a Hungarian and a Russian, Bolyai and Lobachevski, established irrefutably that this demonstration is impossible; they have almost rid us of inventors of geometries 'sans postulatum'; since then the Académie des Sciences receives only about one or two new demonstrations a year.
The question was not exhausted; it soon made a great stride by the publication of Riemann's celebrated memoir entitled: Ueber die Hypothesen welche der Geometrie zu Grunde liegen. This paper has inspired most of the recent works of which I shall speak further on, and among which it is proper to cite those of Beltrami and of Helmholtz.
The Bolyai-Lobachevski Geometry.—If it were possible to deduce Euclid's postulate from the other axioms, it is evident that in denying the postulate and admitting the other axioms, we should be led to contradictory consequences; it would therefore be impossible to base on such premises a coherent geometry.
Now this is precisely what Lobachevski did.
He assumes at the start that: Through a given point can be drawn two parallels to a given straight.
And he retains besides all Euclid's other axioms. From these hypotheses he deduces a series of theorems among which it is impossible to find any contradiction, and he constructs a geometry whose faultless logic is inferior in nothing to that of the Euclidean geometry.
The theorems are, of course, very different from those to which we are accustomed, and they can not fail to be at first a little disconcerting.
Thus the sum of the angles of a triangle is always less