Beview of the Cambridge Course of Mathematics. 317 



11^. Now, to form a more simple idea of the fraction 

 m, we endeavour to compare it with a part of unitjv 

 that we may consider but one term, and for this we divide 

 the two terms by 216 ; we find 1 for the quotient of the nu- 

 merator, and 4/y\ for that of the denominator; this last 

 quotient, wliich is contained between 4 and 5, shews also 

 that the fraction ||f is between ^ and I. By stopping at 

 this point, we see that the second approximate value of 

 the expression ^Vy is 1 and |, or f. But this value is too 

 great, for the true value would be equal to 1 plus 1 divi- 

 <led by 4 and ^r\. which is written thus : I — 1 — 



' 4JL3_ 



lo form an exact idea of the expression I — L_, it is nc- 



eessary to consider it as indicating the quotient of the 

 whole number 1 divided by the whole number 4 accom- 

 panied by the fraction fj\. 



If we divide the two terms of //e ^J 23, the quotient 

 will be -J_; neglecting the /^ which accompany the whole 



conse- 



a 



"23 



number 9, there will be I only instead of ^Vei ^^^ cons 



quentlj, l-J— will a third approximate value of VVV^ 



4i 



value which will be too small, since 9 being less than the 

 true quotient of 216 by 23, the fraction | will be greater 

 than that which ought to accompany 4, and consequently 

 the division 4^ will be greater than the exact division 

 ^tj\i and the quotient -1- smaller than the true quotient. 



By reducing the whole number 4 with the fraction which 

 accompanies it, and performing the division accordiiig to 

 the process of Art. 80, we obtain /y ; and we have 1 and 

 3*7 or if for the third approximate value of ViV- 



The exact expression of this value being ^-~f 



9 A, if we 



divide the two terms of jV by 9, we shall have I—L. 



4 



9 



neglecting the fraction f, there will remain l--i-^ 



2fi 



4- 



9 4, a value 



too great ; for the fraction i being greater than Jy, whose 



place it occupies, will form, by being joined with 9, 

 Vol. V. 41 



