SPACE AND GEOMETRY 41 



straight line falling on two straight lines, etc." 

 This postulate which seems to have been and 

 probably was inserted without much polish, at 

 the last moment, as though it had been hoped to 

 dispense with it, is nevertheless one of the great- 

 est of Euclid's creations, for without such a pos- 

 tulate Euclidean geometry could not exist. 

 There is nothing more interesting in the his- 

 tory of science than the record of the repeated 

 attacks made on this postulate in the hope of 

 reducing it to a proposition deducible from the 

 other four, and which finally resulted in the 

 discoveries by Lobachevski,^ Bolyai and Rie- 

 mann of other geometries, all of which are de- 



2 Those who are interested in the problem of the objec- 

 tive and subjective will find it worthy of note that the 

 two geometries which were published independently and 

 almost simultaneously, one by the Russian Lobachevski, 

 and the other by the Hungarian Bolyai, were so nearly 

 alike that they seem like different drafts of the same com- 

 position. Similarly Hamilton and Grassmann wrote at the 

 same time those papers which were to become the founda- 

 tion of modern vector analysis. We cannot avoid the 

 thought that having embarked upon a certain line of mathe- 

 matical inquiry, while we appear to have preserved the 

 utmost of personal freedom, we seem bound to follow cer- 

 tain paths and to make and remake certain discoveries, 

 just as we do in physics or chemistry. Is there then almost 

 as much an objective world of mathematics as there is an 

 objective world of physics? The views of a number of scien- 

 tists and philosophers upon this subject will be found in the 

 interesting book of E. Meyerson, La Deduction Relativiste. 



