284 LIMITS AND FLUXIONS 



that to every distance corresponds a number and to 

 every number there corresponds a distance. Number 

 was thus given a geometrical basis. This situation 

 continued into the nineteenth century. This metrical 

 view involved the entire theory of measurement, 

 which assumed greater difficulties with the advent 

 of the non-EucHdean geometries. The geometrical 

 theory of number became less and less satisfactory 

 as a logicai foundation. Hence the attempts to 

 construct purely arithmetical theories.^ 



A good share of those difficulties arose from 

 irrational numbers, which could not be avoided in 

 analytical geometry. This occurrence is not merely 

 occasionai ; irrational ratios are at least as frequent 

 as rational ones. What is an irrational number ? 

 How do we operate with irrational numbers ? What 

 constitutes the sum, difìference, product or quotient, 

 when irrational numbers«are involved? No explicit 

 answer was given to these questions. It was tacitly 

 assumed without fear, that it is safe to operate with 

 irrational numbers as if they were rational. But 

 such assumptions are dangerous. They might lead 

 to absurdities. Even if they do not, this matter 

 demands attention when mathematical rigour is 

 the aim. 



242. Perhaps it may be worth while to recali to 

 the reader's mind illustrations of the danger result- 

 ing from taking operations known to yield consistent 



^ For a historical account of the number concept and the founding 

 of the theory of transfinite numbers during the nineteenth century, 

 read Philip E. B. Jourdain's " Introduction " to Cantor's Transjinite 

 Numbers, The Open Court Publishing Co,, 191 5. 



