Category: Almagest

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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 eclipses1but 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”

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Almagest Book IV: Observations Necessary to Examine Lunar Phenomena

So far, Books I & II covered the motions of the sky and how to find the rising times of various points along the ecliptic. This was a good start because, in Book III, we explored the motion of the Sun which is confined to that ecliptic. So while the sun was somewhat complex because of its anomaly, it was still relatively simple. In Book IV, we’ll work on deriving a model for the motion of the moon.

Unfortunately, this is going to be a more complex model. Initially we could be concerned about the complexity of the model because the moon is not confined to the ecliptic – it bobbles above and below it by about 5º, but aside from discussing this briefly, we’ll safely ignore this for now and instead only worry about the moon’s motion in ecliptic longitude, that is to say, its projection onto the ecliptic.

However, what will complicate things is that one of the main things we consider regarding the moon, its phase, is also dependent on the sun. Thus, to consider the moon’s phases, we’ll need to be taking into consideration the sun’s anomalies at the same time we consider those of the moon. In addition, the points at which the moon is at apogee and perigee is not consistent as it was for the sun2.

The good news is that we’ve already explored the two models that Ptolemy uses to explain anomalies from the mean motion. As such, there will be far less exposition in this book and we’ll be able to dive in much more quickly. Continue reading “Almagest Book IV: Observations Necessary to Examine Lunar Phenomena”

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Almagest Book III: On the Inequality of Solar Days

Finally, we’ve arrived at the end of Book III where we’ve arrived at a well developed model of the solar motion. But before closing out, Ptolemy has one last chapter to discuss the inequality of the solar day. Ptolemy states the problem as follows:

the mean motions which we tabulate for each body are all arranged on the simple system of equal increments, as if all solar days were of equal length. However, it can be seen that this is not so.

What Ptolemy is really getting at here is that the term “day” is somewhat ambiguous. As such, the different ways by which we might measure a “day” are explored in this chapter. Continue reading “Almagest Book III: On the Inequality of Solar Days”

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Almagest Book III: On the Calculation of the Solar Position

This chapter is easily the shortest one in Book III. It literally consists of a paragraph (which I’ll quote in its entirely but break into two for ease of reading) that gives a very quick description of how one calculates the position of the sun at any given time from the epoch derived in the last chapter.

So whenever we want to know the sun’s position for any required time, we take the time from epoch to the given moment (reckoned with respect to the local time at Alexandria), and enter with it into the table of mean motion. We add up the degrees [and their subdivisions] corresponding to the various arguments [18-year periods, years, months, etc.], add to this the elongation [from apogee at epoch], 265;15º, subtract the complete revolutions from the total, and count the result forward from Gemini 5;30º rearwards through [i.e. in the order of] the signs. The point we come to will be the mean position of the sun.

Next we enter the same number, that is the distance from apogee to the sun’s mean position, into the table of anomaly, and take the corresponding amount in the third column. If the argument falls in the first column, that is if it is less than 180º, we subtract the [equation] from the mean position; but if the argument falls in the second column, i.e. is greater than 180º, we add it to the mean position. Thus we obtain the true or apparent [position of] the sun. Continue reading “Almagest Book III: On the Calculation of the Solar Position”