C ) 
And thfe ttiay fuffice to -ftew how to derive the true Me- 
ridian Line front the Sines, Tangents or Secants fappofed 
ready made ; but we are not deftituce of a Method for de- 
ducing the fame independently, from the Arch it fe'f h 
the Latitude from the Equator be eftimated by the length of 
its Arch Radius being Unity, and the Arch put tor an 
Integer be a, as before ; the Meridional Parts anfvering to 
that Latitude will be 
which converges much fv^ifcer than any of the jbrmer Series, 
and befides has the advantage ofy^encrea:ling in Arithmetical 
progreffion, which would be of greateafe, if any fliould un- 
dertake ile novo to m^ke the Logarithm Tangents^ or the Me- 
ridian Line to many more places than now we hire them. 
The Logarithm Tangent to the Arch of 45- + f ^ being no 
other than the aforefaid Series A i A* -^t^ , &c. in 
JSlapeirs form, or the lame multiplied into 0,43429, &c. for 
Briggis. 
But becauie all thele Series towards the latter end of the 
Quadrant do converge exceeding flowly, fo as to render 
this Method almoft ufelefs , or at leaft very tedious. It 
will be convenient to apply fome other Arts, by alTuming 
the Secants of fome intermediate Latitudes 5 and you may for 
J or the Sine of <t the Arch of half the difference of Lati- 
tudes, fubftitute — i 6t' +Tis>t» — T»V + Tmp>*% &c. 
according to Mr. JSIe-wton's Rule for giving the Sine from the 
Arch ; And if ctbe no more than a degree, a very few 
fteps will fuffice for all the accuracy that can be defired. 
And if A be commenfurable to a, that is, if it be a certain 
number of thofe Arches which you make your Integer ^ 
then will — be that number : which if we call ». the parts 
of the Meridional Line will be found to be» 
fn 
