MATHEMATICS n 



The same system serves equally well for subtraction. 

 Thus, if from a group of 204 objects, 20 have to be taken 

 away, we write 



204 



20 



Here it is evident (since there is a zero at the right end 

 of the lower number) that no single unit has to be 

 taken away from 204, so the figure 4 must remain 

 unchanged at the right hand of the sum expressing the 

 result. In the second column (which denotes groups 

 of 10) two such have to be taken from zero. This 

 difficulty, as schoolboys know, is evaded by borrowing 

 ten groups of ten from the set of the next higher 

 denomination, then taking two sets of ten from the ten 

 sets thus borrowed, there will remain 8 sets of ten, and 

 8 will therefore be the second figure of the sum denoting 

 the result. The ten groups of ten, which have been bor- 

 rowed, have now to be taken away from the third figure, 

 which from its position shows that it denotes groups of 

 ten times ten. This third figure is 2, from which one 

 being deducted, we have, of course, i as a remainder, 

 and so we express the process thus : 



204 



20 

 184 



The correspondence of this process with the real 

 relations which exist between substantial things, can 

 again be most simply shown by taking 20 marbles from 

 204, and counting the number left. In an analogous 

 manner we can (by practical, material tests) establish 

 the correspondence with reality of the other processes of 



