If it were possible to deduce Euclid's postulate from the several axioms, it is evident that by rejecting the postulate and retaining the other axioms we should be led to contradictory consequences. This little work has inspired most of the recent treatises to which I shall later on refer, and among which I may mention those of Beltrami and Helmholtz. But the question was not exhausted, and it was not long before a great step was taken by the celebrated memoir of Riemann, entitled: Über die Hypothesen welche der Geometrie zum Grunde liegen. They have nearly rid us of inventors of geometries without a postulate, and ever since the Académie des Sciences receives only about one or two new demonstrations a year. Finally, at the beginning of the nineteenth century, and almost simultaneously, two scientists, a Russian and a Bulgarian, Lobachevsky and Bolyai, showed irrefutably that this proof is impossible. It is impossible to imagine the efforts that have been spent in pursuit of this chimera. For a long time a proof of the third axiom known as Euclid's postulate was sought in vain. (3) Through one point only one parallel can be drawn to a given straight line.Īlthough we generally dispense with proving the second of these axioms, it would be possible to deduce it from the other two, and from those much more numerous axioms which are implicitly admitted without enunciation, as I shall explain further on. (2) A straight line is the shortest distance between two points (1) Only one line can pass through two points
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