As we saw for the sun, to be able to use the tables of mean motion to predict the position of the moon, we’ll need to know where the moon was at a specific point in time. Ptolemy chooses as that point in time as the beginning of the first year of the Nabonassar reign. To determine the position of the moon on this date, Ptolemy starts with the second eclipse of the Babylonian triple we discussed in the last chapter and then calculates backwards. Continue reading “Almagest Book IV: The Epoch of the Mean Motions of the Moon in Longitude and Anomaly”
Almagest Book IV: Correction of the Mean Motions of the Moon in Longitude and Anomaly
Towards the beginning of Book IV, Ptolemy went through the methodology by which the various motions in the lunar mean motion table could be calculated. But if you were paying extra close attention, you may have noticed that the values that ended up in the mean motion table didn’t actually match what we derived. Specifically, for the daily increment in anomaly, we derived $13;3,53,56,29,38,38^{\frac{º}{day}}$. But in the table, we magically ended up with $13;3,53,56,17,51,59^{\frac{º}{day}}$. Identical until the 4th division.
So what gives? Why did Ptolemy derive one value and report another?
In Chapter 7, he gives the explanation: He found the value needed to be corrected and did so before he put it in the table. But before he could explain to us how, we needed to cover the eclipse triples we did in Chapter 6. So how do we apply them to check the mean motions? Continue reading “Almagest Book IV: Correction of the Mean Motions of the Moon in Longitude and Anomaly”
Almagest Book IV: Alexandrian Eclipse Triple Geometry – Equation of Anomaly & The Mean Moon
We’re almost finished with chapter 6. All that’s left is to determine the position of the mean moon during one of the eclipses which will tell us the equation of anomaly at that point. To do so, we’ll add a few more points to the image we ended the last post with:
Almagest Book IV: Alexandrian Eclipse Triple – Radius of the Epicycle
Continuing on with Ptolemy’s check on the radius of the epicycle, we’ll produce a new diagram based on the positions of the Alexandrian eclipses. However, instead of doing it piece-by-piece as I did when we explored the Babylonian eclipses, I’ll drop everything into a single diagram since we already have some experience and the configuration for this triple is a bit more for forgiving on the spacing:
Continue reading “Almagest Book IV: Alexandrian Eclipse Triple – Radius of the Epicycle”
Almagest Book IV: Alexandrian Eclipse Triple – Solar/Lunar Positions & Epicyclic Anomaly
Modern commentary on Ptolemy often downplays the Almagest because it is certainly a work that relied heavily on the work that astronomers before him. While we no longer have a thorough record of those predecessors, it seems that few historians think much of the Almagest was truly novel1. But I would hasten to remind that, while Ptolemy stood on the shoulders of those who came before, he certainly climbed there on his own, not simply accepting their results, but doing his best to validate them.
And we’re about to get a big dosing of that, because all the work we’ve done in the past three posts, we’ll be redoing with a new set of eclipses observed by Ptolemy himself, allowing for an independent check on the important value of the radius of the epicycle.
Almagest Book IV: Babylonian Eclipse Triple Geometry – Equation of Anomaly & The Mean Moon
In the last post, we were able to determine the radius of the epicycle when the radius of the deferent is $60^p$. It took a lot of switching between demi-degrees contexts, but in the wake of all that math, we’re left with a mess of lines and arcs that we’ve already determined. So in this post, we’ll use that starting point to go just a little further and determine the position of the mean moon, specifically for the second eclipse. To do so, we’ll need to add a bit more to the configuration we ended with last time:
Almagest Book IV: Babylonian Eclipse Triple Geometry – Radius of the Epicycle
In the last post, we introduced three eclipses from Babylonian times which we used to build a couple intervals: The first eclipse to the second, and the second to the third. Using those, we used lunar and solar mean motion tables to figure out the solar position, as well as its change. Since the moon must be opposite the sun in ecliptic longitude for an eclipse to occur, we used the change in solar position to determine the true change in lunar position in these intervals. From that, we could compare that to the mean motion to determine how much of it must be caused by the lunar anomaly. But while we’ve determined this component, we haven’t done anything with them yet.
So in this post, we’ll start using these to answer several questions that will build out the details of the model. Specifically, we want answers to questions like what is the radius of the epicycle? Where, in relation to the ecliptic was the mean moon during these eclipses and what was the equation of anomaly? That’s a lot of information to extract so I’m going to try to break it up a bit and in this post, we’ll only tackle the radius of the epicycle2
To begin, let’s sketch out the epicyclic lunar model3 with the three eclipses drawn on it.
Continue reading “Almagest Book IV: Babylonian Eclipse Triple Geometry – Radius of the Epicycle”
The SCAdian Astronomical Epoch – Solar Position
In Book III of the Almagest, Ptolemy developed a methodology by which the solar position could be predicted. This method had a few key components. This included determining the mean motion, the equation of anomaly, and calculating a start date/position to which those could be applied.
Unfortunately, trying to use Ptolemy’s solution would no longer work in the present day. While the mean motion and equation of anomaly are still fine, precession of the equinoxes and other small effects over the past ~1,900 years mean too much has changed. So while we can’t use the exact results of Ptolemy’s it should be possible to recreate his methodology with a more recent starting date again allowing for reasonably accurate predictions of the solar position.
So for my entry to this year’s Virtual Kingdom Arts & Science competition, I did. Continue reading “The SCAdian Astronomical Epoch – Solar Position”
Almagest Book IV: The Babylonian Eclipse Triple – Solar/Lunar Positions & Epicyclic Motion of Anomaly
So far in this book we’ve covered the ancient Greek values for the various motions of the moon. For the most part Ptolemy has accepted them as authoritative, but to demonstrate some of the methodology, Ptolemy wants to walk us through one: The lunar anomaly.
We shall use, first, among the most ancient eclipses available to us, three [which we have selected] as being recorded in an unambiguous fashion, and, secondly, [we shall repeat the procedure] using among contemporary eclipses, three which we ourselves have observed very accurately. In this way our results will be valid over as long a period as possible, and in particular, it will be apparent that approximately the same equation of anomaly results from both demonstrations, and that the increment in the mean motion [between two sets of eclipses] agrees with that computed from the above periods.
Almagest Book IV: Uneven Ratios in Epicyclic and Eccentric Model
In the last post, we explored how the eccentric and epicyclic models could produce the same result even if you didn’t have the same period of anomaly and ecliptic longitude. This was done by allowing the center of the eccentre to rotate around the Earth. In this post, we’ll explore how they can still produce the same result even
if [the members of] the ratios are unequal, and the eccentre is not the same size as the deferent…provided the ratios are similar Continue reading “Almagest Book IV: Uneven Ratios in Epicyclic and Eccentric Model”