Triangles in the Sky: Trigonometry and Early Theories of Planetary Motion - Determining r/R

In addition to the periods of the planets, ancient astronomers observed two other gross features of their motions—the lengths and times (durations) of their retrograde arcs. For a single planet the length and time of its retrograde arcs vary somewhat; the following table gives approximate average values that appear to be close to those known to Ptolemy [18].

Table 2

Revisiting Figure 6, we see that fixing T and S and varying the ratio r/R changes the lengths of the retrograde arcs. Larger values of r/R result in larger retrograde arcs, while smaller ones result in smaller arcs or sometimes none at all. This suggests that data like that found in Table 2 might be used to determine the ratio r/R.

According to [18], ancient astronomers may have used such data, plus a little trigonometry, to find an initial estimate of the ratio r/R for each planet. For the outer planets—Mars, Jupiter, and Saturn—this method of determining r/R is somewhat similar to the method actually used by Ptolemy in the Almagest. For the inner planets—Mercury and Venus—Ptolemy in the Almagest computed this parameter by a quite different method which involved using the planet’s so-called “greatest elongation” from the sun.

To compute r/R by using the data in Table 2, we start at a time when the epicycle center, the planet, and the earth are all in a straight line, with the planet P at the point P[0] on the epicycle (see Figure 7).

Figure 7

In the figure, C₀ is the location of the epicycle center at this time and A₀ is the location of the apparent planet on the ecliptic. At this point the planet is in the middle of its retrograde arc. Animating Figure 7 shows the planet’s motion from the middle of the retrograde arc until the end. At that point C represents the new location of the epicycle center and A represents the new location of the apparent planet. Trigonometry now enters the picture, as r is side CP and R is side EC of triangle ECP; thus to find the ratio r/R, we must solve the triangle ECP for the ratio of its sides CP/EC.

Since the planet moves uniformly on the epicycle, the arc P₀P has measure equal to the synodic rate of the planet (see Table 1 in Section 6) multiplied by half the time of its retrograde arc given in Table 2 above. Since arc P₀P and angle P₀CP have the same measure, this determines angle C in triangle ECP. Similarly, since the epicycle center moves uniformly on the deferent, arc C₀C has measure equal to the sidereal rate multiplied by half the time of the retrograde arc. This determines angle C₀EC which forms part of angle E in triangle ECP. Finally, arc A₀A is by definition half of the planet’s retrograde arc; thus its measure is known from Table 2 above. This determines angle A₀EP; adding this to angle C₀EC gives angle E in triangle ECP. Altogether we have found angles C and E in triangle ECP; subtracting their sum from 180° determines angle P. Using the Law of Sines we can now solve the triangle for the ratio CP/EC.

As an illustration of this method we show the computations for the planet Mars, using modern trigonometry and notation.

Example: Mars

From Table 2, we get the retrograde arc data: (1/2)length = 8° and (1/2)time = 36 days.

From Table 1, we find that the synodic rate is 0.462°/day and the sidereal rate is 0.524°/day.

Angle ECP = arc P₀P = (synodic rate) x (1/2)time = 16.6°.

Angle C₀EC = arc C₀C = (sidereal rate) x (1/2)time = 18.9°.

Angle A₀EP = arc A₀A = (1/2)length = 8°.

Angle PEC = the sum of angles C₀EC and A₀EP = 26.9°.

Angle CPE = 180° minus the sum of angles PEC and ECP = 136.5°.

By the Law of Sines, r/R = CP/EC = sine(PEC)/sine(CPE) = 0.66.

Exericise: Use the data in Tables 1 and 2 to find r/R for Mercury, Venus, Jupiter, and Saturn. (Answers are given in Table 3 of Section 8.)