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 sun1.
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.
But before starting in on the math, Ptolemy spends his first chapter doing some preliminary work, laying out what observations we’ll need to be able to produce a mathematical model. Specifically he notes is that eclipses are the key to an accurate model. The reason for this is that an eclipse requires a very specific geometric configuration: The moon must be exactly opposite or between the Earth and sun, neither above or below on the ecliptic. Trying to compare it to background stars or derive the location from a solar eclipse introduces error due to lunar parallax due to the Earth’s diameter being a non-negligible portion of the moon’s distance2. Ptolemy states this quite nicely stating,
the straight line drawn from the center of the earth (which is the center of the ecliptic_ through the center of the moon to a point on the ecliptic, which determines the true position ([as it does] for all bodies), does not in this case always coincide, even sensibly, with the line drawn from some point on the earth’s surface, that is, the observer’s point of view, to the moon’s center, which determines its apparent position. Only when the moon is on the observer’s zenith do the lines from the earth’s center and the observer’s eye through the moon’s center to the ecliptic coincide.
Thus, only lunar eclipses can serve as the foundation for a true theory of the moon’s motion
for the ordered and regular must necessarily precede and serve as a foundation for the disordered and irregular.
Specifically, Ptolemy states that we will use lunar eclipses. This chapter contains a short discussion about the cause of lunar eclipses:
[The moon], when it is diametrically opposite to the sun, it normally appears to us as lighted over its whole surface, since the whole of its illuminated hemisphere is turned towards us as well [as towards the sun] at that time. However, when its position at opposition is such that it is immersed in the earth’s shadow-cone (which revolves with the same speed as the sun, but opposite it), then the moon loses the light over a part of its surface corresponding to the amount of its immersion, as the earth obstructs the illumination by the sun. Hence it appears to be eclipsed for all parts of the earth alike, both in size [of the eclipse] and the length of the intervals [of the various phases].
Essentially, Ptolemy states that because the progression of a lunar eclipse looks the same no matter where you are on the earth, this is the preferred type to use because,
only lunar eclipse observations since, [only] in these does the observer’s position have no effect on the determination of the moon’s position. For it is obvious that, if we find the point on the ecliptic which the sun occupies at the time of mid-eclipse (which is, as accurately as we can determine, the moment at which the moon’s centre is diametrically opposite the sun’s in longitude), then at the same time of mid-eclipse the precise position of the moon’s centre will be the point diametrically opposite.
In short, from lunar eclipses we can know the position of the moon with high accuracy as half way through the eclipse, the moon will be exactly opposite the sun which we have developed a complete theory of in the last book3.
- To put it another way, we can say that there is apsidal precession.
- While Greek astronomers, including Ptolemy, attempted to calculate the distance to the moon, they were ultimately unsuccessful in coming up with a correct value. Regardless, even with a correct value, lunar parallax is still significant enough to consider.
- This theory obvious lacked precession of the equinoxes, but the solar theory was certainly good enough for Ptolemy’s time.