VOL. LXXVII.] PHILOSOPHICAL TRANSACTIONS, IQS 



6. If one equation or ratio is affirmed on the supposition that another given 

 one is true, reduce both the equations by the methods given above, and from the 

 principles before delivered the proposition will often be evident. Hence may 

 be deduced demonstrations to propositions of this sort given by Pappus and 

 others. 



Exam. — Let the ratio a -\- b'. bhe. greater than c -\- d: d, then the ratio b : a 

 -F- b will be less than d : c — d. 



For, since the ratio a -{- b : b is greater than c -\- d : d, the ratio b : a -^ b 



will be less than d : c -\- dj and consequently ^-j- (^ + l) = ^—j- (^ + l)-{-k, 



whence -— 1=^— 1-f^j and — v— = — ^ \- k, and the ratio b : a — b 



less than d: c — d. 



Exam. 2. — Let the ratio o( a -\- b : c -{- d he greater than the ratio of a : c, 

 then will the ratio of b : dhe greater than the ratio of a + ^ : c -f- ^« By the 



preceding method convert these ratios into equations, and there result -f- k 



=z - and - -\- k' = —^TT '» a"d the proposition asserts, that if k be an affirma- 

 tive quantity, k' will also be an affirmative quantity. Reduce these two equations, 

 so as to take away their denominators, and the resulting equations will be 

 ac -\- ad-\- a X {a -{■ b) X k = ac -\- be, and ad -\- bd -\- {a -\- b) . bk' = be 



-\- bd, whence k = ^7^-r> a"^ ^ = ^. , ,y and the proposition is evident. 



Exam. 3. — ^Let a be greater than c, and b, and (a + ^) X (a — ^) = (c -j- fQ 

 X (c — d), that is, d^ — b^ = c^ •— d^ ; then will b be greater than d: for a in 

 the equation a^ ^ b"^ = c^ — d^ write c -{• h, and there results 2ck -f ^^ = b'^^d^, 

 whence P — d^ is an affirmative quantity, and consequently b greater than d. 



Exam. 4. — Let, as in Ex. 1, the ratio a -\- b : b he greater than c -{- d: d, 

 then will b : a —■ b he less than the ratio d: c — d. By the preceding method 



translate these ratios into the two equations — •— r + ^ = — -— ; and — ~- = ^ T 



^ a + b c + a b d 



-\- A'; reduce these equations, so as to take away their denominators, and there 

 result be -Y bd -{■ {a -\- b) X {c -\- dk) = ad -\- bd and da — db = be —• bd -{- 



bdk', and consequently k = ^. -, ^. , and k' = — ^^ — ; but these two frac- 



tions, which express the values of k and k% have the same numerators, and their 

 denominators both affirmative ; therefore if one k be affirmative, the other k' 

 will also be affirmative. 



Corol. From these principles can easily be deduced innumerable propositions of 

 this sort. Assume 2 or more ratios, of which let some be supposed greater than 

 others ; then, from the above-mentioned transformation, by addition, subtraction, 



cc 2 



