[ 27 7 ] 
no occafion for the repetition thereof here. Note, 
The above tradt of Dr. Halley’s is printed with the 
explication of Sherwin’s tables of logarithms (fol. io) 
with many examples of the ufe thereof annexed. 
Secondly, The terms of one of the bed: feries, for 
the re&ification of the circle, are compofed of the 
odd powers of the tangent of any arc, not exceeding 
45- degrees, feverally divided by their refpeCtive in- 
dexes ) of which the fird, third, fifth, &c. terms are 
affirmative j and the fecond, fourth, fixth, &c. terms 
are negative ; and the difference, between the fums of 
the affirmative and negative terms, is the length of that 
arc, of which the tangent, and its powers, conffitute 
the feries. 
Now a mathematician, who under dands the na- 
ture and management of feries, altho’ wholly igno- 
rant of duxions, might inveftigate this feries in the 
following manner : 
It has been geometrically demondrated, that, the 
radius of a circle being unity, if the double of the 
tangent of any arc, be divided by the difference be- 
tween unity, and the fquare of that tangent, the 
quotient will be the tangent of twice the arc. 
Now if an infinite feries be formed by aCtual di- 
vifion, its terms will confid of the doubles of the 
odd powers of the tangent, and will be all affirma- 
tive 5 which feries will exprefs the length of the tan- 
gent of the double of that arc, whofe tangent and its 
powers conditute the fame. 
If a feries, confiding of the tangent and its powers, 
with unknown coefficients, be affumed (as in the for- 
mer cafe) to exprefs the length of the arc ; then the 
length of the double of that arc may be expreded 
two ways ; viz. either by multiplying each term of 
3 the 
