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.
M. Lionville reminded his audience that many demonstrations of this celebrated proposition had been attempted, but had not succeeded, because there are limits within which human reason ceases to be accepted by all. M. Lionville even proposed that the question of the _postulatum_ should be classed among those whose examination is interdicted by the Academy, such as the quadrature of the circle, and the trisection of the angle. On this point M. Lionville quoted an anecdote relative to Lagrange. That great mathematician, believing that he had found an absolute solution of the _postulatum_, went to the Academy to read his demonstration, but on reflection, he changed his mind, and decided that it would be better not to publish it. He put his manuscript in his pocket, and it never came out.
Several geometricians spoke on this occasion, and confirmed the views of M. Lionville; and when the demonstration submitted by the professor was examined, it was found to be false. We must therefore recognize and proclaim that, in geometry, the axioms cannot be demonstrated.
Many people endeavour to derive an argument from that discussion against the certainty of geometry. Among them is M. Bouillaud, a learned physician and member of the Institute, who declared that he could not get over his astonishment at hearing it said that there were several geometries, and that even the bases of that science were doubtful. Reassure yourself, great and good physician, geometry has nothing to lose and nothing to hide, and the certainty of its methods is not imperilled in this question. That which really was at stake was the methodical, classical teaching of geometry. That which was discussed was the best means of instilling the principles of science into the mind. But, as to the truths of geometry, as to the facts themselves, they are secure from all uncertainty, all these disputes upon the truth which must be recognized as axioms, or demonstrated as theorems, are only fancies of the rhetoricians, as vain as they are subtle. No trace of them remains when they are transported into the practice of facts and of mathematical deductions. Ask the astronomers who calculate the orbit of the stars, who fix the moment of an eclipse with unerring precision, ask those who have calculated the parallaxes, whether they trouble themselves by inquiring how it may be demonstrated that the angles of a triangle are equal to two right angles. All the scholastic subtleties are gotten rid of in the course of practical work.