A study on the stellar configurations in Almagest VII 1

Introduction.

Book VII of the Almagest presents one of the most enduring parts of Hellenistic astronomy, that is, the theory of precession. Ptolemy points to Hipparchus of Nicaea (c. -190 – c. -120) as the one responsible for its discovery. As part of the discussion surrounding precession Ptolemy introduces us to a discussion which, distant as it may be from our –and possibly Ptolemy’s– theoretical perspective, appeared as a genuinely interesting debate in Hipparchus’ times: are all the stars subjected to precession, or does it only affect the zodiacal ones? Implicit in this question is another, more general one: are the so-called fixed stars truly fixed? In order provide an answer to them, Ptolemy tells us that Hipparchus determined many stellar configurations. While most of the ones Ptolemy refers to are straight lines formed by three or more stars, it is possible that the original set was much more varied. After presenting the Hipparchian configurations and declaring that they remained, about three centuries later, as they were in Hipparchus’ times, Ptolemy gives the reader a new set of configurations of his own, in order to provide future astronomers with a more secure basis to check the hypothesis that the stars are all fixed in position relative to each other.⬈#p1

In this paper I will give a detailed study of these configurations. As far as I know, this is the first complete analysis of the stellar configurations in the Almagest. Nevertheless, some important work has already been done. First and foremost, we have the notes and commentaries in both Manitius’ (1913) and Toomer’s (1984) modern translations. As will become obvious, they served as the main sources on which my work rests, even if it is to –occasionally– criticize them. Secondly, (Pedersen, 2010) and (Evans, 1998), even if they are works with a much wider scope, both provide some important insight into what role the stellar configurations played in Ptolemy’s Almagest.⬈#p2

The first section of the paper deals with some necessary clarifications regarding the methods I use to evaluate the stellar configurations. After that, the second section goes into the list of Hipparchian configurations that Ptolemy refers in the Almagest. As I said, I will consider the configurations one by one, and accompany each one with the corresponding illustration. In the third section I will do the same with the configurations that Ptolemy presents as his own. The following two sections are a general analysis of the alignments given by both astronomers, and a comparison between some alignments located in the same signs, which could shed some light into the question of the origin of some Ptolemaic alignments.⬈#p3

Methodological clarifications.

The Almagest indicates various kinds of stellar configurations. From the 26 configurations Ptolemy attributes to Hipparchus, 22 are alignments or quasi-alignments between three or four stars. There is one description of an isosceles triangle, one description of two lines being parallel, and two descriptions of a line bisecting another.⬈#p4

Regarding his own configurations, I counted 23, 21 of which are stellar alignments or quasi alignments. The rest consists of one description of an isosceles triangle, and one of a line bisecting another.⬈#p5

Hence, by far the majority of the configurations described in the Almagest are descriptions of three or four stars being in the same line, or close to it. This should not come as a surprise, since Ptolemy himself says that he has given the configurations “[…] which are most suitable for easy comprehension and also for giving an overview of the whole method of comparison […]” (Toomer, 1984: 322). The line is, of course, the simplest geometric configuration he could have chosen. Because he is looking at a sphere, we can confidently say that what Ptolemy is referring to when speaking about an εὐθεῖα, is a great circle of that sphere.⬈#p6

When he describes the alignments, Ptolemy uses different expressions. Many times he indicates that the stars lie on a straight line, in some occasions qualifying the positions as “almost on a straight line”. In other numerous occasions, though, he first points to two stars, and then indicates how far the line they determine passed from a given third star. This distance is never great, at least according to his records. This ambivalence in the expressions used in the descriptions must be considered carefully. The descriptions of a star being at a certain distance from a line determined by two stars can be checked directly. In fact, in (Meeus, 1998: 124) there is a formula for calculating exactly that distance. However, when checking the configurations that the Almagest describes as “straight lines” or “almost straight lines”, one must take into consideration another datum in order to make a fuller assessment of the accuracy of the description.⬈#p7

Refer to Figure 1. Lines AB and DE represent two lines between pairs of stars. In the first case, a third star C is at a given distance from line AB. This distance is equal to the one between point F to line DE in the second case. However, ACB is more straight-like than CFE. This shows that the distance of the middle star to the line between the two extreme stars cannot, by itself, be a good measure of the accuracy of the description. Instead, I will also measure the angle at the middle star, $\angle \mathit{ACB}$ $\angle \mathit{DFE}$ in Figure 1. To do it, I will use the formula given in (Meeus, 1998: 125).⬈#p8

The measurement of the distance from the middle star to the line determined by the other two has it place, though. In Figure 1, if line AB is not great, then it is likely that the distance of C to that line is negligible or very small. But if AB is large, then the distance from C to it may be noticeable. So, even if the angle at C is the same in two different cases, one may be a good alignment and the other one not.⬈#p9

Because of these ambiguities in what it constitutes a good alignment, in the cases described as “straight lines” or “almost straight lines”, I will give both data. If in some case I depart from this criterion, the reasons will be made clear as needed.¹⬈#p10

Another difficulty is presented in the case where Ptolemy describes two lines in the celestial sphere as being parallel. Because there are no parallel great circles in spherical geometry, the natural interpretation is that, because the lines he indicates are so short, they can be seen as two straight lines on a plane. So, to analyze that case, I will assume this, and determine the angle at which both lines will intersect in order to determine how close they were to being parallel. The method I used to do that is described in Appendix 1.⬈#p11

The rest of the configurations offer no special difficulties: in order to analyze the isosceles triangle I simply compare the distances between the apex and the two stars at the base, using the formula given in (Meeus, 1998: 109). Finally, the cases of a line bisecting another are studied by comparing the size of both parts of the bisected line. The method I used to do that is described in Appendix 2.⬈#p12

I will use as reference Toomer’s identifications of the stars, as indicated in the notes to his translation of the Almagest. In the vast majority of the cases, Toomer follows Manitius’ identifications. So, when I discuss Toomer’s identifications, as a rule the reader should understand that I am referring to an identification by Manitius that Toomer has found satisfactory. In some cases, though, Toomer offers (with various degrees of confidence) alternative identifications. In these instances I have done my best to support one or the other with arguments not found in either work. Finally, in some few cases I offer my own identification (also with various degrees of confidence).⬈#p13

In all cases, the positions of the stars were computed using the data provided by the Hipparcos Catalogue. Stellar proper motion was accounted for. The years I used as reference for the Hipparchian and Ptolemaic stellar positions are -150 and 140, respectively. The magnitudes of the stars were provided by the Tycho Catalogue.⬈#p14

The images that go with each configuration were made using Stellarium 0.19.2. There is some variation between the manner in which the Almagest describes the positions of some stars within a given constellation, and how the program depicts them in the constellation drawing. Thus, the images should only be taken as orientative depictions.⬈#p15

Hipparchus’ configurations.

As I noted in the introduction, Ptolemy presents Hipparchus’ stellar configurations in the context of his exposition on the “so called fixed stars” in Almagest VII 1. There, he tells us that the stars are fixed in the sense that their relative positions remain unchanged through time, but that they are not fixed with respect to the equinoctial points. Both hypotheses are explicitly attributed to Hipparchus, and the chapter is merely intended to serve as a confirmation based on a more secure observational basis. As Ptolemy puts it, “[…] our examination is conducted [with material taken] from a longer time-interval, and because the fixed-star observations recorded by Hipparchus, which are our chief source for comparisons, have been handed down to us in a thoroughly satisfactory form.” (Toomer, 1984: 321).⬈#p16

The comparison of the stellar positions is aimed at supporting the first of these tenets, that is, that the stars remain fixed in their relative positions. The second one, which he calls the rearward motion of the stars, Hipparchus called the “displacement of the solsticial and equinoctial points” (Toomer, 1984: 327), and is modernly called precession of the equinoxes, will be dealt with in a following section.⬈#p17

Ptolemy’s text, however, reveals that in the course of his study on the relative positions of the stars Hipparchus changed his mind regarding the extension of their hypothesized relative immobility. When Ptolemy asserts that the stars on the entire celestial sphere are in fixed positions to one another, he says that “This is true not only of the positions of the stars in the zodiac relative to each other, or of the stars outside the zodiac relative to other stars outside the zodiac (which would still be the case if only stars in the vicinity of the zodiac had a rearward motion, as Hipparchus proposes in the first hypothesis he puts forward); but is also true of the positions of stars in the zodiac relative to those outside it, even those at considerable distances.” (Toomer, 1984: 322, italics are mine)⬈#p18

Ptolemy is implying that Hipparchus initially though that the rearward motion of the stars only affected those within the boundaries of the zodiac. Thus, the relative immobility would only apply within the set of zodiacal stars on the one hand, and within the set of extra zodiacal stars on the other, but not to sets of stars comprised of the two kinds. It is clear that Ptolemy does not hold this position, but instead had understood that all the stars undergo the rearward motion (i.e., a motion in the direction of increasing longitude), and thus that all their relative positions remain unchanged. The fact that he indicates that the former was only “the first hypothesis” Hipparchus had put forward already suggests that Hipparchus himself had, at some point, abandoned it for a more comprehensive understanding of the effects of precession. The very fact that Hipparchus’ work about precession was called “On the displacement of the solsticial and equinoctial points” confirms that his final understanding regarding this aspect of the problem was identical to Ptolemy’s: if the objects that moved where those points instead of the stars, then all the stars will move relative to them. As we will see, the Hipparchian alignments Ptolemy quotes confirm this interpretation.⬈#p19

In the following sections I will provide a study of each of the Hipparchian configurations given in the Almagest with a commentary, and the corresponding images. The order will be the same as the Almagest’s. All positional values will only be given for -150, i.e., the times of Hipparchus.⬈#p20

Cancer.

