For the last chapter of Book IV, Ptolemy diverges from his own narrative to address why his predecessor, Hipparchus, evidently had derived two different values for the lunar anomaly which did not match the value Ptolemy derived, or each other.
If you recall, Ptolemy previously showed that we should be able to get the same results using either the eccentric or epicyclic model. However, this assumed some ratios were consistent. Specifically, we needed the ratio of the offset of the eccentre to its radius to be the same as the ratio of the radius of the epicycle to the deferent.
Ptolemy notes that
the first ratio [Hipparchus] found, using the eccentric hypothesis, differs from the second, which was calculated from the epicyclic hypothesis. For, in his first demonstration he derives the ratio between the radius of the eccentre and the distance between the centers of the eccentre and the ecliptic as about $3144 : 327 \frac{2}{3}$ (which is the same as $60 : 6;15$), while in the second he finds the ratio between the line joining the center of the ecliptic to the center of the epicycle and the radius of the epicycle as $3122 \frac{1}{2} : 247 \frac{1}{2}$ (which is the same as $60 : 4;46$).
That’s a fairly large discrepancy and Ptolemy notes that it will have a sizeable effect. The former produces a maximum equation of anomaly of $5;49º$ while the latter only produces a maximum equation of anomaly of $4;34º$. Ptolemy’s derivations split the difference and come up with a ratio of $60 : 5 \frac{1}{4}$ which produces a maximum equation of anomaly of about $5º$.
So how did Hipparchus arrive at such discrepant numbers? Ptolemy argues it was no fault of the two different models, but rather, that Hipparchus used different inputs, specifically pairs of eclipses, for each calculation. And in doing so, Ptolemy states that, while the observations themselves were reliable, Hipparchus incorrectly determined the period between them. And demonstrating the discrepancies is the topic of this chapter.
While it’s a relatively long chapter, there’s it’s mostly descriptions of the eclipses Hipparchus used: Three for the eccentric model and three for the epicyclic. Ptolemy does give the values from his model to compare with Hipparchus, but spares us the math. I’ll do the same1.
First, Ptolemy gives the details of the first triad of eclipses:
[T]he first occurred in the archonship of Phanostratos at Athens, in the month of Poseidon2 [where] a small section of the moon’s disk was eclipsed from the summer rising-point [i.e., the north-east] when half an hour of night was remaining. He adds that it was still eclipsed when it set.
Ptolemy then gives this in the Egyptian calendar:
Now this moment is in the $366$th year from Nabonassar, in the Egyptian calendar… Thoth $26$/$27$ [$-382$ Dec $22$/$23$], $5 \frac{1}{2}$ seasonal hours after midnight.
And converting the seasonal hours to equinoctial3:
[On this date] when the sun is near the end of Sagittarius, one hour of night in Babylon is $18$ time-degrees (for the night is $14 \frac{2}{5}$ equinoctial hours long). So $5 \frac{1}{2}$ seasonal hours produce $6 \frac{3}{4}$ equinoctial hours. Therefore, the beginning of the eclipse was $18 \frac{3}{5}$ equinoctial hours after noon on the $26$th.
Using his knowledge of how long eclipses Ptolemy estimates the time of mid-eclipse:
And since a small section [of the disk] was obscured, the duration of the whole eclipse must have been about $1 \frac{1}{2}$ hours. So the middle of the eclipse must have been $19 \frac{1}{3}$ equinoctial hours after noon.
Making the adjustment from Babylonian time to Alexandrian time4:
Therefore, mid-eclipse at Alexandria was $18 \frac{1}{2}$ equinoctial hours after noon on the $26$th.
Next, Ptolemy want to take this and puts this through his model to determine the positions of the sun and moon. But first, he’ll need to know the interval since the beginning of the epoch which he gives as $365$ Egyptian years, $25$ days, $18 \frac{1}{4}$ hours5.
Using that interval, and the motions since epoch from our solar and lunar mean motion tables with the solar anomaly table, Ptolemy states that the true position of the sun was $28;18º$ into Sagittarius, the mean moon was $24;20º$ into Gemini, and the true position of the moon was $28;17º$ into Gemini being $227;43º$ around the epicycle.
Whew. Done with the first eclipse. On to the second. Fortunately this time he skips straight to the Egyptian calendar:
[T]he next eclipse occurred in the archonship of Phanostratos at Athens, in the month of Skirophorion, Phamenoth $24$/$25$ in the Egyptian calendar, and [the moon] was eclipsed form the summer rising-point [i.e., the north-east] when the first hour [of night] was well advanced.
Determining the interval from this:
This moment is in the $366$th year from Nabonassar, Phamenoth [VII] $24$/$25$ [$-381$ June $18$/$19$], about $5 \frac{1}{2}$ seasonal hours before midnight.
