Review of the Cambridge Course of Mathematics. 317 



8 7 



Now, to form a more simple idea of the fraction 

 fif, we endeavour to compare it with a part of unity, 

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

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

 merator, and ^fjQ for that of the denominator; this last 

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

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

 this point, we see that the second approximate value of 

 (he expression VVy is 1 and \.) or f . But this value is too 

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

 ded by 4 and gVei which is written thus : 1 — i — 



To form an exact idea of the expression 1 — ^, it is ne- 



'^ 4_2_3_' 



2\ 6 



cessary to consider it as indicating the quotient of the 

 whole number 1 divided by the whole number 4 accom- 

 panied by the fraction /y^. 



If we divide the two terms of jVe '^J 23, the quotient 

 will be -i--, neglecting the ^^ which accompany the whole 



number 9, there will be i only instead of //g, and conse- 

 quently, 1^— will a third approximate value of VW, a 



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 

 4 aVe ' ^^^ ^^^ quotient -^ smaller than the true quotient. 



By reducing the whole number 4 with the fraction which 

 accompanies it, and performing the division according to 

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

 rj\ or If for the third approximate value of VsV- 



The exact expression of this value being 1 — L_ 



"9^, if we 

 divide the two terms of /g by 9, we shall have 1. 



neglecting the fraction |, there will remain 1 



4 



9 ^, a value 



too great ; for the fraction | being greater than ^ whose 



2f ^ 



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

 Vol. V. 41 



