Tag: Ptolemy

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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 epicycle1

To begin, let’s sketch out the epicyclic lunar model2 with the three eclipses drawn on it.

Continue reading “Almagest Book IV: Babylonian Eclipse Triple Geometry – Radius of the Epicycle”

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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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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”

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Almagest Book IV: In the Simple Hypothesis of the Moon, the Same Phenomena are Produced by Both the Eccentric and Epicyclic Hypotheses

Our next task is to demonstrate the type and size of the moon’s anomaly.

In chapter 2 of this book, we spent quite a bit of time talking about the moon’s anomaly, describing a method by which Hipparchus could have used periods of eclipses to determine the anomaly’s period. While we never actually completed the method, Ptolemy still gave us the period Hipparchus supposedly derived. Now we’re going to put that to use to start building our first lunar model. Continue reading “Almagest Book IV: In the Simple Hypothesis of the Moon, the Same Phenomena are Produced by Both the Eccentric and Epicyclic Hypotheses”

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Almagest Book IV: Lunar Mean Motion Tables

The fourth chapter of Book IV takes what we worked on in the last post and expands it for convenient reference. As with the solar mean motion tables we created back in Book III, Ptolemy lays this one out in several intervals: 18 year periods, single year periods, months, days, and hours.

These tables essentially answer the question: “If the moon’s mean position was as X, if I waited Y interval of time, where would it be then?” Continue reading “Almagest Book IV: Lunar Mean Motion Tables”

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Almagest Book IV: Favorable Positions for Lunar Eclipse Pairs

Now that we’ve covered the positions the sun needs to be in to avoid its anomaly influencing things, and the positions to avoid for the moon, so its anomaly doesn’t influence things, we’ll look into some positions which would make it the most obvious if the above were. Ptolemy states this saying,

we should select intervals [the ends of which are situated] so as to best indicate [whether the interval is or is not a period of anomaly] by displaying the discrepancy [between two intervals] when they do not contain an integer number of returns in anomaly.

So which are those? Continue reading “Almagest Book IV: Favorable Positions for Lunar Eclipse Pairs”

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Almagest Book IV: The Lunar Anomaly and Eclipses

In the last post, we covered how the sun’s anomaly impacts things, but

we must pay no less attention to the moon’s [varying] speed. For if this is not taken into account, it will be possible for the moon, in many situations, to cover equal arcs in longitude in equal times which do not at all represent a return in lunar anomaly as well.

I’ll preface this section by saying this is, to date, by far the hardest section I’ve grappled with. I believe a large part of the difficulty came from the fact that Ptolemy is exceptionally unclear about what his goal is with this section. My initial belief was that it was to find the full period in which a the position of the sun and moon would “reset” as discussed in the last post. However, that’s something we’re going to have to work up to.

For now, we’re going to concentrate on just one of the various types of months. Namely, the “return in lunar anomaly” which is another way of saying the anomalistic month. Continue reading “Almagest Book IV: The Lunar Anomaly and Eclipses”

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Almagest Book IV: The Solar Anomaly and Lunar Periods

In the last post, we explored various lunar cycles from astronomers predating Ptolemy in which the moon reset its ecliptic longitude and anomalistic motion to define a full lunar period. These ancient astronomers did this by studying pairs of lunar eclipses3but Ptolemy notes that this method

is not simple or easy to carry out, but demands a great deal of extraordinary care

The reason for this difficulty is that, without careful consideration there can essentially be false positives of eclipses separated equally in time, but do not in fact, result in the moon returning to the same ecliptic longitude or same speed.

One of the reasons is that the conditions necessary to produce a lunar eclipse are also dependent on the sun, which has anomalistic motion. As such, it could be entirely possible that the moon could not have yet returned to the same ecliptic longitude as a previous eclipse, but the sun’s anomaly could cause an eclipse anyway.  Thus, a pair of eclipses may be equally separated in time, but

this is no use to us unless the sun too exhibits no effect due to anomaly, or exhibits the same [anomaly] over both intervals: for if this is not the case, but instead, as I have said, the equation of anomaly has some effect, the sun will not have travelled equal distances over [the two] equal time intervals, nor, obviously, will the moon.

To illustrate this, Ptolemy starts with an example. Continue reading “Almagest Book IV: The Solar Anomaly and Lunar Periods”