Early Greek Astronomy

  • Otto Neugebauer
Part of the Studies in the History of Mathematics and Physical Sciences book series (HISTORY, volume 1)


It is the purpose of Book IV to present a survey of the fragmentary data from the early stages of Greek astronomy. About six centuries have to be covered by such an attempt, beginning with the calendaric cycles of Meton and his school in the fifth century B.C. to Ptolemy in the second A.D. Only in two areas is our information substantial enough to make a separate discussion desirable: early planetary and lunar theory on the evidence of tables preserved on papyri of the hellenistic and Roman period (cf. below V A) and the work of the direct predecessors of the Almagest, Apollonius. and Hipparchus (above I D and I E, respectively). For the material left to be included in Book IV we must frequently operate with fragmentary data transmitted by authors of very limited technical competence. The little one can extract from these sources hardly deserves the name “history.”


Solar Eclipse Summer Solstice Lunar Eclipse Lunar Theory Zodiacal Sign 
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  1. 1.
    For the date of Cleomedes (4th cent. A.D.) cf. below V C 2.Google Scholar
  2. 2.
    That Hipparchus observed in Alexandria is by no means certain; cf. above I E 1, p. 276.Google Scholar
  3. 3.
    We are a little better informed about mathematics, thanks to the extant mathematical works of Archimedes and Apollonius.Google Scholar
  4. 4.
    Wilamowitz in his brilliant lecture on “Die Locke der Berenike” (Reden and Vorträge I, 4th ed., Berlin 1925) states categorically (p. 213, note 1) that at the time of Conon “man arbeitete auf der Sternwarte Alexandreias an einem Fixsternkataloge.” This is, of course, pure nonsense. Similarly it is only a modern invention to make Conon a “court-astronomer”; no such rank existed in Ptolemaic Egypt (cf., e.g., the list of officials discussed by Cumont, Ég. astr., Chap. 1 and 2). — What we really know about Conon is little enough. That he was a competent mathematician is evident from the way Archimedes mentions him — in spite of Apollonius’ adverse remarks in the preface to Book IV of the “Conics.” From Conon’s parapegma we have only two data through Pliny (Nat. Hist. XVIII 74, Loeb V, p. 384/5) and seventeen in Ptolemy’s “Phaseis.” Seneca (first cent. A.D.) says (Quaest. nat. VII, III 3, Loeb II, p. 233) that Conon was a careful observer and that he “recorded solar eclipses observed by the Egyptians” — a story difficult to take seriously in view of what we know of Egyptian astronomy (cf. above p. 561). Probus (end of 1st cent. A.D.) ascribes seven books to him “de astrologia” (Thilo-Hagen, Servius, In Vergilii carmina comm., Vol. 3,2, p. 330, 14). — That Conon’s name as astronomer was familiar to Virgil and Propertius (first cent. B.C.) may be due only to Callimachus’ poem on Berenice’s Lock.Google Scholar
  5. 5.
    Ptolemy, Phaseis, Opera II, p. 67; cf. also below p. 929.Google Scholar
  6. 1.
    For a similar misunderstanding concerning the “Regula Philippi Aridaei” cf. Neugebauer [1959, 2].Google Scholar
  7. 1.
    The corresponding latitudinal difference amounts to about 15°.Google Scholar
  8. 2.
    Cf., e.g., Geminus, Isag. V, 2 (ed. Manitius, p. 44, 4–9) and below p. 582 and Fig. 2.Google Scholar
  9. 3.
    E.g., Geminus, Isag. III, 15 (ed. Manitius, p. 42, 3–8); Vitruvius, Archit. IX V, 4 (ed Krohn, p. 213, 24–28; Loeb II, p. 244/245); Manilius I 216 and notes by Housman (Vol. I, p. 17); Theon Smyrn., Astron. II (Dupuis, p.201). Cf. also Kepler, Werke II, p. 135, 15–136, 19.Google Scholar
  10. 4.
    For references to the sun’s zenith positions cf. below p. 937.Google Scholar
  11. 5.
    Cf., e.g., below V B 8, 3.Google Scholar
  12. 6.
    Aristotle, De caelo II, 14 (Budé, p. 100) and II, 11 (Budé, p. 80); for the error in this argument cf. below p. 1093.Google Scholar
  13. 7.
    Diogenes Laertius IX, 21 (Loeb II, p. 430/431). Frank, Plato, p.198–200, denies the validity of this statement in Diogenes and explains it as a misunderstanding of the original formulation by Theophrastus. Cf. also Heidel, Greek Maps, p. 70–72. Both scholars agree that Plato’s Phaedo (≈400 B.C.) must be very near to the time of discovery (Frank, p. 184ff, Heidel, p. 79ff).Google Scholar
  14. 8.
    Diogenes Laertius IV, 58 (Loeb I, p. 434/435).Google Scholar
  15. 9.
    Cf., e.g., Euclid, Phaen., Introd. (ed. Menge, p. 6, 5); Cicero, De Nat. Deorum II, 47 (Loeb, p. 166/169); Theon Smyrn., Astr. I (Dupuis, p. 201); Achilles, Isag. 6 (Maass, Comm. Ar. rel., p. 37, 8); Almagest I, 4; Cleomedes I, 8.Google Scholar
  16. 10.
    Cf., e.g., Neugebauer-Parker EAT I Pl. 8 (column V), Pl. 34, II Pl. 7, etc.Google Scholar
  17. 11.
    For a list of Normal Stars with coordinates cf. Sachs [1952, 2]; also above p. 546.Google Scholar
  18. 12.
    For Babylonian constellations cf. Weidner [1967].Google Scholar
  19. 13.
    Cf. above I E 2, 1 A and I B.Google Scholar
  20. 14.
    Vitruvius, Archit. IX V, 4 (ed. Krohn, p. 213, 15–17; Loeb II, p. 242/243).Google Scholar
  21. 15.
    Boll, Sphaera, p. 60–72 (concerning, p. 5, 23–14, 4 in the ed. Kroll).Google Scholar
  22. 16.
    Rehm [1899], p.252–254; CCAG9, 1, p. 189f.; cf. above p.285.Google Scholar
  23. 17.
    The “accepted” title is only a re-translation by Rehm into Greek of a Latin version (cf., e.g., Keller, Erat., p. 23) which does not contain the word “Catasterism.”Google Scholar
  24. 18.
    Cf. for this literature Knaak in RE 6, 1, col. 277–381 (1909) and Keller, Erat. (1946), p. 18–28.Google Scholar
  25. 19.
    Some always visible stars presuppose the latitude of Alexandria; cf. Rehm [1899], p. 268.Google Scholar
  26. 1.
    Nothing is gained by the countless speculations published on this subject. This includes also the hypotheses, hesitatingly proposed by Manitius (p. 247–252), about excerpts and commentaries supposedly made in Constantinople.Google Scholar
  27. 5.
    Isagoge Chap. III, §8 and 13, Manitius, p.38, 16 and p.40, 15 and 19.Google Scholar
  28. 6.
    Manitius, p. 8/9 and p. 14–19; cf. below p.584.Google Scholar
  29. 1.
    Geminus, ed. Manitius, p. 108, 4–10.Google Scholar
  30. 2.
    Cf. above p. 579.Google Scholar
  31. 3.
    Isagoge, p. 108, 3 Manitius.Google Scholar
  32. 4.
    Porter-Moss VI, p. 94 and p. 97. — This is the same chapel which had the famous “round zodiac of Denderah” on the ceiling.Google Scholar
  33. 5.
    A summary is given by Loret [1884], p.99–102.Google Scholar
  34. 6.
    Grenfell-Hunt, The Hibeh Papyri I, p. 144, p. 148 lines 58 to 62.Google Scholar
  35. 7.
    For this date cf. Griffiths, Plutarch, p. 17.Google Scholar
  36. 8.
    Chap. 13 (Griffiths, p. 138/139; Loeb, Moralia V, p. 36/37), Chap. 39 (Griffiths, p. 178/179; Loeb, p. 94–99); Chap. 42 (Griffiths, p. 184/185; Loeb, p. 100/101). Cf. also the commentaries Griffiths, p. 64 f., p. 312, p. 448 f.Google Scholar
  37. 10.
    This has been seen by Grenfell-Hunt (p. 153 to line 60). — The Roman “Isia” were celebrated in November. Cf. for references Daremberg-Saglio III, p. 583; also Lydus, De mens. IV 148 (ed. Wuensch, p. 1666, 16f.) who gives November 2 and 3 as date for the Isis festival. The Egyptian Khoiak 12 corresponds in A.D. 120 to Alexandrian Athyr 6 = November 2, the date given by Lydus, and in A.D. 140 to Alexandrian Athyr 1=October 28, the beginning date of the “Isia” in the Philocalus Calendar (cf. Stern, Cal. pl. XI, XII).Google Scholar
  38. 12.
    Manitius, p. 281. All these arguments come from Böckh, Sonnenkr, p. 22–26.Google Scholar
  39. 13.
    The other authorities mentioned are Meton, Euctemon, Democritus from the 5th century and Eudoxus and Callippus from the 4th. — Incidentally: no “Egyptians” are mentioned; they appear first in a parapegma from Miletus about 100 B.C. (cf. below p. 588; also above p.561f.).Google Scholar
  40. 2.
    Manitius, p. 181ff.;.; cf. also p. 191ff. on Sirius.Google Scholar
  41. 3.
    Manitius, p. 32, 4; 32, 19; 36, 2; 80, 13–19; 84, 25, etc.Google Scholar
  42. 4.
    Manitius, p.6, 17; 8, 3; 70, 16.Google Scholar
  43. 5.
    Manitius, p.70, 17; cf. also below p.711, n. 26.Google Scholar
  44. 6.
    Manitius, p.70, 19f.; nowhere is recognition of a doctrine of seven climata discernable.Google Scholar
  45. 7.
    Manitius, p. 90, 13–18.Google Scholar
  46. 8.
    In a formulation also found in the “Enoptron” of Eudoxus; cf. Hipparchus, Aratus Comm., ed. Manitius, p. 22, 20–22.Google Scholar
  47. 9.
