12 MONISM IN ARITHMETIC. 



the number which added to b gives a, according as the one or the 

 other of the two operations inverse to addition is meant. From the 

 formula for subtraction, and from the rules which hold for addition, 

 follow now at once the rules which refer to both addition and sub- 

 *"~action. These rules we here omit. 



From the foregoing it is plain that the minuend is necessarily 

 larger than the subtrahent. For in the process of addition the minu- 

 end was the sum, and the sum grew out of the union of two natural 

 number-pictures.* Thus 5 minus 9, or n minus 12, or 8 minus 8, 

 are combinations of numbers wholly destitute of meaning; for no 

 number, that is, no result of counting, exists that added to 9 gives 

 the sum 5, or added to 12 gives the sum n, or added to 8 gives 8, 

 What, then, is to be done? Shall we banish entirely from arith- 

 metic such meaningless combinations of numbers ; or, since they 

 have no meaning, shall we rather invest them with one? If we do 

 the first, arithmetic will still be confined in the strait-jacket into 

 which it was forced by the original definition of number as the re- 

 sult of counting. If we adopt the latter alternative we are forced 

 to extend our notion of number. But in doing this, we sow the 

 first seeds of the science of pure arithmetic, an organic body of 

 knowledge which fructifies all other provinces of science. 



What significance, then, shall we impart to the symbol 



5-9? 



Since 5 minus 9 possesses no significance whatever, we may, of 

 course, impart to it any significance we wish. But as a matter 

 of practical convenience it should be invested with no meaning 

 that is likely to render it subject to exceptions. As" the form of the 

 symbol 5 9 is the form of a difference, it will be obviously con- 

 venient to give it a meaning which will allow us to reckon with it as 

 we reckon with every other real difference, that is, with a difference 

 in which the minuend is larger than the subtrahent. This being 

 agreed upon, it follows at once that all such symbols in which the 

 number before the minus sign is less than the number behind it by 



the same amount may be put equal to one another. It is practical, 







* See page 2, supra. 



