MONISM IN ARITHMETIC. 15 



to be repeatedly summed is called and how often it occurs. We 

 thus reach the notion of multiplication. To multiply a by b means 

 to form the sum of b numbers each of which is called a. The num- 

 ber conceived summed is called the multiplicand, the number which 

 indicates or counts how often the first is conceived summed is called 

 the multiplier. 



It appears hence, that the multiplier must be a result of count- 

 ing, or a number in the original sense of the word, but that the mul- 

 tiplicand may be any number hitherto defined, that is, may also be 

 zero or negative. It also follows from this definition that though 

 the multiplicand may be a concrete number the multiplier cannot. 

 Therefore, the commutative law of multiplication does not hold 

 when the multiplicand is concrete. For, to take an example, though 

 there is sense in requiring four trees to be summed three times, 

 there is no sense in conceiving the number three summed "four 

 trees times." When, however, multiplicand and multiplier are un- 

 named results of counting, (abstract numbers,) two fundamental 

 laws hold in multiplication, exactly analogous to the fundamental 

 laws of addition, namely, the law of commutation and the law of 



association. Thus, 



a times b=b times a, 



and, a times (b times c) = (a times b} times c. 



The truth and correctness of these laws will be evident, if keeping 

 to the definition of multiplication as an abbreviated addition of equal 

 summands, we go back to the laws of addition. Owing to the com- 

 mutative law it is unnecessary, for purposes of practical reckoning, 

 to distinguish multiplicand and multiplier. Both have, therefore, a 

 common name : factor. The result of the multiplication is called the 

 product; the symbol of multiplication is a dot (.) or a cross ( 

 which is read " times." Joined with the fundamental formula above 

 written are a group of subsidiary formulae which give directions how 

 a sum or difference is multiplied and how multiplication is performed 

 with a sum or difference. I need not enter, however, into any dis- 

 cussion of these rules here.. 



As the combination of two numbers by a sign of multiplication 

 has no significance according to our definition of multiplication, 



