Property of Vulgar Fractions. 323 



r 7) a 



Now if there be a fraction - (s < b) lying between *- and r, 



then 

 therefore 



that is, 



r p a p m 

 — ~~ < j — j 

 s q o q 



rq—sp aq—bp 

 ~~s~~ < ~~b 

 1 



rq-sp< p 



viz. a positive integer is less than a proper fraction, which is 

 impossible. Therefore - is the nearest fraction less than ? 

 which has a denominator less than b. 



Also, ? — - is the nearest fraction greater than T which has 



' b—q fe b 



a denominator less than b ; for if - (s < b) lie between 7 and 



7 — — , then 

 b—q 7 



a—p r a—p a. 





b — q 8 ^ b — q b' 



viz. 







as —ps — br + qr aq — bp 

 s < b 





1 . 



or 



as —ps — br + qr< -, 

 b 



which is impossible. 



The proof is exactly similar if the last convergent - be 

 greater than j ; and therefore, always, if - be the last con- 

 vergent to 7, the nearest fractions to j y on each side, are - and 



a ~~r) 



-y — -, whence the truth of the first part of the lemma is evident. 



b—q 7 r 



Also since 



a ___p _ 1 a—p^a__ 1 



b q~~~~bq 'b — q 5"~"~~ (b — q)b 



