Having computed tables that give information about eclipses for the sun and moon at both greatest and least distance, Ptolemy now turns to creating a table to estimate the impact of the moon being at other positions about the epicycle besides apogee and perigee.
This is to be done by a table that uses the distance from apogee about the epicycle as the input and returns what proportion of the difference between apogee and perigee one should use, expressed in sixtieths. The good news is that we’re not really going to have to do any calculations here because we’ve actually already done them when we were putting together our Parallax Table.
Specifically, we had this post in which we calculated how much larger the parallax would be based on the moon not being at its mean position, again in sixtieths. These results were put into column $7$ – Sixtieths for Epicycle at Apogee.
However, in that table, the maximum effect when the moon was at at $90º$ or $270º$ from apogee. In this case, it’s going to be when the moon is $180º$ from apogee. Which means that the arguments need to get multiplied by $2$.
For example, in the Parallax Table, we have the entry for $6º$ which has a value of $0;00,42$ in column $7$. In our new table, we can look at the row corresponding to $12º$, where we again find that value of $0;00,42$1.
This new table will be in $6º$ intervals. This means that every other entry in the new table will fall between entries in our Parallax Table. For example, let’s say we want to look up the value in our new table of $18º$. We should then look up half of that in the Parallax table which would be $9º$. However, that table is in $2º$ increments so we’d need to interpolate.
And that’s all there is to this table.
There is one more table we’ll explore in this chapter which has to do with the relationship between the linear amount of the coverage in an eclipse and the total area being obscured.
- As a quick note, I have diverged from Ptolemy’s notation, both in the Parallax table as well as in recent posts. Ptolemy has often changed over to displaying his values in arcminutes instead of degrees. Here, this is a ratio so there aren’t units, but the same idea applies: Ptolemy displays the units as minutes so $0;00,42$ would instead get displayed as $0;42$. It’s basically shifting the sexagesimal decimal point over one place. It doesn’t matter so long as we note that this is what we’ve done and Ptolemy does indeed have a tick mark indicating that he has done so. But I have shied away from this for consistency.