And here we are. Final chapter of Book VI. Here, we’ll finish out Ptolemy’s discussion of the inclinations of eclipses. In the previous post, we discussed a few things we’ll need to determine that1, but we really haven’t discussed is what the “inclination” means. So I’ll begin by doing that as Ptolemy doesn’t really do a clear job of doing so. Continue reading “Almagest Book VI: Determining the Inclinations”
Almagest Book VI: Table of Angles of Inclination at Eclipses
As usual, after completing a few example calculations, Ptolemy lays out the table discussed in the last post. I’ve again made this table available as a Google Sheet.
Continue reading “Almagest Book VI: Table of Angles of Inclination at Eclipses”
Almagest Book VI: Horizon Diagram
In the previous chapter, we mentioned that one of the things we’ll need for Ptolemy’s weather prediction is the angle between the ecliptic and celestial equator as measured along the horizon. This is something we calculated back in II.2 and only now are we getting around to displaying a diagram to summarize it. So I’ll display the diagram here2 and discussion about it will be beneath the fold.
Almagest Book VI: Angles of Inclination at Eclipses
The final few chapters of Book VI are rather odd. Now that we’ve completed the discussion of eclipse prediction, Ptolemy wants to do an “examination of the inclination which are formed at eclipses.” However, he doesn’t appear to provide any motivation for doing so. Toomer and Neugebauer both indicate that the actual reason was likely weather prediction3, but the Almagest doesn’t contain any information on how this is to be used. Neugebauer indicates that,
[T]he technical term connected with this problem is “prosneusis”…developed from the original meaning of the verb νευειν (to nod, to incline the head, etc…). According to the terminology of hellenistic astrology, the planets or moon can, e.g., give their consent by “inclining” toward a certain position, i.e., by being found in a favorable configuration.
However, aside from these astrological purposes, these last few chapters are essentially left as a free-floating bit of material. Continue reading “Almagest Book VI: Angles of Inclination at Eclipses”
Almagest Book VI: Predicting Solar Eclipses
Now that we understand how to predict lunar eclipses, we’ll turn our attention towards solar eclipses. However, Ptolemy warns us that these will be
more complicated to predict because of lunar parallax4.
Toomer again provides an example that we can follow along with5. This will be Example $12$ from Appendix A. Surprisingly, nowhere in the Almagest does Ptolemy describe the details of a solar eclipse. As such, Toomer has selected his own example. In this case, we are to determine the details of the solar eclipse of June $16$, $364$ CE (Nabonassar $1112$ in the month of Thoth), which was observed by Theon of Alexandria6. Upon observing the eclipse, Theon then followed Ptolemy’s methods in the Almagest and Handy Tables to compare the predictions against observations and his calculation are what Toomer follows as an example using Ptolemy’s methods7. Continue reading “Almagest Book VI: Predicting Solar Eclipses”
Almagest Book VI: Predicting Lunar Eclipses
Having set out the above as a preliminary, we can predict lunar eclipses in the following manner.
As Ptolemy states in opening this chapter, we’re finally done with the preliminary work and we’re ready to start diving into how to actually use everything we’ve done to predict eclipses. As usual, Ptolemy walks us through the steps, but does not provide an example, so I will follow my usual procedure of using example $11$ in Appendix A of Toomer’s translation8.In that example, Toomer invites us to examine lunar eclipses around Nabonassar 28, in the month of Thoth (the first month of the Egyptian year). Continue reading “Almagest Book VI: Predicting Lunar Eclipses”
Almagest Book VI: Eclipse Tables
Having spent five posts building eclipse tables, here’s the full tables. As usual, I’ve placed them in a Google Doc for easy access.
Do note that this table is broken up into four tabs.
Toomer notes that there are a number of errors in the table, but it’s not clear whether they originate with Ptolemy’s calculations or are a result of later transcription errors. Ones that can be confirmed as scribal errors he notes were corrected in the translation.
Almagest Book VI: Table for Magnitudes of Solar and Lunar Eclipses – Lunar Eclipse Example
Having completed an example calculation for converting linear digits to area digits in the previous post, we’ll now do the same calculation for a lunar eclipse. The good news is the setup is the same. While I don’t strictly need to redraw the diagram, I’m going to anyway because the earth’s shadow is so much larger than the moon and drawing it as such helps me visualize things mentally although the respective position of the points doesn’t change at all.
Almagest Book VI: Table for Magnitudes of Solar and Lunar Eclipses – Solar Eclipse Example
Previously, when discussing eclipses, we’ve discussed the amount that is obscured in terms of “digits” where each digit is $\frac{1}{12}$ of the diameter of the object. However, Ptolmey indicates that not everyone necessarily estimates the magnitude of eclipses in this way, stating,
most of those who observe [eclipses]… measure the size of the obscuration, not by the diameters of the disks, but, on the whole, by the total surface area of the disks, since, when one approaches the problem naively, the eye compares the whole part of the surface which is visible with the whole of that which is invisible9.
To deal with this Ptolemy provides “another little table” which will allow us to convert between the linear diameter obscured and the area of either the sun, or moon. Continue reading “Almagest Book VI: Table for Magnitudes of Solar and Lunar Eclipses – Solar Eclipse Example”
Almagest Book VI: Table of Eclipse Correction
Having computed tables that give information about eclipses for the sun and moon at both greatest and least distance, Ptolemy now turns to creating a table to estimate the impact of the moon being at other positions about the epicycle besides apogee and perigee.
This is to be done by a table that uses the distance from apogee about the epicycle as the input and returns what proportion of the difference between apogee and perigee one should use, expressed in sixtieths. The good news is that we’re not really going to have to do any calculations here because we’ve actually already done them when we were putting together our Parallax Table. Continue reading “Almagest Book VI: Table of Eclipse Correction”