in London as a young man, he took the last bus and fell into a reverie, in which he saw atoms whirling before his eyes, and for the first time, they behaved in their 'giddy dance' in an way which gave him the clue he needed to form his molecular theory.
Later, in 1865, another dream in the series enabled him to make what is widely regarded as the most brilliant piece of prediction to be found in the history of organic chemestry. In the dream, the atoms began twisting in a snakelike motion when suddenly, one of the 'snakes' caught hold of its own tail. 'As if by flash of lightning I awoke', he realised that the molecules of certain compounds must be closed 'rings'.

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