    Manitius, p. 50, 12–52, 2. Manitius restores in the corrupt text “Hellespont,” in spite of the fact that the Latin version, made from an Arabic version, has “regionis grecorum.”Google Scholar
  48. 10.
    Aratus, Phen., verses 497–499; Hipparchus, Aratus Comm., ed. Manitius, p. 26, 3–28, 8. Cf. also Ptolemy, Phaseis, Opera II, p. 67, 16–21 (ed. Heiberg) and Cleomedes I, 6 (ed. Ziegler, p. 52, 1–2 ).Google Scholar
  49. 11.
    E.g. Manitius, p. 168, 17.Google Scholar
  50. 12.
    Cf. below p. 590.Google Scholar
  51. 13.
    E.g. Manitius, p. 168, 12. The same definition again in Hyginus (2nd cent. A.D.; Astron. I, 6 ed. Bunte, p. 24; Chatelain-Legendre, p. 5), in Achilles (3rd cent.; Isag. 26, Maass, Comm. Ar. rel., p. 59), and in Cleomedes (around 400; I, 2 ed. Ziegler, p. 20/22, et passim).Google Scholar
  52. 15.
    Cf. below p. 865 f.; also p. 871.Google Scholar
  53. 16.
    Cf below and in general Schlachter, Globus, p. 46.Google Scholar
  54. 17.
    Manitius, p.28, 11–14. This happens at about φ =46;30° (between climates VI and VII).Google Scholar
  55. 18.
    Manitius, p. 166, 1 implies that 6°=4200 stades.Google Scholar
  56. 19.
    Cf. below p. 935.Google Scholar
  57. 20.
    Geminus, p. 5, Hipparchus, p. 127 (ed. Manitius).Google Scholar
  58. 21.
    Cf. above p. 278. The same terminology is found in Strabo II 5, 42 (Loeb I, p. 515), in Pliny N.H. II, 178 (Budé II, p. 78) and in Cleomedes I, 10 (ed. Ziegler, p. 94, 10) who explains the measure of “1/4 sign” for the altitude of the star Canopus as “1/48 of the zodiac”; similar in Geminus, Isag. III, 15 (Manitius, p. 42, 7). Hyginus (2nd cent. A.D.) says that all five important great circles on the sphere are “divided into 12 parts” (ed. Bunte I, 6, p. 25, 20f.; ed. Chatelain-Legendre, p. 6).Google Scholar
  59. 22.
    E.g Manitius, p.26, 8f.; p.82, 20ff.; p.92, 6ff., etc., incorrectly rendered as “1°” by Manitius. Cf. CCAG 1, p. 163, 21 f.: “30° of Cancer or 1st degree of Leo.” For Hipparchus cf. above p. 278f.Google Scholar
  60. 23.
    Isagoge, Chap. III (Manitius, p. 36–43); for the text of this chapter see also Maass, Comm. Ar. rel., p. XXV to XXVIII. Cf. also Vitruvius, above p. 577.Google Scholar
  61. 24.
    Cf. above p. 286.Google Scholar
  62. 25.
    Isagoge V, 53 (Manitius, p.62, 8f.).Google Scholar
  63. 26.
    Pliny N.H. 66 (Budé II, p. 29; Loeb I, p. 215); Hyginus (2nd cent.) ed. Bunte, p. 104,20–23; Chalcidius (4th cent.) ed. Wrobel, p. 136, 17–20; Martianus Capella (5th cent.) ed. Dick, p.438, 15/16, P. 456, 22; Remigius of Auxerre (9th cent.) ed. Lutz, p. 258 (ad 434, 14); Sacrobosco, De spera, ed. Thorndike, p. 88, trsl. p. 128. A passage in Manitius (Astron. I, 682) is doubtful; cf. ed. Housman I, p. 61.Google Scholar
  64. 27.
    Chalcidius, e.g., says (ed. Wrobel, p. 136, 17–20) that “according to the opinion of the old ones” the latitudinal amplitude for the moon and for Venus is 12°. Cf. also below IV B 1, 3 p.Google Scholar
  65. 28.
    E.g. Pliny; cf. note 26.Google Scholar
  66. 29.
    Cf. above p.582 and below p. 590.Google Scholar
  67. 30.
    Cf. above p.515; also Neugebauer-Sachs [1968/1969], p. 203 and note 27 there.Google Scholar
  68. 31.
    Schott-Schaumberger [1941], p. 109, note 1.Google Scholar
  69. 32.
    Cf. below IV A 4,3 A.Google Scholar
  70. 33.
    Cf. for this type of astrological doctrines Bouché-Leclercq AG, p. 159ff.Google Scholar
  71. 34.
    Manitius, Geminus Isag., p. 255, note 6; for the Eudoxan norm cf. below p. 599.Google Scholar
  72. 35.
    Cf. above p. 581.Google Scholar
  73. 36.
    Manitius, p. 10/11.Google Scholar
  74. 37.
    Manitius, p. 12/13. At a later occasion (Manitius, p. 194, 24) the three outer planets are called “the greatest (μέγιστοι) of the planets.”Google Scholar
  75. a The same opinion is also held by Proclus (Comm. Rep., trsl. Festugière III, p. 170, 15f.).Google Scholar
  76. 38.
    Manitius, p. 8/9.Google Scholar
  77. 39.
    Manitius, p. 69/71. Theon in his Great Commentary to the Handy Tables, mentions Serapion as being concerned with the equation of time. If this Serapion is the well-known contemporary of Cicero we would have evidence from the first century B.C. for the recognition of such a correction (in tables based on the era Philip). Cf. Rome [1939] and CA III, introd., p. CXXXIII, note (1). Text in Monum. 13, 3, p. 360.Google Scholar
  78. 40.
    Manitius, p. 100, 9 and 17; p. 200, 9.Google Scholar
  79. 42.
    Manitius, p.8, 23; p. 102, 2 et passim.Google Scholar
  80. 43.
    Manitius, p. 102, 4 et passim.Google Scholar
  81. 44.
    Manitius, p. 110, 22; p. 114, 15 et passim.Google Scholar
  82. 45.
    From (4) and (3) it would follow that lm = 2922:99= 48,42:1,39 =29;30,54,32,43,38,...4.Google Scholar
  83. 46.
    Cf. above IE2,2C.Google Scholar
  84. 47.
    Manitius, p. 116, 20–23; cf. also above p.69 (1), p. 310, and p.483 (3).Google Scholar
  85. 48.
    Manitius, p. 118 to 123.Google Scholar
  86. 49.
    Rome CA III, p. 839, note (1) suggests an emendation of the text on the basis of Theon’s version of the history of the 19-year cycle. Note, however, below p. 623, note 12.Google Scholar
  87. 50.
    Above p. 584 (1).Google Scholar
  88. 51.
    Above note 47.Google Scholar
  89. 52.
    Manitius, p. 200, 10; p. 204, 23/24.Google Scholar
  90. 53.
    Denoted above II B 2, 3 as F*; cf. also below p. 602.Google Scholar
  91. 54.
    Almagest IV, 2 (Manitius, p. 195f.), obtained by multiplication with 3 of the famous “Saros” relation (cf. above p. 502 (1) and p. 310 (5)). Surprisingly Geminus ignores the further equivalence with 726 = 3·242 draconitic months which is the key to the theory of eclipses.Google Scholar
  92. 55.
    Manitius, p. 205.Google Scholar
  93. 56.
    Cf. above p.480 (1) and (2 a). This fictitious derivation of standard Babylonian parameters is reminiscent of Ptolemy’s procedure in motivating the value (6) of p. 585 (cf. above p. 310).Google Scholar
  94. 57.
    Manitius, p. 205–211.Google Scholar
  95. 58.
    Cf. above p. 480 (1) or below p. 602.Google Scholar
  96. 59.
    The arithmetical rule that in a linear progression the mean value is half the total of the extrema is reminiscent of (less trivial) statements by Hypsicles (cf. below IV D 1,2 A).Google Scholar
  97. 3.
    The Greek term παράπηγμα belongs to a verb meaning to “fix beside, or near.” The German term for these inscriptions is “Steckkalender” (Diels-Rehm [1904], p. 100).Google Scholar
  98. 5.
    Cf. Rehm R. E. Par. col. 1300, 5 and 42. For two other small fragments, one from Athens, the other from Pozzuoli, cf. 1.c. col. 1301/2 A 3 and A 4. Cf. also Degrassi, Inscr., p. 299, p. 306–311.Google Scholar
  99. 6.
    For reasons unknown Manitius (p. 211 ff.) translated neither the title nor the headings but gave only a paraphrase.Google Scholar
  100. 7.
    The term “parapegma” does not occur in the title of the literary versions. Geminus, however, in the Isagoge, uses the term as freely as the moderns (e.g. Manitius, p. 182, etc.). Manitius’ rendering “calendar” is quite appropriate.Google Scholar
  101. 9.
    Cf., e.g., Rehm Parap., p. 7.Google Scholar
  102. 10.
    Cf. above p.581, n. 13.Google Scholar
  103. 11.
    In Book II of the “Phaseis”; cf. below V B 8, 1 B.Google Scholar
  104. 12.
    In the Geminus parapegma Meton is only mentioned once (cf below p. 623, n. 12).Google Scholar
  105. 13.
    Rehm RE Par. col. 1300, 23.Google Scholar
  106. 14.
    Diels-Rehm [1904], p. 102/3.Google Scholar
  107. 15.
    In Diels-Rehm [ 1904 ] the julian date is incorrectly given as June 27. Merritt, Ath. Cal., p. 88 gives by mistake Payni 14 instead of 11. All dates are correct in Dinsmoor, Archons, p. 312.Google Scholar
  108. 16.
    Cf below p. 622.Google Scholar
  109. 17.
    Cf. below p. 929 and above p. 562 f.Google Scholar
  110. 18.
    Diels-Rehm [1904], p. 108/9, note 1.Google Scholar
  111. 19.
    Cf. above p. 581 and p. 583. Also Columella, De re rustica XI 1, 31 (Loeb III, p. 68/69), opposes the doctrine of fixed definite dates for the changes of air. Similarly Ptolemy has his doubts about the prognostications associated with the stellar phases (cf. below p.926, n. 4).Google Scholar
  112. 20.
