XXI.] THEORY OF APPROXIMATION. 481 



Arithmetic of Approximate Quantities. 



Considering that almost all the quantities which we 

 treat in physical and social science are approximate only, 

 it seems desirable that attention shonld be paid in the 

 teaching of arithmetic to the correct interpretation and 

 treatment of approximate numerical statements. We seem 

 to need notation for expressing the approximateness or 

 exactness of decimal numbers. The fraction 'Q2^ may 

 mean either precisely one 40th part, or it may mean 

 anything between '0245 and '0255. I propose that when 

 a decimal fraction is completely and exactly given, a 

 small cipher or circle should be added to indicate that 

 there is nothing more to come, as in -0250. When the 

 iirst figure of the decimals rejected is 5 or more, the first 

 figure retained should be raised by a unit, according to a 

 rule approved by De Morgan, and now generally recog- 

 nised. To indicate that the fraction thus retained is more 

 than the truth, a point has been placed over the last figure 

 in some tables of logarithms ; but a similar point is used 

 to denote the period of a repeating decimal, and I should 

 therefore propose to employ a colon after the figure ; thus 

 •025: would mean that the true quantity lies between 

 ■0245" and •025° inclusive of the lower but not the higher 

 limit, Wh(!n the fraction is less than the truth, two dots 

 mi^ht be ])laced horizontally as in -025. . which would 

 mean anything between "025° and '0255° not inclusive. 



When ap]iroximate numbers are added, subtracted, mul- 

 tiplie'j, or divided, it becomes a matter of some complexity 

 to determine the degree of accuracy of the result. There 

 are few persons who could assert off-hand that the sum 

 of the approximate numbers 3470, 52'693, 8o"i, is 167*5 

 withiii less than 'oy. Mr. Saudeman has traced out the 

 1 ules of approximate arithmetic in a very thorough manner, 

 and liis directions are worthy of careful attention.^ The 

 third part of Sonnenschein and Nesl)itt's excellent book 

 on arithmetic ^ describes fully all kinds of ajijiroximate 

 calculations, and shows both how to avoid needless labour 



' .Sanfleinan, PeiicntdicK, p. 214. 



'^ 'I'liK Science and Art of Arithmetic for the Use of l^'chooU. 

 Whilaker and Co.) 



I I 



