Having completed the preliminary models for the inferior planets, Mercury and Venus, Ptolemy now turns his attention towards the superior ones: Mars, Jupiter, and Saturn.
We’ve already covered this topic some back in Book IX. There Ptolemy told us that their models are the same as the one we just explored for Venus, having an equant about which the constant angular speed takes place, and a separate center around which the center of the epicycle will maintain a fixed distance.
He affirms that this center of motion will again be half way between the earth and the equant and tells us that this is affirmed because
the eccentricity one finds from the greatest equation of ecliptic anomaly turns out to be about twice that derived from the size of the retrograde arcs at greatest and least distances of the epicycle.
Presumably, we’ll be going through the necessary calculations to demonstrate this shortly. But before we do, Ptolemy issues a warning:
However, the demonstrates by which we calculate the amounts of both anomalies and [the positions of] the apogees cannot proceed along the same lines for these [superior] planets as for the previous [inferior] two, since these [superior planets] reach every possible elongation from the sun and it is not obvious from observation, as it was from the greatest elongations for Mercury and Venus, when the planet is at the point where the line of our sight is tangent to the epicycle.
What Ptolemy is stating here is that our previous methods for Mercury and Venus largely relied on knowing that these planets were at greatest elongation. When that happens, the line of sight is tangent to the epicycle which is a fact we could exploit to perform our calculations.
But because there is no greatest elongation for these superior planets, we can’t do that in this case. So we will need to develop some new tricks.
So what will we use instead?
So, since that approach is not available, we have used observations of their oppositions to the mean position of the sun to demonstrate, first of all, the ratios of their eccentricities and [the positions of] their apogees. For only in such positions, considered from a theoretical point of view, do we find the ecliptic anomaly isolated, with no effect from the anomaly related to the sun.
So we’ll be using oppositions. To me, this feels a lot like how we used eclipses (which occur at conjunctions and oppositions) for the moon to calibrate that model.
To help us understand how we’ll proceed, Ptolemy produces some diagrams.
In this figure, we again have circle $ABG$ as the eccentre with center $D$ and diameter $\overline{AG}$ with the observer on earth at $E$ at the center of the ecliptic and $Z$ as the equant.
The epicycle is circle $H \Theta KL$ with center $B$.
We’ll then draw $\overline{HM}$ which goes through both $B$ and $E$ as well as $\overline{\Theta Z}$ which also goes through $B$.
Ptolemy now tells us about a new feature of this model:
when the planet is seen along $\overline{EH}$ through the epicycle $B$, then the mean position of the sun, too, will always be on the same line, and that when the planet is at $H$, it will be in conjunction with the mean sun (which will also, in theory, be seen towards $H$), and when the planet is at $K$ it will be in opposition to the mean sun (which will be seen, in theory, towards $M$).
What Ptolemy is doing here is again tying the motion of Jupiter to that of the sun.
If we step out of the Ptolemaic context for a moment and think about modern astronomy, it is quite clear why this should be the case. Opposition happens when the earth is directly between Jupiter and the sun. By and large, this is also when we are at our closest distance to Jupiter1.
Contrariwise, when Jupiter is at conjunction, it is opposite the sun from us and thus at its greatest distance.
While Ptolemy certainly couldn’t tell distance2, he and astronomers before him could certainly see that the superior planets were certainly brightest near opposition and dimmer the closer they got to conjunction3.
Translating this to Ptolemy’s model, this would mean that Jupiter and the other superior planets must be at point $K$ during opposition and at $H$ during conjunction.
Neugebauer states that a slightly more explicit form of this relationship was known to the Babylonian astronomers in which the number of occurrences [of a given phenomenon, such as opposition] plus the number of sidereal rotations of the planet is the number of years.
Ptolemy states this slightly differently:
the sum of the mean motions in longitude and anomaly, counted from the apogee [of eccentre and the epicycle respectively], equals the mean motion of the sun counted from the same staring-point.
Ptolemy explains:
[T]he difference between the angle at centre $Z$ [i.e., $\angle AZB$] (which comprises the mean motion of the planet in longitude), and the angle at $E$ [i.e., $\angle AEB$] (which comprises the apparent motion in longitude is always the angle at $B$4 (which comprises the mean motion on the epicycle).
Ptolemy now states this somewhat more mathematically:
[W]hen the planet is at $H$, it will fall short of a return to the apogee $\Theta$ by $\angle HB \Theta$; but $\angle HB \Theta$ added to $\angle AZB$ produces the angle comprising the sun’s mean motion, namely $\angle AEH$, which is the same as the apparent motion of the planet.
