VOL. LVr.] PHILOSOl'HICAL TRANSACTIONS. 945 



p sin. (4^A + -T^) sill- (-5-A — -1-d), if a >■ d, and hence we may easily compute 

 sin./) = sin. p sin. a cos. e; but if A < d, we shall find cos. f^ = 4 sin. p sin. 

 (iA + iD) Sin. (4.A — iD) ; and hence sm. p = — j-— _. 



6. Similar formulae may be found for cos. p. But as the angle p is pretty 

 small, we might easily fall into sOme erfor by the usual tables of logarithms. 

 He does not say what would be the amount of this error of p, having furnished 

 the manner of avoiding it ; but this remark has not, he thinks, as yet been made 

 in astronomical calculations; and he has found it of great consequence in com- 

 puting eclipses, where the distances to be found are very small arches. 



7. It may also be observed, that if a = d, sin. p = sin. p sin. A; hence p' ^= p 

 in the same case, and p" > p, which seems very odd; but the moon then is be- 

 low the sensible horizon. ;, !..,^ 



Theory of the apparent Diameters of the Moon — 1. First the expression of 

 horizontal diameter of the moon, or of the diameter seen at the horizon, seems 

 to Mr. M. too vague; for we ought to understand by it the diameter seen at the 

 centre of the terrestrial sphere, rather than the apparent diameter at the horizon, 

 which is not affected by refraction. Without this, if the one was confounded 

 with the other, an error would arise for the latitude of Paris from 0''.25 to 0".32. 



2. Let us keep the same denominations of p, p, and a, and call d the apparent 

 semidiameter of the moon at the centre of the -sphere, and d= the apparent 

 semidiameter of the moon at the zenith distance = a. We shall have sin. a : 

 sin. (a -H P) :: tang, d : tang, d, or if we will, sin. a : sin. (a +/>):: d : rf: the 

 error not exceeding an 100th part of a second. ,1; ,v, :<. 



3. We had above sin. p = sin. p sin. (a -f p). Hence sin. p sin. a : sin. p :: 

 (tang. D -. tang, d) :: d : d, or because sin. p = 



£2!JL!L"iIi!llii, it is 1 - sin. p cos. a : cos. p :: d : d, and d = ^'^ . 



1 — gin. P COS. A ■* I — sin. p cos. a 



4. Mr. Euler, in the Memoirs of the Academy of Berlin, 1747, p. 175, 

 makes this same value = ■ ■ ■ . , and aixording to him, v = d. m = sin. ?■ 

 sin. h = COS. {\ -\- p); whence it appears that the true value of the apparent 



-diameter of the moon, is not more difficult to be computed than the approxi- 

 mate one of Mr. Euler, the exact and geometrical formula being tang, d = 

 Sr°S^. -1 "-' "f Mr. EuW d = ,W;-4-,;^^; for in both, the 

 values of D, a and p must be employed. 



5. It also appears to Mr. M. that since 



cos p _ s,n.p ^^j therefore tang, d = ^='"g-°""-P 



sin. A C) . i;;r. T. oir. . » 



SI 



astronomers 



1— sin. P COS. A sin. p sin. A ° sin. p sin. a 



ought no less to employ this last formula, than any other more troublesome, in 



practical computation. The simplest is tang, d = — ^iPi^"" ^* '*' ^ , on the sup- 

 position of an exact table of the parallaxes of altitudes ready made; and he be- 



VOL. XII. Y \ 



