Ptolemy lays out the arrangements described by Hipparchus by going through stars in various constellations:
Stars in Cancer. [Hipparchus] records that the star in the southern claw of Cancer [α Cnc], the bright star which is in advance of the latter and of the head of Hydra [β Cnc], and the bright star in Procyon [α CMi] lie almost on a straight line. For the one in the middle lies $1 \frac{1}{2}$ digits to the north and east of the straight line joining the two end ones, and the distances [from it to each of them] are equal.
First, we should discuss the identification of the stars. The designations of the stars in the brackets are the modern Bayer designations first created in $1603$. For Ptolemy’s part, he described the positions of the stars based on how they should appear in the representation of the constellation. For example, we see him here describing α Cnc as the “southern claw of Cancer”. However, because the depiction of the figures in constellations is somewhat subjective, this leads to some ambiguity in which stars are which, making assessing the validity of Ptolemy’s statements somewhat difficult.
In the star catalog he gives later in this book and the next, there are some $1,022$ stars, many going down to sixth magnitude. While the identity of many of them can be accurately determined, it is not true in every case. The matter is further complicated by the fact that, despite Ptolemy’s claim in this chapter, stars do exhibit motion on the sphere itself, known as proper motion. As such, trying to identify stars by Ptolemy’s description looking at the constellations as they appear today can result in incorrect identifications. To correctly identify the stars, researchers must take into consideration the proper motion over the past $\approx 2,000$ years to understand where the stars were at that time. Several different authors have done this and, by and large, there is good agreement. Only a dozen or so have much debate about them.
We’ll have a lot more to say on this topic as we get further into this book, but for the current context, it makes it difficult to illustrate what Ptolemy is discussing here. I tried doing so in Stellarium which should take proper motion into account, but I quickly found that the arrangements don’t quite match what Ptolemy has stated here. I suspect the error lies with Stellarium as the positions it reports are notably different than those given in Grasshoff’s The History of Ptolemy’s Star Catalogue.
Also worth noting here is the use of “digits”. Oddly enough, this is not the same measurement as the one from the previous discussion on eclipses. There, a digit was $\frac{1}{12}$ of the diameter of the moon. Since the moon has an angular diameter of about $\frac{1}{2}º$, this means that a lunar digit is $0;02,30º$. However, in this context, a digit is twice that value or $0;05º$.
Ptolemy then goes on to describe several more alignments from Hipparchus. There’s not really much more to say so I’ll just quote Ptolemy at length for the remainder of this post.
Stars in Leo. [He records] that the easternmost two [μ, ε Leo] of the four stars in the head of Leo [μ, ε, κ, λ Leo], and the star in the place where the neck joins [the head] of Hydra [ω Hya], lie on a straight line. Also, that the line drawn through the tail of Leo [β] and the star in the end of the tail of Ursa Major [η UMa] cuts off the bright star under the tail of Ursa Major [α UMa] $1$ digit to the west [i.e., passes $1$ digit to the east of it]. Similarly, [he records] that the line through the star under the tail of Ursa Major and the tail of Leo passes through the more advanced of the stars in Coma [Berenices].
Stars in Virgo. [He records] that between the northern foot of Virgo [μ Vir] and the right foot of Bootes [ζ Boo] lie two stars; the southern of these [$109$ Boo[, which is equally bright as the [right] foot of Bootes, lies to the east of the line joining the feet, while the northern one [$31$ Boo], which is half-bright, lies on a straight line with the feet. Furthermore, of these two stars, the half-bright one is preceded by two bright stars, which form, together with the half-bright one, an isosceles triangle of which the half-bright one is the apex. These [two bright stars] lie on a straight line with Arcturus [α Boo] and the southern foot of Virgo [λ Vir]. Also, that between Spica [α Vir] and the second star from the end of the tail in Hydra [γ Hya] lie three stars, all on one straight line [$57$, $63$, and $69$ Vir]. The middle one of these [$63$] lies on a straight line with Spica and the second star from the end of the tail in Hydra.
Stars in Libra. [He records] that the star [μ Ser], which is very nearly on a straight line towards the north with the [two] bright stars in the claws [α, β Lib] is bright and triple: for on both sides of it lie single small stars [$36$ and $30$ Ser].
