Despite the second chapter of Book X ostensibly being about the epicycle of Venus, Ptolemy instead continues discussing the line of apsides. Specifically, we determined that the line of apsides runs through the points $25º$ into Taurus and Sagittarius, but have not determined which is the apogee and which is the perigee1.
In this post, we’ll follow along as Ptolemy determines which of these two points is which.
To do so, Ptolemy again gives us two observations:
[$1$] In the [observations] given to us by Theon, we find that in the thirteenth year of Hadrian, Epiphi [XI] $2/3$ in the Egyptian calendar [Mat $19/20$ $129$ CE], Venus was at its greatest elongation from the sun as morning star, and was $1 \frac{2}{5}º$ in advance of the straight line through the foremost of the three stars in the head of Aries and the star on the hind leg, while its distance from the foremost star of those in the head was approximately double its distance from the star on the leg. Now, at that time, according to us, the foremost star of the three in the head of Aries had a longitude of $6 \frac{3}{5}º$ into Aries and is $7\frac{1}{3}º$ north of the ecliptic, while the star in the hind leg of Aries had a longitude of $14 \frac{3}{4}º$ and is $5 \frac{1}{4}º$ south of the ecliptic. Therefore, the longitude of Venus was $10 \frac{3}{5}º$ into Aries and it was $1 \frac{1}{2}º$ south of the ecliptic. Hence, since the longitude of the mean sun at that time was $25 \frac{2}{5}º$ into Taurus, the greatest elongation from the mean was $44 \frac{4}{5}º$.
[$2$] In the twenty-first year of Hadriang, Tybi [V] $2/3$ in the Egyptian calendar [November $18/19$ $136$ CE], in the evening, we observed Venus at its greatest distance from the sun: When sighted with respect to the stars in the horns of Capricorn, it was seen to occupy [a position] $12 \frac{5}{6}º$ into Capricorn, while the longitude of the mean sun was $25 \frac{1}{2}º$ into Scorpio. Hence in this position, the greatest elongation from the mean comes out as $47 \frac{1}{3}º$.
What Ptolemy has presented is an observation when Venus’ mean position2 is near the two endpoints of the line of apsides, $25º$ into both Taurus and Scorpio.
What we can see is that the observation in Scorpio has the greater elongation meaning this is perigee making the point in Taurus apogee.
Hence, it is clear that the apogee lies $25º$ into Taurus, and the perigee $25º$ into Scorpio.
Ptolemy then tells us he has checked these values against other points along the ecliptic to rule out a double perigee like that of Mercury:
Furthermore, it has also become plain to us that the eccentre of Venus carrying the epicycle is fixed, since nowhere on the ecliptic do we find the sum of the greatest elongations from the mean on both sides to be less than the sum of both in Taurus, or greater than the sum of both in Scorpius.
Finding no instances in which the sum of the elongations higher than these two points, he concludes that the eccentre is not moved as it was for Mercury (although it still precesses with the sun).
I’ll break here as it’s a nice stopping point, but in the next post, we’ll get to the main purpose of this chapter – determining the size of the epicycle.
- Recall that Mercury, only had an apogee and a double perigee. However, this is not the case for Venus which has only a single perigee.
- tracked by that of the sun since Venus’ mean is always linked to that of the sun.