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Anaximander’s Numbers

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When the Earth Was Flat

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Abstract

In 1995, when I discussed several visualizations of Anaximander’s world picture and added my own attempt, I took the interpretation of Anaximander’s numbers by Tannery and Diels for granted. Later, I got my doubts about the right understanding of the doxography on the distances of the heavenly bodies. In 2001, I wrote about my own interpretation: “Even if some parts of this reconstruction might be wrong. I think the conclusion still stands that Anaximander is the originator of the Western world picture or, in other words, the discoverer of space.” In 2003 I wrote: “Any interpretation entailing unacceptable observational consequences that were easy for Anaximander himself to observe must be wrong. In other words, Anaximander’s numbers cannot be in flagrant discrepancy with observational data, for otherwise he would have noticed it.” In 2009, in an article, entitled Problems with Anaximander’s Numbers, I discussed a number of interpretive problems and proposed solutions for understanding them. I concluded that even a symbolical interpretation of the numbers such as “far,” “farther,” “farthest” would not help, because on a flat earth the heavenly bodies are not far away. I tried to save the standard interpretation of the numbers by suggesting that Anaximander did not make a three-dimensional model of his conception of the cosmos. In 2011, this conclusion was repeated, although in Chap. 10 of Heaven and Earth, although I felt increasingly uneasy about it. Since then, my doubts as to whether Anaximander’s numbers as transmitted in the sources are in any of their current interpretations consistent with the conception of the celestial bodies being close by have only become more serious. In this chapter, several interpretations will be discussed and especially one that no longer takes the numbers as indications of distances in the universe but as a kind of calculator for a lunar-solar calendar. At the end of this chapter, I will make up my mind and outline my current position.

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Notes

  1. 1.

    Couprie (1995).

  2. 2.

    See Tannery (1887, 94–98) and Diels (1897).

  3. 3.

    Couprie (2001, 47), see also Couprie (2003, 238).

  4. 4.

    Couprie (2003, 217).

  5. 5.

    Diogenes Laërtius, Vitae Philosophorum 2.1.1 = DK 12A1 = LM ANAXIMAND. R14 = Gr Axr1 = TP2 Ar92 = KRS 94. Eusebius, Preparatio Evangelica 10.14.11 = DK 12A4 = LM ANAXIMAND. R15 = GR Axr2 = TP2 Ar102; not in KRS. Suda, Lexicon alpha = DK12A2 = Gr Axr4 = TP2 Ar237 = KRS 95. These lines omitted in LM ANAXIMAND. D3.

  6. 6.

    Simplicius, In Aristotelis De Caelo Commentaria 471.1–11 = DK 12A19 = LM ANAXIMAND. R17 = TP2 Ar185, not in Gr and KRS.

  7. 7.

    See, e.g., Kahn (1994, 63): “Anaximander’s concern for the sizes and distances of the heavenly bodies is cited by S(implicius) not from Theophrastus but from the ἀστρολογικὴ ἱστορία of Eudemus.” White (2008, 100): “the sizes and distances of the sun and moon (fr. 146),” and 105: “According to Eudemus (fr. 146), he was the first to specify sizes and distances for the heavenly bodies.” Couprie (2011, 118): “Anaximander was the first to gain insight into the measurements and distances (of the celestial bodies), as Eudemus reports,” repeated at 128–129. Hahn (2010, 60): “Simplicius reports, not on the authority of Theophrastus but rather Eudemus (…), that Anaximander was the first to describe the sizes and distances (…) of the heavenly bodies.” (all italics are mine). See also Gemelli Marciano [2007, 43 (12)].

  8. 8.

    P 2.15.6 (≈S 1.24.1) = DK 12A18 = LM ANAXIMAND. D22 = MR 476 = TP2 Ar55 and 148, not in Gr and KRS.

  9. 9.

    Hippolytus, Refutatio Omnium Haeresium 1.6.5 = DK 12A11(5) = LM ANAXIMAND. D7(6) = GrAxr20(4) = TP2 Ar75(5) = KRS 125.

  10. 10.

    See Couprie (2011, 116–117).

  11. 11.

    P 2.24.2 (=S 1.25.1) = DK 12A21 = LM ANAXIMAND. D25 = Gr Axr24 = MR 563 = TP2 Ar59 and 150; not in KRS.

  12. 12.

    See Simplicius, In Aristotelis De Caelo Commentaria 471.1–11 = DK 12A19 = LM ANAXIMAND. R17 = TP2 Ar185, not in Gr and KRS.

  13. 13.

    This was the title of my contribution to Couprie, Hahn, and Naddaf (2003, 165–254).

  14. 14.

    Pseudo-Plutarch, Stromata 2 = DK 12 A10 = LM ANAXIMAND. D8 = Gr Axr19 = TP2 Ar101 = KRS 122(A).

  15. 15.

    P 2.20.1 (=S 1.25.1) = DK 12A21 = LM ANAXIMAND. D23 = Gr Axr22 = MR 514 = TP2 Ar57 and 150 = KRS 126.

  16. 16.

