MONISM IN ARITHMETIC. 17 



plier is sought we call the division measuring, and if the multipli- 

 cand is sought, we call it parting. In both cases the number which 

 was originally the product is called the dividend, and the result the 

 quotient. The number which originally was multiplicand is called 

 the measure; the number which originally was multiplier is called 

 the parter. The common name for measure and parter is divisor. 

 The common symbol for both kinds of division is a colon, a hori- 

 zontal stroke, or a combination of both. Its definitional formula 

 reads, 



(a ~- ) . b = a, or, -- . b = a. 



Accordingly, dividing a by b means, to find the number which mul- 

 tiplied by b gives a, or to find the number with which b must be 

 multiplied to produce a. From this formula, together with the 

 formulae relative to multiplication, the well-known rules of division 

 are derived, which I here pass over. 



In the dividend of a quotient only such numbers can have a 

 place which are the product of the divisor with some previously de- 

 fined number. For example, if the divisor, is 5 the dividend can 

 only be 5, 10, 15, and so forth, and o, 5, 10 and so forth. Ac- 

 cordingly, a stroke of division having underneath it 5 and above it 

 a number different from the numbers just named is a combination 

 of symbols having no meaning. For example, | or -^ 2 - are meaning- 

 less symbols. Now, conformably to the principle of no exception 

 we must invest such symbols which have the form of a quotient 

 without their dividend being the product of the divisor with any 

 number yet defined, with a meaning such that we shall be able to 

 reckon with such apparent quotients as with ordinary quotients. 

 This is done by our agreeing always to put the product of such a 

 quotient form with its divisor equal to its dividend. In this way we 

 reach the definition of broken numbers or fractions, which by the 

 application of the principle of no exception spring from division ex- 

 actly as zero and negative numbers sprang from subtraction, 

 latter had their origin in the impossibility of the subtraction ; the. 

 former have their origin in the impossibility of the division. Putting 



