73 Dr. Waring's Obfcrvations 
tios into equations, and there refult — ~ and ~+k' — 
and the proportion afferts, that if k be an affirmative 
quantity, k' will alfo be an affirmative quantity. Reduce thefe 
two equations, fo as to take away their denominators, and the 
refulting equations will be ac + ad+a x a + b x k — ac + be and 
lc— ad 
ad + bd + a -f b , bk' ~ be q- bd, whence k = — and k ' — 
a[a + b) 
i‘ ~ aTT ' * anc ^ t ^ ie P ro P°^^ on evident. 
Ex. 3. Let a be greater than c, and b, and (a + b') 
x (<2 — K) — (c + n 7 ) x (<: - d), that is , a 2 ~ b 1 — c 1 — dr , then will 
b be greater than d ; for a in the equation a 1 — b r — c~ — d 2 write 
c + k, and there refults 2 ck q- k 2 — b 1 — d *, whence b 2 — d 1 is an 
affirmative quantity, and confequently b greater than d. 
Ex. 4. Let, as in Ex. i. the ratio a+b : b be greater than 
c-\-d : d , then will b : a — b be lefs than the ratio d : c-d. 
By the preceding method tranflate thefe ratios into the 
two equations + k = — and “ClA — — + k\ reduce thefe 
a + b 
equations, fo as to take away their denominators, and there 
refult bc-{-bd+a + b x c-\-dk — ad+ bd and da 
ad — be 
db — be - bd 4- 
bdk' , and confequently k = 
t ,AW and k ' - a -~rr'' but thefe 
two fraflions which exprefs the values of k and k' have the 
fame numerators, and their denominators both affirmative ; 
therefore, if one k be affirmative, the other k' will alfo be affir- 
mative. 
Cor. From thefe principles can cafily be deduced innumera- 
ble propofitions of this fort. Aflame two or more ratios, of 
which let fome be fuppofed greater than others; then, from 
the above-mentioned transformation, by addition, fubtradlion, 
multiplication, divifion, &c. can be found fuch functions of 
the 
