MERITS AND DEFECTS 287 



mctical theory vve have reached a development of 

 rational numbers (integers and rational fractions). 

 We wish, next, to define //;//// and also irrntional 

 number. An early nineteenth-century definition of 

 limit was : '' VVhen the successive values attributed 

 to a variable approach a fixed value indefinitely so 

 as to end by difìfering from it as little as is wished, 

 this fixed value is called the limit of ali the others. " 

 Since, according to our supposition, we are stili in 

 the field of rational numbers, this limit, unless it 

 happens to involve only rational numbers and to be 

 itself only a rational number, is, in our case, non- 

 existent and fictitious. 



If now, as stated above, an irrational number is 

 defined as the limit of certain sequences of rational 

 fractions, trouble arises. The existence of such a 

 Hmit is often far from evident. But aside from that 

 general consideration, the difficulty of the situation 

 in our case is apparent : Irrational numbers are 

 limits, but limits themselves are non-existent or 

 fictitious, unless they are rational numbers. To 

 avoid this breakdown in the logicai development, 

 it was found desirable to define irrational number 

 without using limits. 



245. Wìth the view of avoiding the use of limits 

 in the definition of irrational number, and at the 

 same time avoid inelegant and difficult assumptions, 

 involving complicated considerations relating to the 

 nature of space, ^ devices were invented by several 



^ On this point consult the article " Geometry " in the Encyclopitdia 

 Britannica^ iith edition, the part on Congruence and Measurement. 