    P 2.21.1 (≈S 1.25.1) = DK 12A21 = LM ANAXIMAND. D24 = Gr Axr23 = MR 534 = TP2 Ar58 and150 = KRS 127.

  17. 17.

    Hippolytus, Refutatio Omnium Haeresium 1.6.5 = DK 12A11(5) = LM ANAXIMAND. D7(5) = GrAxr20(5) = TP2 Ar75(5) = KRS 125.

  18. 18.

    P 2.25.1 (=S 1.261) = DK 12A22 = LM ANAXIMAND. D26 = Gr Axr25 = MR 572 = TP2 Ar60 and 151; not in KRS, but see p. 136.

  19. 19.

    Qusṭā ibn Lūqā in Daiber (1980, 155), not in DK, LM, Gr, TP2, MR, and KRS, but see Bottler 404, and Thibodeau (2017, 95).

  20. 20.

    Pseudo-Galen, De Historia Philosopha 67.1 = TP2 Ar224, not in DK, LM, Gr, and KRS.

  21. 21.

    Gruppe (1851, 45 n): “Dagegen ist in der Stelle des Origenes (Philos. Cap. 6) [the work, in the mss. ascribed to Origenes, is usually ascribed to Hippolytus, although this ascription is probably also not right, DC], ein offenbarer Fehler, wenn er von die Sonne dieselbe Zahl meldet, aber als Einheit nicht die Erde sondern den Mond nimmt. Röper (1852, 608–609): “(…) und das urtheil Gruppe’s, kosm. Syst. D. griech. S. 45, dass bei unserem verfasser ein fehler obwalte, indem er als einheit anstatt der erde den mond annehme, ist gewiss richtig.” See also Gregory (2016, 261 n. 21), although his explanation of this emendation, “the circle of the sun is now 27 times that of the earth,” (my italics) is dubious.

  22. 22.

    For S 1.25.1, see TP2 Ar150, and cf. Dox. 348. For Eusebius, Preparatio Evangelica 15.23.1, see TP2 Ar105. For pseudo-Galen, De Historia Philosopha 62.1–3, see TP2 Ar221. Both items not in LM, Gr, and KRS.

  23. 23.

    For S 1.25.1, see TP2 Ar150, and cf. Dox. 351. For Eusebius, Preparatio Evangelica 15.24.1, see TP2 Ar106. For pseudo-Galen, De Historia Philosopha 63,1–2, see TP2 Ar222, with a miswriting: πόλος instead of κύκλος. Both items not in LM, Gr, and KRS.

  24. 24.

    Cf. Guthrie (1962, 95): “We may assume that the rings are one earth-diameter thick.”

  25. 25.

    Tannery (1887, 91–92, 119). For a survey of the interpretations, based on Tannery’s suggestion, see Couprie (2011, 121–136). For a more recent survey, see Gregory (2016, 173–192).

  26. 26.

    See Diels (1897, 236), Couprie (2003, 213; 2011, 130).

  27. 27.

    Cf. Tannery (1887, 91): “La double épaisseur du cerceau est ainsi égale au diamètre de la terre;” Diels (1897, 232): “so ist die die Breite dieser Ringe auf einen Erdradius zu veranschlagen.”

  28. 28.

    See KRS, p. 136 n. 1.

  29. 29.

    Diels (1923, 72, my italics). See also Diels (1897, 232). Obviously, Diels has in mind the real distance from the earth to the sun, 149,597,870.7 km, which would equal, given the diameter of the spherical earth of 12,756.32 km, to a distance of 11,727 spherical earth diameters.

  30. 30.

    See Gregory (2016, 173–192).

  31. 31.

    See Claudius Aelianus, Varia Historia 3.17 = DK 12A3 = LM ANAXIMAND. P8 = TP2 Ar 78; not in Gr and KRS, but see p. 105.

  32. 32.

    Diogenes Laërtius, Vitae Philosophorum 2.1.1 = DK 12A1 = LM ANAXIMAND. R14 = Gr Axr1 = TP2 Ar92 = KRS 94: “Anaximander first discovered the gnomon and set one up at the sundials in Sparta, as Favorinus says in his Miscellaneous Studies, to mark solstices and equinoxes; and he constructed hour-indicators.”

  33. 33.

    Cf. Strabo, Geographica 1.1.11 = DK 12A6 = LM ANAXIMAND. D4 = Gr Axr7 = TP2 Ar32 = KRS 99.

  34. 34.

    Cf. Diels (1897, 233).

  35. 35.

    Cf. Burkert (1963, 106–112, esp. 111).

  36. 36.

    Cf. Diels (1897, 232).

  37. 37.

    Hesiod, Theogonia 722–723.

  38. 38.

    Cf. Hesiod, Theogonia 517–520 and 746–747.

  39. 39.

    Cf. Hesiod, Theogonia 778–779.

  40. 40.

    Cf. Diels (1897, 232–233). Diels does not mention here the number 18.

  41. 41.

    See Couprie (2001, 40–41; 2003, 215; 2011, 136).

  42. 42.

    Cf. Gregory (2016, 203): “I take this as a poetic expression of something exceedingly heavy dropping exceedingly fast.”

