322 fHILOSOPHICAL TRANSACTIONS. [ANNO 1708. 



planet, whose radius cb is equal to the horizontal parallax of the planet from the 

 sun; and let ab be the projection of the celestial meridian, in which is either the 

 north pole p, or the south p, according as the conjunction of the planet may be 

 at y or at Si . Also let c be the common centre of projection of the earth and 

 sun viewed from the earth's centre; and l the projection of any given place. Put 

 the latitude of the place l = l ; the complement of the sun's declination cp or 

 c/; = D; the horary angle cpl or cpi. ■= a; radius = 1; the sun's altitude at 

 the given place and time = c ; and the cos. A. cot. l = tang, g; then will sin. 



c = — '—^ — "— ~ ~ (i), where the lower signs are to be taken when d -< g, 

 otherwise the upper signs, except when a >■ 90°, when the sign — of g becomes 

 +, and consequently the sum of d and G is to be taken. Lastly put the paral- 

 lactic angle pcl or /jcl = a, then will sin. a = "" •**^° — (u). 



§ III. As to CL, it will be at least nearly equal to the parallax of the planet's 

 altitude from the sun (§ i), unless the difference of the altitudes of the centres of 

 the sun and planet be neglected ; in which case the ratio of this difference is to 

 be had, as the parallax is a little different from cl. Yet, even in this case, the 

 parallax of the planet's altitude from the sun, may be estimated in cl, without 

 sensible error. For this end, the difference of the altitudes of the sun's centre 

 and the star is now to be found. Therefore let ab, fig. 9, be the celestial me- 

 ridian ; PM parallel to the equator; zn, zn vertical circles corresponding to the 

 place and time; l/, any given places of the planet before and after the ascensional 

 conjunction ; sr, sr, the differences in right ascension of the sun's centre s and 

 the planet, which call a; and lr, Ir differences of declination, which call d; also 

 let the angle made by a parallel of the equator, and the line joining the sun's 

 centre and planet, viz. the angle lsr or /sr, be called f; then will the tangent 



p = -, and SL or s/ = -. the distance of the centres of the sun and star. If 



a sin. F 



now, from l, /, the perp. ld, Id be drawn to the vertical circles zn, zn, then sd 

 or ?,d will be the difference of altitude sought, which call e ; to find which, we 

 have the formula e = ' ''"'. ^ ^ ~ '^' (in), in which t will be given by d and a, 



sin. V ^ '' a J 



which are easily derived from the observations, and a from the equation (11). 

 With regard to the signs, observe the fpllowing rules: viz. the sign -j- takes 

 place when the observation is made before noon and before the conjunction, or 

 after noon and after the conjunction in right ascension: but the sign — , on the 

 contrary, is used in observations before noon and after conjunction, or after noon 

 and before the same conjunction. These signs are to be changed, using the 

 sign — in the former, and -|- in the latter case of the rule, when the planet's 

 path is to the north of the sun's centre, as in the next transit of Venus. There 

 will then be now given, for the given place and time, by the equations (i) and 