    Sinān’s “parapegma” is preserved through Birini in Chap. XIII of his “Chronology” (trsl. Sachau, p. 233–267). Cf. JAOS 91 (1971), p. 506.Google Scholar
  113. 21.
    Geminus, Elementa Astronomiae, p. 210–233 with German translation and notes.Google Scholar
  114. 22.
    Ptolem. Opera II, p. 14–67; p. III-V; p. CL-CLXV. No translation.Google Scholar
  115. 1.
    Contrary to a widespread belief the sexagesimal system did not originate from any astronomical concept. Its beginnings go back to the earliest Mesopotamian civilization, more than a millennium before any computational astronomy existed. Its origin can be found in the norms for weights and measures in combination with palaeographical processes which lead to the place value notation which is the most characteristic element of this number system; cf. Neugebauer [1927] and ThureauDangin SS.Google Scholar
  116. 2.
    Strabo, Geogr. II 5, 7 (Loeb I, p. 438/9: with different reading: Budé I, 2, p. 85). The same norm is still used in Geminus (Isag., Manitius, p. 58, 23ff.; p. 183, 3ff. etc.; 1st cent. ≈ A.D.), in Manilius (Astron. I, 561ff.; Housman I, p.53; Breiter, p.21/22; 1st cent. A.D.), in Plutarch (Moralia 590 F, Loeb VII, p.464/465; zA.D. 100), by Galen (2nd cent., cf. Rehm [ 1916 ], p.82), in Hyginus (Astron., I,6, ed. Bunte, p. 24; Chatelain-Legendre, p. 5; 2nd cent.), in Achilles (Isag., Maass, Comm. Ar. rel., p. 59, 5; p. 70, 12; 3rd cent.), by Macrobius (≈400: Comm. II 6, 2–5, ed. Eyssenh., p. 606, 24–607, 16; ed. Willis, p. 116, 15–117, 3; trsl. Stahl, p. 207), and by Severus Sebokht (A.D. 660: Nau, Const., p. 93 ).Google Scholar
  117. Achilles (Maass, p. 59, 24ff.) adds the remark that “some” divide the circle not in 60 but in 360 degrees (μοίας) “because the year has 365 days.” In the subsequent description of the angles shown in Fig. 1 (below p.1351) he makes several mistakes. Martianus Capella (De nupt. VIII 837, ed. Dick, p.439; 5th cent.) makes 1 quadrant=18 parts thus 1p=5°. This is obviously absurd since it implies ε=4p=20°. Thus one must emend the (8 + 6 + 4)° = 90° of the text to the same norm (6 + 5 + 4)p =90° found in the above mentioned sources.Google Scholar
  118. a Heath, Arist., p. 352/3 (Hypot. 4); cf. also below p. 773, notes 6 to 9.Google Scholar
  119. 3.
    Cf below p. 699.Google Scholar
  120. 4.
    Huxley [1963], p. 103 suggests the middle of the second century B.C.Google Scholar
  121. 5.
    De Falco-Krause-Neugebauer, Hypsikles, passim.Google Scholar
  122. 6.
    Observations by Timocharis mentioned in the Almagest range between —294 and —271.Google Scholar
  123. 7.
    Alm. VII, 3, Heiberg II, p. 19–23; Manitius II, p. 18–20.Google Scholar
  124. 8.
    The translation of Manitius is misleading in so far as he gives always minutes of arc where the text has only unit-fractions of degrees.Google Scholar
  125. 9.
    First century A.D.; cf. above (p. 579f.).Google Scholar
  126. 10.
    Such is the case already in the Commentaries to the Almagest by Pappus and Theon; cf. Mogenet [1951] or Rome CA II, p. 452–462.Google Scholar
  127. 11.
    Cf., e.g., Diophantus, Opera II, ed. Tannery, p. 3–15, or Pachymeres, Quadrivium, ed. Tannery, p. 331–363 (written about A.D. 1300 ).Google Scholar
  128. 12.
    Rome, CA I, p. 186, 13.Google Scholar
  129. 13.
    Cf., e.g., ACT I, p.39 and Neugebauer-Sachs [1967/1969] I, p. 204/205.Google Scholar
  130. 15.
    Cf. above p. 159f.Google Scholar
  131. 16.
    In both cases one finds that the longitude of Mercury (as morning star) was almost exactly 1° greater than the longitude of the star (the latitudinal intervals are 1;35° and 1;5°, respectively). Since Ptolemy is only interested in the longitudinal component of the distance between the planet and the mean sun, the term brim, literally “above,” seems here to mean “ahead (in longitude).” This is reminiscent of the terminology of Theodosius, where, however, άνώτερον denotes the point ahead in the direction of the daily rotation (cf. below p. 758).Google Scholar
  132. 17.
    Strabo, Geogr. II 1, 18; cf. above p. 304.Google Scholar
  133. 18.
    Alm. VII, 1 Heiberg II, p. 4, 16; 5, 1; 7, 14; 8, 2 and p. 6, 11, respectively.Google Scholar
  134. 19.
    Manitius, p. 186–280; cf. above p. 279.Google Scholar
  135. 20.
    Manitius, p. 186, 11 etc. (50 cases), perversely translated by Manitius by “Mondbreite.”Google Scholar
  136. 21.
    Manitius, p. 206, 4 etc. (9 cases). Otherwise one finds only two more passages which mention cubits, and this only in a loose fashion (Manitius, p. 190, 10 and 272, I).Google Scholar
  137. 22.
    Vogt [1925], col. 30.Google Scholar
  138. 23.
    The only passage, Manitius, p. 272, 2 is an arbitrary emendation; cf., however, above note 18.Google Scholar
  139. 24.
    P. Lond. 130; cf. Neugebauer-Van Hoesen, Gr. Hor., p. 26.Google Scholar
  140. 25.
    Pap. Oslo III, p. 30.Google Scholar
  141. 26.
    Cf., e.g., Gardiner, Eg. Grammar, § 266.Google Scholar
  142. 27.
    Cf., e.g., ACT I, p. 39: 1 finger=6 barley-corns.Google Scholar
  143. 28.
    Cf. Neugebauer-Sachs [1967/1969] I, p. 203; also above II C 2, p. 551.Google Scholar
  144. 29.
    CCAG 8, 3, p. 99, 7f.Google Scholar
  145. 30.
    Text by mistake “circumference.”Google Scholar
  146. 31.
    Isagoge XI, 7 Manitius, p. 134/135; p. 271, note 23.Google Scholar
  147. 32.
    Cf., e.g., below p. 635 (4) and (5); p. 654 (8); p. 667 (1).Google Scholar
  148. 1.
    In a list of solar eclipses, preserved for the years from −474 to −456; cf. Aaboe-Sachs [1969], p. 17. Other early evidence: Neugebauer-Sachs [1967], p. 197f. (≈ −430); Aaboe-Sachs [1969], p. 3ff. (≈ −400).Google Scholar
  149. 2.
    For an example of the earlier terminology (in −418/17) cf. Sachs in Neugebauer, Ex. Sci.(2), p. 140.Google Scholar
  150. 3.
    Cf. Sachs [1952, 3], p. 62.Google Scholar
  151. 4.
    Cf. Sachs [1948], p. 281; also above II C 1, p. 545.Google Scholar
  152. 5.
    Pliny, NH II, 31 (Loeb I, p. 188/9; Budé II, p. 17).Google Scholar
  153. 6.
    Cf., e.g., W. Kroll in RE Suppl. 4 (1924), col. 912f.Google Scholar
  154. 7.
    Cf. below IV D 3, 1.Google Scholar
  155. 8.
    Cf. Neugebauer-Parker, EAT III, p. 204f. Since zodiacs belong naturally to the ceiling decorations the chance of destruction of these monuments is particularly great.Google Scholar
  156. 10.
    The enumeration of the signs, however, always begins with Aries and in this sense one can say that longitudes are counted from 0° to 360°. Incidentally, the Babylonian name for the first sign does not mean “the Ram” but “the Hireling” (1ù-hun-gà).Google Scholar
  157. 11.
    Cf. above II Intr. 4, 1 etc.Google Scholar
  158. 12.
    Kugler, BMR, p. 104ff.Google Scholar
  159. 13.
    Neugebauer-Van Hoesen, Greek Horosc., p. 180ff.Google Scholar
  160. 1.
    Cf. above IV A 3, 3 and below p. 929.Google Scholar
  161. 2.
    Pliny NH XVIII 58, 221 (Loeb V, p. 328/329; ed. Jan-Mayhoff III, p. 204, 12).Google Scholar
  162. 3.
    Pliny NH XVIII 58, 264 (Loeb V, p. 356/357; ed. Jan-Mayhoff III, p. 216, 5).Google Scholar
  163. 4.
    Pliny NH XVIII 57, 214 (Loeb V, p. 324/325; ed. Jan-Mayhoff III, p. 202, 1 and 17). On the other hand Lydus, De mensibus IV, 18 (ed. Wuensch, p. 79, 13f. = Caesar, Comm. III, ed. Klotz, p.219) says — about A.D. 550 — that according to Caesar the sun enters Aquarius on Jan. 22. The Philocalus Calendar of A.D. 354 has even Jan. 23 (cf. Stern, Cal., p. 58f.); but the comparison with the other dates in this calendar shows clearly that Jan. 23 falls outside the scheme of the remaining dates and should be emended to Jan. 17. Obviously Jan. 22 or 23 is a later correction, made in order to obtain a solar longitude in agreement with the norm of Aries 0° for the vernal point. Indeed one has for — 50 Jan. 22 for the sun λ≈301°.Google Scholar
  164. 5.
    Gundel HT, p. 148.Google Scholar
  165. 6.
  166. 7.
    Cf. for this date Neugebauer, Ex. Sci.(2), p. 68f.Google Scholar
  167. 8.
    Vitruvius, De archit. IX, 3 (Loeb II, p. 232–235; ed. Krohn, p. 209; also Loeb II, p. 266/267; Krohn, p. 222).Google Scholar
  168. 9.
    From Venosa in Apulia (east of Melfi). Cf. Degrassi, Inscr., p. 55–62 and Tab. IX.Google Scholar
  169. 10.