Stars in Scorpius. [He records] that the straight line drawn through the rearmost of the stars in the sting of Scorpius [λ Sco] and through the right knee of Ophiuchus [η Oph] bisects the interval between the two advance stars in the right foot of Ophiuchus [$36$ and θ Oph] and that the fifth and seventh joints [in the tail of Scorpius [θ and κ Sco]] lie on a straight line with the bright star in the middle of Ara [α Ara]. Furthermore, that the northernmost star [σ] of the two in the base of Ara [σ and θ Ara] lies between and almost on a straight line with the fifth joint and the star in the middle of Ara, being almost equidistant from both.
Stars in Sagittarius. [He records] that to the east and south of the Circle under Sagittarius [i.e., of Corona Australis] lie two bright stars [α and β Sgr], quite some distance (about $3$ cubits) from each other. The southernmost and brighter of these [β Sgr], which is on the foot of Sagittarius, lies very nearly on a straight line with the midmost [α CrA] of the three bright stars in the Circle (which lie furthest towards the east in that [constellation]) [γ, α, β CrA], and with the rearmost [ζ Sgr] of the [two] bright stars [ζ and σ Sgr] at opposite angles of the Quadrilateral [in Sagittarius, ζ, τ, σ, φ]: the two intervals [between these three stars] are equal. The northernmost [of the two stars to the east of the Circle [α Sgr]] lies to the east of this straight line, but is on a straight line with the [two] bright stars [ζ and σ] at opposite angles of the Quadrilateral.
Stars in Aquarius. [He records] that the two stars close together in the head of Pegasus [θ and ν Peg] and the rear shoulder of Aquarius [α Aqr] are almost on a straight line, to which the line from the advance shoulder of Aquarius [β Aqr] to the star in the cheek of Pegasus [ε Peg] is parallel. Also, that the advance shoulder of Aquarius [β Aqr], the bright star [ζ Peg] of the two in the neck of Pegasus [ζ and ξ Peg], and the star in the navel of Pegasus [α And] lie on a straight line, with equal intervals between them. Furthermore, that the line through the muzzle of Pegasus [ε Peg] and the easternmost [η Aqr] of the stars in the vessel [of Aquarius [η, ζ, π, γ Aqr]] bisects, almost at right angles, the line through the two stars [θ and ν Peg] close together in the head of Pegasus.
Stars in Pisces. [He records] that the star [β Tri] in the snout of the southernmost fish [of Pisces], the bright star in the shoulders of Pegasus [α Peg], and the bright star in the chest of Pegasus [β Peg] lie on a straight line.
Stars in Aries. [He records] that the advance star [β Tri] in the base of Triangulum lies $1$ digit to the east of the straight line drawn through the star in the muzzle of Aries [α Ari] and the left foot of Andromeda [γ And]. Also that the most advanced of the stars in the head of Aries [β and γ Ari] and the midpoint of the base of Triangulum [i.e., halfway between β and γ Tri] lie on a straight line.
Stars in Taurus. [He records] that the [two] easternmost stars of the Hyades [α and ε Tau] and that star [π1 Ori] in the pelt held in Orion’s left hand which is sixth counted from the south, lie on a straight line. And that the line drawn through the advance eye of Taurus [ε Tau] and the seventh star from the south in the pelt [ο2 Ori] cuts off the bright star in the Hyades [α Tau] $1$ digit to the north.
Stars in Gemini. [He records[ that the heads of Gemini [α and β Gem] lie on a straight line with a certain star [ζ Cnc] which lies to the rear of the rearmost head by a distance three times that between the two heads, and that the same star also lies on a straight line with the [two] southernmost [θ and δ Cnc] of the four stars [θ, δ, γ, and η Cnc] round the nebula [Praesepe].
This collection of alignments, are well spaced and Ptolemy tells us that they allowed him to
carry out comparisons practically throughout the sphere [of fixed stars]
and that
we see that no change has occurred up to the present time.
This convinced Ptolemy that the stars remain completely static on the surface of the sphere and that Hipparchus’ hypothesis that there was some motion of the zodiac stars was disproven. The period between Hipparchus and Ptolemy was about $260$ years. Over that period, even the star with the highest degree of proper motion, Barnard’s star, would only have moved $\frac{3}{4}º$ and it’s far too faint to see without the use of a telescope. If we limited ourselves to stars visible to the naked eye, the best bet for visible proper motion would be $61$ Cygni would only have moved $0;23º$ over that time period – an amount not easily measurable with the instrumentation of Ptolemy’s time. This star is also not present in Ptolemy’s catalog.
