¹ JHS lxxxvi (1966) 26–40, quoted hereafter as ‘Dicks’. In reflecting on Dicks' discussion of the astronomical problems, I have received valuable advice from Professor Howard Stein of the Department of Philosophy, Case Western Reserve University, and from Dr W. D. Heintz of the Astronomy Department, Swarthmore College. Needless to say, neither is responsible for the views here presented, except where their names are cited.

² Dicks 39.

³ The evidence is conveniently assembled by Heath, , Aristarchus 78 f.Google Scholar Also DK 59A42. 8–10, A76.

⁴ Dicks 39.

⁵ Neugebauer, O., ‘The History of Ancient Astronomy, Problems and Methods’ in Journal of Near Eastern Studies iv (1945) 2.Google Scholar

⁶ Dicks 27 and 29.

⁷ Thus I shall not follow the precedent set by Dicks' article, which accuses me not only of systematic error and ‘lack of historical sense’, but also of selecting and rejecting doxographical material on the basis of personal preference in order to produce a ‘monstrous edifice of exaggeration’, ‘a travesty of the historical truth’, whose key features are ‘entirely unsubstantiated by the available evidence’ (Dicks 31, 35–38). One hopes that this vehemence of tone reflects Dicks' sense of the importance of the questions at issue between us.

⁸ DK 12A22 = Aëtius ii 25.1

⁹ For Aëtius' dependence on Theophrastus in this context see Aëtius ii 20.3 = Theophr. Phys. Op.fr. 16. Eudemus' report on the early investigation of the ecliptic, as given by Theon of Smyrna, is also reflected (and distorted) in Aëtius ii 12.2: (DK41A7; see Wehrli, , Eudemos fr. 145 Google Scholar with commentary, p. 120.) Theon's version is where Diels read on the basis of the parallel in Aëtius. Now if Eudemus regarded Oenopides as the first to observe the fact that the sun's annual path is set at an angle to the line of diurnal motion, or to the equator, he cannot have ascribed the same discovery to Anaximander. But there are other possible interpretations of Theon's report: (1) What Eudemus attributed to Oenopides was not the mere discovery that the zodiac is inclined but a precise determination of the angle (so von Fritz in PW xvii 2260 f., followed by Burkert, W., Weisheit und Wissenschaft 285 Google Scholar, n. 42); (2) keeping διάζωσις in Theon's text, we might suppose that what Eudemus (or Theon) meant by the ‘belting’ of the zodiac was a definite description or measurement of the zodiacal signs. However, the whole context in Theon makes a very unreliable impression as a ‘fragment’ of Eudemus. Finally, even if Eudemus did not ascribe knowledge of the ecliptic to Anaximander, the common source of Pliny and Aëtius may have drawn this conclusion—rightly or wrongly—from information in Theophrastus.

¹⁰ Dicks 35 f.

¹¹ Heath, , Aristarchus 36 Google Scholar: ‘the hoops remain at fixed inclinations to the plane of the equator’.

¹² Hence I do not see the point of Dicks' complaint (36) that I ‘cannot envisage the ecliptic without mentioning the equator, although there is not a word about this in the original quotation’. How does Dicks interpret the reference to obliquity in Pliny and Aëtius? Geminus says the zodiac is called ( Elementa astronomiae ed. Manitius, , v 53 Google Scholar). One of the parallels it cuts is of course the equator.

It should also be noted that to attribute to Anaximander the concept of the ecliptic is actually to credit him with more than the celestial sphere and the inclined zodiac. See below, esp. n. 24.

¹³ I owe this suggestion to Howard Stein, who writes: ‘It seems to me that this fits better with Anaximander's attribution of the solstices to meteorological causes: the circle of the sun, remaining always parallel to the circles of the stars, moves as a whole among (or rather above) the stars, towards the north in spring, then, turning, towards the south under the influence of the exhalations; and all the while, the circle turns on itself, one revolution per day.’

