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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.

Continue reading “Almagest Book IV: Alexandrian Eclipse Triple – Solar/Lunar Positions & Epicyclic Anomaly”

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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:

Continue reading “Almagest Book IV: Babylonian Eclipse Triple Geometry – Equation of Anomaly & The Mean Moon”

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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”

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Data: Stellar Quadrant Observations – 8/7/2020

Last night was a few days past a full moon, so without it rising until around 11:30, I figured that gave a good balance of observing while still making it home for some of the B3R Bardic. While the moon wasn’t out, seeing was still poor. Due to the this being the closest weekend to the peak of the Perseids (and next weekend’s weather not looking promising), there were a lot of people out at Broemmelsick. This resulted in lots of flashlights and headlights that prevent me from ever getting fully dark adapted. Similarly, there must have been more humidity than it felt like because the skyglow from St Louis and St Charles washed things out more than I anticipated.

Still, I was able to take about a dozen observations of stars and did a few of both Jupiter and Saturn, hoping the average would give good results for them. The data can be viewed by going to the Google Sheet I’ve set up. Overall, the night averaged out extremely well, with an average error in the RA of 0.18º (the equivalent of ~45 seconds late), and an average error in Dec of 0.02º which is really hard to beat. The standard deviations were a bit high this time so there was certainly some scatter, but overall quite pleased with the results.

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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”

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Data: Stellar Quadrant Observations – 7/13/20

In my last post on observing, I introduced the recent upgrades to the quadrant. Unfortunately, the initial tests weren’t good. There was a significant error in the azimuth averaging $~2º$ at $180º$ azimuth. I’ve since determined the reason behind this and implemented what I hoped was a correction. And this recent observing session tested it. Continue reading “Data: Stellar Quadrant Observations – 7/13/20”

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Data: Summer Solstice Observation 6/20/2020

This past weekend was the summer solstice. Since the sun achieves its highest declination on this day, which is related to the obliquity of the ecliptic, this is a fundamental parameter that I try to observe the altitude on the solstice when possible. I did this back in 2018 using the solar angle dial from Book I of the Almagest. This time, I brought out the quadrant as I did for the autumnal equinox in 2018 as well.

Continue reading “Data: Summer Solstice Observation 6/20/2020”

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Data: Stellar Quadrant Observations – 6/13/20 & 6/17/2020

NOTE: This post is actually being posted in December $2020$, as when I was writing my year end post, I was reminded I’d never posted this. This was largely due to having not decided what to actually do with the data for reasons you’ll see. Ultimately, I’ve decided to exclude the data completely.

In addition, since I hadn’t posted this, the discussion of the enhancements to the quadrant was actually featured in a post from July. But for readers who may read this in chronological order, I have instead moved that portion to this post.

It’s been awhile since I’ve posted about observing. Due to COVID, it’s been extra challenging to do any observations given the quarantine I’ve been doing my best to observe. I do still have a few friends that I see in a very small social circle, but between that and the weather last month, there hasn’t been a lot of opportunity. So when the weather called for a near-perfect night for observing and one of the people I trust was willing to assist, there was no way I was going to miss observing.

In addition, this night was extra special because it was going to allow me to test two new upgrades to the quadrant which I’ll start by introducing. Continue reading “Data: Stellar Quadrant Observations – 6/13/20 & 6/17/2020”

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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.

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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”