34(> Mr.W. II. Walenn on Division-Remainders in Arithmetic. 



Thus, in the ordinary way of considering the operation of di- 

 vision, not only is it necessary to know the quotient in order 

 to obtain the remainder, but the value of the remainder is de- 

 pendent upon the value of the last figure of the quotient. This 

 is shown by the fact that if example I. were carried one step 

 further, the quotient would be 0*4098 and the remainder 

 0-0022. 



If, however, in the equation - =c+ -, c be put out of con- 

 sideration, if a and b be integral, and if successive numerical 

 values (as 1, 2, 3, &c.) be assigned to a, the corresponding 

 values of r will also be successive ; they will moreover be 

 periodic, having for their period a value equal to b. For in- 

 stance, when -x is realized mentally in reference to any 



remainder it may possibly have, the remainder must be one 

 of the numbers or units 1, 2, 3, 4, 5, 6, 7, 8, 9, taking, for 

 convenience, 9 to be the remainder when the dividend is an 

 exact multiple of the divisor, so as to include all instances 

 without the use of the symbol 0. 



This is equally true of the general statement j, when b is 



any whole number ; for the point borne in mind in this way 

 of regarding division is the divisor, in reference to whether it 

 is contained a whole number of times or not in the dividend, 

 quite independent of the quotient. If it be contained a whole 

 number of times, the divisor may be put down as the remainder ; 

 if it be not contained a whole number of times, the remainder 

 (in respect of the number of units that make up the divisor) 

 must be one of those units. Considering the remainder to a 

 division in this light, no remainder can be a fractional quan- 

 tity if the dividend be finite and the divisor integral. 



Manifestly the use of looking upon the remainder to a divi- 

 sor in relation to a function that contains the quotient, as in 



the formula -=c+ 1 , is either to be able to continue the 

 o b 



division to the next figure, or to obtain the result of the divi- 

 sion itself with rigid and absolute accuracy. Viewed in this 

 light, r has no other uses and no other properties. 



If, however, the quantity c be put out of consideration, and 

 the remainder be viewed as being made up of units from unity 

 itself to the divisor itself, both inclusive, the case is quite dif- 

 ferent ; the remainder becomes an entity or function which 

 has properties of its own, that not only apply to the particular 

 instance whence it has its being, but which may be used 



