At the beginning of the last chapter, Ptolemy noted that the celestial sphere appears to have a rearward motion of its own known as precession of the equinoxes. This motion means that the position of the sun in the zodiacal signs slowly increase in advance as time passes. In particular, the sun was at the beginning of Aries in Ptolemy’s time, but since then has advanced such that today it lies near the beginning of Pisces.
We can see this mainly from the fact that the same stars do not maintain the same distance with respect to the solstical and equinocital points in our times as they had in former times: rather, the distance [of a given star] towards the rear with respect to [one of] the same points is found to be greater in proportion as time [of observation] is later.
This is something Hipparchus also noted. Ptolemy cites his work, On the Displacement of the Solstical and Equinoctial Points. There, Ptolemy tells us, Hipparchus determined the distance of Spica from the autumnal equinox, something Timocharis had done a couple centuries earlier. Timocharis had found that Spica was $8º$ before from the equinox, but Hipparchus found it to only be $6º$ which implied that the equinox moved, in advance, by $2º$ over that period. Ptolemy also informs us that Hipparchus also compared other stars in the same manner and found
that their motion towards the rear with respect to the ecliptic is, proportionally, similar to the above amount.
Ptolemy then states he
conducted this type of investigation by means of the instrument which we constructed previously for the observations of individual moon-sun distances.
Here, Ptolemy is describing his astrolabe (modernly known as an armillary sphere). He then provides instructions on how it can be used for observations of this sort:
We set one of the astrolabe rings to the apparent position of the moon (computed for the moment of observation), then adjusted the other astrolabe ring to align it with the star being sighted, so that both moon and star would be sighted simultaneously in the proper positions. Thus, we obtained the position of every one of the bright stars from its distance from the moon.
In short, Ptolemy is relying on an accurate lunar model prediction to be able to know the ecliptic longitude and latitude of the moon and then measuring from the moon to the star in question.
Ptolemy then provides us an example:
In the second year of Antoninus, on Pharmouthi [VIII] $9$ in the Egyptian calendar [$139$ Feb $23$], when the sun was just about to set in Alexandria, and the last degree of Taurus was culminating, i.e., $5 \frac{1}{2}$ equinoctial hours after noon on the ninth, we observed the apparent distance of the moon from the sun (which was sighted at about $3º$ into Pisces) as $92 \frac{1}{8}º$. Half an hour later, the sun now having set, and the [first] quarter of Gemini [i.e., Gemini $7;30º$] culminating, the apparent moon was sighted in the same position [with respect to the astrolabe ring], and the star on the heart of Leo [α Leo, Regulus] had an apparent distance from the moon, [as measured] by means of the other astrolabe [ring], of $57 \frac{1}{6}º$ towards the rear along the ecliptic.
Now, at the first [observation] the true position of the sun was very nearly $3 \frac{1}{20}º$ into Pisces. Hence, the apparent position of the moon, since it was $92 \frac{1}{8}º$ towards the rear [of the sun], was approximately $5 \frac{1}{6}º$ into Gemini, which is also the position it out to occupy according to our hypothesis. Half an hour later, the moon should have moved about $\frac{1}{4}º$ towards the rear, and have a parallax in advance, relative to the first situation, of about $\frac{1}{12}º$. Therefore, the apparent position of the moon half an hour later was $5 \frac{1}{3}º$ into Gemini. Hence, the star on the heart, since its apparent distance from the moon was $57 \frac{1}{6}º$ to the rear [of the moon] had a position of $2 \frac{1}{2}º$ into Leo, and its distance from the summer solstice was $32 \frac{1}{2}º$
Here, Ptolemy has slightly modified the procedure he just gave and instead of determining the ecliptic longitude of the moon from the lunar model, he determines it based on the distance from the sun in ecliptic longitude, having determined the position of the sun from the solar model. This strikes me as a better idea as the solar model is likely more reliable. This observation would obviously be done during the day while the sun was still up.
Then the sun would set and the distance from the moon to the star taken with adjustments being made to account for the lunar motion in that time and parallax.
This observation is then compared to one from Hipparchus:
But in the $50^{th}$ year of the Third Kallipic Cycle [127/128 BCE], as Hipparchus’ records from his own observations, [that star] had a distance to the rear of the summer solstice of $29 \frac{5}{6}º$. Therefore, the star on the heart of Leo has moved $2 \frac{2}{3}º$ towards the rear along the ecliptic in $265$ or so years from the observation of Hipparchus to the beginning [of the reign] of Antoninus [$137/138 CE$], which was when we made the majority of our observations of the positions of the fixed stars. From this, we find that $1º$ of rearward motion takes place in approximately $100$ years, as Hipparchus too seems to have suspected, according to the following quotation of his work, On the Length of a Year: “For if the solstices and equinoxes were moving, from that cause, not less than $\frac{1}{100}^{th}$ of a degree in advance [i.e., in the reverse order] of the signs, in the $300$ years they should have moved not less than $3º$.”
So from Hipparchus’ observations, Ptolemy infers a rate of precession of $1º$ per hundred years which agrees with what Hipparchus found as well. Ptolemy also states he checked this using other stars:
In the same way, we took sightings of Spica and the brightest among those stars near the ecliptic, from the moon, and then [having done that] were in a better position to use those stars to take sightings of the rest. We [thus] find that their distances relative to each other are, again, very nearly the same as those observed by Hipparchus, but their individual distances from the solstical or equinoctial points are, in each case, about $2 \frac{2}{3}º$ farther to the rear than those derivable from what Hipparchus recorded.
This rate is actually a tad low as the modern value for the rate is $1º$ every $72$ years. or $1;23º$ per century but given the instrumentation of the time, is a quite reasonable value. This value would stand as the default value in the western and middle eastern world1 until Al-Battani around the end of the $9^{th}$ century.
