PAET II SPACE 



Non-Euclidean Geometries 



Every conclusion presumes premisses. These premisses are either 

 self-evident and need no demonstration, or can be established only 

 if based on other propositions ; and, as we cannot go back in this way 

 to infinity, every deductive science, and geometry in particular, must 

 rest upon a certain number of indemonstrable axioms. All treatises 

 of geometry begin therefore with the enunciation of these axioms. 

 But there is a distinction to be drawn between them. Some of these, 

 for example, " Things which are equal to the same thing are equal 

 to one another," are not propositions in geometry but propositions in 

 analysis. I look upon them as analytical a priori intuitions, and they 

 concern me no further. But I must insist on other axioms which are 

 special to geometry. Of these most treatises explicitly enunciate 

 three: (1) Only one line can pass through two points; (2) a 

 straight line is the shortest distance between two points; (3) through 

 one point only one parallel can be drawn to a given straight line. 

 Although we generally dispense with proving the second of these 

 axioms, it would be possible to deduce it from the other two, and 

 from those much more numerous axioms which are implicitly ad- 

 mitted without enunciation, as I shall explain further on. For a long 

 time a proof of the third axiom known as Euclid's postulate was 

 sought in vain. It is impossible to imagine the efforts that have been 

 spent in pursuit of this chimera. Finally, at the beginning of the 

 nineteenth century, and almost simultaneously, two scientists, a Eus- 

 sian and a Bulgarian, Lobatschewsky and Bolyai, showed irrefutably 

 that this proof is impossible. They have nearly rid us of inventors 

 of geometries without a postulate, and ever since the Academic des 

 Sciences receives only about one or two new dmonstrations a year. 

 But the question was not exhausted, and it was not long before a 

 great step was taken by the celebrated memoir of Riemann, entitled: 

 Ueber die Hypothesen welche der Geometric zum Grunde liegen. This 

 little work has inspired most of the recent treatises to which I shall 

 later on refer, and among which I may mention those of Beltrami and 

 Helmholtz. 



The Geometry of Lobatschewsky. If it were possible to deduce 



