In the previous chapter, we worked out the parallax for the sun and the moon at 4 different positions. Next, we determined how we might make some corrections for the moon if it were between the positions we described, breaking the corrections into ones for the epicycle and for the eccentre. From that work, Chapter $18$ presents the results in tabular format. Since this one is pretty wide, I’ve again made the decision to put it into a Google Sheet.
In the previous few posts, I noted that Ptolemy had several comments on the construction of this table that I’ve chosen to skip, but now that we have the table in front of us, we can revisit. I feel this makes more sense since we can have the table for context.
First, let’s go back to the parallaxes we calculated without the corrections. This was done for an angle of $30º$ from the zenith and we obtained the following:
$\angle{ADL}_{Sun} = 0;01,25º$
$\angle{ADL}_{1a} = 0;27,09º$
$\angle{ADL}_{1b} = 0;32,27º$
$\angle{ADL}_{2a} = 0;40,00º$
$\angle{ADL}_{2b} = 0;52,30º$
These values get entered into the table on the row where the argument in column $1$ is $30º$. However, if you actually look at this line, you’ll see that for $1b$ and $2b$, these values don’t actually appear.
Instead, what Ptolemy has done is reported the difference $1b – 1a$ in column $4$ and the difference $2b – 2a$ in column $6$. Neither Ptolemy, Toomer, Neugebauer, or Pedersen give a reason for this odd way of entering this data.
Regarding these calculations, Ptolemy also states he only did them in intervals of $6º$ indicating the figures between these multiples were determined by interpolation. Toomer notes that the values for the moon’s parallax for the third and fourth limits (columns $5$ and $6$ respectively) are “drastically” rounded and suggests that this is likely because the intended use for these is computing solar eclipses which must happen near the apogee of the eccentre.
Moving on to the values we found for the corrections for the epicycle, we first did the calculation for the moon $60º$ from apogee and then again $60º$ before perigee which is $120º$ after apogee. However, when we look in the table for the values we calculated, we don’t find them where we’d expect.
Rather, we find the first value on the line for $30$ and the second on the line for $60$. The reason is that Ptolemy’s really working to try to squeeze as much into a single table as possible. Columns $2$ – $6$, are measured in terms of distance from the zenith, which can only go up to $90º$. Meanwhile columns $7$ – $9$ are around the epicycle and eccentre which can go up to $180º$ before doubling back. Thus, the two values don’t really fit into the same table well. If he wanted to, he could have had the table go all the way up to $180º$ but then columns $2$ – $6$ would be blank – a waste of valuable parchment. He could also have broken columns $7$ – $9$ out into a separate table, but chose not to. Instead, his solution was simply to divide all the angles in columns $7$ – $9$ by $2$. Thus, the values on the line for $30$ are for $60º$.
Before moving on, let’s remind ourselves of what these numbers represent. They do not represent an actual angle. Rather, this was the proportion of the diameter of the epicycle, expressed in sixtieths, that the moon was closer to the observer than at the first limit (when the mean moon was at apogee) for column $7$, and closer than the third limit (when the mean moon was at perigee) for column $8$. The values inserted are, in both cases, the angular distance from apogee on the epicycle. We’d also calculated values for the angular distance before perigee on the epicycle, but Ptolemy did not record those in the table.
Lastly we have the correction for the changing distance due to the eccentre which we again calculated for an angular distance from apogee of $60º$ but find on the line for $30º$.
But let’s recall that the angular distance from apogee is the double elongation. So for column $9$ the value we’ll enter here is either half the angular distance from apogee or the mean elongation.
Again, reminding ourselves what this was, it was the ratio of the distance closer or further the moon will be brought to the total possible change, expressed in sixtieths and the argument was measured from apogee of the eccentre.
That covers how the table is laid out. In the next chapter, we’ll explore how to use this to calculate actual parallaxes and that’s the last chapter of Book V.
