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”1. Continue reading “Almagest Book VI: Solar Eclipses Separated by Seven Months”
Almagest Book VI: Eclipse Limits for Solar Eclipses – Longitudinal Parallax
Now that we’ve determined how much further from the nodes parallax can cause solar eclipses to occur due to the latitudinal parallax, we need to consider the longitudinal effect. As with the last post, Ptolemy is absolutely no help in this. He simply tosses out some values with no explanation or work stating
When [the latitudinal] parallax is $0;08º$ northwards1, [the moon] has a maximum longitudinal parallax of about $0;30º$ … and when its [latitudinal] parallax is $0;58º$ southwards2, it has a maximum longitudinal parallax of about $0;15º$…
Seeking some assistance, I again refer to Neugebauer and Pappus, but immediately run into an issue. Neugebauer minces no words and states
Ptolemy is wrong in stating that $p_\lambda = 0;30º$ and $p_\lambda = 0;15º$ are the greatest longitudinal components of the parallax for locations between Meroe and the Borysthenes. It is difficult to explain how he arrived at this result.
Well… this will be interesting to try to untangle then. Continue reading “Almagest Book VI: Eclipse Limits for Solar Eclipses – Longitudinal Parallax”
Almagest Book VI: Eclipse Limits for Solar Eclipses – Latitudinal Parallax
Now that we’ve determined how far away from the nodes a lunar eclipse can occur, we’ll work on doing the same for a solar eclipse1. But before diving in, I want to say that this has been one of the most, if not the most challenging section of the the Almagest so far. One of the primary reasons is that Ptolemy shows no work and gives almost no explanation on how he did this. When such things happen, I often turn to Neugebauer’s History of Ancient Mathematical Astronomy which I did in this case. There, Neugebauer refers to Pappus of Alexandria, a fourth century mathematician who did commentary on the Almagest and walks through a process that arrives at the same values as Ptolemy.
However, there was a very large amount to unpack in just a few pages there and, unlike most cases where I can simply work along with it and see where things are going, this time I had to really understand the whole process before the first steps made any sense. This led me to agonize over what was going on with those first steps, amounting to several days of effort and rewriting this post from scratch several times. The result is twofold. First because I feel this section can only be approached by understanding the methodology before diving into the math, there’s going to be far more exposition than normal and, as a result, this is likely to be one of my longer posts. Second, the struggles I had with trying to understand the method and rewriting this post so many times has left me with a lot of fragments of thoughts in my brain and in the blog editor. I’ve done my best to clean it up, and maybe it’s just those thoughts swirling around in my brain, but this post just doesn’t feel as coherent as I like. Apologies in advance if you struggle to follow. Know I did as well.
Anyway, moving on to the topic at hand.
Normally, I like to start with a quote from Ptolemy to give us some direction, but I think Ptolemy did such a poor job of laying this section out, I’m going to avoid doing so for the majority of the post. Instead, let’s try to understand the process by recalling what we did with the moon and discussing how things will change. Continue reading “Almagest Book VI: Eclipse Limits for Solar Eclipses – Latitudinal Parallax”
Almagest Book V: Components of Parallax – Corrections
At the end of the last post, we noted that Ptolemy wasn’t quite satisfied with what we did previously because we used some rather faulty assumptions.
As Ptolemy states it:
For lunar parallaxes, we considered it sufficient to use the arcs and angles formed by the great circle through the poles of the horizon [i.e., an altitude circle] at the ecliptic, instead of those at the moon’s inclined circle. For we saw that the difference which would result at syzygies in which eclipses occur is imperceptible, and to set out the latter would have been complicated to demonstrate and laborious to calculate; for the distance of the moon from the node is not fixed for a given position of the moon on the ecliptic, but undergoes multiple changes in both the amount and relative position.
The key phrase here is the “at the ecliptic, instead of the moon’s inclined circle.” This got swept under the rug in that post because Ptolemy didn’t really explain why the algorithm he gave us should work. So to understand, let’s start by taking a harder look at what’s actually going on.