1) [Hipparchus] records that the star in the southern claw of Cancer, the bright star which is in advance of the latter and of the head of Hydra, and the bright star in Procyon lie almost on a straight line. For the one in the middle lies 1 ½ 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.⬈#p21

Refer to Illustration 1. Toomer identifies the three stars as α Cnc, β Cnc, and α CMi respectively. He notes (322, note 4) that the north-east position cannot be right, and I agree. He calculates that in Hipparchus’ time β Cnc lay about 5´ to the north-west of the line (Toomer, 1984: 322, note 4). This would give a good coincidence with the 1 ½ digits Ptolemy quotes.²⬈#p22

By my calculations, the line was 20.03° long, and β Cnc lay 25.52´ to the north-west of the line, a considerable difference from the Almagest’s account, and Toomer’s calculations. The distance between α Cnc and β Cnc was 10.6°, while that between β Cnc and α CMi was 9.45°. It should also be noted that already in this first configuration we find a star that is well outside the Zodiac: α CMi’s latitude was 15.55° south.³ This already supports the idea that Hipparchus accepted the universal character of the rearward motion of the stars.⬈#p23

Leo.

1) [He records] that the easternmost two of the four stars in the head of Leo, and the star in the place where the neck joins [the head] of Hydra, lie on a straight line.⬈#p24

Refer to Illustration 2. Toomer identifies the first two stars as μ Leo and ε Leo. The third star in the neck of Hydra is identified by him as ω Hya. The μ Leo-ω Hya line was 23.8° long. The angle at ε Leo was 172.9°, and the distance to the μ Leo-ω Hya line was 18.5´. Although it is not such a good alignment, Toomer has a good reason to identify the third star as ω Hya: the description resembles the one in Ptolemy’s catalogue: “[…] the more advanced of the 2 stars in the place where the neck joins [the head] […]” (Toomer, 1984: 392). However, a much better alignment is reached if the third star is identified as ζ Hya. Not only does it determine an angle in ε Leo of 179.78°, but it also –and crucially– lay just 0.58´ away from the μ Leo-ζ Hya line, in Hipparchus’ times. If this is the case, then the alignment’s size would not vary much, spanning 24.36° long. Such an identification would go against the natural assumption that the Hipparchian alignments follow the same descriptions for the stars than those given in the Ptolemaic catalogue.⁴ This variation, however, is not rare: even Ptolemy describes some stars in his own alignments in a different manner as he does in his catalogue (Toomer, 1984: 322, note 3). To conclude, even if I am not sure that the ζ Hya alternative is the correct one, it should not be discarded.⬈#p25

Whatever is the correct option, the stars in this alignment were located at both edges of the Zodiac: the latitudes for μ Leo and ε Leo were 12.27° north and 9.57° north, respectively, while those of ω Hya and ζ Hya were 11.19° south and 11.14° south, respectively.⬈#p26

[…] the line drawn through the tail of Leo and the star in the end of the tail of Ursa Major cuts off the bright star under the tail of Ursa Major 1 digit to the west.⬈#p27

Refer to Illustration 3. These three major stars are identified as β Leo, η UMa, and α CVn, respectively. Toomer calculates (Toomer, 1984: 322, note 8) that the line passed about 30´ from the middle star. By my calculations, it passed 6.54´ away from it, in good agreement with the Almagest’s register.⬈#p28

The alignment’s size is 42.17°, and it belongs to the set of extra zodiacal stars: while the latitude of β Leo was 12.41° north, those of η UMa, and α CVn were 54.41° north and 40.12° north, respectively.⬈#p29

[…] 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.⬈#p30

Refer to Illustration 4. Toomer identifies (with doubts) the “more advanced” stars as 15 Com and 7 Com. They are indeed, among the brightest stars in a not too bright constellation, the ones that were the closest to the β Leo- α CVn line: 30.44´ and 44.38´, respectively. Still, it is not a very good alignment.⬈#p31

The total distance of the line was 27.81°, and the stars were all extra-zodiacal.⬈#p32

Virgo.

1) […] between the northern foot of Virgo and the right foot of Bootes lie two stars; the southern one of these, which is equally bright as the [right] foot of Bootes, lies to the east of the line joining the feet, while the northern one, which is half bright, lies on a straight line with the feet.⬈#p33

Refer to Illustration 5. The two feet are μ Vir and ζ Boo, respectively. Of the two between those, the southern and brightest is 109 Vir (in a slip of the pen Toomer annotated 109 Boo), and the northern one is 31 Boo. As the Almagest says, 109 Vir lies 1.09° to the east of the μ Vir-ζ Boo line. In modern notation, both ζ Boo and 109 Vir have a 3.7 magnitude, so the text seems to be accurate regarding their equal brightnesses. The alignment between the feet and 31 Boo is remarkable: the angle at 31 Boo is 179.6°, and it lay just 1.35´ away from the μ Vir-ζ Boo line. 31 Boo’s magnitude is 4.9, which means that its apparent brightness is 1/3 that of the 109 Vir’s. Considering the crudeness of the magnitude system used in ancient times, it is not a bad approximation.⬈#p34

As before, this case shows an alignment of a mixed kind: while μ Vir is at the inner edge of the Zodiac with a latitude of 9.99° north, ζ Boo is well outside, with a latitude of 28° north. The total distance of the line is 19.2°.⬈#p35

2) […] 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.⬈#p36

Refer to Illustration 6. Manitius identified the two stars at the base as 43 and 46 Boo, in Heis’ catalog (Manitius, 1913: 5). Toomer says that he did not track this identification, “[…] since any identification seems utterly uncertain.” (Toomer, 1984: 323, note 11). I think that Manitius’ identification is correct. Heis’ 43 and 46 Boo (Heis, 88) are our modern HIP 70327 and HIP 70400. Although they are by no means bright (mags. 4.8 and 4.15 respectively), they are the brightest in the region by a considerable margin, and clearly resemble, together with 31 Boo, an isosceles triangle. The distance from 31 Boo to HIP 70327 was 4.49°, and the distance from 31 Boo the HIP 70400 was 4.89°, with a difference of 24´ between the sides of the triangle. The size of the triangle’s base was 2.64°. The three stars are well outside the Zodiac, with northern latitudes close to 20°.⬈#p37

3) These [two bright stars] lie on a straight line with Arcturus and the southern foot of Virgo.⬈#p38

Refer to Illustration 7. This alignment is doubtful. Toomer identifies the southern foot of Virgo with λ Vir, following the description in the Almagest’s catalog. Arcturus, HIP 70327 and HIP 70400 form a good alignment, with the angle at HIP 70327 being 178.44°, and HIP 70327 lying 3.55´ away from the Arcturus-HIP 70400 line. However, these do not align so well with λ Vir. The angle Arcturus-HIP 70400-λ Vir is 170.48°, and HIP 70400 lay 1.42° away from the Arcturus- λ Vir line, which is 33.77° long. Despite these problems, for the moment Toomer’s identification is the best extant option. These identifications would render the alignment as one of a mixed kind: while λ Vir is almost on the ecliptic with a latitude of 0.62° north, Arcturus had a latitude of 32.17° north.⬈#p39

4) […] between Spica and the second star from the end of the tail in Hydra lie three stars, all on one straight line. The middle one of these lies on a straight line with Spica and the second star from the end of the tail in Hydra.⬈#p40

Refer to Illustration 8. Toomer corrects Manitius, who had identified the three stars as 61, 63, 69 Vir. He proposes, 57 Vir instead of 61 Vir, which is plausible. Such an identification would give an angle at the middle star, 63 Vir, of 172.05°, with 63 Vir lying 9.92´ away from the 57 Vir-69 Vir line. Another possibility is the trio 53, 63, 73 Vir. That identification would give an angle at 63 Vir of 173.48°, and a distance of 63 Vir to the 53 Vir-73 Vir line of 9.33´. Whatever the case, it seems likely that the middle star is 63 Vir, which the text tells us was aligned with Spica and the second star from the end of Hydra’s tail, i.e, γ Hya. The angle at 63 Vir for that alignment was 174.09°, and 63 Vir lay 18.66´ away from the Spica- γ Hya line. The entire line was 12.1° long. All the stars discussed here were within the Zodiac, with the exception of 57 Vir (lat. 10.92° south) and γ Hya (lat. 13.62° south).⬈#p41

Libra.