    Degrassi, Inscr., p. 58f. Since the entry of the sun into Gemini is given for May 18 the date of entry into Cancer should perhaps be emended to June 18. Such variations by one day are, of course, explicable by the ambiguity of the use of the term “first degree” The Philocalus Calendar of A.D. 354 gives June 15 for the entry, June 24 for the solstice (as Pliny) but the differences suggest an emendation to 16 or 17; cf. Stern, Cal., p. 58 f. and Degrassi, Inscr., p. 248/249.Google Scholar
  170. 11.
    Cf. for this date Rehm, RE Par. col. 1309, 50.Google Scholar
  171. 12.
    Varro, De re rust. I, XXVIII (Loeb, p. 248–251).Google Scholar
  172. 13.
    Columella, De re rust. IX, XIV 10–12 (Loeb II, p. 480–489; also Wachsmuth, Lydus, De ost., p. 303).Google Scholar
  173. 14.
    Mommsen, Chron.(2), p. 58, p. 60. The term is misleading: no peasant constructed this calendar which is simply the Roman version of a Greek parapegma. Mommsen only intended to underline the usefulness and need for agricultural work of a calendar based on a solar year, recognizable by fixed star phases.Google Scholar
  174. 15.
    Mommsen, Chron.(2), p. 62 or Ginzel, Hdb. II, p. 282.Google Scholar
  175. 16.
    Cf., however, above p. 594, note 4.Google Scholar
  176. 17.
    For additional references cf. Rehm [1927]; also Rehm RE Par. col. 1324 (B 18), col. 1352, 57–1353, 45 and Rehm Parap. Chap. III. 1 never succeeded separating facts from mere hypotheses in this vast literature.Google Scholar
  177. 18.
    Columella, De re rust. XI, II 94 (Loeb III, p. 124/125).Google Scholar
  178. 19.
    Cf., e.g., Rhetorius (≈A.D. 500) who says “at the 30th degree of Cancer, that is at the 1st degree of Leo” (CCAG 1, p. 163, 12f.). The traditional mixup between “first degree” and 0° or 1° mars ancient as well as modern interpretations; cf. also above p. 278 and below p. 600, n. 23.Google Scholar
  179. 21.
    Rehm, Parap., p. 33f. and RE Par. col. 1343, 48ff. thought that Eudoxus’ parapegma could be more accurately dated to 370 B.C. because the symmetry of the seasons should have been adopted under Plato’s influence who was opposed to an anomaly of the solar motion.Google Scholar
  180. 23.
    Columella (IX, XIV 12): “antiquorum fastus astrologorum” (Wachsmuth, Lydus De ost., p. 303, 26f.; Loeb II, p. 488/489).Google Scholar
  181. 25.
    Rehm, for reasons that escape me, follows Columella (cf. below p. 599, note 10).Google Scholar
  182. 27.
    I am not convinced of the customary association of the Roman “rustic calendar” with Callippus (Rehm [1927], p. 216; Parap., p. 44 and references given there).Google Scholar
  183. 28.
    Ed. Housman III, p. 23, p. 68; ed. Breiter, p. 73, p. 88 and p. 87, p. 106. Manilius does not follow, however, a consistent system; in I 622 and 625, e.g., he assumes the beginnings of the signs (cf. Breiter, p. 106).Google Scholar
  184. 30.
    NH II: Loeb I, p. 224/225; Budé II, p. 35 (with antiquated notes on p. 169f.); NH XVIII: Loeb V, p. 328/329.Google Scholar
  185. 32.
    Mich. Pap. III, p. 76; transl., p. 114.Google Scholar
  186. 34.
    Associated with the manuscripts of Censorinus, De die natali (written in 238/9) but probably much older since a similar text appears as scholion to the Aratea of Germanicus who died in A.D. 19. Cf. RE 3, 2 col. 1910, 13–19 and RE 10, 1 col. 461, 67; cf. also C. Robert, Eratosthenis catasterismorum reliquiae (Berlin 1878, reprinted 1963 ), p. 203.Google Scholar
  187. 35.
    Date uncertain; 3rd century or before Firmicus.Google Scholar
  188. 37.
    For the date of the completion of the Mathesis (about A.D. 355) cf. Thorndike [1913], p. 419, note 2.Google Scholar
  189. 38.
    Ed. Kroll-Skutsch II, p. 306f.; cf. also Boll, Sphaera, p. 246f.Google Scholar
  190. 40.
    Ed. Dick, p. 434–438; the essential passage also occurs at the end of a manuscript of Hyginus, Astron. (ed. Hasper, p. 31 f., corresponding to ed. Dick, p. 437, 11–438, 9 ).Google Scholar
  191. 41.
    Etym. V 34; Nat. rer. VIII 1 (ed. Fontaine, p. 204/205). In both works the cardinal points are placed at the 8th calends of April, July, October, January, respectively. The same dates are found in two parallel inscriptions from Rome, known as “calendarium Colotianum” (first cent A.D.) and “calendarium Vallense,” the latter (now lost) combined with sun dials. Cf. for these texts Degrassi, Inscr., p. 284–287 and Pl. 81–86; also Wissowa [ 1903 ].Google Scholar
  192. 42.
    For the intricate questions of authorship and sources of these commentaries cf. Stahl [1965], p. 107 ff.Google Scholar
  193. 45.
    Scientia Petri Ebrei, cognomento Anphus, de dracone, quam dominus Walcerus prior Maluer-nensis ecclesie in latinam transtulit linguam”; cf. Millas Vallicrosa [1943], p. 88. On Petrus Alphonsi (≈ 1100) cf. Cutler [ 1966 ], p. 190, n. 16.Google Scholar
  194. 46.
    Cf. Kaltenbrunner [1876], p. 294 on a version, composed in 1396, of a computus of 1200.Google Scholar
  195. 1.
    Ed. Housman III, p. 68. Rehm, Parap., p. 30, n. 1 suggests Meton for the 10° norm, considering the rhetorical question “who else...?” as a proof. Cf. also above p.496.Google Scholar
  196. 4.
    The name of a canonical collection of texts, arranged in a definite “series,” is taken from its initial words (as one refers to papal bulls) and mentioned in the colophon of each tablet in the series (cf. the colophon given in Bezold-Kopff-Boll [1913], p. 36/37).Google Scholar
  197. 5.
    The names of the months are the usual ones of the civil calendar, i.e. I = Nisan, etc.Google Scholar
  198. 6.
    Cf. Pritchett-van der Waerden [1961], p.43f.; also Bezold-Kopff-Boll [1913]. The date of this collection is about 700 B.C., based on observations made in Babylon about 1000 B.C., according to van der Waerden [ 1949 ], p. 20f.Google Scholar
  199. 7.
    Cf., e.g., van der Waerden [1949], p. 19; [1951], p. 22.Google Scholar
  200. 8.
    Hipparchus, ed. Manitius, p. 128, 21–27.Google Scholar
  201. 9.
    Manitius, p. 20, 4–17; p. 22, 1–9; p. 132, 20–134, 2.Google Scholar
  202. 10.
    Manitius. p.48, 7–10; p.56, 15f. It is amazing to see that the ample testimony of Hipparchus, who still had the writings of Eudoxus at his disposal, is explained away in favor of one sentence in a Roman work on agriculture (Columella; cf. above p. 596), four centuries after Eudoxus. In order to rescue the norm Aries 8° for a “genuine” Eudoxus Rehm postulates a “false” Eudoxus, or at least a “aegypjisierende Überarbeitung” of his parapegma in which the vernal point was moved (why?) to Aries 15° (Rehm RE Par. col. 1308, 35–39; col. 1343, 20–24; Parap., p. 18, p. 35–37). Why Hipparchus used such a version remains unexplained. Rehm also speaks about a “anderweitig erschlossene” false Eudoxus by mentioning doubts cast on the genuineness of a work on the octaeteris, although it has probably nothing to do with the parapegma or the work used by Hipparchus. Böckh (Sonnenkr., p. 192ff.) tried to reconcile our sources by postulating different norms for Eudoxus’ calendaric and “astrognostic” writings.Google Scholar
  203. 16.
    Column XXII, 21. Blass, p. 25; Tannery HAA, p. 294, No. 54.Google Scholar
  204. 18.
    Cf. below p. 1453, Fig. 12 on PL VII (col. X).Google Scholar
  205. 19.
    Turner-Neugebauer [1949], p. 7.Google Scholar
  206. 21.
    Geminus, Isag. II, 27ff. (Manitius, p. 31); cf. above p. 583.Google Scholar
  207. 22.
    Cf. Bouché-Leclercq, AG, p. 187, Fig. 23.Google Scholar
  208. 24.
    Hipparchus, ed. Manitius, p. 132, 7–9.Google Scholar
  209. 25.
    Cf. Rehm [ 1913 ]; Pritchett-van der Waerden [1961], p. 32–36.Google Scholar
  210. 27.
    Cf. below VI A 2, 4 and above p. 159.Google Scholar
  211. 1.
    Version A: from Vat. gr. 191 fol. 170v (CCAG 5, 2, p. 127, 17–19 = Vettius Valens, Anthol. IX 11, ed. Kroll, p. 353,10–13). Version B: from Vat. gr. 381 fol. 163v, published Maass, Aratea, p. 140. For the relationship between these two codices cf. also Maass [1881].Google Scholar
  212. 2.