Continue reading “Almagest Book V: Components of Parallax – Corrections”
Almagest Book V: Components of Parallax – A First Approximation
So far in this chapter, we’ve explored how to use the table of parallax to calculate the parallax of the moon and sun by knowing its distance from the zenith. But this in and of itself isn’t particularly useful. For example, when we did a sample lunar parallax calculation, we determined in that situation, the moon was about $1;10º$ off from its true position. But $1;10º$ in which direction?
Thus, the next step will be to break that down into its components, determining how far off in both ecliptic latitude and longitude the parallax makes the moon appear. Ptolemy again gives steps, but no example, so we’ll continue the previous example we did for the moon, following Neugebauer1. Continue reading “Almagest Book V: Components of Parallax – A First Approximation”
Almagest Book V: Calculating Solar Parallax Along a Great Circle Through the Zenith
Having computed the lunar parallax,
the sun’s parallax for a similar situation [i.e., as measured along an altitude circle] is immediately determined, in a simple fashion (for solar eclipses) from the number in the second column corresponding to the size of the arc from the zenith [to the sun].
Well that sure sounds easy. Let’s look at a quick example. Continue reading “Almagest Book V: Calculating Solar Parallax Along a Great Circle Through the Zenith”
Almagest Book V: Calculation of Lunar Parallax
So far in this book, we’ve refined our lunar model, shown how to use it to calculate the lunar position1, discussed a new instrument suitable for determining lunar parallax, as well as an example of the sort of observation necessary to make the calculation.
Now, Ptolemy walks us through an example of how to calculate the lunar distance using an example entirely unrelated to the one we saw in the last post.
In the twentieth year of Hadrian, Athyr [III] $13$ in the Egyptian calendar [135 CE, Oct. $1$2], $5 \frac{5}{6}$ equinoctial hours after noon, just before sunset, we observed the moon when it was on the meridian. The apparent distance of its center from the zenith, according to the instrument, was $50 \frac{11}{12}º$. For the distance [measured] on the thin rod was $51 \frac{7}{12}$ of the $60$ subdivisions into which the radius of revolution had been divided, and a chord of that size subtends an arc of $50 \frac{11}{12}º$.
Continue reading “Almagest Book V: Calculation of Lunar Parallax”
Almagest Book V: Lunar Parallactic Observations
In the last post we followed along as Ptolemy discussed the construction and use of his parallactic instrument, which he would use to measure the lunar parallax. To do so, Ptolemy waited for the moon to
be located on the meridian, and near the solstices on the ecliptic, since at such situations, the great circle through the poles of the horizon and the center of the moon very nearly coincides with the great circle through the poles of the ecliptic, along which the moon’s latitude is taken.
That’s pretty dense, so let’s break it down with some pictures, First, let’s draw exactly what Ptolemy has described above:
Continue reading “Almagest Book V: Lunar Parallactic Observations”
Almagest Book V: Size of the Second Anomaly
So far, what we know about Ptolemy’s second anomaly is that it doesn’t have an effect at conjunction or opposition. Its at its maximum at quadrature, which is to say, a $\frac{1}{4}$ and $\frac{3}{4}$ of the way through each synodic month1. Its effect is to re-enforce whatever anomaly was present from the first anomaly. Ptolemy laid out a conceptual model in Chapter 2, but to determine the parameters of the model, we’ll need to first explore how much this second anomaly impacts things.
Continue reading “Almagest Book V: Size of the Second Anomaly”
Almagest Book II: Table of Zenith Distances and Ecliptic Angles
Finally we’re at the end of Book II. In this final chapter1, Ptolemy presents a table in which a few of the calculations we’ve done in the past few chapters are repeated for all twelve of the zodiacal constellations, at different times before they reach the meridian, for seven different latitudes.
Computing this table must have been a massive undertaking. There’s close to 1,800 computed values in this table. I can’t even imagine the drudgery of having to compute these values so many times. It’s so large, I can’t even begin to reproduce it in this blog. Instead, I’ve made it into a Google Spreadsheet which can be found here.
First, let’s explore the structure.
Continue reading “Almagest Book II: Table of Zenith Distances and Ecliptic Angles”