¹⁴ See, for Anaxagoras, DK 59A1.9 similarly in 59A67. The former passage is from Diogenes Laertius, the latter from Aëtius. The same two authors ascribe a similar doctrine to the atomiste (Aëtius iii 12 = DK 67A27 and 68A96), and once to Empedocles (Aëtius ii 8.2 = DK 31A58); in Aëtius it is generally the earth rather than the heavens which is said to be tilted. The ancient sources do not clearly distinguish such ἔγκλισις from the obliquity of the zodiac, but an outright confusion of the two nowhere occurs except in the proposed emendation of Diogenes ix 1 printed in DK 67A 1.33

¹⁵ See Arist. Met. 1073b20, Euclid, , Phaenomena p. 6, 21 ff. (ed. Menge, )Google Scholar, Geminus, , Elementa v 51.Google Scholar Even technical authors do not always make a clear terminological distinction between the zodiac and the sun's circle, but use an expression for the former in asserting something true only of the latter (e.g. that it is a great circle, Euclid op. cit. 8, 15; that it touches the tropic at a solstitial point, Achilles, Isagoge xxv 4 Google Scholar, ed. Maass p. 57.21). It seems to have been understood that in referring to celestial κύκλοι that are really bands, such as the zodiac and the Milky Way, the astrono mers normally take the middle circle of the band as its geometrical representation.

¹⁶ See Weidner, E. F., Amer. Journal of Semitic Languages and Literatures xl (1924) 192–5Google Scholar; Schaumberger, J., in Sternkunde und Sterndienst in Babel iii (1935) 319 Google Scholar; and now van der Waerden, B. L., Erwachende Wissenschaft Band ii: Die Anfänge der Astronomie (1966) 77.Google Scholar This last work will be quoted below as ‘van der Waerden (1966)’.

¹⁷ See Weidner, op. cit., and van der Waerden, B. L., Journal of Near Eastern Studies viii (1949) 6–26.CrossRefGoogle Scholar The same tablet divides the year into four periods of three months each during which the sun is described as located in the path of Anu (twice, in spring and fall), in the path of Enlil (summer), and in the path of Ea (winter). Since the path of Anu must thus represent an equatorial belt some 30 degrees wide, the statement that the sun moves in and out of this path can be interpreted as showing that the Babylonians knew (by Assyrian times, and perhaps much earlier) ‘that the sun moves in an oblique circle’ (van der Waerden, op. cit. 24). This is, however, a very Greek way of describing their knowledge. As far as I can judge, the Babylonian texts quoted do not refer to the sun's path as a circle; there is no mention of the equator, and the concept of obliquity is at best implicit in the recognition that the sun passes back and forth across the (presumably parallel) borders of the way of Anu.

In his 1966 book van der Waerden has developed the consequences of his own description and explicitly assigned not only the obliquity of the ecliptic but also the celestial sphere to the Babylonian astronomers of the ‘Mul apin’ period (78 f. with fig. 10; and 134).

If this view were correct, the whole question of the originality of Greek astronomy would have to be regarded in a new light. But it seems unlikely that van der Waerden's conclusion will prove acceptable to other historians of Babylonian science. I have the impression (from a conversation with Dr Heintz) that the paths or zones of the ‘Mul apin’ text can be understood in purely observational terms (e.g. as designating which stars rise and set together) without reference to any geometric model.

¹⁸ See van der Waerden (1966) 125; and compare Neugebauer, O., The Exact Sciences in Antiquity (2nd ed. 1957) 140.Google ScholarPubMed I regret that I helped confuse matters by quoting from the ist edition (1952) of Neugebauer, 's book (in Anaximander 92)Google Scholar; and indeed my entire quotation there is regrettable since, in either edition, Neugebauer is referring to a level of astronomical refinement at the end of what he calls the ‘prehistory’ of Babylonian astronomy (dated to ‘about 400 B.C.’ in the second edition, 103) which was certainly not reached in sixth-century Miletus, and perhaps nowhere in Greece before the time of Meton.

