26 THE ANATOMY OF SCIENCE 



we keep on increasing the number of terms. We 

 may assume with Dedekind^ that the sum of 

 such an infinite series is itself a number, and 

 that in this particular case the number is 1. 



However, if we write the series %-[-% + 

 Ke + Hs -[- . . . we find that the sum of this 

 infinite series converges somewhere in the neigh- 

 borhood of %, but it is not exactly this number, 

 nor indeed is it exactly any one of the numbers 

 that we have hitherto discussed. If we include 

 such sums of infinite series among our numbers 

 then it is evident that we have once more gen- 

 eralized and extended the meaning of the word 

 "number." For a long time, indeed, the new 

 numbers were not formally admitted into the 

 family, but finally were grudgingly adopted 

 and stiU go by their old names "surd" (= ab- 

 surd) and "irrational." 



These irrational numbers may be further 

 illustrated by means of the accompanying table. 

 Any number which is less than 1 can be ex- 

 pressed by crossing out some of the terms in 

 the infinite series % -f- ^ + %> etc. The first 

 row in the table represents that whole infinite 

 series, the dots indicating that the table should 



9 Dedekind, Was sind und was sollen die Z allien? ; Essays 

 on the Theory of Numbers. 



