In our last post, we showed how it is possible to determine the equation of anomaly by knowing the motion around the epicycle and the double elongation. This, combined with the position of the mean moon1 gives the true position of the moon. As usual, Ptolemy is going to give us a new table to make this relatively easy to look up. But before doing so, Ptolemy wants to explain what this table is going to look like. Continue reading “Almagest Book V: Constructing the Lunar Anomaly Table”
Almagest Book V: Determining True Position of the Moon Geometrically From Periodic Motions
Now that we’ve revised our lunar model to include the position of the “mean apogee” from which we’ll measure motion around the epicycle, we need to discuss how we can use this to determine the true position of the moon.
As a general statement, we know how to do this: Take the position of the mean moon, determined by adding the motion since the beginning of the epoch, and add or subtract the equation of anomaly. The problem is that our revisions in this book mean the table for the lunar equation of anomaly we built in Book IV is no longer correct.
Instead, to determine the equation of anomaly, we’ll start with the motion around the epicycle2 and need to factor in the double elongation of the moon from the sun.
To see how to do so, let’s get started on a new diagram:
Almagest Book V: Second Determination of Direction of Epicycle
In the last post, we followed along as Ptolemy determined that the position of “apogee” used for calculating the motion around the epicycle is not the continuation of the line from the center of the ecliptic or center of the eccentre through the center of the epicycle. Rather, motion should be measured from the “mean apogee” which is defined from a third point opposite the center of the ecliptic from the center of the eccentre.
Ptolemy doesn’t give a rigorous proof for this and instead relies on proof by example. So in that last post, we went through one example, but in this post, we’ll do a second one
in order to show that we get the same result at the opposite sides of the eccentre and epicycle.
Continue reading “Almagest Book V: Second Determination of Direction of Epicycle”
Almagest Book V: The Direction of the Moon’s Epicycle
When we built the first lunar model, it was done using observations only at opposition, which is to say, during eclipses which only happen during the full phase. In the last few chapters, we looked at quadrature, which is to say, during first and third quarter moon and derived a second anomaly. But what happens if we consider the moon when it’s somewhere between those phases?
Ptolemy gives the answer:
[W]e find that the moon has a peculiar characteristic associated with the direction in which the epicycle points.
So what does that mean? Continue reading “Almagest Book V: The Direction of the Moon’s Epicycle”
Data: Stellar Quadrant Observations – 10/9/2020 & 10/13/2020
NOTE: This post is actually being posted in December since, as I was writing my year end summary, I realized I never posted these observations!
One of the Big Goals of this project is eventually follow Kepler’s methods to be able to derive the orbit of the planets. While I’m still nowhere near being ready for that as I’m still working on the Almagest, I know enough that I know the key observations are those taken when the planets are at opposition. Thus, the quadrant was originally built in time for the 2018 Mars opposition. But Mars only comes to opposition a little over once every two years.
And it’s now time for another opposition. Since I’ve been having some issues with the recently added azimuth ring, I wanted to make sure they were resolved before the opposition. Thus, I went out a few nights before opposition to see if I’d successfully resolved the issues. That night (10/9), I observed $29$ stars plus the three visible planets (Jupiter, Saturn, and Mars). As usual, a few of these observations got tossed, but the overall data looked pretty good. The average right ascension averaged $0.20º$ low and the declination came out $0.16º$ low which is fairly average. Unfortunately, I didn’t catch any new stars as this portion of the sky is pretty well mapped.
Then came the $13^{th}$ which was the important date of opposition. While waiting for Mars to get to a decent altitude, I was able to take observations of $20$ stars that I kept as well as Jupiter and Saturn.
Coming to Mars, I ended up taking $10$ readings in hopes it would average out well. Ultimately, the average for Mars came out with the right ascension being low by $0.11º$ but $0.44º$ low on the declination.
The overall data for the night wasn’t much better, but actually went the other way: The average right ascension was $0.41º$ high and the declination averaged $0.14º$ low.
As usual, the data is available in the Google Sheet.
Almagest Book V: Second Anomaly Eccentricity
So far, we’ve stated that the effect of the second anomaly is to magnify the first anomaly. In the last chapter, we worked out how much larger. Since this second model works by bringing the moon physically closer and further by offsetting the center of the lunar orbit with an eccentric and having that eccentre orbit the Earth, we can determine how far that center must be. In other words, the eccentricity of this second anomaly. Continue reading “Almagest Book V: Second Anomaly Eccentricity”
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 month3. 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”
Data: Stellar Quadrant Observations – 9/18/2020
Last time I went out observing, I met a Will Day, an photographer with quite a bit of experience in astrophotography. He asked if he could photograph Yseult and I as we worked and it ended up with one of the coolest images to come out of this project. But as cool as it was, it didn’t feel quite right since we were using the red headlamps and wearing mundane clothes with the period instrument.
Fortunately, Will was game to try something more planned to really capture the period feel. So Yesult and I went out last month, putting on our best late period garb. The results were beyond what I expected.
If you’ve ever wondered what it feels like to do this observing, this is it. Continue reading “Data: Stellar Quadrant Observations – 9/18/2020”
Almagest Book V: Model for the Second Anomaly
In the last chapter, we introduced an instrument capable of determining the ecliptic latitude and longitude of an object so long as the position of the true sun or fixed star is known. Using this on the moon, Ptolemy found
that the distance of the moon from the sun was sometimes in agreement with that calculated from the above [Book IV] hypothesis, and sometimes in disagreement, the discrepancy being at some times small and at other times great.
How so? Ptolemy provides details. Continue reading “Almagest Book V: Model for the Second Anomaly”
Almagest Book V: On the Use of an Astrolabe
Now that Ptolemy has described how to construct an astrolabe, he covers its usage. Specifically, Ptolemy begins when
both sun and moon could be observed above the earth at the same time.
As a brief reminder, what Ptolemy is really discussing here is what was necessary for him to make the observations that revealed the moon’s second anomaly. So what is given here is really in that context. As such, neither the sun nor moon is actually required for use of this instrument. In this case, Ptolemy is using the sun to align the instrument for use, and considering the moon to be the target, hence why the sun and moon are required. But, as we’ll see, a star can be used for alignment and there’s no reason one couldn’t be used as a target either.
Continue reading “Almagest Book V: On the Use of an Astrolabe”