Nonetheless the point which I intended to illustrate by the quotation from Neugebauer is one which I still maintain: that the creation, by the Milesians and their successors, of a theory of geometric world-models different in kind from mythic speculation is to be understood in part as the Greek reaction to new and more extensive contact with astronomical lore from the East. For a more modest estimate of the knowledge available to the Greeks in the sixth century, see below.

¹⁹ DK 6A1. Dicks 26 f. is very contemptuous of Cleostratus, and of those modern historians who take him seriously, on the grounds that our authority, Pliny, is unreliable. But Dicks ignores this reference to Cleostratus in Theophrastus and the surviving verses from his poem which mention the Scorpion. The fact that Atlas is included by Pliny in the same context as discoverer of the celestial globe (by obvious rationalising of a well-known story derived from Hesiod, and often represented in vase-painting and sculpture) shows that Pliny is uncritical of his sources, but it cannot be used to impugn his authority whole sale. Like Diogenes Laertius, Pliny repeats what ever he has found written somewhere, and what he has found is often silly. But each case must be judged on its merits, and in this case, what Pliny tells us about Cleostratus is just what we would expect on the basis of the other evidence.

²⁰ See Rehm, A., Abh. bayer. Akad. (Munich 1941) Heft xix 12–14.Google Scholar Other discussions of Cleostratus are cited in Burkert, W., Weisheit und Wissenschaft 312 nn. 56 and 58.Google Scholar For a comparison of Greek and Babylonian constellations in the zodiac, see van der Waerden, , Journal of Near Eastern Studies viii (1949) 13 f.Google Scholar and now van der Waerden (1966) 256 ff.

²¹ See Cleostratus DK 6B4, Parmenides DK 28A1.23 and A40a. For the Venus observations in early Babylon, see Pannekoek, A., A History of Astronomy (1961) 33 Google Scholar; van der Waerden (1966) 49. For the Babylonian 8-year cycle, Pannekoek 51 f. and van der Waerden (1966) 112. Dicks is sceptical of the 8-year cycle (33, n. 39), but the convergence of Geminus, Censorinus, and the evidence from Babylon should suffice to establish its chronological priority over the Metonic cycle.

²² In the case of κύκλωψ as in the case of ἀίδηλα I see an intentional use of ambiguity or plurisignificance, which is an essential feature of Parmenides' style too often overlooked by commentators. For κύκλωψ there are at least two appropriate meanings: (1) ‘round-faced’ or ‘round-eyed’ (so Diel-Kranz), and (2) ‘cycle-faced’, i.e. changing her appearance according to the monthly cycle. The first is the surface reading, on the level of Parmenides' Homeric diction; the second is the deeper reading, on the level of his astronomical concerns. For further remarks on intentional ambiguity in Parmenides see my review of Bollack, J., Empédocle, in Gnomon xli (1969) 441 f.Google Scholar

²³ The suggestion that the Milesians were not interested in observational astronomy seems to be incompatible with almost every one of the stories concerning Thales, and with the traditional ascription to him of an old ‘Nautical Astronomy’. On this and other early didactic poems with scientific content, see Nilsson, , Rh. M. lix (1904) 180–6.Google Scholar

²⁴ This is an oversimplification, since only the ecliptic is a great circle. As suggested above, we must distinguish between (1) the discovery that the zodiac is inclined to the equator, and (2) the discovery that the sun always moves in the middle of this band on a single great circle. Pliny assigns only (1) to Anaximander; Aëtius combines this with (2) by referring to the distinct circles of sun and moon as ‘lying aslant’. (Howard Stein reminds me that there is in fact no single circle for the moon's path, but this error could be due to Anaximander rather than to the doxography.) I am therefore inclined to suppose that Aëtius has conflated two reports, both of which were given in his sources: (i) the inclination of the zodiac, as reported by Pliny, (ii) the explanation of sun and moon as huge circles or rings, as given by other doxographers such as Hippolytus. If Aëtius is responsible for this confusion, then Anaximander may in fact have discovered the obliquity of the zodiac (or learned of it from Babylon) without discovering the ecliptic in any precise sense. He may have thought of the circles of sun and moon as always lying parallel to those of the stars, and rotating daily, but as pushed north and south in the course of the year (for the moon, in the course of the month). See below, p. 107.

