112 ROYAL SOCIETY OF CANADA 



Lobachevski and Bolyai. These mathematicians showed the existence 

 of perfectly consistent systems of geometry in which the eleventh axiom 

 did not hold. It followed, therefore, that this axiom could not be 

 ,a consequence of the other Euclidean axioms, and that, accordingly, all 

 efforts to prove it must necessarily be fruitless. This axiom is thus 

 shown to be a fact of observation, and geometry becomes a branch of 

 natural science. In our space, the Euclidean space, parabolic space 

 as it has been called, only one straight line in a plane can be drawn 

 through a given point parallel to a given straight line; in the space 

 of Lobachevski, hyperbolic space, an infinite number of such lines can 

 be drawn; and in the space of Eiemann, elliptic space, no such lines 

 can be drawn, — there are no parallels, no lines that do not meet. 



There is a letter written in 1799 by Gauss to the elder Bolyai 

 from which it appears that Gauss was at that time occupied with the 

 foundations of geometry. 



Lobachevski views were first published in a lecture given before 

 the Faculty of Mathematics and Physics of the University of Kasan, 

 February 26th, 1826. 



Gauss in a letter to Bessel (January 27th, 1829) states that the 

 foundations of geometry cannot be established a priori, and there appears 

 reason to believe his researches were along tlie line of those of Loba- 

 chevski and Bolyai. These researches, however, were never published. 



Wolfyang Bolyai in 1832-3 published a two volume work on 

 mathematics, and at the end of the first volume occurred an appendix, 

 a memoir written by Johann Bolyai in 1823, in which the theory of 

 parallels was developed along the same lines as Lobachevski followed. 



Niemann's " Uber die Hypothesen welche der Géométrie zu Grunde 

 liegen " was published in 1854. 



In a letter to his father, written November 3rd, 1823, the younger 

 Bolyai claims that " from nothing he had created another wholly new 

 world," and so in a sense he had. 



The remarkable discoveries of Lobachevski and Bolyai passed 



practically unnoticed until the attention of the mathematical world 

 was directed to them by Eiemann and Baltzer, about 1866; This then 

 may be considered the date at which the vision of mathematicians was 

 cleared in respect to the foundations of geometry so far as the so-called 

 parallel axiom is concerned. 



But all difficulties respecting the foundations of geometry were 

 by no means thus disposed of. The discovery of the space of Loba- 

 chevski made men doubtful of final principles. They found themselves 

 existing between two kinds of space entirely different from their own, — 



