Chapter 11 of Book 5 is one of those rare chapters that’s blessedly free of any actual math. Instead, Ptolemy gives an overview of the problem of lunar parallax, stating that it will need to be considered because “the earth does not bear the ratio of a point to the distance of the moon’s sphere.” In other words, the ratio of the diameter of the earth to the distance of the moon isn’t zero.
However, this does pose an interesting question. We’ve previously given the radius of the eccentre as $49;41^p$, but we haven’t given the radius of the Earth in the same units. Thus, how can this ratio even be taken to know this?
Ptolemy states it can be done from observation:
[T]he only appropriate procedure is, first given some particular parallax, to find the ratio of the distance. For it is possible to make an observation of a parallax of this kind by itself, but quite impossible to determine the amount of distance [by itself].
In other words, we’ll just assume for now that parallax is observable and go out and try to measure it. If we can, then we can calculate the ratio. This is something that, for the moon, should be relatively easy, whereas trying to determine the distance in a more concrete set of units is impossible.
Ptolemy then gives a bit of history, discussing Hipparchus’ attempt at this problem, who approached it by trying to find the distance to the sun instead of the moon. Which makes sense because, in the same set of arbitrary units, we’ve said the radius of the solar eccentre is $60^p$. So if you know the distance to the sun, you can determine the distance to the lunar sphere from the ratio. However, he notes that Hipparchus was inconsistent about his observations of the sun, finding sometimes, it had a parallax, but other times it did not. Thus,
the ratio of the moon’s distance came out different from him for each of the hypotheses he put forward.
Ptolemy decides he’s not going to get involved in this argument about the solar parallax at all,
for it is altogether uncertain in the case of the sun, not only how great its parallax is, but even whether it has any parallax at all.
Instead, Ptolemy is indicating that, to determine the true distance to the moon in a set of units that’s meaningful here on earth1, we’ll measure the moon’s parallax directly. But to do so, we’ll need a new instrument which we’ll explore in the next chapter.
Before closing out, I do want to take a moment to preface the remainder of this book with a warning from Neugebauer’s History of Ancient Mathematical Astronomy. Because of the small angles being measured here, there are significant inaccuracies that get rather swept under the rug. The result is “one of the most unsatisfactory topics in the whole Almagest, still further aggravated by quite unnecessary trigonometric inaccuracies in the determination of the components of parallax.”
Yikes!
