Mr. hutton on 
482 
tangent is one of tlie numbers y, j, j, &c. Then if the 
tangent of this difference, juft now found, be taken for t, 
the fame expreflions. will give the tangent of an arc, 
which is equal to the difference between the arc of 45 0 
and the double of the firft arc. Again, if for t we take 
the tangent of this laft found difference, then the fore- 
going expreflions will give the tangent of an arc, equal 
to the difference between that of 45 0 and the triple of 
the firft arc. And again taking this laft found tangent 
for t, the fame theorem will produce the tangent of an 
arc equal to the difference between that of 45 0 and the 
quadruple of the firft arc; and fo on, always taking for t 
the tangent laft found, the fame expreffions will give the 
tangent of the difference between the arc of 45 0 and the 
next greater multiple of the firft arc ; or that, of which 
the tangent was at firft affirmed equal to one of the finall 
numbers j, j, 8cc. This operation, being continued 
till fome of the expreffions give fuch a fit, fmall, and fim- 
ple fraction as is required, is then at an end, for we have 
then found two fuch fmall tangents as were required, 
viz. the tangent laft found, and the tangent firft affumed. 
Here follow the feveral operations adapted to the fe- 
veral values of t . The letters a , c , d, See. denote the 
feveral fucceffive tangents. 
07 1 
1 . T ake t - then the theorem — ; — gives 
zJ 2 + T o 
I 
Therefore the arc of 45 0 , or |th of the circumference, is 
either 