    Apollinarios (listed in RE 1, 2 col. 2845 as Apollinaris No. 12) is mentioned by Vettius Valens (p. 250, 26 Kroll=CCAG 5, 2, p. 38, 17); hence he cannot be much later than about A.D. 150. Hephaestion (about A.D. 380) seems to associate him (CCAG 8, 2, p. 61, 16; p. 63, 21) with Antiochus (of Athens) who lived in the first or second century A.D. Hence a date around A.D. 100 could be assumed for Apollinarios. The “Anonymus of 379” (and, following him, “Palchus”: CCAG 5, 1, p. 205, 5 and CCAG 1, p. 80, 19) treat Apollinarios as parapegmatist like Meton and Euctemon, observing in Athens (suggested by his association with Antiochus of Athens?) Honigmann’s (SK, p. 42) “von Laodikeia” is a simple mistake (mixup with a christian author).Google Scholar
  213. Achilles (Maass, Comm. Ar. rel., p.47, 13f. or Aratea, p. 143, n. 52) mentions Apollinarios as having written on solar eclipses in the seven climates. A long excerpt in an anonymous fragment (CCAG 8, 2, p. 132, 4 to perhaps 133, 28) shows him as being familiar with the technical terminology of lunar theory. Porphyri in his Introduction to the Tetrabiblos (CCAG 5, 4, p. 212, 14 = Riess [1891], p. 334, frgm. 3) refers to him in connection with arithmetical methods of computing oblique ascensions (cf. also CCAG 8, 4, p. 50, fol. 46).Google Scholar
  214. In astrological context the name of Apollinarios appears beside the above mentioned passages in Vettius Valens and Hephaestion (also CCAG 8, 2, p. 62, 1) in the preface to the Isagogika of Paulus Alexandrinus (p. 1, 13, ed. Boer) and in CCAG 6, p. 15 (fol. 341v).Google Scholar
  215. 5.
    It is not difficult to replace the numbers in the text by others which are not astronomically excluded but this kind of simply rewriting a text carries little conviction.Google Scholar
  216. 7.
    This parameter appears, e.g., with Ulugh Beg (Kennedy, Survey, p. 167 sub P), i.e. about 1440. Cf. also for Hipparchus above p. 293.Google Scholar
  217. 10.
    Rome [1926], p. 9, translating a passage published in Rome CA III, p. 838, 26–839, 10.Google Scholar
  218. 12.
    The latter seeks a balance between solar years and lunar months by introducing some new forms of “years” which fit some convenient pattern of intercalations. Cf., e.g., for the 19-year cycle above p. 601 (first line in A and B) or below IV B 1, 2.Google Scholar
  219. 16.
    Cf. for this problem below p. 626.Google Scholar
  220. 17.
    Above IV A 4, 2 A and below IV D 1, 2, respectively.Google Scholar
  221. 18.
    Tannery, Mém. Sci. II, p. 345f.; repeated in Heath, Aristarch., p. 314f.Google Scholar
  222. 19.
    Censorinus, ed. Jahn, p. 57, 6–8Google Scholar
  223. 20.
    The text has by mistake only 365 days; cf., however, below p. 623.Google Scholar
  224. 21.
    It follows from (6) that 40,34y = 2434y = 4,6,57,0d = 889020d gives the smallest number of years corresponding to an integer number of days on the basis of (5). Censorinus (Chap. 18, ed. Jahn, p. 55, 14f.) says that Aristarchus assumed 2484y for the return of all planets to the same position. This is certainly a mistake since no such small common planetary period exists. Tannery suggested emending 80 to 30 (π for λ) and assumes that the exact completion of solar years and of days caused a misinterpretation of (6) as a planetary “great year.”Google Scholar
  225. 2.
    For an early text considering latitudes cf. Neugebauer-Sachs [1968/1969] I, p. 209; cf. also above IIC3,p.554.Google Scholar
  226. 4.
    The Greek order differs also from the Egyptian sequence; the Indian order, however, is derived from the Greek one since it is the order of the days in the planetary week. Cf. below p. 690.Google Scholar
  227. 5.
    Alm. IX, 3; omitting here Ptolemy’s refinements expressed as corrections beyond or below exact returns; cf. above p. 151.Google Scholar
  228. 6.
    Cf. above p. 351. In the astrological literature these parameters are rarely mentioned; an example is CCAG 7, p. 120f. which mentions the values of N listed in (1), with the exception of N = 83 for Jupiter, which is, however, also a Babylonian goal-year parameter (cf. p. 391 (12)). The text in question might be from Heliodorus (around A.D. 500); cf. I.c. p. 119, n. 27. The same set of parameters, again with Jupiter’s 83, is also used in the “Almanac” of Azarquiel (epoch 1088 Sept. 1) which is based, however, on much older sources; cf. Boutelle [1967].Google Scholar
  229. 9.
    Cf. Tannery, Mém. Sci. 4, p. 265, from Cod. Scor. III. Y. 12 = CCAG 11,1 cod. 7, fol. 71 = Catálogo... Biblioteca de el Escorial II, p. 160, No. 282, 3.Google Scholar
  230. 10.
    Cf. above p. 390 (10 a) and (10 b).Google Scholar
  231. 11.
    The text has 309 instead of 720 (error for 309 syn. months = 25 Eg. years).Google Scholar
  232. 13.
    Lydus, De mensibus, p. 56f. ed. Wuensch (error for Mars: 294 instead 284).Google Scholar
  233. 16.
    Cf. Paulus Alex., ed. Boer, p. 12, 15 and p. 14, 15–18; also Ptolemy, Tetrab., Loeb, p. 97 and p. 107.Google Scholar
  234. 18.
    CCAG 1, p. 163 (cf. apparatus for the correct number); Lydus, De mens., p. 57, 6–8 (ed. Wuensch); Psellus, Omnif. doctr. § 161, 9 f. (ed. Westerink, p. 82), also Tannery, Mém. Sci. 4, p. 261 f. and Boll [1898]. The smallest common multiple of these numbers would be 1461 · 200=292 200.Google Scholar
  235. 19.
    Proclus, Comm. Rep., ed. Kroll II, p. 23; trsl. Festugière II, p. 128f.Google Scholar
  236. 21.
    Vettius Valens, Anthol. IV, Chap. 1, 3, and 30 (ed. Kroll, p. 158f. and p.205f.). The total of the minimum periods in (3) is 129; multiplication with the factor 2;50 changes it (exactly) to 365;30. Consequently these new periods are now called “days”; then they are subjected to new arithmetical modifications which are supposed to represent the combined influence of the planets, etc. Similarly Firmicus Maternus II, 25 (Kroll-Skutsch, p. 73 f.) with some errors which can easily be corrected on the basis of the numbers in (3).Google Scholar
  237. 22.
    Collected in Schnabel, Ber., p. 250–275. Schnabel’s own contributions must be taken with great caution as far as Babylonian astronomy is concerned.Google Scholar
  238. 1.
    Explaining the star of Bethlehem by planetary conjunctions, comets, novae, etc., is a classical example.Google Scholar
  239. 2.
    Commentary to Plato’s Timaeus, ed. Diehl I, p. 100, 29–101, 2; transl. Festugière I, p. 143. Cf. also the discussion by Martin [ 1864 ].Google Scholar
  240. 3.
    From Simplicius, commentary to Aristotle’s De cado II, 12 (Comm. in Arist. gr. VII, p. 506, 8–16, ed. Heiberg).Google Scholar
  241. 4.
    One need hardly to point out that no trace of such Babylonian data can be found in Aristotle’s work. All this has been said long ago by Martin [1864], of course, with very little effect.Google Scholar
  242. 5.
    Chronogr. 207 (ed. Dindorf, p.390, 1–5 = Schnabel, Ber., p. 268, 28–33), written A.D. 794 (Dindorf, p. 389, 20 ).Google Scholar
  243. 6.
    Cicero, De divinatione II 42, 87 (Loeb, p.468/471).Google Scholar
  244. 7.
    Cf. Neugebauer-Van Hoesen [1964], p. 66 and Neugebauer-Parker [1968]; also above p. 575.Google Scholar
  245. 11.
    In contrast the astrological geography in Manilius, which reflects conditions in Ptolemaic Egypt at the end of the third century B.C. (cf. Bartalucci [1961]), operates with the association of countries and zodiacal signs; cf. also Cumont [1909].Google Scholar
  246. 12.
    Chaldaeis in predictione et in notatione cuiusque vitae ex natali die minimum esse credendum.”Google Scholar
  247. 13.
    What he says, however, about the geocentric distances of the planets (De divin. II 43, 91) is not part of any astrological theory. Modern scholars act much in the same way by reading into the text words which are not there; cf., e.g., the “translation” in Loeb, p. 471: “from the positions of the stars on the day of his birth” (italics mine).Google Scholar
  248. a Labat, Calendr. Baby’., § 64 (p. 132–135) where the future of a child is predicted from the month in which it was born. For a Hittite translation of a Babylonian text of this type (from the second half of the second millennium B.C.) cf. Sachs [1952, 3], p. 52, note 17a.Google Scholar
  249. b An Egyptian papyrus from the Ramessite period, concerning lucky and unlucky days (published: Bakir, Cairo Cal.) contains also entries of the type “someone born on this day will die by a crocodile,” etc. There is, of course, no trace of astrology in this text of the 12th century B.C. and the only element of prediction or advice is the calendar date with its associated religious feasts or mythological events. By the time of Cicero a text of this type could easily be ascribed to the “Chaldeans.”Google Scholar
  250. c Ed. Diehl III, p. 151, 1–9; trans’. Festugière IV, p. 192. This passage has been discussed many times, e.g. by Kroll [1901], p. 561 or by Cumont [1911], p. 5.Google Scholar
  251. d Rehm, Parap., p. 122–140 argued that much in “De signis” belonged to Euctemon. There is no reason, however, to assume that the passage in question belongs to this “Grundschrift.”Google Scholar
  252. e Cf. also below IV A 4, 4 B.Google Scholar
  253. 14.
    Cf. texts and tabulation in Bilfinger, Bürgerl. Tag, p. 1–16.Google Scholar
  254. 15.
    Cumont [1910]; also Fotheringham [1928].Google Scholar
  255. 16.
    A beautiful example is the diagram in Schnabel, Ber., p. 110. His derivation of all these works from two single sources, Berosus and Posidonius, is about as well established as our descent from Adam and Eve.Google Scholar
  256. 17.
    Strabo XVI 1, 6 (Loeb VII, p. 203); repeated in the Tribiblos of Theodoros Meliteniotes (about 1370): cf. CCAG 5, 3, p. 140, 30–141, 1=Migne PG 149 col. 997/8; Pliny NH VI 30, 121–123 (Loeb II, p. 431); these passages also in Schnabel, Ber., p. 9.−Cf. also above p. 352.Google Scholar
  257. 20.