1) […] the star which is very nearly on a straight line towards the north with the [two] bright stars in the claws is bright and triple: for on both sides of it lie single small stars.⬈#p42

Refer to Illustration 9. Toomer corrects Manitius’ identification, and proposes μ Ser instead of α Ser as the star aligned with α and β Lib. This correction is completely justified. The two small stars are 30 and 36 Ser. The line α Lib-μ Ser was 19.15° long. The angle at β Lib was 168.97°, with a distance of 55.89´ to the α Lib-μ Ser line. As the text indicates the alignment is, truly, just approximate. While α and β Lib were obviously within the Zodiac, μ Ser had a latitude of 16.47° north.⬈#p43

Scorpius.

1) […] the straight line drawn through the rearmost of the stars in the sting of Scorpius and through the right knee of Ophiuchus bisects the interval between the two advance stars in the right foot of Ophiuchus […]⬈#p44

Refer to Illustration 10. Toomer’s identification of the four stars seems undisputable: λ Sco for the sting, η Oph for the knee, and the pair of θ Oph and 36 Oph for the stars in the right foot. The bisecting line λ Sco-η Oph had a size of 21.92°, and the bisected one, θ Oph-36 Oph, was 1.56° long. In this last one, the section on the side of θ Oph had a size of 0.65°, and that on the side of 36 Oph had a size of 0.91°, with a difference of 15.48´ and a ratio of 0.72 between them. With the exception of λ Sco (lat. 13.53° south), all of the stars were zodiacal.⬈#p45

2) […] the fifth and seventh joints lie on a straight line with the bright star in the middle of Ara.⬈#p46

Refer to Illustration 11. The identification by Toomer of the three stars as θ Sco, κ Sco, and α Ara is certain. The total size of the alignment is 10.98°, and all of the stars were extra-zodiacal, with southern latitudes of 19.4°, 15.38° and 26.28°, respectively. The angle at θ Sco was 172.98°, and that star lay 18.94´ away from the κ Sco-α Ara.⬈#p47

3) […] the northernmost star of the two in the base or 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.⬈#p48

Refer to Illustration 12. This alignment was probably detected when determining the previous one. Toomer’s identification of the star at the base as σ Ara (lat. 22.9° south) is certain. The alignment was, as the text indicates, only approximate: the angle at σ Ara was 174.39°, and it lay 10.15´ away from the θ Sco-α Ara line, which was 6.9° long. The distances from σ Ara to θ Sco and to α Ara were 3.5° and 3.41°, respectively. This is a difference of only 5.09´, and the stars were all extra-zodiacal.⬈#p49

Sagittarius.

1) […] to the east and south of the Circle under Sagittarius lie two bright stars, quite some distance (about 3 cubits) from each other. The southernmost and brighter of these, which is on the foot of Sagittarius, lies very nearly on a straight line with the midmost of the three bright stars in the Circle (which lie furthest towards the east in that [constellation], and with the rearmost of the two bright stars at opposite angles of the Quadrilateral: the two intervals are equal.⬈#p50

Refer to Illustration 13. Toomer’s identification of the two stars “to the east and south of the Circle” as α and β Sgr is certain, as is that of the other two as α CrA and ζ Sgr. The only ambiguity lies in the fact that there are two stars at the foot of Sagittarius, and not one: β¹ and β² Sgr. They were distanced by a considerable 19.76´. The general direction of the line they determine, however, is similar to that of the alignment under study, so the choice between one of these two does not have a notable effect in the final measurements. I will choose β¹ Sgr, simply because it is the brightest of the two. The text says that α and β Sgr are separated by “about 2 cubits”. By my calculations, the distance was in fact 3.9°. If we assume a value of 2° or 2.5° for a cubit, then the Almagest’s estimation is about right.⬈#p51

The angle at α CrA was 170.49°, and this star lay 37.69´ away from the β¹ Sgr- ζ Sgr line. Thus, the alignment (which was 15.08° long), is a very poor one. With the exception of ζ Sgr (lat. 6.94° south), the stars were extra-zodiacal.⬈#p52

The text also adds that the intervals are equal. I find that the distances from α CrA to β¹ Sgr and ζ Sgr were 7.06° and 8.08°, respectively. This means a variation of 1.02°, and a ratio of 1.14.⬈#p53

2) The northernmost [of the two stars to the east of the Circle] lies to the east of this straight line, but is on a straight line with the [two] bright stars at opposite angles of the Quadrilateral.⬈#p54

Refer to Illustration 13. This alignment is very close to the previous one. The identification of the third star as σ Sgr is mandatory. The angle at ζ Sgr was 175.81°, with a distance of that star to the α Sgr- σ Sgr line of 12.99´. The entire alignment was 15.46° long. While σ and ζ Sgr were zodiacal, α Sgr’s latitude was 18.07° south.⬈#p55

Aquarius.

1) […] the two stars close together in the head of Pegasus and the rear shoulder of Aquarius are almost on a straight line, to which the line from the advance shoulder of Aquarius to the star in the cheek of Pegasus is parallel.⬈#p56

Refer to Illustration 14. The three stars in the alignment are identified by Toomer as θ and ν Peg, and α Aqr (lats. 16.48° north,15.76° north, and 10.78° north respectively). The other two stars, as β Aqr and ε Peg (lats. 8.77° north and 22.22° north, respectively). The identifications are certain. The alignment is extremely poor. The angle at ν Peg was 138.47°, and that star lay 50.24´ away from the θ Peg-α Aqr line. Curiously, θ and ν Peg form a very good alignment with β Aqr, the “advance shoulder” of Aquarius: in this case the angle at ν Peg is 178.12°, and that star lay just 2.75´ away from the θ Peg-β Aqr line. Given that β Aqr is mentioned afterwards in the same description, is it possible that in the original Hipparchian list of configurations there were two distinct descriptions that used the same stars, and that these were later conflated into one description in a confused way?⬈#p57

The alleged parallelism between the two indicated lines θ Peg-α Aqr (6.56° long), and ε Peg-β Aqr (15.76° long) is notably good, assuming that we are looking at a plane and not a sphere.⁵ On the one hand, θ Peg is 6.92° away from the β Aqr-ε Peg line. On the other, the distance of α Aqr to the same line is 7.34°. Given the sizes of the lines, this means that they will cross at an angle of 3.66°. So, their disposition bears a great resemblance to parallelism.⬈#p58

2) […] the advance shoulder of Aquarius, the bright star of the two in the neck of Pegasus, and the star in the navel of Pegasus lie on a straight line, with equal intervals between them.⬈#p59

Refer to Illustration 15. There are no reasons to dispute the identifications of the stars as β Aqr, ζ Peg and α And. It was a fairly big alignment, spanning 51.2° long the night sky. The angle at ζ Peg was 177.22°, with a distance of 39.8´ from ζ Peg to the β Aqr-α And line. It can be noted that the other, dimmer, star “in the neck of Pegasus”, ξ Peg, determines a better alignment: the angle was 178.67°, and the distance to the β Aqr-α And line was 19.12´. As the text points out, though, ζ Peg is much brighter⁶, which explains that it was the one used as reference.⬈#p60

Regarding the distances from ζ Peg, the value to β Aqr was 23.89°, and to α And, 27.33°. The difference was 3.44°, with a ratio of 1.14. Again, ξ Peg gives a better equivalence in the distances, with a value of 25.87° for the distance to β Aqr, and 25.33° to α And. In this case, the difference is just 32.57´, with a ratio of 1.02. However, the identifications are, as I said, certain, and given that the improvement is not qualitative, it is reasonable to assume that the superior brightness of ζ Peg explains why it was chosen over ξ Peg. With the exception of β Aqr, all the stars were extra-zodiacal.⬈#p61

3) […] the line through the muzzle of Pegasus and the easternmost of the four stars in the vessel [of Aquarius] bisects, almost at right angles, the line through the two stars close together in the head of Pegasus.⬈#p62

Refer to Illustration 16. The four stars are identified with certainty as ε Peg, η Aqr, θ Peg and ν Peg. The line ε Peg-η Aqr was 16.13° long, and the line θ Peg-ν Peg, 1.56°. The two lines intersected at an angle that deviated 2.5° from perpendicularity.⁷ In the bisected line θ Peg-ν Peg, the part on the side of θ Peg was 0.95°, and that on the side of ν Peg, 0.6°, with a difference of 0.35°, and a ratio of 1.57.⬈#p63

With the exception of η Aqr (lat. 8.28° north), all the stars were extra-zodiacal.⬈#p64

Pisces.

