476 Mr. wales on the Refoluibn 
cmifideratiofi, It is not always neceffary to exterminate all the 
unknown quantities but one. 
EXAMPLE VII. 
Suppofe the equation to be refolved were a = .375 = 
i6y'=p:4j/ 4 — 20^ 3 dt:4jv 2 + 5y : and, firft, let the upper ligns 
hate place, and it is manifeft, that the latter fide of the equa- 
tion may bo- divided into two parts; namely, 4)' 2 - 4 v 4 = 
47 s • 1 ~\~y • 1 -y> and 1 6y 5 - 20 y 3 + sy =y 5 - 1 0/ . I +y . T^y 
4- Sy • 1 +jyr • 1 — jy 12 . But the former part is (by tab. I.) the 
fquare of the fine of twice the arc which has y for its fine (radius 
being = 1) and the latter part the fine of five times the fame 
arc. Hence, therefore, the given quantity (~ .375) is equal 
to the fum of the fine of five times the arc (A) which has y 
for its fine, and the fquare of the fine of twice the fame 
arc. Now, as the fquare of the fine of twice the arc (A) 
muft rieceffarily in this inftance be very fmall in comparifon of 
the fine of five times the fame arc (A), it is manifeft, that the 
fine of five times the arc which has y for its fine will be very 
little lefs than .375, and of courfe that arc (5A) can be but 
very little lefs than 22 0 2', the fine of which is next greater 
than that number. Affume it 21 0 , and the fifth part of it, or 
that arc which hasy for its fine, will be 4 0 12', the double of 
which is 8° 24k Now the log. fine of 8° 24' is 9.1645998, 
which being doubled is 8.3291996, the logarithm of .0213403, 
and this number being taken from .375 leaves *3536597, 
which ought to have been *3583679, the fine of 21 0 , and of 
courfe is too fmall by .0047082 : the arc has, therefore, been 
affumed too great. 
Let 
