Finally we’re at the end of Book II. In this final chapter1, Ptolemy presents a table in which a few of the calculations we’ve done in the past few chapters are repeated for all twelve of the zodiacal constellations, at different times before they reach the meridian, for seven different latitudes.
Computing this table must have been a massive undertaking. There’s close to 1,800 computed values in this table. I can’t even imagine the drudgery of having to compute these values so many times. It’s so large, I can’t even begin to reproduce it in this blog. Instead, I’ve made it into a Google Spreadsheet which can be found here.
First, let’s explore the structure.
I’ve chosen to break each latitude into a separate table for ease of use. You can find these tabs along the bottom of the sheet. These seven latitudes actually refer back to the canonical seven standard climata that were commonly referred to in Hellenistic times to divide the known world.
Beneath that, we see sections for each of the 12 zodiacal constellations. The Hour column denotes the number of hours before or after the point in the given constellation is on the meridian.
The Arc column is the angular distance from the first point of the given constellation, to the zenith, as measured by the arc of the great circle that would connect the two.
The next two columns we should pause and discuss a bit because, as we noted previously, the intersection of two great circles actually creates four angles. So we need to be careful to understand which one Ptolemy’s referring to when he said we’ve calculated it for the angle that
lies to the rear of the intersection of the circles and to the north of the ecliptic.
This is because the next two columns, the East and West angles, are what this statement refer to. The East angle is when the point on the ecliptic we’re considering is east of the meridian, and the West angle is when it’s to the west.
The part about “north of the ecliptic” is straightforward enough: The ecliptic divides the celestial sphere into two halves. One of them contains the north celestial pole. Next, we are here defining forwards and rearwards in terms of the motion of the stars which rotate east to west. So if we imagine ourselves inside the celestial sphere, it would be the angle to the north of the ecliptic and more eastwards. In other words, up and to the left as highlighted in this picture.
However, if we imagine ourselves outside the celestial sphere, it is up and to the right as we can see in this illustration from Neugebauer’s History of Ancient Mathematical Astronomy2.
It’s also important to note that, in the picture above, from inside the celestial sphere, the angle in question is exterior to the triangle formed by the three lines. However, if we wait until the same point is on the other side of the meridian, it will become an interior angle.
It is still “up and left” when viewed from inside the celestial sphere.
If you’ve already looked at the table, you’ll notice for the case of Meroe, there’s a bit of an oddity that some of the angles have a N after them. This is because Meroe’s latitude is less than the obliquity of the ecliptic and results in the ecliptic sometimes lying in the northern sky. The translation I’m using (Toomer), indicates that this notation may not be in Ptolemy’s original text since Ptolemy describes the layout of the table, but is suspiciously silent on any such notation. As such, it is entirely possible that this notation was added later. But since we have no extant copies of the original work, it’s impossible to know for sure.
As noted above, this table is an enormous mathematical undertaking. As reproduced in the translation I’m using, the translator notes that the table does contain computational errors, stating
recomputation of the data using modern formulas reveals considerable inaccuracies in Ptolemy’s values. The zenith distances are generally correct to within 2′, although occasional errors of up to 10′ occur, but the angles regularly show errors of 10′ and occasionally as much as 1º (e.g. Parallel through Middle Pontus, Gemini, 1 hour from noon, eastern angle: text 99;49º; computed 100;54º).
I have chosen to follow the path that Toomer took and not make the corrections directly in the table, as we will certainly be referring to this table in the future, and if we do not use the same values Ptolemy did, it will certainly make future understanding more difficult.
Ptolemy closes out the chapter with a promise to do one last thing:
the only remaining topic…is to determine the coordinates in latitude and longitude of the cities in each province which deserve note, in order to calculate the [astronomical] phenomena for those cities. However, the discussion of this subject belongs to a separate, geographical treatise.
This promise, is not fulfilled in the Almagest, but is completed in another comprehensive work: Ptolemy’s Geography. Instead, Ptolemy states that, for this work,
For the time being we take the locations [of the cities] for granted, and think it appropriate to add no more than the following. Whenever we are given the time at some standard place, and we undertake to determine what the corresponding time is at another place, then, if they lie on different meridians, we have to take the distance between the two places in degrees, measured along the equator, and determine which of them is to the east or west, and then increase or decrease the time at the standard place by the same number of time-degrees, to get the corresponding time at the required place. We increase if the required place is the further east, and decrease if the standard [place is the further east].
In other words, Ptolemy acknowledges here that the locations used for the cities and climata in the Almagest aren’t all that precise, but provides here a fairly simple correction should the precise latitude and longitude be known.
And there you have it. That is the end of Book II of the Almagest.
Perhaps somewhat surprisingly, we won’t be coming back to this for quite awhile. In Book III, we’ll tackle the solar model. Since its motion defines the ecliptic, we’re not going to have too much cause to be looking at the angles between the ecliptic and much else. Then, in Book IV, we’ll get to the moon but keep things relatively simple. It’s not until Book V that we start discussing parallax, which is dependent on the angle from the zenith which we’ve calculated here.
So it might seem surprising that in Book I and II we’ve done all this work, just to set it aside. But I feel like these books work well as a sort of set of prerequisites before getting into the meat so as not to interrupt the flow later on.
- In reality, the description of the table was in the last chapter and this chapter is only the table and a few closing notes. But it made little sense to me to separate the description of the table from the table itself.
- For this figure, Neugebauer was talking about the angle in an abstract sense at the beginning of II.10 in which we looked at numerous angles. This is why the segment of the great circle, not the ecliptic, extending from $P$ doesn’t point to the north celestial pole.