ax - ab = bx
(a-b)x = ab
x = ab/(a-b).
Fig. 153.
137. The Relative Distance of a Superior Planet.—Let S, e, and m, in Fig. 153, represent the relative positions of the sun, the earth, and Mars, when the latter planet is in opposition. Let E and M represent the relative positions of the earth and Mars the day after opposition. At the first observation Mars will be seen in the direction emA, and at the second observation in the direction EMA.
But the fixed stars are so distant, that if a line, eA, were drawn to a fixed star at the first observation, and a line, EB, drawn from the earth to the same fixed star at the second observation, these two lines would be sensibly parallel; that is, the fixed star would be seen in the direction of the line eA at the first observation, and in the direction of the line EB, parallel to eA, at the second observation. But if Mars were seen in the direction of the fixed star at the first observation, it would appear back, or west, of that star at the second observation by the angular distance BEA; that is, the planet would have retrograded that angular distance. Now, this retrogression of Mars during one day, at the time of opposition, can be measured directly by observation. This measurement gives us the value of the angle BEA; but we know the rate at which both the earth and Mars are moving in their orbits, and from this we can easily find the angular distance passed over by each in one day. This gives us the angles ESA and MSA. We can now find the relative length of the lines MS and ES (which represent the distances of Mars and of the earth from the sun), both by construction and by trigonometrical computation.
Since EB and eA are parallel, the angle EAS is equal to BEA.
SEA = 180° - (ESA + EAS)
ESM = ESA - MSA