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 $0;08º$ northwards2, [the moon] has a maximum longitudinal parallax of about $0;30º$ … and when its [latitudinal] parallax is $0;58º$ southwards3, 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 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: 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 month4. 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 chapter5, 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”
Almagest Book II: Angle Between Ecliptic And Altitude Circle – Calculations
In the last post, we started in on the angle between the ecliptic and an altitude circle, but only in an abstract manner, relating various things, but haven’t actually looked at how this angle would be found. Which is rather important because Ptolemy is about to put together a huge table of distances of the zenith from the ecliptic for all sorts of signs and latitudes. But to do so, we’ll need to do a bit more development of these ideas. So here’s a new diagram to get us going.
Here, we have the horizon, BED. The meridian is ABGD, and the ecliptic ZEH. We’ll put in the zenith (A) and nadir (G) and connect them with an altitude circle, AEG6. Although it’s not important at this precise moment, I’ve drawn it such that AEG has E at the point where the ecliptic is just rising. Continue reading “Almagest Book II: Angle Between Ecliptic And Altitude Circle – Calculations”
Almagest Book II: Angle Between Ecliptic And Altitude Circle – Relationships
We’re almost to the end of book II. There’s really 2 chapters left, but the next one is almost entirely a table laying out the values we’ve been looking at here recently, so this is the last chapter in which we’ll be working out anything new.
In this chapter, we’ll tackle the angle between the ecliptic and a “circle through the poles of the horizon”. If you imagine standing outside, the zenith is directly overhead which is the pole for your local horizon. Directly opposite that, beneath you, is the nadir. If these two points are connected with a great circle, that’s the great circle we want to find the angle of with respect to the ecliptic. Because we measure upwards, from the horizon, along an arc of these great circles, to measure the altitude of a star, these are often called altitude circles.
But while we’re at it, Ptolemy promises that we’ll also determine “the size of the arc…cut off between the zenith and…the ecliptic.” In other words, because the ecliptic is tilted with respect to the horizon, the arcs between the two will be different.
To get us started, Ptolemy begins with the following diagram.
Continue reading “Almagest Book II: Angle Between Ecliptic And Altitude Circle – Relationships”