On Ptolemy's observations, see BrittonJohn P., Models and precision: The quality of Ptolemy's observations and parameters (New York, 1992), 1–11. The accurate value of the obliquity for Ptolemy's time was significantly lower, approximately 23°40′46″.
2.
A selection from the more recent scholarship would include RawlinsDennis, “Eratosthenes' geodesy unraveled: Was there a high-accuracy Hellenistic astronomy?”, Isis, lxxiii (1982), 259–65; FowlerDavid H., “Eratosthenes' ratio for the obliquity of the ecliptic”, Isis, lxxiv (1983), 556–62 (with a reply by D. Rawlins); GoldsteinBernard R., “The obliquity of the ecliptic in ancient Greek astronomy”, Archives internationales d'histoire des sciences, xxxiii (1983), 3–14; TaisbakC. M., “Eleven eighty-thirds: Ptolemy's reference to Eratosthenes in Almagest I.12”, Centaurus, xxvii (1984), 165–7; and ToomerG. J., Ptolemy's Almagest (London, 1984), 63 n. 75.
3.
This is pointed out, for example, by NeugebauerO., A history of ancient mathematical astronomy (3 vols, Berlin, 1975), i, 101 n. 1. Alexandria's correct latitude is 31°13′; Ptolemy should have been able to detect a discrepancy of a quarter degree.
4.
Eratosthenes is not known to have divided the circle into 360°; Strabo (2.5.7) attributes to him a division into sixty units, which would of course translate easily into degrees.
5.
The clearest source for (2) and (3) is Cleomedes 1.7; for (4), which involves the rounding up of Eratosthenes's estimate of the earth's circumference as 250000 stades to the nearest multiple of a thousand divisible by sixty, see Strabo 2.5.7.
6.
For numbers within about half a minute of (23°51′20″/180) either way, approximation by continued fractions (or mathematically equivalent methods) yields after four steps. While the specific mathematical tools available in Antiquity for obtaining close fractional representations of given quantities are a matter for conjecture, there is no question that such tools did exist.
7.
Neugebauer, op. cit. (ref. 3), ii, 304–6, attempted to explain the numbers as generated by an arithmetical sequence; but he failed to show how Hipparchus could have found a sequence matching so accurately the theoretically correct latitudes.
8.
DillerA., “Geographical latitudes in Eratosthenes, Hipparchus and Posidonius”, Klio, xxvii (1934), 258–69; RawlinsDennis, “An investigation of the ancient star catalog”, Publications of the Astronomical Society of the Pacific, xciv (1982), 359–73, p. 368.
9.
According to Strabo (2.5.35), Hipparchus stated that on his southernmost parallel, through the “Cinnamon-bearing country”, the star α UMi grazes the horizon, and according to Ptolemy, Geography 1.7, Hipparchus placed α UMi from the celestial pole. Hence Hipparchus should have situated this parallel 8680 stades from the equator, not 8800 as Strabo has it. In 2.5.36 Strabo reports Hipparchus as stating that Syene was both on the tropic circle and on the circle for which the longest day is 13½ hours; both cannot be true if Syene is 16800 stades (exactly 24°) north of the equator, whereas both could be approximately true if Syene was about 100 stades further south. In 2.5.38 the shadow ratio 3: 5 is given for Alexandria, hence its latitude should be almost exactly 31°, or 21700 stades north of the equator; but Strabo's intervals yield 21800. Has someone added 100 stades to all Hipparchus's distances in order to put Syene at latitude 24°? Among other inconsistencies are two seriously faulty stade intervals near the northern end of the list, for which I believe the explanation of Neugebauer (op. cit. (ref. 3), ii, 305) is correct.
10.
From 2.5.38: Alexandria, equinoctial shadow ratio 3: 5 (an editorial emendation of 7: 5 in the manuscripts); Carthage, 7: 11. In 2.5.39 Strabo situates Syracuse 400 stades north of the parallel through Rhodes, which would make it 25600 stades, or 36°34′, north of the equator; this is certainly derived from an equinoctial shadow ratio of 3: 4, though Strabo does not mention this.
11.
Other attested shadow ratios for specific localities, e.g. those in Vitruvius 9.7 and Pliny 2.182–3, are always ratios of whole numbers (and all equinoctial). The only list of ratios (equinoctial and solstitial) that I know of that resemble this Hipparchian 41½ 120 in refinement is Ptolemy's in Almagest 2.6, which is of course calculated. (So, at least in part, is the cruder list in Pliny 6.212–18.) It is true that Strabo (1.4.4, cf. 2.1.12 and 2.5.8) reports that Pytheas measured some shadow ratio at Massilia and that Hipparchus measured the same shadow ratio “at the same season” at Byzantium, but it is hardly to be believed that Pytheas's late fourth-century B.C. measurement was to a precision of of the gnomon's length. Nevertheless there is a long scholarly tradition of taking 41½: 120 as an observation by Pytheas: See DicksD. R., The geographical fragments of Hipparchus (London, 1960), 178–9, and SzabóÁrpád, “Strabon und Pytheas — die geographische Breite von Marseille. Zur Frühgeschichte der mathematischen Geographie”, Historia scientiarum, xxix (1985), 3–15. Goldstein (pp. cit. (ref. 2), 10–12) recognizes that the ratio must be the product of calculation, and infers as I do that Hipparchus must have had a value for the obliquity close to Ptolemy's.
after four steps. While the specific mathematical tools available in Antiquity for obtaining close fractional representations of given quantities are a matter for conjecture, there is no question that such tools did exist.
from the celestial pole. Hence Hipparchus should have situated this parallel 8680 stades from the equator, not 8800 as Strabo has it. In 2.5.36 Strabo reports Hipparchus as stating that Syene was both on the tropic circle and on the circle for which the longest day is 13½ hours; both cannot be true if Syene is 16800 stades (exactly 24°) north of the equator, whereas both could be approximately true if Syene was about 100 stades further south. In 2.5.38 the shadow ratio 3: 5 is given for Alexandria, hence its latitude should be almost exactly 31°, or 21700 stades north of the equator; but Strabo's intervals yield 21800. Has someone added 100 stades to all Hipparchus's distances in order to put Syene at latitude 24°? Among other inconsistencies are two seriously faulty stade intervals near the northern end of the list, for which I believe the explanation of Neugebauer (
of the gnomon's length. Nevertheless there is a long scholarly tradition of taking 41½: 120 as an observation by Pytheas: See