of adfefted Equations . 
469 
EXAMPLE III. 
To find the value of x in an equation of the form 
x 3 *f r z x -<u. 
Let us take as examples of this equation x 3 + 3V = .04, 
4~ j.y = .o8 ? - and x 3 + 3V — .1 2, which are three of the inftances 
given by Dr. halley, in his Synopfis of the Aftronomy of 
Comets, to illustrate the mode of computation that he purfued 
in confruiting his general table for calculating the place of a 
comet in a parabolic orbit : and it is obvious, a being put for 
the known fde of the equation, that it may be transformed to 
vfr x \/3 -f x — s/a : where, if x. be confidered as- the tangent 
of an arc, the radius of which is v/3, v^3 -M* will be the fecant 
©f that arc ; and, consequently, by what is Shewn in the frft 
example, an arc mud: be found, fuch, that the fum of the tabular 
log. fecant and half the tabular log. tangent may be equal to the 
excefs of half the log. of a above | of the log. of 3. In the 
firff of the above three inftances this excefs will be found, 
18.943x891, in the fecond 19,09.37041, and in the third 
19.1817497; and by running the eye along Gardiner’s 
Tables of Logarithmic Sines and Tangents, it will be found, 
that the firfl falls between o° 20 ;/ and o° 26' 30", the fe- 
cond between o° 52' 50" and o° 53' o 7 , and the third between 
r° 19' 20" and. j° if 30 // and, bypurfuing the mode which 
has been deferibed' in the two former examples, the exact arcs 
will be found o° 26' 27", 7,. o° 54' 51", 7, and i° 19' 20T1? 
and their refpedlive tangents, to the radius \/ 3, .01333248, 
.0266611, and .039978 7, the three values of x fought. And' 
in this manner Dr. halley’s table may be extended to any 
length with the utmoft eafey expedition, and accuracy. 
3 
Thus- 
