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_. 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 than two right angles, and the difference between this sum and two right angles is proportional to the surface of the triangle. It is impossible to construct a figure similar to a given figure but of different dimensions. If we divide a circumference into _n_ equal parts, and draw tangents at the points of division, these _n_ tangents will form a polygon if the radius of the circle is small enough; but if this radius is sufficiently great they will not meet. It is useless to multiply these examples; Lobachevski's propositions have no relation to those of Euclid, but they are not less logically bound one to another. RIEMANN'S GEOMETRY.--Imagine a world uniquely peopled by beings of no thickness (height); and suppose these 'infinitely flat' animals are all in the same plane and can not get out. Admit besides that this world is sufficiently far from others to be free from their influence. While we are making hypotheses, it costs us no more to endow these beings with reason and believe them capable of creating a geometry. In that case, they will certainly attribute to space only two dimensions.