of the Heptagon* 129 



Admitting then the known value, 



7r= 3-14159 26535 89793 . . ., 

 or the deduced expression, 



lff = «-rL-n =0-00000 48481 36811 095 . . ., 



I infer as follows 



PM 



^-^ = + 0"-12653 31307 822, 

 y -<f> = + 0"-03615 23230 806, 



^ = 25° 42' 51"-39241 91054 91, 



and think that these twelve decimals of a second, in the value of 

 the angle </>, may all be relied on, from the care which has been 

 taken in the calculations. 



6. The following is a quite different way, as regards the few 

 last steps, of deducing the same ultimate numerical results. 

 Admitting (comp. Art. 3) the value*, 



2 cos ~ =*= 1-24697 96037 17467 06105, 



as the positive root, computed by Horner's method, of the cubic 

 equation 



^ + ^_ 2 ^-l=0, 



and employing the lately calculated value/ of 1 + cos 2$, I find 

 by square roots the following sines and cosines, with the same 

 resulting error of the angle ijfr as before : 



sin £ = \s/H^z = 0-43388 37391 17558 1205 ; 



7 A 



cos^ = l«y2 + ~z = 0-90096 88679 024191262; 



7 A 



1 



(B) J 



sin<£ = W4-2/ = 0-43388 35812 03469 1138; 

 cos </> = \ \/2f= \g = 0-90096 89439 49819 2956 

 sin(y -$\ = +0-00000 01752 71408 3339; 



- -6= +0"-03615 23230 806. 



7 r 



* Compare a preceding note. 

 Phil Mag. S. 4. Vol. 27. No. 180. Feb. 1864. K 



