J 
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Converging Series for the Circle. 481 
fome multiple of thofe arcs fhall differ from an arc of 
45% tlie tangent of which is equal to the radius, by other 
fmall arcs, which alfo fhall have tangents denoted by 
other fuch fmall and limple vulgar fractions. For it is evi- 
dent, that if Inch a fmall arc can he found, fome multiple 
of which has fuch a propofed difference from an arc of 
45 0 , then the lengths of thefe two fmall arcs will beeafily 
computed from the general feries, becaufeof the fmallnefs 
and fimplicity of their tangents ; after which', if the pro- 
per multiple of the firft arc be increafed or diminifhed 
by the other arc, the refult will be the length of an arc 
of 45 0 , or |th of the circumference. And the manner in 
which I difcover fuch arcs is thus : 
Let t, t, denote any two tangents, of which t is the 
greater, and t the lefs ; then it is known, that the tangent 
of the difference of the correfponding arcs is equal to~qq* 
Hence, if /, the tangent of the fmaller arc, be fuccef- 
fively denoted by each of the fimple fractions q, ~ y . 
&c. the general expreffion for the tangent of the 
difference between the arcs will become refpe&ively 
7^7’ TFT’ ^77’ "f+T’ f° that T ke expounded 
by any given number, then thefe expreffions will give 
the tangent of the difference of the arcs in known num- 
bers, according to the values of /, feverally affumed re- 
flectively. And if, in the firft place, t be equal to 1, the 
tangent of 45 °, the foregoing expreffions will give the 
tangent of an arc, which is equal to the difference be- 
tween that of 45 0 and the firft arc; or that, of which the 
5 tangent 
