Modern commentary on Ptolemy often downplays the Almagest because it is certainly a work that relied heavily on the work that astronomers before him. While we no longer have a thorough record of those predecessors, it seems that few historians think much of the Almagest was truly novel1. But I would hasten to remind that, while Ptolemy stood on the shoulders of those who came before, he certainly climbed there on his own, not simply accepting their results, but doing his best to validate them.
And we’re about to get a big dosing of that, because all the work we’ve done in the past three posts, we’ll be redoing with a new set of eclipses observed by Ptolemy himself, allowing for an independent check on the important value of the radius of the epicycle.
To get us started, let’s look at the eclipses Ptolemy observed:
The first occurred in the seventeenth year of Hadrian, Pauni [X] 20/21 in the Egyptian calendar [133 May 6/7]. We computed the exact time of mid-eclipse as $\frac{3}{4}$ of an equinoctial hour before midnight. It was total. At that time, the true position of the sun was about $13 \frac{1}{4}º$ into Taurus [$43 \frac{1}{4}º$ ecliptic longitude].
The second occurred in the nineteenth year of Hadrian, Choiak [IV] 2/3 in the Egyptian calendar [134 Oct 20/21]. We computed that mid-eclipse occurred 1 equinoctial hour before midnight. [The moon] was eclipsed $\frac{5}{6}$ of its diameter from the north. At that time, the true position of the sun was about $25 \frac{1}{6}º$ into Libra [$205 \frac{1}{6}º$ ecliptic longitude].
The third eclipse occurred in the twentieth year of Hadrian, Pharmouthi [VIII] 19/20 in the Egyptian calendar [136 Mar 5/6]. We computed that mid-eclipse occurred 4 equinoctial hours after midnight. [The moon] was eclipsed half of its diameter from the north. At that time the position of the sun was about $14 \frac{1}{12}º$ into Pisces [$344 \frac{1}{12}º$ ecliptic longitude].
Now we’ll determine the solar motion and the period for these pairings. From the first to the second, this is a change of 161;55º and takes place over the course of 1 year, 166 days, and $23 \frac{5}{8}$ equinoctial hours if converted to solar days.
Doing the same for the second eclipse to the third we get a change of 138;55º in solar position in a period of 1 year, 137 days, and $5 \frac{1}{2}$ equinoctial hours.
Next, we turn to the lunar mean motion table and look up the change in longitude corresponding to the intervals in question in both longitude (motion on the deferent) and anomaly (motion around the epicycle). Doing so we find that for the first to the second eclipse we have an increase of 169;37º in ecliptic longitude and 110;21º in anomaly. For the second to third eclipse, we find an increase of 137;34º in ecliptic longitude and 81;36º in anomaly.
This then gets compared to the solar motion2. Thus, for the first to second eclipse, the 110;21º motion around the epicycle produces a 7;42º difference3 in ecliptic longitude. For the second to third eclipse, the 81;36º motion around the epicycle produces a 1;21º difference4 in ecliptic longitude.
Up next, we’ll again walk through the process of determining the radius of the epicycle to make sure it lines up with the values from Hipparchus!
- One of the few places I know the consensus is that it is novel is the second anomaly of the moon which will get explored in Book V.
- Which is equal to the lunar motion over the same interval since eclipses can only occur at opposition.
- Reason: $161;55º – 169;37º = 7;42º$. I should also note that if I were worrying about the signs here, it would be negative. This indicates that the apparent motion between the first eclipse and second would have had a rearwards effect along the ecliptic compared to the mean motion.
- Reason: $138;55º – 137;34º = 1;21º$