318 Review of the Cambridge Course of Mathematics. 



a denominator too great, the fraction joined to 4 will con- 

 sequently be too small, and the ]ast denominator being too 

 small, will render the last fraction too great. 



By reducing, at first, 9^ to a fraction, we have — ; -^ 



will be then f^, and the approximate value will become 

 1— L. ; now -L- gives -J^f, which joined to unity becomes 



m, or ^1 for a fourth approximate value of VVV- 

 Resuming the expression, 1—-^ — 



4 



9 



1 



2 f , we divide the two 

 terms of the last fraction f by 6, and obtain 1—, and 



4—^ 



l| 



9 



I 



2 



1 



1 j; neglecting the fraction |, there will remain 

 1-1- 



4 



1 



9-^ 



_ } ; and we see as above, that this value is smaller 

 than the true value. 



The fraction -L reduces itself to \ ; and since the pre- 



C)l 3 ' ^ 



ceding — gives ^\, the next preceding becomes _i— , equal 

 • 91 4/9 



to yVj 5 so that the fifth approximate value is lyVa ^^ 



113" 



Dividing, finally, by 4 the two terms of the fraction | 

 which was neglected above, we have for a quotient -i-; and 



by suppressing the fraction 1, we obtain the new value 

 1— !_ 



4 



9 



1 



2. 



1 I, greater than the true value. If we reduce, suc- 

 cessively, all the denominators to a fraction, to obtain the 

 simple fraction which it represents, we shall find lj\\ or 

 jH- By restoring the fraction | to the side of the last 

 denominator, we form the expression 1— i- 



9 



2 



1 



1 



1 1, which be- 



ing reduced as the preceding, reproduces the fractional 

 number VW« 



A 



