In the third chapter of Book IV, we’re going to do some more work with the lunar motion. We already did a fair amount of that at the beginning of the last chapter in which Ptolemy essentially quoted some numbers for various lunar periods from his predecessors. Now, we’ll extend those. Continue reading “Almagest Book IV: On the Individual Mean Motions of the Moon”
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”
Almagest Book IV: Mean Periods of the Moon
When we developed the model for sun, one of the first things we did was to determine the rate of mean motion for the sun to which the anomalistic motion could be added. However, the moon is going to be more complicated for several reasons. As such, before going further, we’ll need to spend some time examining the various lunar periods. Continue reading “Almagest Book IV: Mean Periods of the Moon”
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”
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”
Almagest Book III: On the Epoch of the Sun’s Mean Motion
We’ve come a long way in this book establishing a working model for solar motion. In fact, we’ve explored two models and derived a table that shows how far the sun would be away from its mean motion based on the mean position. However, at this point, everything has been done in terms of apogee and perigee.
In this chapter, we’ll be defining the “epoch“. What that means is that Ptolemy is going to pick a point in time, and define where the sun was on that date. Then, applying the tools we have developed in this book, we’ll be able to determine where the sun is at any other given date using the epoch as the starting point. For those that like things in a bit more mathy terms, it’s the location of the sun on the eccentre at time = 0, wherein Ptolemy will decide what that date is. To get there, we’ll first establish precise point on the eccentre at a known point in time, and then use the methods from this chapter to go backwards until we get to the chosen epoch date. Continue reading “Almagest Book III: On the Epoch of the Sun’s Mean Motion”
Almagest Book III: Table of the Sun’s Anomaly
In this chapter, Ptolemy lays out the Table of the Sun’s Anomaly. He actually discusses the table layout at the end of the last chapter, but it seems to me to make more sense to group it with the table so I’ll do that here. Continue reading “Almagest Book III: Table of the Sun’s Anomaly”
Almagest Book III: Equation of Anomaly from Perigee using Epicyclic Hypothesis
As the final post for this chapter, we’ll examine the equation of anomaly for angles measured from perigee in the epicyclic model. Again, we’ll start with a basic diagram for the epicyclic model and this time, we’ll drop a perpendicular onto $\overline{DA}$ from H to form $\overline{HK}$.
Continue reading “Almagest Book III: Equation of Anomaly from Perigee using Epicyclic Hypothesis”
Almagest Book III: Equation of Anomaly from Perigee using Eccentric Hypothesis
In the past couple Almagest posts, we’ve demonstrated that you can derive the equation of anomaly if you know the angle from apogee of either the mean or apparent motion using either the eccentric or epicyclic hypothesis3. Now, we’ll do the same using the angular distance from perigee again using 30º as our example. Continue reading “Almagest Book III: Equation of Anomaly from Perigee using Eccentric Hypothesis”
Almagest Book III: Equation of Anomaly from Apogee using Epicyclic Hypothesis
Despite Ptolemy demonstrating the equivalence of the eccentric and epicyclic hypotheses many times now, he sets out to prove that we can arrive at the same equation of anomaly using the epicyclic hypothesis as we do the eccentric. As we did for the eccentric, we’ll again use the case when the mean motion is 30º from apogee. Continue reading “Almagest Book III: Equation of Anomaly from Apogee using Epicyclic Hypothesis”