In the last post, we followed Ptolemy as he reviewed three eclipses Hipparchus used to determine the parameters for his model showing they were different from Ptolemy’s. In this post, we’ll repeat the procedure for the second set of three eclipses, again showing that Hipparchus’ calculations did not match those of Ptolemy.
This set of eclipses, were observed in Alexandria, which is convenient since it means we won’t need to convert from Babylonian time to Alexandrian. So jumping right in,
the first [eclipse] occurred in the $54$th year of the Kallippic Cycle, Mesore [XII] $16$ in the Egyptian calendar [$-200$ Sept. $22$]. In this eclipse the moon began to be obscured half an hour before it rose, and its full light was restored in the middle of the third hour [of the night]. Therefore mid-eclipse occurred at the beginning of the second hour1, $5$ seasonal hours before midnight, and also $5$ equinoctial hours, since the sun was near the end of Virgo2. So mid-eclipse at Alexandria occurred $7$ equinoctial hours after noon on the $16$th
Now understanding the date and time of mid eclipse, we can determine the interval since the beginning of the epoch which Ptolemy gives as $564$ Egyptian years, $345$ days, and $6 \frac{1}{2}$ hours. This can be used to determine the solar and lunar positions wherein the true position of the sun is $26;6º$ into Virgo, the mean moon is at $22º$ into Pisces, and the true moon is $26;7º$ into Pisces being $300;13º$ around the epicycle.
That completes the first eclipse. The second
occurred in the $55$th year of the same cycle, Mechir [VI] $9$ in the Egyptian calendar [$-199$ Mar. $19$] [and] began when $5 \frac{1}{3}$ hours of night had passed, and was total. So the beginning of the eclipse was $11 \frac{1}{3}$ hours after noon on the $9$th (since the sun was near the end of Pisces3), and mid-eclipse was $13 \frac{1}{3}$ equinoctial hours after [noon].
From this, Ptolemy determines the interval as $547$ Egyptian years, $158$ days, and $13 \frac{1}{3}$ equinoctial hours. This results in positions of the sun being $26;17º$ into Pisces, the mean moon being $1;7º$ in Libra, and the true position being $26;16º$ into Virgo being $109;28º$ around the epicycle from apogee.
Now that we’ve gotten the time and position for these two eclipses, we can determine the interval between them and the change in position. The difference in time was $178$ days and $6 \frac{5}{6}$ equinoctial hours over which time the position of the sun (and thus the moon) would have changed by $180;11º$ according to Ptolemy’s model.
In contrast, Hipparchus determined an interval of $178$ days and $6$ equinoctial hours during which time he calculated the position of the sun to have changed by $180;20º$.
Moving onto the third eclipse of this triad, it
occurred in the same ($55$th) year of the Second Cycle, on Mesore [XII] 5 in the Egyptian calendar [$-199$ Sept $11$] and it began when $6 \frac{2}{3}$ of the night had passed and was total. [Hipparchus] also says that mid-eclipse occurred at about $8 \frac{1}{3}$ [seasonal] hours into the night, which is $2 \frac{1}{3}$ seasonal hours after midnight. Now, when the sun is near the middle of Virgo, one hour of the night in Alexandria is $14 \frac{2}{5}$ time-degrees. So $2 \frac{1}{3}$ seasonal hours will produce about $2 \frac{1}{4}$ equinoctial hours. So mid-eclipse was $14 \frac{1}{4}$ equinoctial hours after noon on the $5$th.
Ptolemy gives the interval from epoch as $547$ years, $334$ days and $13 \frac{3}{4}$ hours at which point the sun would have been $15;12º$ into Virgo, the mean moon would have been $10;24º$ into Pisces, and the true moon being $15;13º$ into Pisces being $249;9º$ around the epicycle.
Looking at the interval between the second and third eclipse, this is a difference of $176$ days and $\frac{2}{5}$ of an equinoctial hour over which the sun, according to Ptolemy, would have moved $168;55º$.
In contrast, Hipparchus calculated an interval of $176$ days and $1 \frac{1}{3}$ hours over which the sun moved $168;33º$
Ptolemy, taking his values as correct, recaps stating that Hipparchus has shown miscalculated the interval by $\frac{5}{6}$ of an hour between the first and second eclipse and $\frac{14}{15}$4 of an hour between the second and third. Similarly, this resulted in an error of $\frac{1}{6}º$ and $\frac{1}{3}º$ in the change in solar and lunar positions respectively.
Circling back around to the thesis of this chapter, in which Ptolemy stated that these discrepancies were the reason Hipparchus ended up with two contradicting sets of ratios of the epicycle to deferent or offset of the eccentre to observer, I don’t feel like Ptolemy has really proved the point here. He’s walked us through the steps of getting many of the values needed to calculate the ratio, but without actually going through that next step things feel a little incomplete in my opinion.
However, I think what Ptolemy was essentially trying to get at, albeit obliquely, is that the two eclipse Triads that Ptolemy calculated, with greater care, resulted in a consistent set of ratios. Since being able to repeat a procedure and get the same result is a key requirement in good science, I think Ptolemy is hinting that this is why we can place more faith in his results. But to completely prove this out, I think Ptolemy could have better made his point by using the parameters he calculated using his intervals and mean motions to show that again, a more careful treatment does indeed result in a set of consistent values. So Ptolemy leaving off here really feels like he didn’t complete the proof despite claiming
we have plainly displayed the reason for the above discrepancy.
Regardless, that’s where Ptolemy ends Book IV. But that’s not the end of the chapters on the moon! As we’ve noted several times, Ptolemy claims to have discovered a “second anomaly” of the moon. So in the next book, we’ll be exploring that!
- Ptolemy is again assuming that a full eclipse takes four hours. So if the eclipse ended three hours before midnight, two hours prior was five hours before midnight or seven hours after noon.
- Essentially what he’s saying here is that since this is very close to the equinox, the seasonal hours and the equinoctial hours are the same. So no conversion is necessary.
- Again, stating that this was near an equinox so the seasonal hours and equinoctial hours are roughly the same.
- Which he states as $\frac{5}{6} + \frac{1}{10}$ of an hour.