We have just said that geometry has its axioms. Let us remember that an entire school of geometricians amused themselves by disputing the axioms, under the pretext that it was impossible to demonstrate them. We were present, in December, 1866, at a curious sitting of the Institute, during which M. Lionville, a celebrated mathematician, and professor at the Sorbonne, explained this strange polemic with great skill. In attempting to demonstrate the propositions of geometry, certain axioms, _i.e._, self-evident truths, must be admitted in the first place. Otherwise, the primary reasoning will have no basis. But, among the numerous propositions of this kind which present themselves to the mind, and which result from the admission of one of their number, which is the most evident? That depends on the nature of the mind of each of us, and therefore it is that there is not, and that there never will be, an argument on this question. There is a school of geometry which pretends to demonstrate everything. There is another, the true and good school, which, recognizing that the human mind has limits, and that everything is not accessible by our thoughts, lays down, under the name of axioms, certain truths which do not require proof, or, which is often the same thing, are incapable of proof. Among the number of self-evident truths, or truths difficult of demonstration, we find the question of parallel lines. What are two parallels? Two lines which never meet each other. But how can we prove this property of two lines by reasoning? That is not, exactly speaking, possible, since the notion of the infinite is not admitted, or not understood by everybody, and cannot, therefore, serve as the basis of an absolutely rigorous argument. It was for this reason that Euclid, the founder of geometry in ancient times, laid down this truth as a simple axiom, requiring (hence the _postulates_ of Euclid, from the Latin verb _postulare_, to demand), that the truth of this principle, which he acknowledged himself unable to prove by logical demonstration, should be granted. A hundred geometricians, since Euclid, who renounced the attempt to demonstrate it, have tried to prove this theory of parallels, but not one has succeeded. It was on the occasion of a fresh attempt at demonstration by a mathematician in the provinces, that M. Lionville spoke before the Academy, to recall the principles almost unanimously professed by geometricians on this subject. The question is, in reality, thoroughly understood; it is treated on all works on geometry, and has been for a long time a settled matter. But certain minds are tempted by the subtlety of certain subjects, and the question of the _postulatum_ turns up periodically before the learned societies, as it does in the conversations between the teachers of mathematics.