²⁵ Compare the passage in the pseudo-Platonic Erastai 132–b (DK 41.2) where two boys are said to be disputing ‘about Anaxagoras or Oenopides; for they appeared to be drawing circles and representing certain inclinations (ἐγκλίσεις), by inclining their hands relative to one another (ἐπικλίνοντε), all in great earnestness’. Our question is: who was the first scientist to do for the zodiac, or for the ecliptic, what our author imagines schoolboys doing in the latter part of the fifth century?

²⁶ In favour of the spherical shape for Parmenides' οὐρανός see Heath, , Aristarchus 69 Google Scholar; and now Tarán, L., Parmenides (1965) 241 Google Scholar: ‘a solid sphere of Night’. Burnet's case against the spherical heaven (EGP 188 with n. 2) relies entirely upon the untrustworthy wording of Aëtius, who compares the outer circle to a τεῑχος or city-wall. I have argued (Anaximander 116 f.) that spherical shape is first assigned to the earth (by Parmenides or by Pythagoreans) as a generalisation of the principle of cosmic symmetry.

After Parmenides, the spherical shape is reasonably well attested for Empedocles. See fr. 38.4 A spherical οὐρανός for Empedocles is presupposed by the reference to hemispheres in DK 31A51 (Aëtius); that the fixed stars are ‘bound’ to it is stated in 31A54 (ako Aëtius). The conflicting view reported by Aëtius in another context (31A50), that Empedocles' heaven is egg-shaped, must be mistaken, and is in fact contradicted by other evidence in Aëtius (cf. the σφαῑρα enclosing the sun in A58 with the statement in A50 that the circuit of the sun marks the limit of the κόαμος).

²⁷ If, as I suggest, Parmenides was working with the model of a celestial sphere on which the ecliptic or zodiac is drawn obliquely, there may be a grain of truth to Strabo's report (DK 28A44a), on the authority of Poseidonius, that Parmenides was the inventor of the division into five zones. The only anachronism may lie in Poseidonius' application of this to the earth (although, if Parmenides' earth was spherical, as it is reported to have been, even the projection of the zones on to the earth is not impossible for his time: it may at first have been done a priori, without latitude observations). Three of the five celestial zones are given as soon as the ecliptic is drawn. The other two are easily defined: the arctic region where the stars never set and the antarctic where they never appear (to an observer, say, in Elea). The fact that Parmenides is said to have placed the tropics too far from the equator—i.e. that he had too high a value for the obliquity of the ecliptic—perhaps tells against this being a late fabrication. Note that Parmenides' error could easily be explained by the assumption that the ecliptic, as a single circle, had not yet been distinguished from the wider zodiacal band. In that case, the true discovery of the ecliptic (and its obliquity) would belong after all to Oenopides in the middle of the fifth century. This discovery would naturally be connected with an Anaxagorean investigation of the precise circumstances of lunar eclipse. See n. 25 above.

²⁸ Evidence and discussion in Anaximander 53–5. On this point even Dicks is not sceptical, though he doubts whether the argument from symmetry should be regarded as a mathematical insight (36 n. 53).

²⁹ The Presocratic Philosophers 134 f.

³⁰ DK 12A1.2 (Diogenes Laertius). Also Pliny nat. hut. vii 56.203 cited in Anaximander 60: sphaeram invenit.

³¹ DK 13A14 (Aëtius):

³² Heath, , Aristarchus 45 Google Scholar, following Tannery.