1) […] the star in the snout of the southernmost fish [of Pisces], the bright star in the shoulders of Pegasus, and the bright star in the chest of Pegasus lie on a straight line.⬈#p65

Refer to Illustration 17. The identification of the stars as β Psc, α Peg and β Peg is certain. The alignment is fairly good, with an angle of 177.74° at α Peg, and a distance of 14.46´ to the β Psc-β Peg line. The size of the alignment was 24.17°. The set of stars is of a mixed kind: while α and β Peg had latitudes of 19.47° north and 31.11° north, respectively, β Psc was just within the Zodiac, with a latitude of 9.1° north.⬈#p66

Aries.

1) […] the advance star in the base of Triangulum lies 1 digit to the east of the straight line drawn through the star in the muzzle of Aries and the left foot of Andromeda.⬈#p67

Refer to Illustration 18. Toomer (324, note 26) says that “There is no doubt about the identification of the stars” as β Tri, α Ari and γ And. He is right. He also indicates that the Almagest’s distance of one digit is too low, and says that he has calculated that β Tri lay “[…] well over a degree to the east of the line connecting α Ari and γ And”. By my calculations, the distance was 50.29´ in Hipparchus’ time.⬈#p68

The quasi-alignment was 18.82° long, and of a mixed kind: the southernmost of its stars, α Ari, had a latitude of 9.93° north. The one in the Triangulum and the one in Andromeda were extra-zodiacal, with latitudes of 20.46° and 27.67°, respectively.⬈#p69

2) […] the most advanced of the stars in the head of Aries and the midpoint of the base of Triangulum lie on a straight line.⬈#p70

Refer to Illustration 19. In this case, Toomer proposes two possibilities. The first, which is Manitius’, identifies the stars in Aries as β and γ Ari, and the “midpoint of the base of Triangulum” as the middle point between β and γ Tri. The second is presented by Toomer himself (1984: 324, note 27), and identifies the stars in Aries as λ and β Ari, and the midpoint as a star, δ Tri. Given that in his catalog Ptolemy puts γ Ari “on the horn” of Aries (Toomer, 1984: 360), it is possible that Hipparchus had done the same. If this is the case, then it would not be “in the head”, and β Ari would take its place as the most advanced in the head, followed by λ Ari.⬈#p71

While Toomer (albeit with doubts) ultimately favoured Manitius’ identification over this second possibility, I think that his alternative idea is actually correct. Not only it is much better than he seems to realize (he qualifies the alignment it determines as “approximate”), but Manitius’ is much worse.⬈#p72

Manitius’ β Ari-γ Ari line passes way off the middle point between β and γ Tri. In fact, it passed 41.12´ to the west of β Tri, the most advanced of the stars at the base of Triangulum. Toomer’s alternative is much better. Because of the fact that δ Tri has a high proper motion⁸, it is very helpful to have modern satellite data to calculate a good estimate of its position over 21 centuries ago. In this case, the angle at the middle star, λ Ari, was 179.47°. For comparison with the previous alternative, in this case the β Ari- λ Ari line passed just 6.25´ to the east of δ Tri.⬈#p73

Assuming the “Toomer alternative”, the alignment was 14.19° long, and it was of a mixed kind. While β Ari was zodiacal (lat. 8.43° north) and λ Ari was right in the limit (lat. 10.68° north), δ Tri (lat. 19.49° north) was well outside the Zodiac.⬈#p74

Taurus.

1) […] the [two] easternmost stars of the Hyades and that star in the pelt held in Orion´s left hand which is sixth, counted from the south, lie on a straight line.⬈#p75

Refer to Illustration 20. This alignment is a fairly good one. Toomer’s identification of the stars as ε Tau, α Tau, and π¹ Ori is correct. Both stars in Taurus were zodiacal, but π¹ Ori was outside, with a latitude of 12.52° south. The alignment was 11.01° long. The angle at α Tau was 177.58°, and that star lay 5.68´ away from the ε Tau-π¹ Ori line.⬈#p76

2) […] the line drawn through the advance eye of Taurus and the seventh star from the south in the pelt cuts off the bright star in the Hyades 1 digit to the north.⬈#p77

Refer to Illustration 21. There is no doubt that the “bright star in the Hyades” is α Tau. It seems certain, as well, that the star in the pelt is ο² Ori. Manitius (Manitius, 1913: 8) identifies the star in the “advance eye of Taurus” as δ Tau, presumably our δ¹ Tau. Toomer challenges this last identification, proposing that we should instead prefer ε Tau. He gives two reasons for this (Toomer, 1984: 324, note 29): first, that the description for ε Tau in Ptolemy’s star catalog puts it as being “on the northern eye”, while δ¹ Tau is “the one between this [γ Tau] and the northern eye”. The second reason is that he calculates that the ο² Ori-δ¹ Tau line passed about 1° to the north of α Tau, while the ο² Ori- ε Tau passed about 8´ to the north, much closer to the Almagest’s description.⬈#p78

There are several things to point out in this case. First that the Almagest’s expression is indicating that the line is passing to the south of α Tau, not the north. Also, by my calculations the ο² Ori-ε Tau line passed, in Hipparchus’ times, 53.18´ to the north of α Tau, and not the 8´ calculated by Toomer. Manitius’ identification is indeed better. The ο² Ori-δ¹ Tau line passed to the south of α Tau (as the Almagest says). The distance is also better, being 30.09´ away from the line.⬈#p79

However, it is possible that neither of those stars was the one referenced by the text. While the Hipparchian description distinguishes the eyes in Taurus as being one “the easternmost” and the other the “more advanced”, Ptolemy’s catalog describes them as being “the southern” and “the northern” (Toomer, 1984: 362). This allows for us to suspect that there could be small variations in the details of their respective constellation drawings, at least regarding the Hyades. If we assume that the description of the alignment is referring to δ³ Tau, a star that is not in Ptolemy’s catalog but that is nevertheless quite bright (mag. 4.32), then we obtain that the ο² Ori-δ³ Tau passed 5.42´ to the south of α Tau, almost precisely the value given in the Almagest.⬈#p80

If this is the case, the alignment was 8.73° long. All the stars were zodiacal.⬈#p81

Gemini.

1) […] the heads of Gemini lie on a straight line with a certain star which lies to the rear of the rearmost head by a distance three times that between the heads […]⬈#p82

Refer to Illustration 22. The identification of the two heads of Gemini as α and β Gem is beyond doubt. Toomer’s identification of the third as ζ Cnc is also certain. The alignment is very good, with an angle at β Gem of 178.62°, and a distance of 4.9´ to the α Gem- ζ Cnc line. The alignment was 16.52° long, and all the stars were within the boundaries of the Zodiac.⬈#p83

The distance between the two heads of Gemini was 4.67°, and that between β Gem and ζ Cnc was 11.85°. This is a ratio of 2.54, not so far from the 3 indicated by the text.⬈#p84

2) […] the same star [i.e., ζ Cnc] also lies on a straight line with the [two] southernmost of the four stars round the nebula [Praesepe].⬈#p85

Refer to Illustration 22. These last two stars are identified with certainty as θ and δ Cnc. The alignment is quite good, with an angle at θ Cnc of 179.49°, and a distance of 0.99´ to the δ Cnc- ζ Cnc line. It was a small alignment, just 7.77° long. All the stars were zodiacal.⬈#p86

Ptolemy’s configurations.

