Figure 1.27

How Copernicus  calculated the distance to the planets. (A) When an inner planet appears in the sky at its farthest point from the Sun, the planet's angle on the sky away from the Sun, A, can be measured. You can see from the figure that at the same time angle B is 90°. The planet's distance from the Sun can then be calculated with geometry, if one knows the measured value of angle A and the fact that the Earth-Sun distance is 1 AU.
(B) Finding the distance to an outer planet requires determining how long it takes the planet to move from being opposite the Sun in the sky ( the planet rises at sunset) to when the Sun-Earth-planet angle is 90° (the planet rises at noon or midnight). Knowing that time interval, one then calculates what fraction of their orbits the Earth and planet moved in that time. Multiplying those fractions by 360° gives the angles the planet and Earth moved. The difference between those angles gives angle B. Finally, using geometry and the value of angle B as just determined, the planet's distance from the Sun can be calculated.