With both the planetary mean motion tables, and the tables of anomaly, we’re now ready to calculate planetary longitude.
As with before, Ptolemy explains how this is to be done, but doesn’t provide an example. As such, I’ll follow along with example $14$ from Toomer (in Appendix A).
To being, here’s the sample problem:
Compute the longitude of Mars on Nabonassar $886,$ Epiphi [XI] $15/16$ at $9$pm1.
Step 1: Calculate the Increase in Longitude and Anomaly from Epoch using the Mean Motion Table
From the tables for mean motion, we compute the mean positions in longitude and anomaly for the moment required (by addition, and casting out complete revolutions).
The time period between epoch and the date given for the example is $885$ years, $314$ days, $9$ hours.
So we begin by looking up those intervals in the mean motion tables and adding those to the epoch positions:
| Longitude | Anomaly | |
| Epoch Position | $3;32º$ | $327;13º$ |
| $810$ years | $138;15,13º$ | $24;48,59º$ |
| $72$ years | $92;17,21º$ | $250;12,21º$ |
| $3$ years | $213;50;43º$ | $145;25,31º$ |
| $300$ days ($10$ months) | $157;13;04º$ | $138;28,22º$ |
| $14$ days | $7;20,13º$ | $6;27,43º$ |
| $9$ hours | $0;11,47º$ | $0;10,23º$ |
| Total | $612;40,21º$ | $892;46,18º$ |
| Total – Full Revolutions | $252;40,21º$ | $172;46,18º$ |
Step 2: Determine the Angle of the Epicycle Center from Apogee (Taking into Account Precession)
Then, taking as argument the distance from the apogee of the eccentre at that moment to the mean position in longitude…
Ptolemy’s instructions here are quite subtle. They state that we need to determine the “distance from apogee of the eccentre at that moment“. This indicates we need to adjust the position of apogee for precession.
Taking the interval described above and a rate of $1º$ per century, I find a precession of $8;52º.$
This gets added to the apogee position from epoch of ♋︎ $16;40º$ ($106;40º$ ecliptic longitude) for a final epoch position for apogee of $115;32º$ ecliptic longitude or ♋︎ $25;32º.$
We subtract this from the position of the planet determined above to determine the distance from apogee:
$$252;40º – 115;32º = 137;08º.$$
Toomer refers to this value as the “mean centrum”, but as Ptolemy has not used this terminology, I will avoid it.
Step 3 – Enter this Value into the Anomaly Table and Look Up the Results in Columns 3 & 4 (Interpolating as Necessary) and Add them Together
…we enter the anomaly table belonging to the planet in question, and take the value for the longitudinal correction corresponding to that argument in the third column, together with the value (in minutes) in the fourth column (which has to be added or subtracted).
We’ll enter this into our table of anomaly and look up the results in columns $3$ and $4$. As usual, we’ll need to do some interpolating between rows.
For the third column, I find a value of $9;03º.$
For the fourth column, the values in this column barely change between rows for this example, and since our value we’re looking up is much closer to the row for $138º$ we can estimate that the value in this column should be $-0;41º$ without having to bother with the interpolation.
This value gets either added or subtracted from the one from column $3$ depending on the sign. Since this is negative, we subtract to find our total value from this step to be $8;22º.$
Step 4 – Add or Subtract this from the Longitude and Subtract or Add it to the Anomaly
We subtract the result from the mean longitude and add it to the anomaly if the above-mentioned argument for the longitude [i.e., the mean centrum] falls in the first column, but if it falls in the second column, we add the result to the longitude and subtract it from the anomaly, to get both positions corrected.
The value we found for the “above-mentioned argument for longitude” was $137;08$ which was found in the first column. Thus, we will subtract the result to the longitude and add it to the anomaly.
Subtracting the above from the mean longitude:
$$252;40º – 8;22º = 244;18º.$$
Adding to the anomaly:
$$172;46º = 8;22º = 181;08º.$$
This is our corrected anomaly.
Step 5 – Enter the Corrected Anomaly into Column 6
Then, we enter, with the corrected anomaly [counted] from the [epicyclic] apogee, into [one of] the first two columns, take the corresponding amount in the sixth column (the equation for mean distance), and write it down separately.
So, we take the $181;08º$ and look it up in the table from column $6$. Again, with a bit of interpolation, I find $2;10º$ as the result.
This gives us the equation of anomaly if the epicycle were at mean distance.
Step 6 – Determine the Correction to the Equation of Anomaly
Similarly, we enter with the amount for the mean longitude [i.e., the mean centrum] (which we used as argument at the beginning) into the same argument [columns].
