VOL. LXVI.] PHILOSOPHICAL TRANSACTIONS. 87 



for the tangent of the diiference between the arcs will become respectively 



T-+T' sTT' 4+T' '5~Hrf' ^'^^ ' ^" ^'^^"^ '^ '^ ^^ expounded by any given 

 number, then these expressions will give the tangent of the difference of the 

 arcs in known numbers, according to the values of t, severally assumed res- 

 pectively. And if, in the first place, x be equal to 1, the tangent of 43°, the 

 foregoing expressions will give the tangent of an arc, which is equal to the dif- 

 ference between that of 43° and the first arc ; or that of which the tangent is 

 one of the numbers i, ■}■, -l, ±, &c. Then if the tangent of this diflTerence, 

 just now found, be taken for x, the same expressions will give the tangent of an 

 arc, which is equal to the difference between the arc of 45° and the double of 

 the first arc. Again, if for t we take the tangent of this last found difference, 

 then the foregoing expressions will give the tangent of an arc, equal to the dif- 

 ference between that of 45° and the triple of the first arc. And again taking 

 this last found tangent for x, the same theorem will produce the tangent of an 

 arc equal to the difference between that of 45° and the quadruple of the first arc; 

 and so on, always taking for x the tangent last found, the same expressions will 

 give the tangent of the difference between the arc of 45° and the next greater 

 multiple of the first arc ; or that of which the tangent was at first assumed equal 

 to one of the small numbers J, i, J-, -l, &c. This operation, being continued 

 till some of the expressions give such a fit, small, and simple fraction as is re- 

 quired, is then at an end, for we have then found two such small tangents as 

 were required, viz. the tangent last found, and the tangent first assumed. 



Here follow the several operations adapted to the several values of t. The 

 letters a, b, c, d, &c. denote the several successive tangents. 



] . Take t = i, then the theorem gives a = -, 6 = — -. Therefore 



the arc of 45° or -l of the circumference, is either equal to the sum of the two 

 arcs of which -^ and ^ are the tangents, or to the difference between the arc of 

 which the tangent is 4-, and the double of the arc of which the tangent is -i ; 

 that is, putting a = the arc of 45°, then 



The former of these values of a is the same with that before-mentioned, as o-iven 

 by M. Euler; but the latter is much better, as the powers of ^^ converge much 

 faster than those of J . 



Corol. From double the former of these values of a subtracting the latter 

 the remainder is, 



A = 3 X (1-^:5+195-7-95 &c.) -I- - X (1 - 3— + ^. - ^ &c.). 

 which is a much better theorem than either of the former. 



