290 DEVELOPMENT' of a certain 



19. 1 sh al L conclude this appendix with an example of the ap- 

 plication of our formula; and in the determination of the loga- 

 rithms of e, e° t &c. 1 -f <?', 1 -f e" y &c. by means of the trigonome- 

 trical tables, I mall avail myfelf of the remarks which have been 

 made for that purpofe at Art. 15. of the preceding paper. As 

 to the quantities fin 2<p\ lin 4^, &c. it is eafy to fee, from their 

 relation to each other, that their logarithms may be readily de- 

 termined by means of the fame tables. 



Example. It is required to determine the length of an el- 

 liptic arch, when e, the excentricity, =v ^ — .7071068, and <p f 



the arch of the circumfcribing circle zz 70 °, the femi-tranfverfe 

 being fuppofed zz 1. 



1. Calculation of the logarithms of e\ 1 4- ^', &c. 



Logarithms. 



e zz 



fin 



a = fin 45 , 0, 0", 



1.8494850 



e = tan— z= 

 2 



fin 



a' = fin 9% 52', 45^.41. 



1.2344486 



1 -f- <?' = fee - 







0.0687694 



• =tan- = 

 2 



fin 



ufzz fin o°, 25'. 40^.7 



3.8733008 



l + /=fec^ 





_ 



0.0032320 



*'" = tan — 

 2 





. 



5- I 444734 



i-fV"=fedf 





- 



0.0000060 



Because e xv is nearly equal to one-fourth of the fquare of 

 c'" t (Art. 7.) it is evident that e lv , 1 + *»v &c. may be all ne- 

 glected. 



2. Calculation 



