Having laid out the model for all of the planets in the last book, Ptolemy is ready to dive right into Venus without much for exposition. However, Ptolemy immediately runs into problems.
The available ancient observations did not supply us with exact pairs of positions [suitable] for this purpose, but we used contemporary observations for our approach, as follows.
Ptolemy is going to be using precisely the same technique he used to determine the line of apsides for Mercury: taking a pair of observations when the planet is at greatest elongation on either side of the sun (thus, both as a “morning” and “evening” star) as this exploits a symmetry in which the line of apsides will fall directly between these two observations.
Unfortunately, Ptolemy states that he couldn’t find any observations from the ancient (i.e., Babylonian) astronomers to fulfill this purpose. Instead, he is forced to use observations that are from Theon of Alexandria.
Conceptually, this shouldn’t pose a problem as there’s no strong need to be have observations separated by great spans of time as we would for determining mean motions. However, there’s serious problems with the observations – the dates he uses are off by as much as three weeks and the observations aren’t really at greatest elongation.
Unfortunately, the sources that seem to really explore this are the same ones I mention at the end of this post1 for which I have been unable to source a copy.
So, for now, I’ll just have to take Ptolemy at face value:
[$1$] Among the observations given to us by the mathematician Theon, we found one recorded in the sixteenth year of Hadrian, on Pharmouthi [VIII] $21/22$ in the Egyptian calendar [March $8/9$ $132$ CE], at which, he says, the planet Venus was at its greatest elongation as evening-star from the sun, and was the length of the Pleiades in advance of the middle of the Pleiades; and it seemed to be passing it a little to the south. Now, according to our coordinates, the longitude of the middle of the Pleiades at that time was $3º$ into Taurus, and its length is about $1 \frac{1}{2}º$: so clearly Venus’ longitude at that moment was $1 \frac{1}{2}º$ into Taurus. So, since the longitude of the mean sun at that moment was $14 \frac{1}{4}º$ into Pisces, the greatest distance from the mean as an evening star was $47 \frac{1}{4}º$.
[$2$] In the fourth2 year of Antoninus, Thoth [I] $11/12$ in the Egyptian calendar [July $29/30$ $140$ CE], we observed Venus at its greatest elongation from the sun as morning-star. It was [the breadth of] half a full moon to the north-east of [the star in] the middle knee of Gemini. At that moment, the longitude of the fixed star, according to us, was $18 \frac{1}{4}º$ in Gemini, so Venus was about $18 \frac{1}{2}º$ [into Gemini]. And the mean sun was $5 \frac{3}{4}º$ into Leo. So the greatest distance as morning-star was the same amount as before, $41 \frac{1}{4}º$.
If we convert these both to ecliptic longitude, for ease of discussion, we get that the first observation was at $344 \frac{1}{4}º$ and the second at $125 \frac{3}{4}º$ in the second.
That’s a difference of $218 \frac{1}{2}º$, half of which is $109 \frac{1}{4}º$. Adding that to the second observation, we find that the line of apsides must then go through an ecliptic longitude of $235º$ or $25º$ into Scorpio with the other side of the line being $180º$ away or $25º$ into Taurus.
Ptolemy agrees, stating,
Therefore, since the mean position was $14 \frac{1}{4}º$ into Pisces at the first observation and $5 \frac{3}{4}º$ into Leo at the second, and the point on the ecliptic half way between these falls [at either] $25º$ into Taurus [or] $25º$ into Scorpio, the diameter through the apogee and perigee must got through the latter [points].
To check this, Ptolemy adds another set of observations:
[$3$] Similarly, in the [observations we received] from Theon, we found that in the twelfth year of Hadrian, Athyr [III] $21/22$ in the Egyptian calendar [October $11/12$ $127$ CE], Venus, as a morning-star, had its greatest elongation from the sun when it was to the rear of the star on the tip of the southern wing of Virgo by the length of the Pleiades, or less than that amount by its own diameter; and it seemed to be passing the star one moon to the north. Now, the longitude of the fixed star at that time, according to us, was $28 \frac{11}{12}$ into Leo: hence, the longitude of Venus was about $0 \frac{1}{3}º$ into Virgo. And the mean sun was at $17 \frac{26}{30}º$ into Libra. So the greatest elongation from the mean as morning-star was $47 \frac{16}{30}º$.
[$4$] In the twenty-first year of Hadrian, Mechir [VI] $9/10$ in the Egyptian calendar [December $25/26$ $136$ CE, in the evening, we observed Venus at its greatest elongation from the sun. It was in advance of the northernmost star of the four which almost form a quadrilateral (behind the star to the rear of and on a straight line with the [two] in the groin of Aquarius): [its distance from the star was] about two-thirds of a full moon, and it seemed about to obscure the star with its light. Now, the longitude of the fixed star at that time, according to us, was $20º$ into Aquarius, and the mean sun’s longitude was $2 \frac{1}{15}º$ into Capricorn.
Again, I’ll convert these observations to modern ecliptic longitude for convenience. Doing so, we find the first observation placed the sun at $197 \frac{26}{30}º$ and the second at $272 \frac{1}{15}º$.
The difference, then, is $75 \frac{1}{5}º$, half of which is $37 \frac{1}{10}º$.
Adding that to the first observation, we find that the midpoint is located at $235º$ ecliptic longitude(after some slight rounding) or $25$ into Scorpio, in agreement with the previous observation.
And that’s all for the first chapter.
