28 THE ANATOMY OF SCIENCE 



bination of I's and O's in such infinite frame- 

 works as are represented in our diagram; in 

 other words, when we have added to the integers 

 all fractions rational and irrational, then, and 

 only then, is our system of "real" numbers com- 

 plete. The addition of the irrationals has made 

 the number system not more cumbrous, but 

 really more simple, and the system now has in 

 every sense the properties of a continuum. This 

 makes it possible to represent numerically a 

 continuous change such as the growi:h of a tree 

 with time, or the change of volume with tem- 

 perature, so also it brings the number system 

 into full correspondence with the positions of 

 points on a geometrical line. 