Ptolemy relates, after giving the Hipparchian list, that he has compared the descriptions left by Hipparchus with the layout of the stars in his own times, and that he has found that “[…] no change has occurred up to the present time.” (Toomer, 1984: 325). This supports the idea that precession affects the entirety of the celestial sphere, and not just the zodiacal belt. However, so as to allow his successors to make new comparisons through longer time intervals, he gives a new list of configurations “[…] which were not previously recorded.”⬈#p87

In the following section, I will continue with the analysis as I have done with the Hipparchian cases.⬈#p88

Aries.

1) The two northernmost of the three stars in the head of Aries and the bright star in the southern knee of Perseus and the star called Capella lie on a straight line.⬈#p89

Refer to Illustration 23. An independent identification of the two stars in Aries is not so simple, since the description of the stars here does not fit easily into the way he describes the constellation in the catalog. It is certain that the southern knee of Perseus is ε Per. Toomer prefers α and β Ari for the stars in Aries, and given the good alignment they produce there is no good reason to dispute this identification. This alignment was quite big, spanning 48.07° long the night sky. On the one hand, the angle at β Ari-α Ari-α Aur was 179.53°, with a distance of 1.82´ from α Ari to the β Ari-α Aur line. On the other, the angle α Ari-ε Per-α Aur was 176.93°, and there was a distance of 34´ from ε Per to the α Ari-α Aur line.⬈#p90

Taurus.

1) The line drawn through the star called Capella and the bright star in the Hyades cuts off the star in the advance leg of Auriga a little to the east.⬈#p91

Refer to Illustration 23. The identification of the bright star in the Hyades as α Tau, and of Auriga’s knee as ι Aur is certain. The distance from ι Aur to the α Tau-α Aur line was 22.95´. The entire alignment was 30.8° long.⁹⬈#p92

2) […] the star called Capella, the star which is common to the rearmost foot of Auriga and the tip of the northern horn of Taurus, and the star in the advance shoulder of Orion lie on a straight line.⬈#p93

Refer to Illustration 24. There is no doubt that the star shared by Taurus and Auriga is β Tau. Toomer also identifies “the advance shoulder of Orion” as γ Ori, a star which the catalog describes as “the left shoulder of Orion” (Toomer, 1984: 383). So, it is almost certain that this is the correct identification, even if λ Ori (probably “the nebulous star in the head of Orion”, in Ptolemy’s catalog) determines a much better alignment. The angle α Aur-β Tau-γ Ori was 173.73°, and β Tau was 1.12° away from the α Aur-γ Ori line. The entire alignment was 39.9° long.¹⁰⬈#p94

Gemini.

1) […] the two bright stars in the heads of Gemini and the bright star in the neck of Hydra lie very nearly on a straight line.⬈#p95

Refer to Illustration 25. Toomer follows Manitius in identifying “the bright star in the neck of Hydra” as θ Hya. However, given that Ptolemy describes κ Hya as being “after the bend” in the neck of Hydra (Toomer, 1984: 392; emphasis mine), I think that he was referring to α Hya. This option would not only give a much better alignment, but it would also justify him calling the star “bright”: while Toomer’s identification would give an angle at β Gem of 168.83° and a distance of 48.2´ to the α Gem-θ Hya line, my identification would give an angle of 174.63°, and a distance of 24.13´. Assuming my identification, the alignment was 48.73° long, while Toomer’s and Manitius’ was 37.59°.⬈#p96

Cancer.

1) […] the two stars close together in the front leg of Ursa Major, the star on the tip of the northern claw of Cancer, and the northernmost of the ´Aselli´ lie on a straight line.⬈#p97

Refer to Illustration 26. The identification of the four stars is certain: ι UMa, κ UMa, ι Cnc and γ Cnc, respectively. Toomer rightfully points out that the alignment would have been good (“more plausible”) had Ptolemy removed κ UMa from the configuration (Toomer, 1984: 325, note 36). As Toomer suggests, given that ι UMa is to the north of κ UMa, the original should have indicated the extreme of the alignment not as being comprised of the “two stars close together”, but of “the northernmost of the two stars close together”. Because the text uses the term “northern” or “northernmost” several times in the description, in my opinion it is likely that at some early stage in the textual tradition the hypothesized “northernmost” in the Ursa Major section was lost due to a natural confusion on the part of the copyist.⬈#p98

The unlikeliness of the present text representing the original Ptolemaic description can be assessed by looking at the angles it produces: the ι UMa-κ UMa-ι Cnc angle was 136.75°, and κ UMa was 44.93´ away from the ι UMa-ι Cnc line. These are very bad results, even for a crude estimation.⬈#p99

If we ignore κ UMa, we get an angle ι UMa-ι Cnc-γ Cnc of 179.22°, and a distance of ι Cnc to the ι UMa-γ Cnc line of 4.42´. The entire alignment was 26.87° long.⬈#p100

2) […] the southern Asellus, the bright star in Procyon, and the bright star between them (which is in advance of the head of Hydra), lie almost on a straight line.⬈#p101

Refer to Illustration 27. The identification as δ Cnc, α CMi, and β Cnc, is certain. The alignment is not particularly good, even for an “almost” straight line. The angle at β Cnc was 146.82°, and the distance of that star to the δ Cnc-α CMi line was 3°. The alignment spanned 20° long the night sky.⬈#p102

A possible explanation is that Ptolemy was not originally referring to Procyon, but to Sirius. The Almagest indicates the star as being ὁ ἐν τῷ Πρόκυνι (Heiberg, vol II, 9). If the original instead read ὁ ἐν τῷ κῠνί, “the one in the dog”, then it would be indicating α CMa, and the alignment would be much better. Ptolemy’s catalog calls Sirius “the Dog” (Toomer, 1984: 387). The name Procyon literally means before or in front of, the dog. Given the similar names and their associations, and since they are in the same general direction, a confusion by the copyist cannot be discarded. As we will see in the last section, a comparison between Ptolemy’s and Hipparchus’ alignments in Cancer may provide additional support for this option. If this is the case, it would be one of just three mentions to Sirius in the entire Almagest.¹¹ By my calculations, in this case the angle at β Cnc was 174.61°, and the distance of that star to the δ Cnc-α CMa line was 49.89´. The entire alignment would have been 45.06°.⬈#p103

Leo.

1) […] the straight line drawn from the midmost star of the bright stars in the neck of Leo to the bright star in Hydra cuts off the star on the heart of Leo a little to the east.⬈#p104

Refer to Illustration 28. Toomer’s identification of the stars as γ Leo, α Hya and α Leo is certain. The line passed 1.06° to the west of α Leo, and was 31.3° long.⬈#p105

2) The [line] from the bright star in the rump of Leo to the bright star in the back of the thigh of Ursa Major (which is the southernmost star on the rear side of the quadrilateral), cuts off, a little to the west, the two stars which are close together in the rear paw of Ursa Major.⬈#p106

Refer to Illustration 29. The four stars can be securely identified as δ Leo, γ UMa, and the pair ν and ξ UMa, respectively. The δ Leo-γ UMa was 33.96° long, and passed 1.3° to the east of ν UMa, and 56.79´ to the east of ξ UMa.⬈#p107

Virgo.

1) […] the line from the star in the back of the thigh of Virgo to the second star in the tip of Hydra’s tail cuts off the star called Spica a little to the west.⬈#p108

Refer to Illustration 30. The first two stars can be identified with confidence as ζ Vir and γ Hya. The line they determine was 22.88° long, and passed 38.08´ to the east of Spica.⬈#p109

2) The line from Spica to the star in the head of Bootes cuts off Arcturus a little to the east.⬈#p110

Refer to Illustration 31. The star in the head of Bootes is β Boo. The Spica-β Boo line was 56.2°, and passed 15.23´ to the west of Arcturus.⬈#p111

3) Spica and the stars in the wings of Corvus lie on a straight line.⬈#p112

Refer to Illustration 32. The stars in the wings are δ and γ Crv. The Spica-γ Crv line was 17.92°. The angle at δ Crv was 178.75°, and the distance of thar star to the line was 3.76´. This is a very good alignment. It should be noted that while Toomer and Manitius (Manitius, 1913: 10) only refer to the two stars I mentioned, it is possible that Ptolemy was thinking of a set of three: η, δ and γ Crv. Not only is η Crv described in Ptolemy’s catalog as the “the rearmost of the two stars in the rear wing” (Toomer, 1984: 394), but is also forms a good alignment with the other two stars in Corvus.⬈#p113

4) Spica and the star in Virgo’s thigh, and the northernmost, bright star of the three in the advance knee of Bootes lie on a straight line.⬈#p114

Refer to Illustration 33. This is a very good alignment. The star in Virgo’s thigh has already been identified as ζ Vir. The bright star in the advance knee of Bootes is η Boo. The Spica- η Boo line was 30.61° long. The angle at ζ Vir was 179.38°, and the distance of that star to the line was just 4.64´.⬈#p115

Libra.

