58 On a Method of Checking Calculations. 



a ly a 2 , a 3 , being the digits in N. 2. There is a check 



upon the place of any figure, so far as to its being in an even 

 or odd place. 3. There is a check upon the figures themselves, 

 except when their sum (in the odd places or in the even places) 

 = 11 or any multiple of 11. 



One eleventh, only, of reciprocals or fractions are unable to 

 be expressed by whole-number unitates. In 100 there are 9 

 fractional unitates to reciprocals. 



Therefore the method of checking calculations that this 

 paper advocates is the combination of the formulation of work 

 and of results with casting out the elevens, in preference to 

 . casting out the nines. 



The shortest way to cast out the elevens is to add together 

 all the digits in the odd places of the given number to form 

 one sum total, and all the digits in the even places to form 

 another sum total, then to subtract the latter sum from the 

 former. In the process of obtaining each sum total, as soon 

 as any number higher than 11 is obtained its unitate should 

 be substituted. In obtaining the unitate to a decimal, it must 

 be remembered to take all the alternate digits from the unit's 

 digit as additive. 



In checking additions, the one operation only need be used, 

 and that without putting pen to paper. All the digits in the 

 odd places may be added together, reducing from time to time, 

 from which sum the sum of the digits in the even places, simi- 

 larly obtained, is subtracted. 



In checking subtractions, it is only necessary to be sure that 

 the unitate of the subtrahend is less than that of the minuend ; 

 this is easily done, if necessary, by the addition of eleven to 

 the unitate of the minuend. 



Multiplications only require the reduction of the unitate of 

 the product to a number less than 11. In multiplications into 

 which decimals enter, the operation must be without contrac- 

 tion. 



In divisions, the division may be finished at any predeter- 

 mined place of decimals, taking into account the fraction at 

 the end which includes the remainder. The fractions are dealt 



with (as if of the form a . j)bj the table of unitates of UnN -1 , 

 which is 