³³ Burnet, , EGP 77 Google Scholar with n. 4. Similarly Kirk and Raven, 155. For a different view, see Guthrie, W. K. C., A History of Greek Philosophy i (1962) 135–8.Google Scholar

³⁴ For the tilting, see above, n. 14. My comments on Anaximenes are based upon a suggestion of Howard Stein, who writes: ‘If I were trying to convey to a student the basic notion, and the chief difference from the naïve view, the first point I should make is that the stars do not, in the naïve sense, “rise” and “traverse the sky” and “go down” and “return underneath”, but in a more accurate view they go around; the ones near the pole go around remaining always in view, while the ones farther from it are hidden part of the time by the northerly parts of the earth, since in their turning they sink behind it.’ Anaximenes' metaphor of the cap revolving on our head ‘approaches the metaphor of the celestial sphere. A cap, after all, sits back on the head: it goes low in back and high in front. If a cap is twisted around on the head in the most natural way, it stays “on” the head (does not go “under” it), but the part that goes around in back does also sink down low “behind” the head. The very fact that later commentators seem mystified by this phrase prevents any suspicion here of a backwards attribution of later ideas.’

³⁵ See σφαιροειειδῆ in DK 21A33.2 (Hippolytus), with the parallels cited by Diels in note on Doxographi graeci 481.9.

³⁶ I have not touched upon the possibility of significant work in astronomy by Pythagoras or Pythagoreans in the period between Anaximenes and Parmenides, since the evidence for such is practically non-existent. This does not mean that there was no Pythagorean astronomy around 500 B.C.; only that we can scarcely hope to know anything about it. van der Waerden, B. L., in Die Astronomie der Pythagoreer (1951) 28 f.Google Scholar and PW xxiv 290–4, reconstructs an earlier and a later Pythagorean world-system, but he does not attempt to specify how old the early system is. If we could trust Aëtius' statement (DK 24A4) that Alcmaeon recognised that the movement of the planets was in the reverse direction to that of the fixed stars, and if we were sure of the early date of Alcmaeon, we would have a faint glimmer of pre-Parmenidean astronomy in Magna Graecia.

³⁷ Heath, , Aristarchu 45.Google Scholar

³⁸ The Exact Sciences in Antiquity 155.

³⁹ Dicks 39 n. 64. Cf. the remarks of Neugebauer cited over n. 45. But it now seems that van der Waerden would dissent; see his new thesis on the geometric elements in Babylonian astronomy (1966), 134, and my comment above, n. 17.

⁴⁰ See Anaximander 115. Neugebauer's statement that the discovery of the sphericity of the earth was ‘recent’ in the time of Eudoxus is one for which I can find no evidence (The Exact Sciences in Antiquity 153). A reader points out that Phaedo 97d8 would not be decisive if the words could refer to a choice ‘between the two kinds of flat earth, the traditional one, round like a penny, and an oblong rectangular one, such as Herodotus believed in’. At the dictionary level, perhaps, the words might mean this (just as ‘the earth is round’ in English might mean ‘the earth is disk-shaped’), but in the context of the Phaedo such an interpretation is implausible, to say the least. The words quoted express the first burning question in cosmology which Socrates wished to have Anaxagoras decide on the basis of a rational order for the whole universe; and the immediately following questions are concerned with the position of the earth and the position and movements of the astral bodies. In this connection, the hypothesis of an oblong earth is of no interest, but the question of spherical shape is of very great interest indeed, and is, with the position of the earth, the first question to be decided when the subject is taken up again at Phaedo 108e5.

⁴¹ Neugebauer, , Proc. Am. Philos. Soc. cvii (1963) 530.Google Scholar

⁴² ibid. Neugebauer dates this discovery in the time of Theaetetus and Eudoxus. But the most powerful form of geometric proof, by reductio, is brilliantly exemplified in Zeno's paradoxes and clearly embedded in the argument of Parmenides' poem. The systematic generalisation of the notion of proof in mathematics must have been a slow development, perhaps not completed before Euclid. But the origins can be traced back to the early fifth century, in fact, to nearly the same time (c. 500 B.C.) when the celestial sphere is first attested.