1) […] the bright stars in the claws, and the star on the tip of Hydra’s tail are very nearly on a straight line.⬈#p116

Refer to Illustration 34. The two stars in Libra are α and β Lib, and the one on the tip is π Hya. The entire β Lib-π Hya line spanned 24.05° long the night sky. The angle at α Lib was 178.04°, and the distance of that star to the line was 11.84´.⬈#p117

2) The bright star in the southern claw, Arcturus, and the midmost of the three stars in the tail of Ursa Major lie on a straight line.⬈#p118

Refer to Illustration 35. This is an extremely large alignment, and a very good one. The α Lib-ζ UMa line spanned 73.33° long the night sky. The angle at Arcturus was 179.9° (just 5.9´ from a perfect alignment!), and that star was located only 2.2´ away from the α Lib-ζ UMa line.¹²⬈#p119

3) The bright star in the northern claw, Arcturus, and the star in the back of thigh of Ursa Major lie on a straight line.⬈#p120

Refer to Illustration 35. With this alignment, Ptolemy produces a nice set of two descriptions that use the stars in both claws, Arcturus, and a star in Ursa Major. The β Lib-γ UMa line spanned 76.27° long the night sky. The angle at Arcturus was 177.07°, and β Lib-γ UMa passed 1.13° away from Arcturus.⬈#p121

Scorpius.

1) […] the star on the rear shin of Ophiuchus, the star in the fifth tail-joint of Scorpius, and the more advanced of the two stars close together in its sting lie on a straight line.⬈#p122

Refer to Illustration 36. Toomer identifies “rear shin of Ophiuchus” as ξ Oph, and the two stars in Scorpius as θ and υ Sco, respectively. The identification seems certain. Thus, we get an alignment that was 22.27° long. The angle at υ Sco was 176.73°, and distance of that star to the θ Sco-ξ Oph line of 14.89´.⬈#p123

2) The most advanced of the three stars in the breast of Scorpius, and the two stars in the knees of Ophiuchus, form an isosceles triangle, the apex of which is the most advanced of the three stars in the breast.⬈#p124

Refer to Illustration 36. Toomer’s identification of the stars as σ Sco, ζ Oph, and η Oph, respectively, is certain. The distance between σ Sco and ζ Oph was 15.47°, and that between σ Sco and η Oph was 15.08°. The difference between the two sides was just 23.23´, and the ratio 1.03: an almost perfect isosceles triangle.⬈#p125

Sagittarius.

1) […] the star on the front, southern hock of Sagittarius (which is of second magnitude), the star on the arrow-head, and the star in the rear knee of Ophiuchus lie on a straight line.⬈#p126

Refer to Illustration 37. As before¹³, here we have to decide which star in the southern hock of Sagittarius Ptolemy is referring to: β¹ or β² Sgr? The better fit seems to be β¹ Sgr. With this star, we have an alignment with γ Sgr and η Oph which is fairly good. It had a size of 40°. The angle at γ Sgr was 177.33°, and that star lay 29.18´ from the β¹ Sgr-η Oph line.⬈#p127

2) The star in the knee of the same leg of Sagittarius (which lies near Corona [Australis]), the star on the arrow-head, and the star in the advance knee of Ophiuchus lie on a straight line.⬈#p128

Refer to Illustration 37. As he had done with the alignments between Libra, Arcturus and Ursa Major, here Ptolemy is using two stars from the front leg in Sagittarius, γ Sgr, and both knees in Ophiuchus to produce a set of two related alignments. In this second alignment, the stars are identified with certainty as α and γ Sgr, and ζ Oph. The alignment was 47.37° long. The angle at γ Sgr was 179.18°, and that star lay 10.32´ from the α Sgr-ζ Oph line.⬈#p129

Capricorn.

1) […] the line drawn from the bright star in Lyra to the stars in the horns of Capricorn cuts of the bright star in Aquila a little to the east.⬈#p130

Refer to Illustration 38. The stars in Lyra and Aquila are, with certainty, Vega and α Aql. Toomer gives good reasons (Toomer, 1984: 326, note 44) to identify “the stars in the horns of Capricorn” not as α and β Cap, but as the general direction of α, β, ν, and ξ Cap, i.e., the stars that Ptolemy locates in the horns in his catalog (Toomer, 1984: 375). I am not convinced that Ptolemy would give such a vague reference in his description of the alignment. However, I am not able to give any further argument other than the fact that this is not what he did in any other case.⬈#p131

For the analysis I will use as a reference β Cap, simply because its direction is approximately in the center of this set of stars. Assuming this, the alignment would have been 58.74° long. The β Cap-Vega line passed 2.07° to the west of α Aql.⬈#p132

2) The line from the bright star in Aquila to the first-magnitude star in the mouth of Piscis Austrinus bisects, approximately, the interval between the two bright stars on the tail of Capricorn.⬈#p133

Refer to Illustration 39. The four stars can be identified with certainty as α Aql, α PsA, and γ and δ Cap. The two lines determined by them were 59.06° and 1.76°, respectively. The section of γ Cap-δ Cap on the side of δ Cap was 1.34°, and the one on the side of γ Cap was 0.42°, with a ratio of 3.17: hardly an “approximate” bisection.⬈#p134

Aquarius.

1) […] the line from the first-magnitude star in the mouth of Piscis Austrinus to the the star in the muzzle of Pegasus cuts off the bright star in the rear shoulder of Aquarius, a little to the east.⬈#p135

Refer to Illustration 40. The stars are identified with certainty as α PsA, ε Peg, and α Aqr. The α PsA-ε Peg line was 43.14° long, and α Aqr lay 56.4´ to the east of it.⬈#p136

Pisces.

1) […] the stars in the mouths of Piscis Austrinus and the southern fish and the advanced stars of the quadrilateral in Pegasus lie on a straight line.⬈#p137

Refer to Illustration 41. There can be no doubt that the stars are α PsA, β Psc, and α and β Peg. The alignment is fairly good. If we take the α PsA-β Psc-β Peg alignment (57.57° long), the angle at β Psc was 177.03°, and that star was located 47.59´ away from the α PsA-β Peg line. If we take the α PsA-α Peg-β Peg alignment, we have an angle at α Peg of 176.39°, and a distance between α Peg and the α PsA-β Peg line of 40.01´.⬈#p138

General analysis.

The first thing that strikes the scholar when comparing both lists of stellar configurations is the fact that Hipparchus’ begins in Cancer, while Ptolemy’ does so in Aries. It may be that this variation has no special meaning. However, a possible reason for Hipparchus to put the origin of the list in the summer point rather that in the vernal point may lie in the fact that he used the Kallippic Cycle¹⁴ as the main reference for dating astronomical observations. Now, the First Kallippic Cycle began with the summer solstice of -329, so this may be an indication that his epoch had, at least in this respect, a greater significance.⬈#p139

In my study of the Hipparchian configurations I have noted the zodiacal or extra-zodiacal nature of the stars that determine them. As Evans suggests (Evans, 1998: 259) it may very well be the case that Hipparchus’ discovery of precession came while developing his lunar model. In fact, Ptolemy says (Toomer, 1984: 327) that Hipparchus used, as a reference for comparison, the longitude of Spica Timocharis had obtained during a lunar eclipse. Even if Timocharis had used the distance between Spica and the moon as a way of determining the position of the moon, Hipparchus could reverse the method. Given that he had an accurate solar model, he could determine the position of the moon independently, and thus obtain the longitude of Spica at the time of Timocharis. Because the previous reference for stellar positions was the moon, Hipparchus must not have had many positions of extra-zodiacal stars which he could use to check the extent of the precession phenomenon. Thus, his research on this question must have been tentative.⬈#p140

However, when we look at the list of configurations Ptolemy preserved in the Almagest, we can see that Ptolemy’s suggestion that Hipparchus ended up concluding that precession was universal was certainly grounded in Hipparchus’ research. Out of the 24 sets of stars¹⁵ he uses in the configurations that have come down to us, 14 are of a mixed nature. The rest are equally divided: 5 are sets of all extra-zodiacal stars, and 5 are sets of all zodiacal ones. In fact, almost 6 out of 10 of the stars the configurations refer to had a latitude greater than ± 10°. If Ptolemy’s is a list which is representative of the original work, then it seems that Hipparchus made sure to cover all the possibilities in his research.⬈#p141

