MONISM IN ARITHMETIC. H 



ations do differ, and it is accordingly desirable in a logical investi- 

 gation of the structure of arithmetic, to distinguish the two by dif- 

 ferent names. As in all probability no terms have yet been sug- 

 gested for these two kinds of subtraction, I propose here for the 

 first time the following words for the two operations, namely, de- 

 traction to denote the finding of the augend, and subtertraction to 

 denote the finding of the increment. We obtain these terms simply 

 enough by thinking of the augmentation of some object already ex- 

 isting. For example, the cathedral at Cologne had in its tower an 

 augend that waited centuries for its increment, which was only 

 supplied a few decades ago. As the cathedral had originally a 

 height of one hundred and thirty metres, but after completion was 

 increased in height twenty-six metres, of the total height of one 

 hundred and fifty-six metres one hundred and thirty metres is clearly 

 the augend and twenty-six metres the increment. If, now, we wished 

 to recover the augend we should have to pull down (Latin, dctrahcre) 

 the upper part along the whole height. Accordingly, the finding of 

 the augend is called detraction. If we sought the increment, we 

 should have to pull out the original part from beneath (Latin, subter- 

 trahere}. For this reason, the finding of the increment is called sub- 

 tertraction. Owing to the commutative law, the two inverse opera- 

 tions, as matters of computation, become one, which bears the name 

 of subtraction. The sign of this operation is the minus sign, a hori- 

 zontal stroke. The number which originally was sum, is called in 

 subtraction minuend ; the number which in addition was increment 

 is now called detractor ; the number which in addition was augend 

 is now called subtertractor. Comprising the two conceptually dif- 

 ferent operations in one single operation, subtraction, we employ 

 for the number which before was increment or augend, the term sub- 

 trahend, a word which on account of its passive ending is not very 

 good, and for which, accordingly, E. Schroder proposes to substi- 

 tute the word subtrahent, having an active ending. The result of 

 subtraction, or what is the same thing, the number sought, is called 

 the difference. The definition-formula of subtraction reads 



a b -f b = a, 

 that is, a minus b is the timber which increased by b gives a, or 