Converting those season hours to equinoctial:
When the sun is near the end of Gemini, one hour of the night at Babylon is $12$ time-degrees. Therefore the $5 \frac{1}{2}$ seasonal hours produce $4 \frac{2}{5}$ equinoctial hours. So the beginning of the eclipse was $7 \frac{3}{5}$ equinoctial hours after noon on the 24th.
Next, he determines the time of mid-eclipse:
[S]ince the duration of the whole eclipse is recorded as three hours, mid-eclipse… occurred $9 \frac{1}{10}$ equinoctial hours after [noon].
Converting to Alexandrian time:
So in Alexandria it must have occurred about $8 \frac{1}{4}$ equinoctial hours after noon on the $24$th.
From that Ptolemy gives the interval since the beginning of epoch as $365$ Egyptian years, $203$ days, and $7 \frac{5}{6}$ equinoctial hours. Again popping that through all our tables, we determine the position of the sun as $21;46º$ into Gemini, and the mean moon to be at $23;58º$ into Sagittarius with the true position after anomaly of $21;48º$ into Sagittarius, being $27;37º$ around the epicycle.
As we did previously, we can now determine the difference in time between these two dates to be $177$ days, $13 \frac{3}{5}$ equinoctial hours. In that interval the sun would have moved $173;28º$ in longitude according to Ptolemy’s calculations.
This is then compared to Hipparchus’ calculations wherein he determined an interval of $177$ days and $13 \frac{3}{4}$ equinoctial hours wherein the sun moved $172 \frac{7}{8}º$.
Turning to the third eclipse, again thankfully already in the Egyptian calendar:
[T]he third eclipse occurred in the archonship of Euandros at Athens, in the month Poseidon I, Thoth $16$/$17$ in the Egyptian calendar [when the moon] was totally eclipsed, beginning from the summer rising-point [i.e., the north-east], after four hours [of night] had passed.
Giving this as the interval from the beginning of epoch:
This moment is in the $367$th year from Nabonassar, Thoth [I] $16$/$17$ [$-381$ Dec. $12$/$13$], about $2 \frac{1}{2}$ hours before midnight.
Again, the seasonal hours must be converted to equinoctial:
[W]hen the sun is about two-thirds through Sagittarius, one hour of night at Babylon is about $18$ time-degrees. So $2 \frac{1}{2}$ seasonal hours produce $3$ equinoctial hours. Therefore the beginning of the eclipse was $9$ equinoctial hours after noon on the $16$th.
Knowing that the eclipse was full and that full eclipses last about $4$ hours from the first moment the shadow starts eclipsing the lunar disk until the moment it ends that means mid-eclipse was half that time or $11$ hours after noon on that date.
Then converting to Alexandrian time:
[I]n Alexandria mid-eclipse must have occurred $10 \frac{1}{6}$ equinoctial hours after noon on the $16$th.
From the beginning of epoch, this gives an interval of $366$ Egyptian years, $15$ days, and $9 \frac{5}{6}$ equinoctial hours.
Determining the positions of the sun and moon we get that the true position of the sun was $17;30º$ into Sagittarius, the mean moon was $17;21º$ into Gemini, and the true moon was $17;28º$ into Gemini being $181;12º$ around the epicycle.
Creating an interval between this third eclipse and the second gives us a time period of $177$ days and $2$ hours in which time the sun would have moved $175;44º$.
This is compared to the values Hipparchus determined of $177$ days, and $1 \frac{2}{3}$ hours in which time the sun would have moved $175 \frac{1}{8}º$.
Thus, Ptolemy has laid out the discrepancies between Hipparchus’ calculations and his own. It was about $\frac{1}{6}$ of an hour in the first and $\frac{1}{3}$ of an hour in the second. In both cases, Hipparchus’ numbers varied by about $\frac{3}{5}º$. Without diving further into it, he notes that
Errors of this amount can produce a considerable discrepancy in the size of the ratio.
We could determine the differences this makes in the ratios by repeating the procedure we saw in Book IV.6, but again, since Ptolemy doesn’t, I won’t either.
This post has already gotten quite long, so I’ll leave off here. In the next post, we’ll go ahead and repeat this for the second triad of eclipses Hipparchus used which will complete Book IV!
- Which means this post is going to be a whole lot of me just repeating what he’s said and offering a bit of light commentary.
- Toomer notes that these dates are almost certainly in the Metonic calendar.
- This conversion was covered in Book II.9.
- We first did this in Book IV.6.
- In every case where Ptolemy gives the interval, he actually gives two. One “reckoned simply” and another “reckoned accurately”. Because Ptolemy always uses the second one, I’ve neglected to discuss the former in most posts. We did explore this once in Book IV.6, in which the interval “reckoned accurately” is the mean solar days.