The fourth chapter of Book IV takes what we worked on in the last post and expands it for convenient reference. As with the solar mean motion tables we created back in Book III, Ptolemy lays this one out in several intervals: 18 year periods, single year periods, months, days, and hours.
These tables essentially answer the question: “If the moon’s mean position was as X, if I waited Y interval of time, where would it be then?”
To use it, you look up Y in the table1, find the corresponding value in the table, and add it to X. Subtract out any integer numbers of rotations.
Not only does Ptolemy present this table in many intervals, but it is also listed for all four of the different mean motions we derived in the last chapter: longitude, anomaly, latitude, and elongation.
The Toomer translation, I believe, follows the original source with the longitude and anomaly are on the left page, while latitude and elongation are on the right.
This is also the reason that the periods are not presented in descending order. Rather, the way they are presented2 (which I have chosen to follow) allows them to fit neatly onto as few pages as possible.
Lastly, the first row contains the Epoch, informing where the moon was at t = 0, when the epoch started.
Due to the overall width of this table, it was impractical to put in this post directly, so this table is also being stored in a Google sheet which I have made public. I’m sure it contains some typos, so feel free to reach out to me if you spot any errors!
UPDATE: In addition to the original sexagesimal, I have added columns converting things to decimal.
