METHODS OF SCIENCE; NUMBERS 25 



day a clock strikes on the average six and a 

 half strokes per hour. 



Likewise the negative numbers found easy 

 interpretation in debit and credit, and so frac- 

 tions and negatives were given full standing 

 with integers in the society of numbers. While 

 the integers formed an infinite series, the frac- 

 tions form infinite series even between two in- 

 tegers. Thus if we w^ite 1% between 1 and 2 

 and then interpolate 1% and 1% and then 1%, 

 1%, etc., and this process is kept up indefi- 

 nitely, we get an infinite series so dense that be- 

 tween an}^ two of these fractions, no matter how 

 close together, an infinite number of terms may 

 still appear. 



Yet if this process is carried on indefinitely 

 no one of our numbers will be exactly 1%. 

 When, however, we have interpolated all of our 

 thirds and fifths and sevenths, and so on, into 

 our already dense and infinite series, it might 

 be imagined that our needs would be satisfied 

 and our instinct for abstraction appeased. But 

 we find that we are not yet through. If we write 

 % + %H-% + W6 + %2, and so forth, and 

 add these fractions together as we write them 

 do^\Ti, the total sum never will exceed 1, but will 

 approach this number more and more closely as 