    Cf. above p. 601; from Anthol., p. 353, 12 (Kroll)=CCAG 5, 2, p. 127, 19.Google Scholar
  258. 21.
    Anthol., p. 354, 4–6=CCAG 5, 2, p. 128, 14–16; cf. Cumont [1910], p. 161–163.Google Scholar
  259. 22.
    One also would like to know which Apollonius is meant; cf. above p. 263.Google Scholar
  260. 24.
    Strabo III 5, 9 (Loeb II, p. 153). For a Seleucia on the Persian Gulf cf. Cumont [1927]; also Tarn-Griffith, Hellenistic Civilization (3rd ed., 1952 ), p. 158.Google Scholar
  261. 26.
    Cf., e.g., Susemihl, Griech. Litt. I, p. 764, n. 265; RE Suppl. 5 col. 962f. (Kroll).Google Scholar
  262. 27.
    Bergk, Abh., p. 170 suggests, with no trace of a proof, that observations mentioned in the Almagest using the “Chaldean era” were therefore made by Seleucus. This era is nothing but the Syrian form of the Seleucid Era; cf. above p. 159.Google Scholar
  263. 29.
    Pseudo-Plutarch, De plac. II, 1 (ed. Bernardakis V, p. 297, 10) and Stobaeus, Ecl. phys. I, 21 (ed. Wachsmuth I, p. 182, 20f.). Pines [1963] discovered in Arabic sources (Rāzi, died about 925) arguments in support of this theory, probably belonging to Seleucus.Google Scholar
  264. 30.
    Iamblichus, The Egyptian Mysteries VIII, 1 (Budé, p. 195). Manetho reported 36525 books (a number which represents 25 Sothic periods of 1461 years; cf. Manetho, Loeb, p.227, p.231). Of no interest is the enumeration of 7 phases of the moon (Clemens Alex., Stromata VI 16, 143, 3, ed. Stählin, p. 505, 1–5 ).Google Scholar
  265. 31.
    Cf. Susemihl, Griech. Litt. I, p. 861f.; Kroll in RE 4 A, 1 col. 563; Esther V. Hansen, The Attalids of Pergamon (Cornell Studies in Classical Philology 29, 1947), p. 370; Cumont [1910], p. 162. Also Wellmann [ 1935 ], p. 427, 433, 438; Bidez-Cumont, Mages I, p. 193. The name Sudines poses some problems; an Old-Babylonian name Suddänu is attested (Ranke PN, p. 166 ). One also could think of a name ending in -idin or -idina but the first half should then contain more than su (or shu?). In our material of astronomical texts the name does not occur.Google Scholar
  266. 37.
    Pliny NH II 6, 39 (Loeb I, p. 193, Budé II, p. 19). One MS has 23° instead of 22° (cf. ed. Jan-Mayhoff I, p. 139, 1 ).Google Scholar
  267. 38.
    E.g. P. Mich. 149 (2nd cent. A.D.) X, 31 (Mich. Pap. III, p. 75 and p. 102) and Maass, Comm. Ar. rel., p. 601 (from Anonymus Sangallensis).Google Scholar
  268. 39.
    Cf. Ptolemy’s discussion in Alm. XII, 10 (above I C 3, 1 ).Google Scholar
  269. 42.
    Pliny NH XVIII 57, 211 f. (Loeb V, p. 322–325). The relevant passages are collected by Wachsmuth in Lydus, De ost., p. 321–331.Google Scholar
  270. 43.
    NH XVIII 57, 215f. (Loeb V, p. 325/7); cf. also above p. 562.Google Scholar
  271. 44.
    This terminology is attested, e.g., in Parker, Vienna Pap., p. 6, a text belonging to the Persian period, around 500 B.C.Google Scholar
  272. 45.
    Columella, De re rustica XI 1, 31 (Loeb III, p. 69).Google Scholar
  273. 46.
    For another reference to Chaldeans by Columella cf. above p. 595 f. (concerning the winter solstice).Google Scholar
  274. 47.
    That parapegmata could have been abstracted from Normal Star Almanacs (cf. above p.553) is not impossible but not very plausible.Google Scholar
  275. 48.
    Cf. on these origins Sachs [1952, 3], introduction.Google Scholar
  276. 49.
    This should not be taken too literally; no element in the theory is sufficiently sensitive to require a distinction, e.g., between Babylon and Uruk.Google Scholar
  277. 1.
    Schiaparelli, Scritti II, p. 85f. (=[1877], p. 172f.).Google Scholar
  278. That the Athenian calendar (the only one about which we have ample information) shows no relation to the Metonic 19-year cycle has been stated repeatedly, e.g. in Meritt, Ath. Cal., p. 4/5. From a tabulation made by Pritchett (Ch. M. Tables 8 and p. 62) I reproduce here in Fig. 2 A the epigraphically secure evidence for intercalary (*) and ordinary (o) years, in B the corresponding numismatic results (from M. Thompson, Coinage, p. 612/613) for the last cycles in the same period (cycle 6, 18 = −319 to cycle 18, 9= −100).Google Scholar
  279. 7.
    Diodorus XII 36, 3 (Loeb IV, p. 448/449).Google Scholar
  280. 8.
    The έπισημασίαι; cf. below p..Google Scholar
  281. 9.
    Fotheringham [1924]; revived by van der Waerden [1960].Google Scholar
  282. 10.
    Cf. above p. 588 and below p. 622, n. 2. The text of the parapegma is given also in Merits, Ath. Cal. p. 88 but should be corrected following Dinsmoor, Archons, p. 312, n. 1.Google Scholar
  283. 10a.
    Added in proofs. My denial of evidence for a calendar expressly constructed for astronomical purposes was based on the implicit assumption that the Athenian dates which are given as the equivalents of Egyptian dates by Ptolemy and in the parapegma in Miletus are dates in the civil calendar, thus useless for astronomical purposes. To this my colleague G. J. Toomer objects that no motive can be seen for the use of Athenian dates in Miletus (which has a calendar of its own) or by Timocharis in Alexandria. He therefore assumes that the cycles of Meton, and then of Callippus, contained definite schematic rules for the lengths of the months and for intercalations, of course independent of the local calendar, although using the Athenian names. Cf. for details the article “Meton” in DSB, vol. 9, p. 337–340.Google Scholar
  284. 11.
    Geminus, Isag. VIII, 50–56 (Manitius, p. 120/123).Google Scholar
  285. 12.
    For the number of years any era (e.g. Olympiads, or lists of archons, etc.) would suffice to establish the correct distance.Google Scholar
  286. 16.
    Remigius, Comm. in Mart. VIII, ed. Lutz, p. 284, 2f. Cf. for this cycle below p. 624.Google Scholar
  287. 18.
    E.g. by van der Waerden [1952, 2] and [1970].Google Scholar
  288. 1.
    Censorinus, De die nat. 18, 8, ed. Hultsch, p. 38, 13–15 (also Diels, VS(5), p. 404, 18f.).Google Scholar
  289. 2.
    Censorinus 19, 2, p. 40, 14f. (also Diels, VS(5), p. 404, 19f.).Google Scholar
  290. 3.
    Cf., e.g., the Eudoxus Papyrus, below p. 624.Google Scholar
  291. 4.
    Cf. Ideler, Chronol. I, p. 309 and [ 1810 ] p. 410.Google Scholar
  292. 5.
    Cf., e.g. the Eudoxus Papyrus, below p. 624.Google Scholar
  293. 6.
    Cf. the sequence of the powers of 3 mentioned by Plutarch, De animae procreatione ( 1028 B), ed. Hubert, p. 183, 23–25.Google Scholar
  294. 7.
    elianus, Varia historia X, 7 (ed. Herscher, p. 109, 15–18; also Diels, VS(5), p. 394, 15–17). Censorinus 19, 2 (Hultsch, p. 40, 19f.).Google Scholar
  295. 8.
    Aaboe-Price [ 1964 ], p. 5. For another reconstruction cf. Tannery, Mém. Sci. II, p. 358f.Google Scholar
  296. 8a.
    G. J. Toomer considers the reference to a specific number of days in the cycle to be a later addition. Cf. DSB, vol. 10, p. 180.Google Scholar
  297. 9.
    Censorinus 18, 8 (ed. Hultsch, p. 38, 16f.).Google Scholar
  298. 10.
    The error is perhaps caused by contamination with the directly preceding Callippic cycle of 76 years with 28 intercalations.Google Scholar
  299. 14.
    Most numbers given by Tannery are restored (without warning); cf. for the text Blass, p. 20 col. XIII, 12-XIV, 6.Google Scholar
  300. 15.
    This rule implies synodic months slightly shorter than 29 1/2 days, since 48,40:1,39 ≈ 29;29,42.Google Scholar
  301. 16.
    Censorinus, De die nat. 18 (ed. Hultsch, p. 36–40).Google Scholar
  302. 17.
    Translation by Heath (Aristarchus p. 291) of “cuius maxime octaeteris Eudoxi inscribitur”, whatever this should mean. Dositheus was a friend of Archimedes and is mentioned for his observations by Ptolemy in the “Phaseis” (cf. below p. 929); also above p. 581.Google Scholar
  303. 18.
    This seems to follow from Geminus, Isag. VIII, 24 (Manitius, p. 110, 2 ).Google Scholar
  304. 20.
    Censorinus, De die nat., p. 39, 12f. and p.40, 16f., ed. Hultsch.Google Scholar
  305. 22.
    That is to say: during these 4 years the Egyptian Thoth 1 is the same as the Alexandrian Thoth 1 (i.e. August 30 in −25, August 29 in −24 to −22). In −21 Alex. Thoth 1= Egypt. Thoth 2 (= August 30). Cf. for the “Era Augustus” below p. 1066.Google Scholar
  306. 2.
    Diodorus XII 36, 2 (Loeb IV, p. 446/447). Ptolemy, Alm. III, 1 (Manitius, p. 143) says that the summer solstice in this year was observed, but only superficially recorded, by the school of Meton and Euctemon for the morning of Phamenoth 21 (of the year Nabonassar 316, i.e. −431 June 27; cf. above p. 294 and p. 617).Google Scholar
  307. 3.
