Mr. W. H. Walenn on Division-Remainders in Arithmetic. 347 



wherever calculations have value. More than a thousand 

 years ago the Arabians used the tarazu or balance to check the 

 operations of multiplication and division ; the operation called 

 tarazu is that of casting out the nines, or ascertaining the 

 remainder to 9 that any given number has, and with these 

 remainders performing the same operations as with the num- 

 bers whence they are derived. 



Lucas de Borgo, in the fifteenth century, used the remain- 

 ders to division by 9, as well as to division by 7, obtained for 

 the various data in a numerical calculation (each datum having 

 its corresponding remainder), to check the operations of addi- 

 tion, subtraction, multiplication, and division. This employ- 

 ment of the remainder to a divisor is perhaps the first in- 

 stance of the use of a corresponding or factitious number or 

 function in computations; for logarithms were invented by 

 Baron Napier long after — namely, in 1614. 



Mathematicians of recent times have extended the isolated 

 observations of Lucas de Borgo, and have proved the still 

 more general principle that all direct and some inverse opera- 

 tions upon the remainders to divisors are respectively analo- 

 gous to the same operations upon the dividends. This now 

 appears evident from the parcelling out of numerical values 

 which division by a constant divisor affords. The difference 

 between these remainders, as factitious numbers or functions, 

 and logarithms, is, that in the first case the same operation — 

 but in a shorter and more condensed form — is used to obtain a 

 result ; whereas, in the second case, the factitious numbers are 

 capable of being dealt with by an operation one degree lower 

 in the scale of operations. Moreover, in the case of remainders, 

 the application is possible in consequence of their periodicity, 

 and the original result of the calculation is not obtainable, but 

 is simply analogized ; whereas the logarithms being continuous 

 functions, give results quite parallel to those of the intended 

 calculation, and therefore translatable into the answer by suit- 

 able calculations or references. 



Looking upon these remainders as functions of numbers, it 

 is important that their use in performing analogous calcula- 

 tions to a primary one should be recognized and made possible 

 by means of Tables, also that a method should be obtained 

 for their easy discovery. 



From what has been published in the Transactions of the 

 Sections of the British Association for the Advancement of 

 Science (see the i Report ' for 1870), it appears that the nu- 

 merical theorem therein put forth by the author of the present 

 paper gives a means of ascertaining the remainder to a divisor 

 without knowing any thing of the value of the quotient. This 



