the Interminable Decimals. 19 



given, the proper multiplier is at once found by observing 

 what is carried from the wth figure of the development ; and 

 it is plain that what is thus carried may be increased by any 

 multiple of the divisor. And nothing further in the way of 

 explanation can be required. 



A word or two is however necessary, in reference to the 

 foregoing rule, for finding the diameter of a circle from its 

 circumference. By the rule, the expression for the diameter 

 from the circumference, c, is 



^(1 + -0004)0. 



But the true formula, or rather that which replaces the use 



of3'14.16, is 



7 7 I +'0004 



C=^r-'' c. 



22(1 — -0004) ~ 22 1— -00000016 

 So that in the work of the second example above, the divisor 

 1— -00000016 has been regarded as unit; an error which 

 cannot affect the first six places of figures, and which can 

 affect the seventh only by a unit ; for 



1 + -0004 _ (l + -0004-)(l+ -000000 16) 

 1- -00000016 ~ 1— -0000000000000256 

 And this error is of no practical importance, in reference at 

 least to a comparison between the new and the old rules ; for 

 the seventh figure will always be erroneous, even when 3'1416 

 is employed, the result can be correct only to the sixth place 

 in the most favourable circumstances. 



I would here further add, as arithmetical facility, in calcu- 

 lations of very frequent occurrence in practice, is my more 

 especial object of consideration in the present paper, that the 

 result of any multiplication by 3-1416 may with but little 

 trouble be converted into what the more accurate multiplier 

 3-141593 would have produced. For since 



3-1416 

 diminished by 3*141593 



is -000007 

 it follows that if we only push the number multiplied by 3*1416 

 six places back, multiply it so depressed by 7, and subtract 

 the result from the product furnished by 3-1416, we shall get 

 the same figures as the more correct multiplier above would 

 have produced. For instance, from the circumference deter- 

 mined above, namely, 



2774723-9520 

 subtract 6-182540 



#i and we shall have 2774717-769460-ii Ju.aiqot 



C2 



