of the Heptagon, 

 let X . . . 6 be six auxiliary angles, such that 



127 



r' = 2rtan0 1 , 



y=psin2<£ 2 , p — # = rtan 2 £ 



p — q + r=lp tan 2cj) 4} s = r tan 20 5 , r{r n — r) = r , r" sin0 6 ; 



we shall then 1 have the following system of equations, to which 

 are annexed the angular values, deduced by interpolation from 

 Taylor's seven-figure logarithms, only eleven openings of which 

 are required, if the logarithms of two and four be remembered, 

 as they cannot fail to be by every calculator. 



IT 



cot0j =2cos — 







^ = 31° 43 2-91 



sin20 2 = cos 2 x 







2 = 23 10 35-52 



7T 



tan 2 3 = 4 sin 2 2 tan — 







03=33 51 31-90 



7T 



cot 20 4 = cos 2 3 tan - 







4 =31 41 39-37 



tan 20 5 = 2 sin 2 4 sec 20 4 



tan 



7T 



5 



5 =2O 55 15-93 



sin 6 = sm 2 (j) 5 cot(f) l 







6 =11 54 22-60 



_cos 2 <p = cos 6 sec 20 5 



tan 



*i 



=25 42 51-4. 



It is useless to attempt to estimate hundredths of seconds in 

 this last value, because the difference for a second, in the last 

 logarithmic cosine, amounts only to ten units in the seventh 

 place of decimals, or to one in the sixth place. But if we thus 

 confine ourselves to tenths of seconds, a simple division gives 

 immediately that final value, under the form 



^ = !^_° = 25 42'51"-4; 



7 7 



it appears therefore to be difficult, if it be possible, to decide by 

 Taylor's tables, whether the equations (B), deduced from Roberts 

 construction, give a value of the angle 0, which is greater or less 

 than the seventh part of two right angles. (It may be noted that 

 tan 20j = 2; but that to take out X by this equation would 

 require another opening of the tables.) 



5. To fix then decisively the direction of the error of the ap- 

 proximation, and to form with any exactness an estimate of its 

 amount, or even to prove quite satisfactorily by calculation that 

 any such error exists, it becomes necessary to fall back upon 

 arithmetic ; and to carry at least the first extractions to several 

 more places of decimals, — although fewer than those which have 

 been actually used in the resumed computation might have suf- 

 ficed, except for the extreme accuracy aimed at in the resulting 



