( xp8 ) 
f^tio of Radius to luch Tangent will be 
^^into t {tt + im — Jf/fr + &c. 
wh^ji the, arch is greater than 45^', or 
^into-r+ \tt + -f ij-^ + &c. 
vvhen it is lefs than,4jS'. And by the Qme do^^rine put- 
ting T for the Tangent of any arch, and t for the difFerence 
thereof from the Tangent of another arch, the LogaiiLhm 
qf their ratio will be 
I . t . tt , t* . t^ ■ 
- into -f +^j--h + , + y-y.^ &c. 
when T is. the greater Term^ or 
I - . t t t , t ^ t"^ - t ^ 
■^ into - + _ +yy, ,6cc. 
when T is the leffer Term ; 
And if be.fuppofed .0002908^82, &c. = its reci- 
P^'^^^^^ will be, 3437,74677o78493925'26, &c. which nval- 
tiplied into the aforefaid 5'm>/, fliall give precifely the diffe- 
rence of Meridional parts, between the, two Latitudes to 
whofe half com.plements the alTumed Tangents "belong. 
Nor is it material from whether Pole. you, eftimate the Com- 
plements, whether the .elevated or depreffed ; the Tangents 
being, to one another in. the fame r/??fo as their Comple- 
ments, but inverted. 
In the fame Difcourfe I alfo fiiewed that the Series .might 
be made to converge twice as fwifc,all the even Powers being 
omitted : and that putting t for thefum of the two. Tan- 
gents the ftme Logarithm would be. 
m a T ' 3 ^ 5* T* ' 7t^ ' 9T^ > 
but the rjfif of t to or of thefum of two Tangents to 
their difference, is the fanie as that of the fine of the fum 
of the. arches, tothe /;;e of their difference. Wherefore if 
S be put for the /we Complement of the Middle Latitude, 
aa^ J for the of half the difference of Latitudes , the. 
Series will be 
