of adfeSied Equations .. 477 
Let 20 0 45' be next affumed ; the fifth part of which 
is 4 0 9/, and twice this laft number is 8° 18% of which 
the log. fine is 9.1594354; and this being doubled is 
8.3188708, the log. of .0208387; and this being taken 
from .375 will leave .3541613: lefs hill than the fine of 
20 0 45' by .0001297. 
Take now 20° 443 the fifth of which is 4 0 8' 48", and 
two-fifths is 8° 17' 36" ; and the log. fine of this is 9.1590889, 
which being doubled gives 8.3181778, the logarithm of 
.0208055 ; and this being taken from .375, leaves .3541945 ; 
more than the fine 20° 44' by .0000795. Now 2092 (the 
fum of the laid two lafi: errors) is to 60" as 795 (the 
laid error) is to 23". Which being added to 20° 44', the 
lafi: afiumption, gives 20° 44' 23" for five times the arc of 
which y is the fine : y is therefore the fine of 4 0 8' 5 2". 6, 
or .07233202. 
When the lower figns in the equation have place, the 1 
given quantity a will be equal to the excefs of the fine of five 
times an arc above the fquare of the fine of twice that arc : and' 
the operation, after affuming, from the circumftances of the 
queftion, or equation, an arc which is nearly five times that 
having y for its fine, is this. Find the logarithmic fine of 
two-fifths of that arc, double it, find the number correfponding 
to this logarithm, and add to it the value of a, which Ihould 
then be equal to the fine of the arc firft affumed ; and if it 
is not, to repeat the operation until an error is obtained on 
each fide, and not very diftant from the truth, as is done above,,, 
and wliich may always be done with three afiumptions. 
A multitude of examples might be added from the writings- 
©f different authors, who have either left their conclulions 
1 - 
imex- 