⁴³ ibid. 534.

⁴⁴ ibid. 530.

⁴⁵ ibid. 535.

⁴⁶ See n. 43. For some suggestions on fifth-century contacts, see Burkert, W., Weisheit u. Wissenschaft 295.Google Scholar My metaphor of the three waves is of course a simplification, and is not intended to exclude more or less continuous contacts with Eastern science on the part of individual Greeks. Thus I leave open the question whether a distinct ‘wave’ should be recognised in Plato's old age and associated with the alleged voyage of Eudoxus to Egypt. (See, e.g. Solmsen, F., Plato's Theology 96, n. 23Google Scholar; van der Waerden [1966] 129–31.) The Epinomis (986e–988a) recognises that the names and knowledge of the planets originated in Eastern lands, and Aristotle in de Caelo 292a6–9 speaks of the elaborate observational data accumulated in Egypt and Babylon. What is not clear, however, is how much information was made available to Eudoxus and his contemporaries that was essentially different from the knowledge of astral cycles that had reached Meton and others in the fifth century. In any case, the really important astronomical innovation of Plato's old age, the theory of uniform circular paths for the ‘planets’ (mentioned at Laws 822a), is surely a Greek and not a Babylonian achievement. What is suggested by the tradition concerning Eudoxus' travels is that the specifically Greek interest in a simple cosmic model led him to seek more detailed information on the relative lengths of the cycles. But that these planetary phenomena were characterised by cyclical recurrence had been known—in the case of Venus—since early Babylonian times, and in Greece since Parmenides. Some of the details were presumably recorded in Democritus' book on the planets.

⁴⁷ Dicks 33 f., following Heath, , Aristarchus 294.Google Scholar

⁴⁸ DK 6A1.

⁴⁹ DK 12A1, A2, A4 (Diogenes Laertius, Suda, Eusebius, all referring to Anaximander). Exactly the same list of achievements is assigned by Pliny to Anaximenes, apparently as a result of a copying error (DK 131A4a). A systematic interest in solstices (together with eclipses) is ascribed to Thales by Diogenes Laertius, on the authority of Eudemus (Eudemus fr. 145 Wehrli = DK 11A1 = D.L. i 23). In this context, Diogenes mentions as witnesses to Thales' competence in astronomy not only Herodotus but also Xenophanes, Heraclitus and Democritus. This suggests (1) that Eudemus is the source for the consistent core of the late doxography concerning practical astronomy in Miletus (whereas Favorinus is quoted only for an implausible variant, the anecdote about the sundial at Sparta), and (2) that Eudemus in turn appealed to the four authors cited as his own source of information.

Dicks' attempt (33, n. 35) to discredit this doxographical tradition by deriving it all from Diogenes' text, as if Eusebius had Diogenes in front of him as he copied out the entry on Anaximander, will not recommend itself to anyone familiar with the usual standards of doxographical scrutiny. In fact Diogenes, although the earliest, is the least restrained of our three Greek sources here: he alone cites the improbable anecdote from Favorinus. All we can infer is that Diogenes, Suda and Eusebius have a common source, who is earlier than Favorinus and Pliny (DK 13A14a).

⁵⁰ This seems to be the sense of in Homer, Od. xv 404, though we are not told which solstitial rising (or setting) point is intended. Compare in Od. xii 4 (‘mean rising point’ — due east); in both cases a place or direction is named for the phenomena which occur there.

⁵¹ ch. 11 (the most dangerous changes of the seasons are): ‘both solstices, and especially the summer one, and the equinoxes, both of which are generally believed to be dangerous ( sc. ) but the fall equinox is especially so’. Dicks' comment on these words, ‘the equinoxes, which as a less familiar concept require an explanatory description’ (33, n. 38), is unintelligible to me. As the term νομιζόμεναι shows, the author can assume that most people, or most doctors, are perfectly familiar with the concept of equinoxes.

