346 PHILOSOPHICAL THAWSACTIGNS. [aNNO I766. 



lieves it will be as easy to compute with tangents as with arches, by means of 

 logarithms; and therefore this simplification in putting arches instead of tangents 

 is unnecessary. 



6. To try the consequences of this theory, he made A = 30°, d = 15', and 

 taking the vertical of Upsal to the terrestrial axis for the radius of the sphere, 

 he found p = 55' 10".3, supposing that the axis of the earth, is to the diameter 



of the equator, as ] 99 to 200, and by the formulae tang, c/ = . 



tang.D8in.(A+p) ^ Jang^co^^ ^ ^ng^n^ ^^ ^^^^^ ^^^^, ^^„ ^^ ^^ 

 sin. A 1 — sin. p COS. a sin. p sin. a ' 



by the formula d = ?i2»^ — j^g had rf = 15' 11". 67 5% and lastly by that 



•' 1 — sm P COS. A J J 



of Euler d = : ; — ; — ; we have d=z\b' 12*.635; whence it appears 



1 — sin. p cos, (a + ;>) ' ~ 



that the error is very small, but that with the same trouble one may avoid any 

 error whatever. 



7. The present case did not give an error of ©".OOl in substituting 1 or the 



radius instead of cos. p. Hence he concludes that d =. : will be a 



' 1 — sin. p COS. A 



more exact formula than that of Euler d = •; — -: — ~ ; — - — r. 



I — sin. p COS. (A + p) 

 _ , . , D i_ J D sin. p COS. A 



8. By takuiff d = : , we have d — d = : = 



■' o 1 — sin. p COS. A 1 — sm. p cos. a 



Dsin.ycos^ _ i> tang, p ^^^^^ ^^^^^^ ^^ elegant theorem, to find the increase 



sin. A cos. p tang, a ° 



of the apparent diameter of the moon. 



9. He found others by the following methods. Since sin. A : sin. (a + p) :: 



tang. D : tang, d, and sin. A : sin. (a + />) — sin. a :: tang, d : tang, d — tang, d 



:: sin. d cos. d : sin. (d — d)-, but cos. d = cos. d without any sensible error, and 



sin. D cos. D = i sin. 1 d, and sin. (a + />) — sin- a = 2 sin. -^p cos. (a + 4-J&), 



, „, • /J \ sin. 2 Dsin. J;>cos. (a + Jp) , , 

 we shall have sm. (d — d) = -^^^ — . In the same manner, 



as he before found sin. p' = sin. p sin. a, and sin. p sin. a : sin./) :: tang, d : tang. 

 d, hence sin. />' : sin. j& — sin. />' :: sin. p' : 2 sin. {-^p — ^ p') cos. {^p -f -j. />) :: 



• ^ ■ tJ ^\ sin. 2 D (^p - Ip') cos. (If 4- Ip) 

 i. sm. 2 D : sin. (d-Ti)= -^.l^^A • 



10. Lastly let l = the distance of the moon from the centre of the sphere, 

 / its radius, that of the sphere being = i, we have i : l :: sin. p : i, and l : / :: i : 

 tang. D or I : / :: sin. p : tang, d = / sin. p; hence / = -^~— being once found, 

 since sin. a : sin. (a + />) :: tang, d : tang, d, and sin. (a + />) : sin./> :: i : sin. p, 

 we shall have sin. a : sin. p :: tang, d : sin. p tang. d:\l: tang, d = -^/- He 

 found the logarithm of / = 9.4343965 at Upsal, by putting 10 for that of the 

 radius of the sphere determined as before. 



