Towards the beginning of Book IV, Ptolemy went through the methodology by which the various motions in the lunar mean motion table could be calculated. But if you were paying extra close attention, you may have noticed that the values that ended up in the mean motion table didn’t actually match what we derived. Specifically, for the daily increment in anomaly, we derived $13;3,53,56,29,38,38^{\frac{º}{day}}$. But in the table, we magically ended up with $13;3,53,56,17,51,59^{\frac{º}{day}}$. Identical until the 4th division.
So what gives? Why did Ptolemy derive one value and report another?
In Chapter 7, he gives the explanation: He found the value needed to be corrected and did so before he put it in the table. But before he could explain to us how, we needed to cover the eclipse triples we did in Chapter 6. So how do we apply them to check the mean motions?
Ptolemy does so by again pairing up some eclipses, this time doing the second eclipse from each triple. We’ll recall that for the first triple, the second eclipse occured when the mean moon was $14;44º$ into Virgo and $12;24º$ past apogee1 on the epicycle.
For the second eclipse of the second triple we showed the mean moon to be $29;30º$ into Aries with it being $64;38º$ past apogee.
If we take the difference between this pair of eclipses, this means that the mean moon travelled $224;46º$ in longitude and 52;14º around the ecliptic2.
Ptolemy calculates that the time from the first of these two eclipses was 854 years, 73 days, 23 hours, and 20 minutes. Converting the years to days, it’s 311783 days, 23 hours, and 20 minutes.
I’ll pause here a moment that in the calculations we’re about to do, if you actually use a value of 20 minutes, you won’t get the same answer as Ptolemy. Rather, Toomer suggests that Ptolemy used a value of 18 minutes, so to make sure my final result agrees with Ptolemy, I’ll use 18 minutes as well.
Now, let’s apply the uncorrected rates for both longitude and anomaly to that interval by multiplying them together and then subtracting out full revolutions.
For longitude:
$$(311783 + (23+\frac{18}{60}) / 24) days \cdot 13;10,34,58,33,30,30^{\frac{º}{day}} = 224;46…º$$
This agrees perfectly with what we calculated for the motion in longitude between the two eclipses described above. Thus, the value we derived in Chapter 3 was accurate.
For anomaly:
$$(311783 + (23+\frac{18}{60}) / 24) days \cdot 13;3,53,56,29,38,38^{\frac{º}{day}} = 52;31…º$$
We could carry on to seconds and further divisions, but since we only calculated the change for these two eclipses in this post to the minutes, we can stop since we can easily see there is a discrepancy of about 17 minutes.
This indicates that the value we determined in Chapter 3 was ever so slightly off. By 0;17º in 311783 days, 23 hours, and 18 minutes. Ptolemy divides this error by the number of days it took to accumulate it, to arrive at a correction of $0;0,0,0,0,11,46,39^{\frac{º}{day}}$.
This was then subtracted from the original, uncorrected value to arrive at $13;3,53,56,17,51,59^{\frac{º}{day}}$ as reported in the lunar mean motion table.
