DEVELOPMENT OF MATHEMATICAL THOUGHT. 735 



we apply the counting process to the needs of geometry 

 and physics. We are here confronted with notions 

 which require to be arithmetically defined — the in- 

 finite and the continuous. The same notions at the 

 beginning of the century attracted the attention of 

 eminent analysts like Cauchy. It is now clear, thanks 

 to the labours of Prof. Georar Cantor of Halle, that 66. 



'^ Geor-Can- 



for mathematical purposes we must distinguish between Qf''yjg*''*'°''y 

 the indefinitely great and the actually infinite in the ^'■'^'^'^""'te- 

 sense of the transfinite. To deal with the actually 

 infinite, as distinguished from the immeasurably or 

 indefinitely great, we have to introduce new notions and 

 a new vocabulary. For instance, in dealing with infinite 

 aggregates, the proposition that the part is always less 

 than the whole is not true. Infinities, indeed, differ, 

 but not according to the idea of greater and smaller, of 

 more or less, but according to their order, grade, or 

 power (in German Mdchtigkeit). Two infinities are 

 equal, or of the same power, if we can bring them into 

 a one-to-one correspondence. Prof. Cantor has shown 

 that the extended range of numbers termed algebraic 

 have tlie same power as the series of ordinary integers — 

 one, two, three, &c. — because we can establish a one-to- 

 one correspondence between the two series — i.e., we can 

 count them. He has further shown that if we suppose 

 all numbers arranged in a straight line, then in any 

 portion of this line, however small, there is an infinite 

 number of points which do not belong to a countable or 

 enumerable multitude. Thus the continuum of numeri- 

 cal values is not countable — it belongs to a different 



