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Thales' Determination of the Diameters of the Sun and Moon

Published online by Cambridge University Press:  23 December 2013

A. Wasserstein
A. Wasserstein
Affiliation:
University of Glasgow
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Extract

Cle omedes, De motu circulari corporum caelestium. II. 75 (p. 136 Ziegler). Ἐλέγχεται δὲ καὶ διὰ τῶν ὑδρολογίων τὸ εὔηθες τοῦ λόγου [viz. ὅτι ποδιαῖός ἐστιν ὁ ἥλιος]. Δείκνυται γὰρ δι' αὐτῶν, ὅτι, ἂν ᾖ ποδιαῖος ὁ ἥλιος, δεήσει τὸν μέγιστον τοῦ οὐρανοῦ κύκλον ἑπτακοσίων πεντήκοντα ποδῶν εἶναι. Διὰ γὰρ τῶν ὑδρολογίων καταμετρούμενος εὑρίσκεται μέρος ἑπτακοσιοστὸν καὶ πεντηκοστὸν τοὑ οἰκείου κύκλου. Ἐὰν γὰρ, ἐν ᾧ αὐτὸς ἀνέρχεται πᾶς ἐκ τοῦ ὁρίƷοντος ὁ ἥλιος, κύαθος, φέρε εἰπεῖν, ῥεύσῃ, τὸ ὅδωρ ἀφεθὲν ὅλῃ τῇ ἡμέρᾳ καὶ νυκτὶ ῥεῖν εὑρίσκεται κυάθους ἔχον ἑπτακοσίους καὶ πεντήκοντα. Λέγεται δὲ ἡ τοιαύτη ἔφοδος ὑπὸ πρώτων τῶν Αἰγυπτίων ἐπινοηθῆναι.

Type
Research Article
Information
The Journal of Hellenic Studies , Volume 75 , November 1955 , pp. 114 - 116
Copyright
Copyright © The Society for the Promotion of Hellenic Studies 1955

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References

My thanks are due to my friend, Dr. T. R. Tannahill of the Department of Astronomy, University of Glasgow, who kindly read the manuscript of this article and made valuable suggestions in matters of astronomical information.

1 This figure is repeated at II. 82 (p. 150 Ziegler).

2 E.g. Heraclitus: see fragment 3 (Diels6, p. 151) Cf. Diog. L. IX. 7 (Diels 6 I, p. 141) and Theodoretus Graec. Aff. Cur. IV. 22, p. 106 (Raeder):

A useful résumé of various interpretations of the Heraclitus fragment is given by G. S. Kirk, The Cosmic Fragments of Heraclitus, 1954, pp. 280 sq. (He mentions only Cic. de fin. I. 6. 20 and Acad. II. 26. 82 as suggesting that Epicurus compared the size of the sun to that of the human foot. Cleomedes is, of course, arguing primarily against Epicurus; and he confirms much more certainly than Cicero that Epicurus held this opinion. Both the Cicero passages do indeed mention the foot measure in connexion with Epicurus. But on both occasions the comparison seems to be that of the speaker, while what is actually ascribed to Epicurus is only the opinion that the sun is about as big as it seems.)

3 For the use of the κλεψύδρα, etc., in astronomical observation cf. Mart. Cap. VIII 847, 860 (pp. 446 and 452 Dick); Macrobius, , Comm. in Somn. Scip. I. 21, 12Google Scholarsq. (for which see now the excellent translation with notes, etc., by W. H. Stahl, Columbia University Press, 1952); Ptolem. Synt. V. 14.; Theon Alex, in Ptol. Synt. ed. Camerarius, Basileae 1583 quoted by Manitius, Procli Hypotyposis (Teubner), pp. 309 sq.

4 It does not matter whether we are dealing here with Thales himself or one of his successors. What is important is the fact, if it is a fact, that the tradition embodies a piece of knowledge known to pre-Alexandrian science.

5 See Tannery, P., Pour l'histoire de la Science Hellène, 1930, p. 71.Google Scholar

6 Hultsch, F., Winkelmessungen durch die Hipparchische Dioptra, Abh. Gesch. Mathem. IX. 1899, p. 193.Google Scholar

7 Cf. Seneca, , Apocolocyntosis 2Google Scholar, ‘horam non possum certam tibi dicere: facilius inter philosophos quam inter horologia convenid’.

8 In the same chapter Ptolemy gives the angle subtended by the moon as 31⅓ sixtieths of a degree. He obtained this, not by measuring with the dioptra, but by calculation, (ibid.).

9 The angle subtended by the sun and the moon as given by Thales is ½°; as given by Cleomedes ( of the circle) = 28′ 48″; the best modern measurements are: for the sun ca. 32′ 35″ max., 31′ 30·8″ min., 31′ 59·2″ mean; for the moon ca. 33′ 32″ max., 29′ 20″ min., 31′ 5·2″ mean.

10 Hultsch, , Winkelmessungen etc., p. 193.Google Scholar

11 Cf. Proclus, , Hypotyp. iv. 74, p. 120Google Scholar Manitius, and ibid. pp. 309 sq.; Pappus ap. Theon. in Ptolem. Synt. p. 262, given by Manitius, loc. cit.

12 Verhandlungen der Berl, anthropol. Ges. 1889, p. 321, quoted by Zimmern, H.. Das Princip unserer Zeit-und Raumteilung, Ber. über d. Verh. Ges. Wiss. Leipzig, Phil. Hist. Kl. 53, 1901, p. 48.Google Scholar

13 This is based on Pappus ap. Theon. Alex, in Ptol. Synt., for which see Manitius, , Procl. Hyp. p. 309.Google Scholar

14 Aristarchus had given the diameter of the moon as of a sign of the zodiac, i.e. as or 2° of the circle. (On the sizes and distances of the sun and moon, hypothesis 6 See SirHeath, Thomas, Aristarchus of Samos, p. 352.Google Scholar) Archimedes says that Aristarchus had given of the orbit as the length of the sun's diameter. (Archimedes, Heiberg, Teubner 1913 II, p. 222.)

Archimedes (ibid.) gives a method for measuring this angle. His result is between and of a right angle, i.e. and of the circle.

15 See Tatius, Achilles, Isagog. in Arat. 18Google ScholarUranolog. Dionys. Petav. p. 137.

16 Nor, incidentally, is it certain that they obtained the result 1: 720 at all. SirHeath, Thomas (Aristarchus of Samos, pp. 22–3)Google Scholar argues that their result may have been 1°, not ½°.

17 Hoppe, , Mathem. und Astron. im. Kl. Altertum, 1911, p. 23Google Scholar, argues convincingly that the measurement of angles was developed by the Sumerians even before the division of the circle.

18 ‘Measuring’ would, of course, not be entirely mechanical; it would involve calculation or geometrical construction, or some such process. See Ptolemy, loc. cit.