As usual, after completing a few example calculations, Ptolemy lays out the table discussed in the last post. I’ve again made this table available as a Google Sheet.
We can briefly follow Ptolemy’s description of this table in which
we computed the sizes of the angles for the other [integer] digits [of magnitude], [always taking] that angle which was less than a right angle, in units where one right angle equals $90ยบ$ (corresponding to the graduation of the quadrant of the horizon). We constructed a table with $22$ lines and $4$ columns. The first column contains the digits of actual obscuration, measured along the diameter, found for mid-eclipse; the second contains the angles occurring at solar eclipses at the moment of the beginning of the eclipse and the moment of the end of emersion; the third contains the angles occurring at lunar eclipses at the moment of the beginning of the eclipse and of the end of emersion; and the fourth contains the angles occurring at lunar eclipses at the moment of the end [of the partial phase of] the eclipse and the moment of the beginning of emersion.
The table is quite straightforward and we can see that values we calculated in the last post.
What might not be immediately obvious is why columns $2$ and $4$ are incomplete. This is because for solar eclipses (in column $2$, they can never be “more than complete” since the sizes of the lunar and solar discs are so close. Thus, an obscuration of $12$ digits is the maximum. For the moon, they can be, so Ptolemy calculates these in column $3$. In column $4$, this is specifically for the first and last moments of totality, so anything less than $12$ digits doesn’t make any sense.
As a quick note, the circular table we explored in the last post was actually part of this chapter as well, but is made more sense to display it there.
