VOL. XI.] PHILOSOPHICAL TRANSACTIONS. 329 



opposition to the sun, be observed ; for then the sun and the earth are in the 

 same right line with Mars, or, which always is the case when it lias latitude, 

 with that point where a perpendicular let fall from Mars meets the plane of the 

 ecliptic. Thus, in the scheme, S, A and P are in a right line ; then after 68/ 

 days Mars returns to the same point P, where it was in opposition to the sun 

 in the former observation ; but since the earth does not return to A till after 

 7304- days, in B it respects the sun in the line SB, and Mars in the line BP; 

 and observing the longitudes of the sun and of Mars, all the angles of the tri- 

 angle PBS are given ; and supposing PS 100000 parts, the length of the line 

 SB is found in the same parts; in the same manner, after a second period of 

 Mars, the earth being in C, the line SC is found, and in like manner the lines 

 SD, SE, SF; and the differences of the observed places of the sun, are the 

 angles at the sun ASB, BSC, CSD, DSE: and thus at length we come to 

 this geometrical problem, viz. Three lines meeting in one of the foci of the 

 ellipsis being given, both in length and position, it is required to find the length 

 of the transverse diameter, together with the distance of the foci : the resolu- 

 tion of which is also extended to the other planets, if, after knowing the theory 

 of the earth's motion, we investigate, according to the method proposed by Dr. 

 Ward, Bishop of Sarum, in his Astronomiae Geometrica, lib. 2, part 2, cap. 5, 

 three distances of any planet from the sun in their positions. But because the 

 doctor supposes a planet to move in its orbit in such a manner, as in equal times 

 to describe equal angles about the other focus, and upon this builds his calcu- 

 lation ; it does not seem improper to show how the same thing may be done 

 without supposition, that which is inconsistent with observation. 



Let S, %. 7, be the sun; ALBK the orbit of the earth; and let P be the 

 planet, or the point in the plane of the ecliptic, where a perpendicular from the 

 planet meets it; let AB be the line of the apses of the earth's orbit. In the first 

 place, let the longitude and latitude of the planet in P be observed, together 

 with the longitude of the sun from the earth in K; and after a revolution of the 

 same planet, the earth being in L, let again the positions of the planet and of 

 the sun be observed as before. Now from the observed longitudes of the sun, 

 and aphelion of the earth, the angle ASK, ASL are given, and consequently the 

 sides SK, andSL: for if the angle of the co-equated anomaly be acute, the 

 proportion is, as the difference of the mean distance and of the cosine of the 

 angle multiplied into the eccentricity, is to the aphelion distance, so is the 

 parhelion distance, to the distance of the planet from the sun in the given ano- 

 maly; but if the angle be obtuse, the first term of the proportion is the sum of 

 the two parts, as it was their difference in the former analogy: now in the 

 triangle KSL, the sides KS, LS, and the angle KSL are given; to find the side 



VOL. II. U u 