But, as we saw, Ptolemy’s role is not limited to recounting some of his predecessor’s configurations, and checking them in his own times. In a bold move, Ptolemy recognizes the possibility that proper stellar motion may be so slow that even the period between himself and Hipparchus is not enough to detect a variation in their relative positions. His list of configurations is thus given as supplementary material for future astronomers to check this feature. One the one hand, this shows us a rather modern approach on Ptolemy’s part. On the other, we should not overestimate the importance of this addendum, because Ptolemy had shown that, if the fixed stars were in fact not fixed and had a proper motion, the speed of this motion was so slow that it would not affect the accuracy of his planetary models. He could trust that both the ancient observations he had received¹⁶ and the predictions his theories could deliver in the foreseeable future would fit well in his model of the cosmos, even if there was in fact some kind of proper motion. If some future astronomer found this, only minor modifications would be needed to correct his work.⬈#p142

As Evans points out, though, these considerations were probably not at the center of Ptolemy’s preocupations when composing this section of the Almagest:⬈#p143

“For Hipparchus it was still an open question whether the precession was confined to the zodiac stars or shared by the whole sphere. […] By Ptolemy’s time the issue had been settled. Ptolemy’s aim was simply to give his readers a number of easily checked rough alignments involving bright and familiar stars.” (Evans, 1998: 250)⬈#p144

As in many other sections of the Almagest, it is likely that Ptolemy is here just offering enough observational evidence to justify going forward in his theoretical constructions without leaving any relevant issue (like the fixedness of the stars) untouched. In this sense, this section is there mainly due to its didactical importance. That explains that Ptolemy used brighter stars: the average magnitude of the stars Hipparchus used was 3.41, with a standard deviation of 1.45.¹⁷ In contrast, Ptolemy’s stars had an average magnitude of 2.5, with a standard deviation of 1.14. This means that the average Ptolemaic star was 2.3 times brighter than the average Hipparchian one. This criterion of brightness likely forced Ptolemy to make his alignments much larger: while the average size for Hipparchus was 19.94° (stand. dev. 11.5), Ptolemy’s was almost double, with a value of 39.94° (stand. dev. 16.87).⬈#p145

However, an analysis of the alignments shows that Ptolemy’s alignments were not “[…] somewhat sloppy […] (Evans, 1998: 250) when compared to Hipparchus’. If we look at the cases which the astronomers described as being “straight lines” without any further qualification (15 for Hipparchus and 13 for Ptolemy, if we don´t count the exceptionally bad case in Hipparchus’ first alignment in Aquarius, θ Peg-ν Peg-α Aqr), we find that the average error in the angle at the middle star was 3.5° for Hipparchus (stand. dev. 2.98), and 2.21° for Ptolemy (stand. dev. 1.74). The distances of the middle stars to the line determined by the extreme ones were 21.42´ (stand. dev. 22.9) for Hipparchus, and 25.22´ (stand. dev. 24.34) for Ptolemy. These results indicate that both astronomers, when pointing at alignments as if they were very close to being perfect lines, produced results with very similar errors.⬈#p146

Many alignments are nevertheless qualified in various ways, as we have seen. Table 1 shows how these different quasi-alignments were given for Hipparchus.⬈#p147

I have no answer as to how Hipparchus could give such a bad estimate in the cases of Aries and Cancer. While the cases for Taurus and Leo are very good approximations to the 1 digit value, the other two are off by several digits.⬈#p148

The cases of approximate alignments are much better. We can see that an approximate alignment was, for Hipparchus, one that was close to 170°. Given that there is a big variation in how that angle relates to the distance of the middle star to the line, it is likely that this second value was of little consequence in the method he used to determine the alignments.⬈#p149

Ptolemy’s quasi-alignments are shown in Table 2.⬈#p150

In Ptolemy’s case, the term μικρόν, when referring to the distance to the line between the two extreme stars, is frankly very vague. We find that it can vary from less than ½° in the cases of Taurus and Virgo, up to more than 2° in the case of Capricorn. In general, it indicates a fairly large distance when compared to the alleged “straight lines” we saw before. ⬈#p151

The last three cases are dissimilar. While the case in Gemini fits with the average error in Hipparchus, the case in Libra is very close to a perfect alignment: it could have been described by Ptolemy as a “straight line”. The case in Cancer is a special one. As we saw earlier, while the text clearly indicates Procyon as one of the extremes, the error is so big, and so out of tune with the average error, that the Sirius option (174.81° and 49.89´) should not be discarded.⬈#p152

These considerations about the errors in both astronomers may point to the question about observational methods: what kind of instrument were Hipparchus and Ptolemy using when determining these alignments? One could even ask if they were using the same. Pedersen (2010: 237) suggests that they used a tout string to determine the alignments. Evans (1998: 249) argues against this possibility, correctly indicating that it is very difficult to focus one’s view on the string and the stars at the same time, and that “One can actually do much better with the unassisted eye than with the aid of a handheld string.” During a recent trip to a location with unpolluted skies, I carried out a simple experiment and asked some laymen to find several sets of three stars that they considered to be on a straight line. I then checked their choices and found that the errors very much coincided with the margin of error which our astronomers’ alignments showed for “straight lines”. The errors in alignments determined by me also fit into this general range. This is, of course, just an informal study, and a more methodical survey might be carried out in the future to determine more precisely how accurately the unaided eye determines great circles in the sky.⬈#p153

Hipparchus’ influence on Ptolemy: some interesting comparisons.

Did Ptolemy use Hipparchus’ configurations in the determination of his own, original ones? Regarding the type of configurations, it is obvious that he was inspired by Hipparchus, or by the subset of Hipparchian configurations he chose to quote. We only find in the Ptolemaic ones cases of straight lines, quasi-straight lines, bisections of one line by another, or an isosceles triangle. In the Hipparchian list we already find examples of these types. So there is no originality in this regard on Ptolemy’s part.⬈#p154

Regarding the star selection, only about 35% of the stars Ptolemy uses were also used by Hipparchus. The five brightest stars used by Hipparchus are also among the stars used by Ptolemy: Arcturus, Procyon, α Tau, Spica and β Gem. Only the last three were used by Ptolemy as reference stars (Pedersen, 2010: 236) in other parts of the Almagest. As was pointed out earlier, though, given his aims it is only natural that he used as many bright stars as he could in his alignments. There are other instances, though, where Ptolemy reused fainter stars, such as β Psc, which is almost the faintest used by Ptolemy. A comparison between the alignments in Hipparchus and Ptolemy explains why Ptolemy reused it.⬈#p155

Refer to Illustration 42. It is clear from the comparison that Ptolemy likely determined his own as a mere extension of the Hipparchian alignment he was checking. Under closer examination, other Ptolemaic examples seem plausibly to have originated during Ptolemy’s study on Hipparchus’ alignments.⬈#p156

Refer to Illustration 43. Ptolemy’s alignment in Gemini indicates that a certain star is in the same line as α and β Gem. This is parallel to Hipparchus’ first alignment in Gemini, which does the same. Again, Ptolemy’s example seems to be an extension of Hipparchus’.⬈#p157

The following comparison, even if it does not indicate a Hipparchian inspiration, does support the idea that Ptolemy had originally referred to Sirius instead of Procyon in his second alignment in Cancer. Refer to Illustration 44.⬈#p158

In red, we can see that Hipparchus determined that α Cnc and β Cnc were almost in the same line with Procyon. If Ptolemy, with good reasons, judged this alignment to be accurate, why would he then say that it is δ Cnc the star in Cancer that aligned with β Cnc and Procyon?⬈#p159

Ptolemy’s study of Hipparchus’ alignments in Virgo also seem to have served of inspiration. Refer to Illustration 45.⬈#p160

As before, the Ptolemaic alignment is nothing more than an extension of the one Hipparchus had left, and which Ptolemy says he checked.⬈#p161

Finally, and although it is not obvious, it is possible that a similar case can be found in Libra. Refer to Illustration 46. It is possible that Ptolemy noticed the β Lib-α Lib-π Hya alignment while checking Hipparchus’ α Lib-β Lib-μ Ser alignment.⬈#p162

The subject of this paper has been deliberately kept exclusively focused on the analysis of the configurations themselves, and the relations I could find between them. However, there are several questions which are central to this section of the Almagest and which deserve new research: what was the method, if there was any, to determine the alignments given here, or the angular distances involved in the descriptions provided by the text? How are the descriptions given in this section related to the star catalog in the Almagest? What was the fate of the list of configurations within the Hellenistic and Islamic astronomical communities?⬈#p163

I hope that this paper will serve as a stepping-stone on which future research can be built upon.⬈#p164

Appendix 1. Determination of parallelism.