  43. 43.

    Cf. KRS, p. 136: “His proportionate distances may have influenced Pythagoras.” See also Zhmud (2012, 292).

  44. 44.

    A somewhat problematic solution would be to accept that Anaximander, when measuring the height of the heaven, made a calculation error similar to that of the Chinese astronomers; see Chap. 13.

  45. 45.

    Cf. West (1971, 92).

  46. 46.

    Cf. Eggermont (1973, 124–125)

  47. 47.

    Eggermont (1973, 128).

  48. 48.

    Eggermont (1973, 128, 127).

  49. 49.

    See Hahn (2001, passim, and especially 156, 158, and 78).

  50. 50.

    For an extensive criticism of Hahn’s attempt, see Couprie and Pott (2002) and Couprie (2011), Chap. 12.

  51. 51.

    See Couprie and Pott (2002), and Couprie (2011, 153–160).

  52. 52.

    Naddaf (1998, 23).

  53. 53.

    Gregory (2016, 74; see also 123): “Anaximander’s system has a strong tendency to symmetry.”

  54. 54.

    Gregory (2016, 124).

  55. 55.

    Gregory (2016, 164).

  56. 56.

    Gregory (2016, 184, my italics). With “this sequence” Gregory means: “the aesthetically pleasing sequence of 9, 18, 27, which can be seen as an extension of the 3:1 ratio of the earth’s width to its depth” (2016, 173).

  57. 57.

    O’Brien (1967, 427).

  58. 58.

    See Corre (2013).

  59. 59.

    O’Brien (1967, 423, n. 4).

  60. 60.

    Cf. O’Brien (1967, 425).

  61. 61.

    Cf. Apuleius, Florida 18.32 = DK 11A19 = LM THALES R13 = TP1 Th178; not in Gr and KRS. See also Diogenes Laërtius, Vitae Philosophorum 1.24 = DK 11A1(24) = LM THALES R14 = Gr Ths1(24) = TP1 Th237(24); not in KRS, but see p. 83.

  62. 62.

    Röper (1852, 608): “(…) nachrichten, wonach Anaximander den kreis der sonne 28 mal oder, vermuthlich nach abzug des der ἐκπνοὴ zukommenden raumes, 27 mal, den kreis des mondes aber 19 mal grosser sein liess als die erde.” Röper does not mention a number 18 for the moon wheel minus its ἐκπνοὴ.

  63. 63.

    Cf. Gregory (2016, 169–219). Exceptions are O’Brien (1967, 423, n. 4) and Naddaf (2001, 11).

  64. 64.

    Conche (1991, 209, my italics).

  65. 65.

    Naddaf (2001, 11). The total number of disks should be 28, as Forbiger’s text (1877, 523, n. 57) shows. It is not clear to me, why Naddaf calls this “an obvious petitio principii argument.”

  66. 66.

    Cf. Graham (2013, 58): “the ring of the sun (…) one earth-diameter in thickness.” See also his drawing at 59, Fig. 2.1. Graham’s drawing is, however, strange in two other aspects: the heavenly wheels are not tilted but lie in the same plane as the earth, as if the situation before the tilt of the celestial axis is rendered, and only one wheel for the stars is drawn.

  67. 67.

    Kahn (1994, 87–88).

  68. 68.

    Kahn (1994, 88).

  69. 69.

    Dicks (1970, 46–47).

  70. 70.

    Cf. Fehling (1985, 222).

  71. 71.

    Cf. Stritzinger (1952, 65–66).

  72. 72.

    Thibodeau (2017, 99).

  73. 73.

    Thibodeau (2017, 102).

  74. 74.

    Rescher (2014).

  75. 75.

    Cf. Thibodeau (2017, 103, 106).

  76. 76.

    Thibodeau (2017, 99, n. 26).

  77. 77.

    Thibodeau (2017, 110).

  78. 78.

    See Forbiger (1877, 523 n. 57): “Diese verschiedenen Angaben lassen sich wohl so erklären, dass Anaxim. den Luftkreis, der den eigentlichen Kern der Sonnen umgab, für 27 mal, also den Kern und den Lichtkreis zusammen, oder die ganze Sonne, für 28 mal grösser als die Erde hielt.”

  79. 79.

    See Hahn (2010, 162, Fig. 6.11).

  80. 80.

    See Couprie (2011, 109–110).

  81. 81.

    Cf. Kočandrle (2017, 2018).

  82. 82.

    P = Aëtius in pseudo-Plutarch, Placita (numbering according to Dox).

    S = Aëtius in Stobaeus, Anthologium (numbering according to Wachsmuth and Hense).

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  1. Department of Philosophy, Faculty of Philosophy and Arts, University of West Bohemia, Pilsen, Czech Republic

    Dirk L. Couprie

  2. Amsterdam, The Netherlands

    Dirk L. Couprie

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Couprie, D.L. (2018). Anaximander’s Numbers. In: When the Earth Was Flat. Historical & Cultural Astronomy. Springer, Cham. https://doi.org/10.1007/978-3-319-97052-3_6

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