CHAPTER III
The Non-euclidean Geometries
Every conclusion supposes premises; these premises themselves either are self-evident and need no demonstration, or can be established only by relying upon other propositions, and since we can not go back thus to infinity, every deductive science, and in particular geometry, must rest on a certain number of undemonstrable axioms. All treatises on geometry begin, therefore, by the enunciation of these axioms. But among these there is a distinction to be made: Some, for example, 'Things which are equal to the same thing are equal to one another,' are not propositions of geometry, but propositions of analysis. I regard them as analytic judgments a priori, and shall not concern myself with them.
But I must lay stress upon other axioms which are peculiar to geometry. Most treatises enunciate three of these explicitly:
1º Through two points can pass only one straight;
2º The straight line is the shortest path from one point to another;
3º Through a given point there is not more than one parallel to a given straight.
Although generally a proof of the second of these axioms is omitted, it would be possible to deduce it from the other two and from those, much more numerous, which are implicitly admitted without enunciating them, as I shall explain further on.
It was long sought in vain to demonstrate likewise the third axiom, known as Euclid's Postulate. What vast effort has been wasted in this chimeric hope is truly unimaginable. Finally, in the first quarter of the nineteenth century, and almost at the same time, 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.