The History of Non-Euclidean Geometry |
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| Introduction
Euclid
Saccheri
Gauss/Bolyai
Lobachevsky Riemann/Klein INDEX |
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In the theory of parallels we are even now not further than Euclid. This is a shameful part of mathematics . . . – Gauss The first person to understand this problem of the fifth postulate was Gauss. In 1817, after many years looking at the problem, he had become convinced it was independent of the other four. Gauss then began to look at the consequences of a geometry where this fifth postulate was not nessecarily true. He never published his work due to pressure of time, perhaps illustrating Kant’s statement that Euclidean geometry requires the inevitable necessity of thought. You ought not to try the road of the parallels; I know the road to its end – I have passed through this bottomless night, every light and every joy of my life has been extinguished by it – I implore you for God’s sake, leave the lesson of the parallels in peace . . . I had purposed to sacrifice myself to the truth; I would have been prepared to be a martyr if only I could have delivered to the human race a geometry cleansed of this blot. I have performed dreadful, enormous labours; I have accomplished far more than was accomplished up until now; but never have I found complete satisfaction . . . When I discovered that the bottom of this night cannot be reached from the earth, I turned back without solace, pitying myself and the entire human race.He ignored his father’s impassioned plea, however, and worked on the problem himself. Like Gauss, he looked at the consequences of the fifth postulate not being nessecary. His major breakthrough, was not his work, which had already been done by Gauss, but the fact that he believed that this ‘other’ geometry actually existed. Despite the revolutionary new ideas that were being put forward, there was little public recognition to be had. |
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