18 Mr. J. R. Young on a Property of 



ciphers prefixed, the remainder will be a figure preceded by 

 two '9s, and soon. Consequently, if unity be divided by any 

 number, giving an interminable decimal, as by 7, lor instance, 

 then if from this unit we take care to subtract such a figure, 

 preceded by ciphers or not, as will cause the remainder to be 

 accurately divisible by 7, it is plain that the subtractive num- 

 ber when divided by 7, must furnish the same interminable 



decimals, after a limited number, as those furnished by-; 



otherwise the difference between the two rows could not be a 

 finite number; that is, the remainder mentioned above could 

 not be exactly divisible by 7. 



This explanation will no doubt prove sufficient: three figures 



were cut off from the end of the partial development of- above, 



and the other figures multiplied by 6, because 1 —•006 = •994', 

 which is exactly divisible by 7. 



But an explanation yet more simple readily offers itself. 



Still reverting to the fraction -, we see that, in prosecuting 



the division, the first remainder that we get is '3 ; therefore 



so that 



y=--r 



7 1 ' 



the first figure in the development of-. Again, the second 



remainder furnished is '02, and 



1 -02 



7 7 

 the first two figures in the development. In like manner, the 

 third remainder being '006, 



7 7 



the first three figures in the development; and, generally, the 

 remainder carried from the «th figure of the development, 



being in the «th place of decimals, -, that is the whole deve- 



j 7 



lopment, diminished by - multiplied by that remainder, must 



leave a result consisting only of the n first figures of the deve- 

 lopment. Hence the number n of figures to be cut off being 



