As promised in the last chapter, Ptolemy’s first task in eclipse prediction is going to be laying out a table of mean syzygies around which eclipses might be possible, so we can check those to see if an eclipse might occur instead of performing useless calculations where the sun and moon are nowhere near a syzygy. In this post, we’ll go over the construction of that table!
The first thing that Ptolemy wants to determine is when the last mean conjunction1 happened before the beginning of the epoch on Thoth 1 in the first year of Nabonassar. Fortunately, the lunar mean motions table has a column that already has the elongation and since the elongation is just the angular distance between the sun and moon, we can ask when this was last zero.
At the beginning of the epoch, the mean elongation was $70;37º$ and that increases by $12;11,26º$ per day. Thus, we can divide to determine the number of days since the last mean conjunction to be $5;47,33$ days before the beginning of the epoch.
So when will the next one be? Back in Book IV, Ptolemy derived the length of the lunar mean synodic month2 to be $29;31,50$ days. Thus, the next conjunction into the epoch will occur $23;44,17$ days into the epoch which will define the first line in the table Ptolemy is constructing.
We can then calculate the motions of each object. The mean sun would have moved $23;23,50º$, the moon would have moved $310;08,15º$ about its epicycle, and $314;02,21º$ through its cycle in latitude. Adding those increments onto the starting positions for each, we get the following:
1. The distance of the mean Sun3 from Apogee: $288;38,50º$
2. Anomaly of Moon from Epicyclic Apogee: $218;57,15º$
3. Latitude of Moon from Northern Limit: $308;17,21º$
The three values get entered into the columns.
But Ptolemy’s not just interested in the mean conjunctions. Because this is a table of syzygies, he’s also interested in mean oppositions which happen exactly half way through the synodic month. In that amount of time ($14;45,55$ days), we can again determine the increments and get that the mean sun would move $14;33,12º$, the moon would move $192;54,30º$ on its epicycle, and $192;20,06º$ in the argument of latitude. These can then be subtracted from the previous values at mean conjunction to determine the positions at the previous opposition and get entered into a second half of the table for just such mean oppositions.
Now, if you recall Ptolemy’s previous tables, he often constructs the tables such that they have different intervals, often using $18$ year intervals as the largest, then years, months, days, etc… But in this table, instead of using $18$ year intervals as his largest, Ptolemy instead uses $25$ years4 for the first table. This is because Ptolemy5 noticed that $25$ years is almost perfectly divisible by an integer number of lunar synodic months – $309$ to be specific.
There is a bit of a discrepancy: The $309$ lunar synodic months is $0;02,47,05$6 days under $25$ years.
In that $25$ years, less the $0;02,47,05$, we can calculate how much the sun and moon would have moved in each of the motions with which we’re concerned7. The mean sun would have advanced $353;52,34,13º$. The moon would have moved $57;21,44,01º$ about the epicycle and its argument of latitude increased by $117;12,49,54º$.
Again, that wasn’t for the full $25$ year period, but the period just under it, so we need to subtract that small difference in days from the number of days into that new line. This gets done for both the conjunction and opposition tables and since this pattern repeats, Ptolemy continues repeating these increments to complete the first part of the tables, which is for $25$ year periods until $1101$ years.
Next up, Ptolemy creates a table for the increments in individual years, giving them for $1$ year through $24$ since the $25^{th}$ year would be on the previous table. However, this gets a bit trickier because the number of synodic months does not fall so evenly into a single year. Most often, there are $13$ synodic months in a year, but sometimes there are $12$ and Ptolemy must carefully determine which each line is so he knows how much to increment.
If we consider the case of a year which has $13$ synodic months, a conjunction or opposition will occur one year plus $18;53,51,48º$ days after the previous one. In that time, the mean sun will have moved $18;22,59,18º$, motion of the moon about the epicycle will be $335;37,01,51º$, and the increase in the lunar argument of latitude is $38;43,03,51º$
In the case of a year with $12$ synodic months, a conjunction or opposition will occur $354;22,01,40$ days later in which time the mean sun will have moved $349;16,36,16º$, the moon about the epicycle by $309;48,01,42º$, and the moon’s argument of latitude increase by $8;02,49,42º$.
That’s where this chapter ends, but I do want to say a bit more as the table Ptolemy’s creating has one more section to it for individual months. Ptolemy doesn’t say how they were calculated but for the sake of completeness, they’re simply the daily increment in each of these values multiplied by the number of days in a synodic month and then added successively.
As a final note, while Ptolemy does state
In the actual tabular entries it will be sufficient to go only as far as the second sexagesimal place.
However, we can clearly see that Ptolemy calculated with higher precision and then rounded the final product as the increments can vary by $0;00,01$ from line to line.
I’m still working on my big paper trying to redo the epoch calculations for the moon to create a modern epoch, but this is taking longer than I like so I’ve decided that I should make sure I don’t get completely stuck and continue making progress on the main text lest I get stuck and frustrated as working on one part can often serve as inspiration for another!
- Here, “mean conjunction” is meaning the conjunction of the mean points for each body, not taking into consideration any of the anomalies.
- The average time for the moon to return to the same phase and since the phases are determined by the elongation, this is the same as asking how long until it returns to conjunction.
- And thus the mean moon since we’re discussing conjunctions.
- Quick reminder that Ptolemy works in Egyptian years which I discussed in detail in this post.
- And certainly astronomers before him.
- If you are following along with this in Neugebauer’s History of Ancient Mathematical Astronomy, there appears to be an error here as he reports a value of $0;02;57,05$ days. Seeing this disagreement, I have checked the calculation myself and confirm that it is as Ptolemy (or at least Toomer’s translation) reported.
- Subtracting out full revolutions.