What Ptolemy is saying is that $\angle AZB + \angle HB \Theta = \angle AEH$.
However, Toomer notes that this is actually incorrect and should be $\angle AZB – \angle HB \Theta = \angle AEH$. He offers the following diagram to explain:
In figure $P1$, we start off with the planet at apogee, both of the eccentre and epicycle with the sun at conjunction5.
Then, a year and some time passes until the next conjunction, illustrated in $P2$ (which is the same as our diagram above but with the angles better drawn in). In that time, the eccentre has rotated $\overline{\kappa}$ while the sun has gone some distance $\kappa$. Meanwhile, the planet about the epicycle has gone $\alpha$.
In this context, the real equation should be that $\kappa + 360º = \overline{\kappa} + \alpha$6.
Toomer suggests that this error was not due to Ptolemy, but was caused by someone later inserting a phrase which muddied the reading.
Ptolemy goes on to give the situation for when the planet is instead at $K$:
And when the planet is at $K$, its motion on the epicycle, again, will be $\angle \Theta BK$, and $\angle \Theta BK + \angle AZB$ equals the mean motion of the sun counted from the apogee, $A$. Thus, the latter comprises $180º + (\angle AZB – \angle LBK) = 180º + \angle GEM$, i.e. the mean position of the sun will be opposite the apparent position of the planet.
Ptolemy then describes another consequence of these requirements: That the line defined by the mean sun as viewed from the earth, will always be parallel to that of the planet about the epicycle.
Or, as Ptolemy puts it:
[I]n such configurations [i.e., mean conjunctions and oppositions], the line joining the epicycle centre $B$ to the planet and the line from $E$, our point of view, to the mean sun, will coincide in one straight line, but at all other [sun-planet] elongations [these lines] will always be parallel to each other, although the direction in which they point will vary.
Ptolemy then offers a proof using the following diagram:
From this and the above requirements, Ptolemy states the following:
$$\angle AEX = \angle AZ \Theta + \angle NB \Theta$$
$$\angle AZ \Theta = \angle AEH + \angle HB \Theta.$$
The first of these can be solved for $\angle AZ \Theta$ and substituted into the second which gives:
$$\angle AEX = \angle AEH + \angle HB \Theta + \angle NB \Theta.$$
But note that $\angle AEX = \angle AEH + \angle HEX$. So we can substitute:
$$\angle AEH + \angle HEX = \angle AEH + \angle HB \Theta + \angle NB \Theta.$$
Then, we can subtract $\angle AEH$ from both sides to get:
$$\angle HEX = \angle HB \Theta + \angle NB \Theta.$$
Lastly, we can add the right side of the equation to get:
$$\angle HEX = \angle HBN$$
which proves that the two lines are parallel as we set out to demonstrate.
But what of it?
The trick of being able to determine the parameters for the epicycle is to know where on the epicycle the planet is. After all, with the exceptions of maximum anomaly, any line of sight to the planet will cross the epicycle twice. How are we to know if the planet is on the front point or the back?
With Mercury and Venus, we were able to exploit the fact that greatest anomaly is effectively the same thing as greatest elongation, and thus, an observation just before greatest elongation would be more clockwise than one just after.
However, because the superior planets aren’t tied to the elongation of the sun, we can’t use this trick. But what Ptolemy is demonstrating is that the motion about the epicycle is still tied to the sun in a way we can exploit.
Thus, we find that in the above configurations of conjunction and opposition with respect to the mean sun, the planet is viewed, in theory, [along the line] through the centre of the epicycle, just as if its motion on the epicycle did not exist, but instead it were itself situated on circle $ABG$ and were carried in uniform motion by $\overline{ZB}$, in the same way as the epicycle centre is. Hence, it is clear that it is possible to isolate and demonstrate the ratio of the ecliptic eccentricity by [both] such types of [planetary] positions, but since the conjunctions are not visible, we are left with the oppositions on which to build our demonstrations.
In short, conjunction puts the planet on the near side of the epicycle, opposition puts it on the far side. We can’t see conjunction, but we can see opposition. So that’s what we’ll make use of as we start our work on determining the models for the superior planets, starting with Mars.
- The non-circular orbits can complicate this a bit, but not too significantly.
- Although it would have made his job a lot easier if he could.
- Although they certainly couldn’t see them during conjunction since they would be lost in the daylight.
- Since we don’t have a defined point for the planet in Ptolemy’s diagram, we can’t explicitly define this angle.
- Toomer has opted to only draw the direction of the sun itself, and not its actual position which should be interior here since we’re talking about the superior planets.
- Toomer here is only counting $\kappa$ as the angle beyond a full revolution. Hence the inclusion of the $+360º$.