16 MONISM IN ARITHMETIC. 



*vhen the multiplier is zero or a negative number, it will be seen 

 that we are again in a position where it is necessary to apply the 

 above explained principle of no exception. We revert, therefore, to 

 what we above established, that zero and negative numbers are sym- 

 bols which have the form of differences, and lay down the rule that 

 multiplications with zero and negative numbers shall be performed 

 exactly as with real differences. Wh'y, then, is minus one times 

 minus one, for example, equal to plus one? For no other reason 

 than that min-.is one can be multiplied with an ordinary difference, 

 as, for example, 8 minus 5, by first multiplying by 8, then multiply- 

 ing by 5, and subtracting the differences obtained, and because 

 agreeably to the principle of no exception we must say that the mul- 

 tiplication must be performed according to exactly the same rule 

 with a symbol which has the form of a difference whose minuend is 

 less by one than its subtrahent. 



As from addition two inverse operations, detraction and subter- 

 traction, spring, so also from multiplication two inverse operations 

 must proceed which differ from each other simply in the respect that 

 in the one the multiplicand is sought and in the other the multiplier. 

 As matters of computation, these two inverse operations coalesce 

 in a single operation, namely, division, owing to the validity of the 

 commutative law in multiplication. But in so far as they are differ- 

 ent ideas, they must be distinguished. As most civilised languages 

 distinguish the two inverse processes of multiplication in the case 

 in which the multiplicand is. a line, we will adopt for arithmetic a 

 name which is used in this exception. Let us take this example, 



4 yards X 3 = 1 2 yards. 



If twelve yards and four yards are given, and the multiplier 3 is 

 sought, I ask, how many summands, each equal to four yards, give 

 twelve yards, or, what is the same thing, how often I can lay a 

 length of four yards on a length of twelve yards? But this is measur- 

 ing. Secondly, if twelve yards and the number 3 are given, and the 

 multiplicand four yards is sought, I ask what summand it is which 

 taken three times gives twelve yards, or, what is the same thing, 

 what part I shall obtain if I cut up twelve yards into three equal 

 parts? But this is partition, or parting. If, therefore, the multi- 



