MATHEMATICS AND LOGIC. 155 



VI. 



The definitions of number are very numerous and 

 of great variety, and I will not attempt to enumerate 

 even their names and their authors. We must not be 

 surprised that there are so many. If any one of them 

 was satisfactory we should not get any new ones. If 

 each new philosopher who has applied himself to the 

 question has thought it necessary to invent another, 

 it is because he was not satisfied with those of his 

 predecessors ; and if he was not satisfied, it was because 

 he thought he detected 2i petitio principii. 



I have always experienced a profound sentiment 

 of uneasiness in reading the works devoted to this 

 problem. I constantly expect to run against a petitio 

 principii, and when I do not detect it at once I am 

 afraid that I have not looked sufficiently carefully. 



The fact is that it is impossible to give a definition 

 without enunciating a phrase, and difficult to enun- 

 ciate a phrase without putting in a name of number, 

 or at least the word several, or at least a word in the 

 plural. Then the slope becomes slippery, and every 

 moment we are in danger of falling into the petitio 

 principii. 



I will concern myself in what follows with those 

 only of these definitions in which \.\\q petitio principii 

 is most skilfully concealed. 



VII. 



Pasigrapiiy. 



The symbolical language created by Signor Peano 

 plays a very large part in these new researches. It is 



