on Converging Sericfes. yy 
trary figtis to P ; and Qj but if one has the lame, and the 
other contrary, then will k and k? have contrary figns. 
5. Let home affirmative quantities be lefs than others, then 
any direct affirmative function of the former, viz. fun 61 ion, 
in which 110 negative or impoffible quantities or indexes are 
contained, will be lefs than the fame function of the latter. 
The contrary happens when the indexes are all negative, and 
the quantities affirmative as before : for example, let two quan- 
tities be lefs than two others, then the product of the two for- 
mer will be lefs than the produ6t of the two latter. 
Cor. Hence foine quantities may often be known to be 
greater or lefs than others from their diredt functions being 
greater or lefs than the fame funftions of the others : for 
example, let a z - b z be an affirmative quantity, then will a be 
greater than b. 
6. If one equation or ratio is affirmed on the fuppofi- 
tion that another given one is true, reduce both the equations 
by the methods given above, and from the principles before 
delivered, the proportion will often be evident. 
Hence may be deduced demonfhatiens to propofitions of 
this fort given by Pappus and others. 
Ex. Let the ratio a + b : b be greater than c + d : d, then 
the ratio b : a — b will be lefs than d : c — d. 
For, fince the ratio a+b : b is greater than c + d : d, the 
ratio b:a + b will be lefs than d:c + d , and confequently 
-r(l +I ) = T(j + 1 ) + ^ 'v lienee - i = c - — l + k, gnd 
a jzl — c — 4- k. and the ratio b : a-b lefs than d : c - d. 
Ex. 2. Let the ratio of a + b : r + r/be greater than the ratio 
of a : c, then will the ratio of bid be greater than the ratio of 
a+b: c + d. By the preceding method convert thefe ra- 
tios 
