Tracing the history of medieval astronomy
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.
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.
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 invisible1.
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”
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”
With the solar eclipse tables complete, we’ll now turn our attention to the lunar eclipse tables. These largely follow the same format and calculations as the solar eclipse tables, but will include an additional column for the “half totality”. Continue reading “Almagest Book VI: Construction of the Eclipse Tables – Lunar Eclipse Tables”
Towards the end of $2021$, I got stuck on a particular calculation and it took me over a month and a half to resolve. Frustratingly, Ptolemy showed no work, Neugebauer made no comment (which generally indicates he found no fault in the calculation), and Pedersen skipped the chapter entirely. It seemed there was little to no help available save a footnote in the Toomer translation I’m using which stated
a somewhat unsatisfactory numerical verification of [the calculation] (using the Handy Tables) is in Pappus’ commentary (Rome[$1$] $232-4$).
The citation here is to a text entitled Commentaires de Pappus et de Theon d’Alexandrie sur l’Almageste(Commentaries of Pappus and Theon of Alexandria) by Adolphe Rome.
Pappus was a $4^{th}$ century astronomer/mathematician so my hope was that the source by Rome that Toomer was citing was a translation. As was obvious from the title, the work was written in French and I’d taken enough French that I hoped I would be able to muddle through a translation and so I requested the text through an interlibrary loan.
Continue reading “Commentaries of Pappus and Theon of Alexandria on the Almagest – A. Rome (1931)”
In the first post on Book VI, I stated that, while we could calculate the position of the sun and moon every day to determine whether an eclipse was happening, we wanted to rule out as much as possible. To that end, we’ve spent most of our time trying to figure out when we do or do not need to worry about there being an eclipse. First we looked at determining mean conjunctions, then showed how to get from mean to true syzygy, then looked at how far away from a mean syzygy an eclipse could occur, and finally, in the last chapter, we looked at numerous periods to see whether or not they would be possible.
However, we’ve now run out of things that Ptolemy wants to rule out. As such, in what’s left, we’ll need to actually go through at least some of the calculations. Specifically, in this chapter we’re going to work on some tables that, if we input the argument of
the moon’s position in latitude [for a given syzygy, we will know] which of those syzygies will definitely produce an eclipse, as well as the magnitudes and times of obscuration for these eclipses.
Continue reading “Almagest Book VI: Construction of the Eclipse Tables – Solar Eclipse Tables”
We have finally reached the final in this run of eclipse timing feasibility checks. In it Ptolemy wants to demonstrate that it is impossible to have two eclipses separated by one month
even if one assumes a combination of conditions which could not in fact all hold true at the same time, but which may be lumped together in a vain attempt to provide a possibility of the event in question happening.
In short, we’re going to assume an overly ambitious “best case” scenario which can’t actually happen because some of these best case conditions contradict one another. Continue reading “Almagest Book VI: Solar Eclipses Separated by One Month”
Having established that two solar eclipses separated by the five months from the same location are just barely possible, Ptolemy then works on whether it will be possible for the same to occur over a period of seven months concluding that it is possible, provided it happen in the “shortest $7$-month interval”2. Continue reading “Almagest Book VI: Solar Eclipses Separated by Seven Months”
Ptolemy next looks at whether or not it is possible for a solar eclipse to occur five months after a previous one. We’ve already done a fair bit of the heavy lifting for this topic as some of the math we did when considering lunar eclipses separated by five months will still apply. In that post, we determined that the moon would have moved on its inclined circle by $159;05º$ between true conjunctions. This does require we adopt the same assumptions of the sun moving its greatest distance and the moon moving its least.
What we’ll need to focus on for this post is redoing the eclipse limits for the situation in question. Continue reading “Almagest Book VI: Solar Eclipses Separated by Five Months”