    The “Uruk scheme” (above II Intr. 3, 2) would give as date of the summer solstice III 11 (not 13); from Parker-Dubberstein BC one obtains III 10 for −431 June 27. Apparently Skirophorion was about 2 or 3 days ahead of the real lunar month.Google Scholar
  308. 6.
    We know from Hipparchus that this was before his time the commonly accepted value (Alm. III, 1, Manitius I, p. 145 ).Google Scholar
  309. 7.
    Geminus, Isag, VIII, 58 (Manitius, p. 122, 13–15). In view of the later 76-year cycle the fraction 5/19 is also put in the form 1/4+1/76 (e.g. Alm. III, 1 Heiberg, p.207, 10; also Geminus, Isag. VIII, 58 Manitius, p. 122, 15 f.), and Theon in his Commentary to Alm. III, 1, quoting Hipparchus (cf. Rome CA III, p. 838 or [1926], p. 9f.).Google Scholar
  310. 8.
    Geminus, Isag. VIII, 52 (Manitius, p. 121).Google Scholar
  311. 9.
    Alm. III, 1 (Manitius I, p. 145, 14f.). Theodosius, De diebus II, 18 (ed. Fecht, p. 152, 2) in a passage for which cf. below p. 754, n. 19.Google Scholar
  312. 10.
    Geminus, Isag. VIII, 50 (Manitius, p. 121).Google Scholar
  313. 11.
    Censorinus, De die nat. 18, 8 (ed. Hultsch, p. 38, 9–11 ).Google Scholar
  314. 12.
    The role of Meton seems not too well defined in the ancient tradition. Ptolemy (Alm. III, 1, Manitius I, p. 143) in discussing the summer solstice of −431 speaks first about observations by the “school of Meton and Euctemon,” later on (p. 144) only about the “school of Euctemon.” In the “Geminus”-parapegma Meton is mentioned only once (in contrast to Euctemon “passim”). The “Eudoxus Papyrus” (P. Par. 1) mentions for the length of the seasons only Eudoxus, Democritus, Euctemon, Callippus (Tannery HAA, p. 294, No. 55). Meton’s name does not appear in Geminus’ Isagoge and is also omitted in Maass, Aratea p. 140, list “B” for the length of the year (in contrast to “A”; cf. above p. 601). Modern scholars have avoided the problem by “emending” versions they did not like; cf. Rome [1926], p. 8 = CA III, p. 839, note (1).Google Scholar
  315. 13.
    Alm. III, 1 (Manitius I, p. 145); Geminus, Isag. VIII, 59 (Manitius, p. 123).Google Scholar
  316. 14.
    For the Metonic cycle one obtains from (4) 1m=29;31,54,53,37,...d; for the Callippic cycle 29;31,51,3,49,...d (cf. above p. 616). Note that neither one of these numbers appears among Babylonian parameters.Google Scholar
  317. 15.
    Cf., e.g., the “years of grace” (Chaîne, Chron., p. 111).Google Scholar
  318. 16.
    Censorinus, De die nat. 18, 9 (ed. Hultsch, p. 38, 18 f.); cf. also above I E 2, 2 C.Google Scholar
  319. 2.
    Cf. Tannery HAA, p. 285 ff. (Nos. 7, 34, 35, 40).Google Scholar
  320. 6.
    Simplicius, Comm., ed. Heiberg, p. 494, 23–495, 16; Schiaparelli, Scritti II, p. 97f.; [1877], p. 184.Google Scholar
  321. 7.
    Also for the sun; cf. below IV B 2, 2 for the solar latitude.Google Scholar
  322. 8.
    For a mistake in Simplicius’ formulation concerning the nodal motion cf. Schiaparelli, Scritti II, p. 21; [ 1877 ], p. 118 (or Tannery, Mém. Sci. I, p. 328–332). Text: Simplicius, Comm., p. 494, 23–495, 16; Schiaparelli, Scritti II, p. 97 (No. 3); [1877], p. 184; Tannery, Mém. Sci I, p. 330.Google Scholar
  323. 9.
    Simplicius, Comm., p. 497, 18–22; Schiaparelli, Scritti II, p. 100/101; [1877], p. 187. For the inequality of the seasons cf. above I B 1, 3.Google Scholar
  324. 10.
    Eudemus in Simplicius, Comm., p. 497, 12–21; Schiaparelli, Scritti II, p. 100/101 and p. 85; [1877], pp. 187 and p. 172.Google Scholar
  325. 12.
    Schiaparelli suggested (Scritti II, p. 85f.; [1877], p. 172f.) about 6° for the maximum equation, thus γ = 6° for the angle between the axes of the two spheres which generate the hippopede (cf. below Fig. 27). From this results a latitudinal width of the curve of about 0;9° (r≈0;0,10) and for the additive or subtractive velocity a maximum of about 1;20o/d (cf. below IV C 1, 2 B (2) and (6) with Δt ≈ 27;30d) — a very reasonable estimate for the equation.Google Scholar
  326. 15.
    For the Babylonian evidence cf., e.g., Neugebauer-Sachs [1968/9] I, p.203 and ACT I, p. 190f. (No. 200 obv. I, 20); above p. 515 and p. 520.Google Scholar
  327. 16.
    Cf. above p. 583; also below p.782.Google Scholar
  328. 17.
    Martianus Capella VIII, 867 (ed. Dick, p. 456/7).Google Scholar
  329. 18.
    Dupuis, p. 313/315.Google Scholar
  330. 19.
    Alm. V, 7 (Man. I, p. 285, 6–17); also Theon, Comm. to Alm. IV, 9 (Rome CA III, p. 1068, 2) or Proclus, Hypot. IV, 63 (Man., p. 116, 22). For the parallaxes cf. above p.101 and p.324.Google Scholar
  331. 1.
    For the theory of solar latitude cf. below IV B 2, 2.Google Scholar
  332. 2.
    Cf. Schiaparelli, Scritti II, p. 23–42; [ 1877 ], p. 120–136.Google Scholar
  333. 4.
    Simplicius, Comm., ed. Heiberg, p.497, 17–22; Schiaparelli, Scritti II, p. 100/101; [1877], p. 187.Google Scholar
  334. 5.
    Assuming a maximum equation of about 2° one obtains according to p.680f. (2), (4), and (6) with γ= 2° a negligible additional latitude (βmax ≈ 0;1° from r ≈ 0;0,1) and ≈ 0;2o/d as a maximum change of velocity, which is a reasonable amount.Google Scholar
  335. 6.
    Simplicius, Comm., p. 503, 11; Schiaparelli, Scritti II, p. 107; [ 1877 ], p. 193.Google Scholar
  336. 7.
    Written around 190 B.C. but based on an earlier version, possibly of the period around 300 B.C.; cf. below IV C 1, 3 A.Google Scholar
  337. 8.
    Tannery, HAA, p. 294, No. 55; text; Blass, p. 25; Not. et Extr. 18, 2, p. 74f.Google Scholar
  338. 9.
    Cf. Schiaparelli, Scritti II, p. 83; [1877], p. 170. From the dates and intervals given in the “Geminus” parapegma (cf. above p. 581) one finds, however, for Callippus s1=92, s2=s3=89, s4 = 95 (cf. below p.1352, Fig.4). The explicit statement in the papyrus seems to me the more reliable source.Google Scholar
  339. 10.
    Tannery (Mém. Sci. 2, p. 236–247) detected in Eudoxus’ parapegma a similar symmetry for the fixed star phases. Cf. also Boeckh, Sonnenkr., p. 110 f. and KI. Schr. 3, p. 343–345. In Ethiopic astronomy, which in many ways depends on hellenistic prototypes, we find a schematic year of 364 days, divided into four seasons of 91 days each.Google Scholar
  340. 11.
    Rehm (RE Par. col. 1343, 48; also Parap., p. 39) calls it a “revolutionäre Tat” to ignore the earlier observations of Euctemon in order to obtain “harmony” as visualized by Plato.Google Scholar
  341. 12.
    The Eudoxus Papyrus, e.g., accepts a strictly symmetric (linear) scheme for the length of daylight simultaneously with the recognition of seasons of unequal length (cf. below p.706).Google Scholar
  342. 13.
    Rehm [1913], p. 9. The Eudoxus Papyrus gives only s1 =s2=90 days, s3 =92 days.Google Scholar
  343. 17.
    It also should be noted that the 32nd day in Taurus is actually attested in the text (Geminus, ed. Manitius, p. 232, 7).Google Scholar
  344. 1.
    For our evidence for this definition of the obliquity of the ecliptic cf. below p. 733 f.Google Scholar
  345. 2.
    Simplicius, Comm., p. 493, 15–17 ed. Heiberg (Schiaparelli, Scritti II, p. 96; [1877], p. 182); Hipparchus Ar. Comm. I, IX (p. 88, 14–22 ed. Manitius) in criticizing the opinions of Eudoxus and Attalus. For the importance of the ortive amplitudes cf. above I A 4, 4 and below p. 977 f.Google Scholar
  346. 3.
    Hipparchus I.c. p. 88, 21 f.; cf. also above p. 278 and below p. 807.Google Scholar
  347. 4.
    Theon of Smyrna (about A.D. 130, mainly based on Adrastus, about a generation earlier) Expositio..., Astron., Chap. 38 (ed. Hiller, p. 194, 4–8; Martin, p. 314/315; Dupuis, p. 212/213); also Chap. 12 (ed. Hiller, p. 135, 12–14; Martin, p. 174/175; Dupuis, p. 222, 10f./223, 8f.). For Adrastus cf. Theon, ed. Hiller, index p.213.Google Scholar
  348. 5.
    Pliny NH II 67 (Loeb I, p. 214/215; Budé II, p. 29).Google Scholar
  349. 6.
    Chalcidius 88, ed. Wrobel, p. 159, 10–12; cf. also Chap. 70, ed. Wrobel, p. 137, 12f.Google Scholar
  350. 7.
    Martianus Capella, De nuptiis VIII 867 (ed. Dick, p. 457, 2–5 ).Google Scholar
  351. 10.
    Cf., e.g., the collection given in Lattin [1947], p. 215, no. 87.Google Scholar
  352. 11.
