Adams. — On Natural Sines. 203 



A 5 sm 9° = A 5 sin 0° + A 6 sin + 0-00008, 11208, 05979, 42968, 24092, 



56 

 A 7 sin 0° = - k . A-> sin 9° - 0-00000, 19974, 63467, 73330, 64528, 



5908 

 A 6 sin 9° = A 6 sin 0° + A 7 sin C - 0-00000, 87752, 2S818, 20181, 30342, 



811 

 A 8 sin 0° = - k . A 6 sin 9° = + 0-00000, 02160, 75256, 81886, 56155, 



99392 

 A 7 sin 9° = A 7 sin 0° + A 8 sin L - 0-00000, 17813, 88210, 91444, 08372, 



5969 

 A y sin 0° = - k . A 7 sin 9° = + 0-00000, 00438, 63689, 84123, 11107, 



46185 

 A 8 sin 9° = a 8 sin 0° + A 9 siu r -- + 0-00000, 02599, 38946, 66009, 67263, 



45577 

 a 10 sin 0° = - k . a 8 sin 9 C - 0-00000, 00064, 00559, 55467, 55455, 



96711, 3 



This exhibits the complete working necessary to obtain the 

 leading differences up to A lu sin 0°. As will be seen from the 

 schedule above, the only operations are addition and multi- 

 plication. Each multiplication was done to the full extent as 

 shown above on the Brunsviga calculating-machine without 

 any intermediate record, and each multiplication was checked 

 by doing it in duplicate : thus, to obtain k . sin 9°, k was first 

 set on the machine and multiplied by sin 9°. then sin 9° was 

 set on the machine and multiplied by k, and no result was 

 accepted unless every figure to the last agreed in each case. 



Having now obtained the leading differences and the initial 

 term (sin 0°), the table is formed in the usual way, with the 

 following results : — 



Sines. Corrections. 



0° 0-00000, 00000, 00000, 00000, 0000 



9° 0-15643, 44650, 40230, 86901, 0105 



18° 0-30901, 69943, 74947, 42410, 2293 



27° 0-45399, 04997, 39546, 79156, 0408 



36° 0-58778, 52522, 92473, 12916, 8706 



45° 0-70710, 67811. 86547, 52440, 0845 - 1 



54° 0-80901, 69943, 74947, 42410, 2295 - 2 



63° 0-89100, 65241, 88367, 86235, 9712 - 3 



72° 0-95105, 65162, 95153, 57211, 6443 - 4 



81° 098768, 83405, 95137, 72619, 0045 - 5 



90° 1-00000, 00000, 00000, 00000, 0006 - 6 



The last value is 6 in excess in the twenty-fourth decimal 

 place ; comparing the value of sin 81° with cos 9° as found 

 direct from the series it is seen that this value is 5 in excess ; 

 and comparing sin 54° with sin 18° the former is 2 in excess : 

 hence it seems reasonable to adjust these values by deducting 

 1, 2, 3, 4, 5, and 6 from sin 45°, sin 54°, sin 63°, sin 72°, 

 sin 81°, and sin 90°. 