Refer to Figure 1. Let A and B, and C and D, be two pairs of stars that determine lines AB and CD respectively. If they are not parallel, they will intersect at a point E. Produce lines CF and DG in such a way that they are perpendicular to AB. Finally, produce line DH in such a way that it is perpendicular to CF.⬈#p165

One way to determine how far away from parallelism AB and CD are, is to determine the angle ∠CEA. The smaller it is, the closer their relation is to parallelism. From (Meeus, 1998: 109) and (Meeus, 1998: 125) we know the value of CF, DG and CD. Because HFGD is a rectangle, we know that DG = HF. But CH = CF - HF, therefore we can get $$\mathit{CH}=\mathit{CF}-\mathit{DG}.$$⬈#p166

So, we have the right-angled triangle CHD of which we know CH and CD. So, we can get $$\mathit{HD}=\sqrt{{\mathit{CD}}^2-{\mathit{CH}}^2}.$$⬈#p167

Then, we obtain $$\angle \mathit{HCD}={\sin }^{-1}\frac{\mathit{HD}}{\mathit{CD}}.$$⬈#p168

Now we have the right-angled triangle FCE, of which we know angle $\angle \mathit{FCE}=\angle \mathit{HCD}$. So, we can get $$\angle \mathit{CEA}=\angle \mathit{CEF}=90°-\angle \mathit{FCE}.$$⬈#p169

Appendix 2. Determination of bisection.

Refer to Figure 1. Let A and B be two stars that determine line AB, and C and D another pair of stars that determine line CD. Let E be the point where AB and CD intersect. Finally, determine line AC. If line AB bisects line CD, then $\mathit{EC}=\mathit{ED}$. If it does not, then one will be greater than the other.⬈#p170

We have the spherical triangle ACE. Thanks to (Meeus, 1998: 109) and (Meeus, 1998: 124), we know side AC and angles $\sphericalangle \mathit{ACD}=\sphericalangle \mathit{ACE}$ and $\sphericalangle \mathit{CAB}=\sphericalangle \mathit{CAE}$, that is, a side and two adjacent angles. Therefore, we get $$\sphericalangle \mathit{CEA}={\cos }^{-1}$$⬈#p171

Once we have that value, we can get $$\mathit{EC}={\cos }^{-1}\left(\frac{\cos \sphericalangle \mathit{CAE}+\cos \sphericalangle \mathit{ACE}\times \cos \sphericalangle \mathit{CEA}}{\sin \sphericalangle \mathit{ACE}\times \sin \sphericalangle \mathit{CEA}}\right)$$ We also know, thanks to (Meeus, 1998: 109), the distance CD. Therefore, we can get $\mathit{ED}=\mathit{CD}-\mathit{EC}$.⬈#p172

Bibliography

European Space Agency et alia. (1997). The Hipparchos and Tycho Catalogues: Astrometric and Photometric Star Catalogues derived form the ESA Hipparcos Space Astrometry Mission (Vols. V-IX). Noordwijk, Netherlands: ESA Publications Division.⬈#reference-1

Evans, J. (1998). The History and Practice of Ancient Astronomy. New York: Oxford University Press.⬈#reference-2

Heis, E. (1872). Atlas Coelestis Novus. Cologne: DuMont Schauberg.⬈#reference-3

Jones, A. (2000). Calendrica I: New Callippic Dates. Zeitschrift für Papyrologie und Epigraphik, 129, 141-158.⬈#reference-4

Jones, A. (2003). A Posy of Almagest Scholia. Centaurus, 45, 69-78.⬈#reference-5

Manitius, K. (1913). Des Claudius Ptolemäus Handbuch Der Astronomie (Vol. II). Leipzig: B. G. Teubner Verlag.⬈#reference-6

Meeus, J. (1998). Astronomical Algorithms (2nd ed.). Richmond, Virginia: Willmann-Bell, Inc.⬈#reference-7

Pedersen, O. (2010). A Survey of the Almagest: with annotation and new commentary by Alexander Jones. (A. Jones, Ed.) New York: Springer.⬈#reference-8

Toomer, G. (1984). Ptolemy's Almagest. Princeton, New Jersey: Princeton University Press.⬈#reference-9

Funding

Research Program “Filosofía e Historia de la Ciencia, etapa II”. Universidad Nacional de Quilmes, Argentina.

Research Project “Astronomía pre-Newtoniana: aspectos históricos (segunda etapa)”. Agencia Nacional de Promoción Científica y Tecnológica. Universidad Nacional de Quilmes, Argentina.

Acknowledgments.

I would like to thank Christián Carman, Aníbal Szapiro, Diego Pelegrin and Alexander Jones for their comments and suggestions to earlier versions of this paper. I would also like to thank Miguel Barzizza Mondria, A. Benjamín Menendez, Sebastián Sarabola and J. Quinto Driollet for their assistance in my informal observational experiments.

Notes

¹ It could be argued that it is arbitrary to measure the distance of the middle star to the line determined by the extremes, instead of measuring the distance of one of the extremes to the line determined by the other two. The only reason I have to decide for the former option is that, in the cases when the Almagest gives any indication of the distance of a star to a line, it gives the distance of the middle one. In my opinion, this says something about the method used by them to measure the alignments, even if it is true that there is no certainty about it. Given that the calculations were done on a computer, it would be easy to give all the values for the deviations in every case. However, this would make the text even more cumbersome than it already is.⬈#footnote-1

² The cubit is an angular unit that, in Ptolemy’s astronomy, represents 2° (Jones, 2003: 76). The digit represents $\frac{1}{24}$^th of a cubit: approximately 5´. See (Toomer, 1984: 322, note 5).⬈#footnote-2

³ I will classify as zodiacal stars the ones which had, in Hipparchus’ times, a latitude less than 10° north or south.⬈#footnote-3

⁴ ζ Hya is already, in Ptolemy’s catalogue, located in the head of Hydra.⬈#footnote-4

⁵ See the “Methodological clarifications” at the beginning.⬈#footnote-5

⁶ The magnitudes are 3.39 for ζ Peg and 4.31 for ξ Peg, meaning that the first was 2.3 times brighter than the second. The Almagest’s catalog also (crudely) indicates this relation with magnitudes 3 and 4, respectively.⬈#footnote-6

⁷ This value was obtained from the angle $\sphericalangle \mathit{CEA}$.⬈#footnote-7

⁸ According to the Hipparcos mission data, it has an annual proper motion of 1151.61 milliarcseconds in right ascension, and of -246.32 in declination.⬈#footnote-8

⁹ This alignment is one of the few Evans refers to in his History and Practice (Evans, 1998: 249), and gives the same distance I obtained.⬈#footnote-9

¹⁰ Evans (1998: 249) also points out that this is a poor alignment, with an error of “[…] more than a degree.”⬈#footnote-10

¹¹ Besides the mention in the star catalog, he uses it as the reference for the construction of the solid globe (Toomer, 1984: 405).⬈#footnote-11

¹² Evans (1998, 249) emphasizes that “This seems to be the only one of Ptolemy’s alignments that can be used today to reveal a proper motion”, correctly indicating that today Arcturus is notably farther from the line than in Ptolemy’s times (46.17´ by my calculations). As Evans says, it is precisely Arcturus’ high proper motion (an annual motion -1093.45 milliarcseconds in RA, and -1999.4 milliarcseconds in declination, according to the Hipparcos mission data). However, the angle at Arcturus is 177.93°. Both the angle and the distance to the line are well within Ptolemy’s margin of error: one should only look to the next one to find similar errors in a similarly large alignment.⬈#footnote-12

¹³ See the first Hipparchian configuration for Sagittarius.⬈#footnote-13

¹⁴ Cf (Toomer, 1984: 138 and 328) for instances of Hipparchus dating observations using this reference. For a more general discussion of Kallippic dates, see (Jones, 2000).⬈#footnote-14

¹⁵ As I said in the beginning, the Almagest describes 26 configurations. In a couple of cases though, the same set of stars is used for two different configurations.⬈#footnote-15

¹⁶ The earliest observation in the Almagest that involves a reference star is from -294 (Toomer, 1984: 337). Thus, from Ptolemy’s point of view, the interval to his own time is not so great that a possible proper motion would affect its usefulness.⬈#footnote-16

¹⁷ In this analysis and the ones that follow, I have mostly assumed Toomer’s identifications of the stars, and preferred my own only when it seemed secure to do so. Given the number of alignments we have received, the results are very much the same regardless of what options I chose.⬈#footnote-17