10 MONISM IN ARITHMETIC. 



guished in some way, for example, by saying a is to be increased by 

 b, or b is to be added to a, and so forth. Here, it is plain, a is al- 

 ways something that is to be increased, b the increase. Accordingly 

 it has been proposed to call the number which is regarded in addi- 

 tion as the passive number or the one to be changed, the augend, 

 and the other which plays the active part, which accomplishes the 

 change, so to speak, the increment. Both words are derived from 

 the Latin and are appropriately chosen. Augend is derived from 

 augere, to increase, and signifies that which is to be increased ; in- 

 crement comes from increscere, to grow, and signifies as in its ordi- 

 nary meaning what is added. 



Besides the commutative law one other follows from the idea ot 

 counting the associative law of addition. This law, which has ref- 

 erence not to two but to three numbers, states that having a certain 

 sum, a -f- b, it is indifferent for the result whether we increase the 

 increment b of that sum by. a number, or whether we increase the 

 sum itself by the same number. Expressed in the symbolic lan- 

 guage of arithmetic this law reads, 



a+(f, + c) = (a + S) + c. 



To obtain now all the rules of addition we have only to apply the 

 two laws of commutation and association above stated, though fre- 

 quently, in the deduction of the same rule, each must be applied 

 many times. I may pass over here both the rules and their estab- 

 lishment. 



In addition, two numbers, the augend a and the increment b 

 are combined into a third number c, the sum. From this operation 

 spring necessarily two inverse operations, the common feature of 

 which is, that the sum sought in addition is regarded in both as 

 known, and the difference that in the one the augend also is regarded 

 as known, and in the other the increment. If we ask what number 

 added to a gives c, we seek the increment. If we ask what number 

 increased by b gives c, we seek the augend. As a matter of reckon- 

 ing, the solution of the two questions is the same, since by the com- 

 mutative law of addition a-\-b = b-}-a. Consequently, only one 

 common name is in use for the two inverses of addition, namely, 

 subtraction. But with respect to the notions involved, the two oper- 



