Stars in Cancer<\/strong>. [Hipparchus] records that the star in the southern claw of Cancer [\u03b1 Cnc], the bright star which is in advance of the latter and of the head of Hydra [\u03b2 Cnc], and the bright star in Procyon [\u03b1 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.<\/p><\/blockquote>\n 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 \u03b1 Cnc as the “southern claw of Cancer”. <\/span>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.<\/p>\n 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<\/em> 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.<\/p>\n 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\u00a0should<\/em> 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.<\/em><\/p>\n Also worth noting here is the use of “digits”. Oddly enough, this is\u00a0not<\/em> 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 $\\frac{1}{2}\u00ba$, this means that a lunar digit is $0;02,30\u00ba$. However, in this context, a digit is twice that value or $0;05\u00ba$.<\/p>\n 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.<\/p>\n Stars in Leo<\/strong>. [He records] that the easternmost two [\u03bc, \u03b5 Leo] of the four stars in the head of Leo [\u03bc, \u03b5, \u03ba, \u03bb Leo], and the star in the place where the neck joins [the head] of Hydra [\u03c9 Hya], lie on a straight line. Also, that the line drawn through the tail of Leo [\u03b2] and the star in the end of the tail of Ursa Major [\u03b7 UMa] cuts off the bright star under the tail of Ursa Major [\u03b1 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].<\/p>\n Stars in Virgo<\/strong>. [He records] that between the northern foot of Virgo [\u03bc Vir] and the right foot of Bootes [\u03b6 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 [\u03b1 Boo] and the southern foot of Virgo [\u03bb Vir]. Also, that between Spica [\u03b1 Vir] and the second star from the end of the tail in Hydra [\u03b3 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.<\/p>\n Stars in Libra<\/strong>. [He records] that the star [\u03bc Ser], which is very nearly on a straight line towards the north with the [two] bright stars in the claws [\u03b1, \u03b2 Lib] is bright and triple: for on both sides of it lie single small stars [$36$ and $30$ Ser].<\/p>\n Stars in Scorpius<\/strong>. [He records] that the straight line drawn through the rearmost of the stars in the sting of Scorpius [\u03bb Sco] and through the right knee of Ophiuchus [\u03b7 Oph] bisects the interval between the two advance stars in the right foot of Ophiuchus [$36$ and \u03b8 Oph] and that the fifth and seventh joints [in the tail of Scorpius [\u03b8 and \u03ba Sco]] lie on a straight line with the bright star in the middle of Ara [\u03b1 Ara]. Furthermore, that the northernmost star [\u03c3] of the two in the base of Ara [\u03c3 and \u03b8 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.<\/p>\n Stars in Sagittarius<\/strong>. [He records] that to the east and south of the Circle under Sagittarius [i.e., of Corona Australis] lie two bright stars [\u03b1 and \u03b2 Sgr], quite some distance (about $3$ cubits) from each other. The southernmost and brighter of these [\u03b2 Sgr], which is on the foot of Sagittarius, lies very nearly on a straight line with the midmost [\u03b1 CrA] of the three bright stars in the Circle (which lie furthest towards the east in that [constellation]) [\u03b3, \u03b1, \u03b2 CrA], and with the rearmost [\u03b6 Sgr] of the [two] bright stars [\u03b6 and \u03c3 Sgr] at opposite angles of the Quadrilateral [in Sagittarius, \u03b6, \u03c4, \u03c3, \u03c6]: the two intervals [between these three stars] are equal. The northernmost [of the two stars to the east of the Circle [\u03b1 Sgr]] lies to the east of this straight line, but is on a straight line with the [two] bright stars [\u03b6 and \u03c3] at opposite angles of the Quadrilateral.<\/p>\n Stars in Aquarius<\/strong>. [He records] that the two stars close together in the head of Pegasus [\u03b8 and \u03bd Peg] and the rear shoulder of Aquarius [\u03b1 Aqr] are almost on a straight line, to which the line from the advance shoulder of Aquarius [\u03b2 Aqr] to the star in the cheek of Pegasus [\u03b5 Peg] is parallel. Also, that the advance shoulder of Aquarius [\u03b2 Aqr], the bright star [\u03b6 Peg] of the two in the neck of Pegasus [\u03b6 and \u03be Peg], and the star in the navel of Pegasus [\u03b1 And] lie on a straight line, with equal intervals between them. Furthermore, that the line through the muzzle of Pegasus [\u03b5 Peg] and the easternmost [\u03b7 Aqr] of the stars in the vessel [of Aquarius [\u03b7, \u03b6, \u03c0, \u03b3 Aqr]] bisects, almost at right angles, the line through the two stars [\u03b8 and \u03bd Peg] close together in the head of Pegasus.<\/p>\n Stars in Pisces<\/strong>. [He records] that the star [\u03b2 Tri] in the snout of the southernmost fish [of Pisces], the bright star in the shoulders of Pegasus [\u03b1 Peg], and the bright star in the chest of Pegasus [\u03b2 Peg] lie on a straight line.<\/p>\n Stars in Aries<\/strong>. [He records] that the advance star [\u03b2 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 [\u03b1 Ari] and the left foot of Andromeda [\u03b3 And]. Also that the most advanced of the stars in the head of Aries [\u03b2 and \u03b3 Ari] and the midpoint of the base of Triangulum [i.e., halfway between \u03b2 and \u03b3 Tri] lie on a straight line.<\/p>\n Stars in Taurus<\/strong>. [He records] that the [two] easternmost stars of the Hyades [\u03b1 and \u03b5 Tau] and that star [\u03c01<\/sup> 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 [\u03b5 Tau] and the seventh star from the south in the pelt [\u03bf2<\/sup> Ori] cuts off the bright star in the Hyades [\u03b1 Tau] $1$ digit to the north.<\/p>\n Stars in Gemini<\/strong>. [He records[ that the heads of Gemini [\u03b1 and \u03b2 Gem] lie on a straight line with a certain star [\u03b6 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 [\u03b8 and \u03b4 Cnc] of the four stars [\u03b8, \u03b4, \u03b3, and \u03b7 Cnc] round the nebula [Praesepe].<\/p><\/blockquote>\n This collection of alignments, are well spaced and Ptolemy tells us that they allowed him to<\/p>\n carry out comparisons practically throughout the sphere [of fixed stars]<\/p><\/blockquote>\n and that<\/p>\n we see that no change has occurred up to the present time.<\/p><\/blockquote>\n 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.\u00a0The 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 $\\frac{3}{4}\u00ba$ 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 $0;23\u00ba$ 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.<\/p>\n
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