Then, if [that argument] falls in the upper lines, which are closer to the apogee than that for the mean distance (this will be clear from the entries in the eighth column), we take the corresponding number of sixtieths in the eighth column, take, from the fifth column (for the [difference at] greatest distance), the entry on the same line as that for the equation at mean distance which was written down separately, form the fraction of that [entry for the] difference corresponding to the above number of sixtieths, and subtract the result from the amount which we wrote down separately.
But, if the argument of the above longitude [i.e., the mean centrum] falls in the lower lines, which are closer to the perigee than that for the mean distance, we take the corresponding number of sixtieths in the eighth column, as before, take from the seventh column (for the [difference at] least distance), the entry corresponding to the equation for mean [distance] which was written down separately, form the fraction of that difference corresponding to the above number of sixtieth, and add the result to the number we wrote down separately. The result will be the corrected equation [of anomaly].
This portion is pretty complicated and can’t really be broken down much as there’s some if/then logic in it. Specifically, we take the mean longitude from above ($137;08º$) and find it in the table.
Ptolemy tells us we need to ask if this falls in the upper or lower part of the table which, he says, will be obvious by looking at the eighth column. Here, Ptolemy is referring here to whether the values are positive or negative as there is a sharp divide. Our value is in the positive portion towards the bottom, so we’ll skip to the last of the three paragraphs2.
We then take the value from the eighth column. Interpolating between rows, I find $+37;09$3. This step is the same regardless of whether the value was positive or negative.
The next step seems to be very poorly worded to me. It’s clear that we’ll be looking a value up in column $7$. What value is not entirely clear. Ptolemy calls it the “equation for mean [distance] which was written down separately.”
If we look up a ways, the value we were instructed to “write down separately” was the “equation for mean distance” we found from column $6$, which we found to be $2;10º.$
However, Toomer’s example makes it clear that this value is not what we use4. Rather, we use the value that we’d entered to find that value ($181;08º$) and look that up in the seventh column.
Doing so, I find the result to be $0;53º.$
The last bit is rather poorly worded too. But, what Ptolemy is trying to say is that we take this value, multiply it by the value (in sixtieths) we found in the eighth column, and add it to the value we “wrote down separately.”
In other words, our final equation of anomaly is
$$c = c_6 + c_7 \cdot c_8 = 2;10º + 0;53º \cdot 0;37,09$$
$$= 2;43º.$$
You may have noticed that the value used here for what we got from column $8$ has changed slightly. Specifically, when we looked it up, it was $+37;09$ but, when I used it above, it was $0;37,09.$ This is because this value in the table is in sixtieths. Thus, we needed to divide by $60$ to put it in the same units as the other terms, which means sliding over each term one sexagesimal place to the right.
Step 7 – Add or Subtract the Correction to the Equation of Anomaly
We now need to determine whether the value we just calculated is additive or subtractive.
If the corrected anomaly is in the first column, we add that corrected equation to the amount for the corrected longitude. But, we subtract it if the corrected anomaly is in the second column. Using the result to count from the apogee of the planet at that moment, we reach its apparent position.
This instruction is quite tricky. We just calculated the corrected equation of anomaly. However, this step tells us to take the corrected anomaly, i.e., the angle about the epicycle from the epicycle’s apogee (after correction for the point of view of the observer), and determine if it is in the first or second column.
Thus, we need to be using the $181;08º$ value.
Since that is in the second column, the equation of anomaly we just calculated is subtractive. Hence, we subtract it from the corrected mean longitude:
$$244;18º – 2;43º = 241;35º.$$
This is in ecliptic longitude. So, if we wanted to translate this into a specific sign, it would be ♐︎ $1;35º$5.
And that’s it!
We now know how to calculate planetary longitudes which completes Book XI.
Looking forward, the entirety of Book XII is dedicated to retrograde motions and Book XIII to finding planetary latitudes.
But, before moving on to those books, I think I’m going to devote a little more time to this chapter. Specifically, when discussing how Ptolemy developed the tables of anomaly, there were several steps Ptolemy didn’t show because that sort of calculation has been explained before. For the time being, I have taken Ptolemy’s word at his values, but I’d like to go back and check them.
Additionally, Mercury’s model being different means that, even when Ptolemy did show some steps, I skipped Mercury since the calculations would be different and I couldn’t use the same formulas in the Google Sheet. Thus, I’d like to go back and compute them.
But other than that, Book XI is finished!
- This is the observation that Ptolemy cited when we calculated the size of Mars’ epicycle.
- Note that I have broken up the instructions into three paragraphs for the above quote. Toomer left it as a single paragraph.
- Note that there are no units here. Ptolemy is using this as a dimensionless factor to scale things.
- We will make use of that value in just a moment.
- Since this sample problem was based on an observation Ptolemy gave, we can compare to the observation to see how we did. Ptolemy reported that Mars was observed at ♐︎ $1 \frac{3}{5}º$ or ♐︎ $1;36º,$ so we have very good agreement.