⁵² Op. 562. Wilamowitz, comments (Hesiodos Erga 106)Google Scholar: ‘Das vollendete Jahr wird durch die Tag—und Nachtgleiche bezeichnet…. Der Verfasser hat an ein Jahr gedacht, das mit Frühling anfing, wie z.B. in Keos … und an die Isemerie’. Friedrich Solmsen suggests to me (in a letter) that the ‘interpolation’ is not likely to be later than the sixth century; and he agrees that the reference must be to the equinoxes.

⁵³ Cf. Dicks 34 f. For the Babylonian practice in the pre-Hellenistic period, see Neugebauer, , The Exact Sciences in Antiquity 102 Google ScholarPubMed: ‘it is the summer solstices which are systematically computed, whereas the equinoxes and the winter solstices are simply placed at equal intervals’.

⁵⁴ This procedure with the gnomon was suggested to me by Howard Stein. Note that by either procedure one has an astronomical determination of due east and due west. It is worth recalling that Anaximander was a cartographer, and that Greek maps were normally oriented by reference to six cardinal points: ‘equinoctial sunrise’ (=east), ‘equinoctial sunset’ (=west), plus summer and winter sunset and sunrise at the four solstitial points. See Aristotle, , Meteorology ii 6 Google Scholar, and W. A. Heidel, The Frame of the Ancient Greek Maps.

Dr W. D. Heintz points out that the gnomon procedure described in the text is exposed to systematic error if the horizon is not perfectly level in east and west, and that more precise measurement is possible on the basis of the sun's altitude at noon (as computed from the shadow length): one dates the equinox by the noon altitude half-way between two solstitial measurements. Here again, no spherical model is required for the measurements, but only for their theoretical justification.

After writing this, I notice that essentially the same two procedures are conjecturally assigned to the Babylonians by Kugler, , Sternkunde und Sterndienst in Babel i 175 f.Google Scholar Tables correlating the length of shadow with the hour of the day at different dates in the year are partially preserved in the second ‘Mul apin’ tablet; see Weidner, , Am. Journal Sem. Lang, xl (1924) 198–201.Google Scholar Both Weidner and van der Waerden regard these tables as confirming Herodotus' report of the Babylonian origin of the gnomon. See van der Waerden (1966) 63, 80.

In his 1966 book, p. 134, van der Waerden suggests a variant of the straight line test which does not depend upon sunrise and sunset observations but makes use of the fact that only at the equinox does the tip of the gnomon shadow describe a straight line on a flat sundial, as the shadow moves throughout the day.

⁵⁵ Professor Stein and Dr Heintz, working on different assumptions as to the procedure used, agree that the accuracy of measurement reflected in the Metonic cycle requires a comparison of observations recorded over at least a century. If Meton and Euctemon discovered the cycle themselves, they must have had access to solstice and/or equinox records going back to the time of Anaximenes, if not earlier. Of course these records, or more likely the cycle itself, may have been introduced directly from Babylon in the fifth century. Hence Meton's achievement cannot guarantee the antiquity of systematic observations in Greek astronomy.

⁵⁶ (cited by Achilles, , Isagoge xxviii, ed. Maass, p. 62 Google Scholar; not included in the numbered fragments):

fr. 163 (Gerytades):

⁵⁷ See A. Rehm, s.v. ‘Horologium’, PW viii 2417: ‘eine hohle Halbkugel, das Gegenbild des Himmelsgewölbes’.

⁵⁸ Dicks 37.

⁵⁹ It may be argued that in India Pāṇini's grammar occupies the position of geometry in Greece, as the paradigm of scientific knowledge. See Staal, I. F., ‘Euclid and Pāṇini’, Philosophy East and West xv (1965) 99–116.CrossRefGoogle Scholar