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ISAW Papers 23 (2022)
A study on the stellar configurations in Almagest VII 1
Gonzalo L. Recio, Universidad Nacional de Quilmes/Universidad Pedagógica Nacional
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.
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.
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.
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.
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.
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.
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.
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, in Figure 1. To do it, I will use the formula given in (Meeus, 1998: 125).
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.
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
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.
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.
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).
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.
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.
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).
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.
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)
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.
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.
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.
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
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.
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.
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.
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.
[…] 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.
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.
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.