    In Chap. 27 (ed. Hiller, p. 172, 15–173, 16; Martin, p. 258/263; Dupuis, p. 278/281); also Schiaparelli, Scritti H, p. 30f.; [1877], p. 126f.Google Scholar
  353. 12.
    Some manuscripts seem to give 365 1/6 but the emendation 365 1/2 is secured by the additional remark that the return occurs every two years at the same hour.Google Scholar
  354. 13.
    Schiaparelli, Scritti II, p. 26; [1877], p. 122. Cf. above p. 625, note 8 for the erroneous interchange of the roles of the second and the innermost sphere.Google Scholar
  355. 14.
    Schiaparelli, Scritti II, p. 33 ([1877], p. 128) suggested a connection between Theon’s latitude theory and the octaeteris. But a luni-solar intercalation cycle has nothing to do with the solar latitude; furthermore one should not separate in (1) the motion of the nodes from the motion of the apogee which is part of the same speculative doctrine.Google Scholar
  356. 3.
    This Commentary was written in the second half of the 4th century A.D.; cf. below p.966f. The historical interest of the section in question was first recognized by Delambre (1817; HAA II, p. 625627); the text was edited by Halma (1822; HT I, p. 53) with a French translation. Nallino (1903; Batt. I, p. 298) gave a Latin translation; Dreyer (1906; Plan. Syst., p. 204) translated it into English. Duhem (1914; SM II, p. 194) again into French. For an Arabic version, found in the Picatrix (second half of the 11th century) cf. the German translation by Plessner-Ritter, p. 82 (1962).Google Scholar
  357. 4.
    Battānī substituted the name of Ptolemy for Theon (Nallino, Batt. I, p. 126, Chap. 52). Also Bīrūnī in his Chronology (written A.D. 1000) refers to a work by Ptolemy “On the spherical art” (trsl. Sachau, p. 322). This is apparently the same work which Sā‘id al-Andalusī in his Tabagāt al-umam (written 1068) ascribes to Theon (trsl. Blachère, p. 86). Birúni in his Astrology (written 1029) properly reverts to Theon’s authorship (trsl. Wright, p. 101, No. 191).Google Scholar
  358. 5.
    The title shown in Halma’s edition is Περί τροπής and is indeed found in Par. gr. 2399 fol. 12v. Par. gr. 2400 fol. 13’, 2423 fol. 140v, and Vat. gr. 208 fol. 80v, however, all have the plural. There is no basis for Halma’s new technical term “De la conversion” which has been repeated everywhere in the modern literature.Google Scholar
  359. 6.
    Vat. gr. 1059 fol. 112r II. All these texts are unpublished, excepting Halma’s Par. gr. 2399.Google Scholar
  360. 7.
    Παλαιοί τών άποτελεσματικων.Google Scholar
  361. 9.
    Delambre gives here the specific number of 77 years since Diocletian and Dreyer includes it in his translation. Neither Halma’s printed text nor any of the unpublished MSS known to me has such a number.Google Scholar
  362. 11.
    Cf. above IV A 4, 2 A and p. 600. Apparently an explanation of this kind was also in Birūnī’s mind (Chronology, trsl. Sachau, p. 322, 15–25; cf. also his Astrology, trsl. Wright, p. 101, No. 191).Google Scholar
  363. 12.
    Cf. above p.276 and Table 28 there. Excepting the summer solstice of −431 June 27 which is the traditional epoch date for the Metonic cycle (cf. above p. 622) and Aristarchus’ summer solstice of −279 reported by Hipparchus (cf. below p. 634), the above listed equinoxes are the earliest ones mentioned in the Almagest.Google Scholar
  364. 14.
    III 54 (Manitius, p. 66/69); expressly mentioned (with an amplitude of 8°) in scholion 316 (Manitius, p. 275). Manitius misinterpreted Proclus (p. 287, note 7) when he related this remark to the Eudoxan hypothesis of a solar latitude (for which see above IV B 2, 2), perhaps misled by Schiaparelli who also connected the theory of trepidation with the theory of solar latitude. Schiaparelli remarked correctly (Scritti II, p. 31/32; [1877], p. 127) that a vibration of the equinoxes (i.e. of the intersections of the solar orbital plane with the equator) is a necessary consequence of the assumption of a rotation of the nodes of the solar orbit (i.e. of the intersections with the ecliptic). Assuming an inclination of 1/2° between orbital plane and ecliptic (cf. above IV B 2, 2) and ε =24° one finds that the true equinoxes deviate from the mean at most about 1;15° with a period of 2922 years (because of (1), p. 630). Obviously these parameters cannot explain Theon’s data, even if it were permissible to combine the relations (1) which imply the existence of a solar anomaly with a homocentric model which excludes anomaly.Google Scholar
  365. 17.
    Brahe, Opera II, p.255f. and Dreyer’s remarks in Opera I, p. XLVI f.Google Scholar
  366. 3.
    Who found it convenient to reduce the distance of the sun to 1179 earth radii (cf. Neugebauer [1968, 2], p. 101).Google Scholar
  367. 4.
    His table of parallaxes, Progymn. I 80 (Opera II, p. 65) gives 0;3° for the horizontal parallax of the sun.Google Scholar
  368. 5.
    Epitome Astron. Cop. IV, 1 (Werke 7, p. 279 ).Google Scholar
  369. 6.
    Flamsteed and Cassini (cf. Houzeau, Vadem., p. 405).Google Scholar
  370. 4.
    Heath, Arist., p. 352–411. Tannery, Mém. Sci. I, p. 371–396 reached conclusions in many respects similar to the following arguments.Google Scholar
  371. 5.
    Cf., e.g., Heath, Arist., p. 412–414 and below p.640.Google Scholar
  372. 6.
    Vitruvius, Archit. IX, 2, ed. Krohn, p. 208, 1–25; Loeb II, p. 228/231; Budé, p. 17f. and notes p. 124–130.Google Scholar
  373. 2.
    For the actual formulation of (1) cf. above p.590.Google Scholar
  374. 3.
    In the text (3) is expressed in the form that the apparent diameter of the moon is 1/15 of a zodiacal sign.Google Scholar
  375. 4.
    Cf below p.643. Surprisingly the enumeration of results at the beginning mentions only the boundaries for the ratios Rs/Rm and ds/dm (Propos. 7 and 9) and for de/dm (Propos. 15) but not for de/dm (Propos. 17).Google Scholar
  376. 2.
    It is not unusual in ancient drawings to find that different cases are superimposed in the same figure; cf., e.g., the diagrams for stereographic projection (below V B 3, 2).Google Scholar
  377. 3.
    Pappus, Coll. VI, ed. Hultsch, p. 560, 11–568, 11; trsl. Ver Eecke II, p. 430–435. The earliest extant manuscript of Aristarchus’ treatise is Vat. gr. 204 of the 10th cent.; cf. Heath, Arist., p. 325.Google Scholar
  378. 1.
    A clear analysis is given, e.g., in Dijksterhuis, Archimedes, p. 370–373.Google Scholar
  379. 5.
    Cf. for details below IV B 3, 3 C, but also below p. 664 (11) and (12).Google Scholar
  380. 6.
    Thus 1°≈833 stades. Aristotle, De caelo II, 14 (Loeb, p.254/255) mentions an estimate of ce=400000 stades (thus 1°≈1100 st.).Google Scholar
  381. 7.
    Lejeune [1947] is undoubtedly right when he renders őψις by the technical term “pupil” and not by the indefinite “eye” (cf. in particular p. 37, n. 3). The same conclusion had been reached by F. Schmidt, Instrum., p. 330 (1935).Google Scholar
  382. 2.
    Hippolytus, Heres. IX 12 gives a vivid picture of the fierce strife in the christian community of Rome at the end of the second century A.D.Google Scholar
  383. 3.
    He was, after all, a contemporary of Clement of Alexandria, Tertullian, Origenes, and other educated theologians (and therefore prone to fall into heresies).Google Scholar
  384. 4.
    Heresies IV, 8–11, ed. Wendland, p.41–44; trsl. Preysing, p. 51–54.Google Scholar
  385. 5.
    Ridiculed by Hippolytus as being useless for the true faith. It is illuminating to compare this position with the attitude toward astronomy expressed in the contemporary epigram conventionally ascribed to Ptolemy (cf. below p. 835).Google Scholar
  386. 6.
    Tannery, Mém. Sci. I, p. 393, but without following up the consequences of his observation.Google Scholar
  387. 7.
    I am using Wendland’s edition which is preferable to Heiberg’s excerpts in Archimedes, Opera II, p. 552–554.Google Scholar
  388. 19.
    The distance from the earth to Saturn thus becomes 54α=299383020 stades; cf. above (16) and (18).Google Scholar
  389. a Macrobius, Comm. II, 3 (ed. Willis II, p. 106; trsl. Stahl, p. 196).Google Scholar
  390. 21.
    Perhaps 1680000 stades; cf. Tannery, Mém. Sci. I, p. 394, note * and Wendland’s apparatus to p. 41, 13.Google Scholar
  391. 1.
    For a detailed discussion of these dates cf. Marie Laffranque, Posidonios d’Apamée (Paris 1964 ), p. 99–108.Google Scholar
  392. 2.
    Of greatest influence were the books by K. Reinhardt, Poseidonios (1921) and Kosmos and Sympathie (1926), summarized in his article in RE 22,1 col. 558–826 (1953). A useful discussion of the sources is found in Gronau, Poseid. (1914). For an excellent study of Posidonius’ personality and influence see A. D. Nock, Posidonius, J. Roman Studies 49 (1959), p. 1–15.Google Scholar
  393. 4.
    Gronau, Poseid., passim, in particular Chap. V and Reinhardt, Pos. p. 185. Cf. also the final (spurious) remarks in Cleomedes (Ziegler, p. 228, 1–5): “The preceding teachings are not the author’s own opinion but collected from older or more recent summaries; much of it is taken from Posidonius.”Google Scholar
  394. 5.
    Cf. above IV A 3, 2. Some scholars tried to make Geminus a pupil of Posidonius. Even if it were not chronologically excluded (above p. 580) I see nothing that supports such a conjecture (cf. above p. 578, also Reinhardt, Pos. p. 178).Google Scholar
  395. 6.
    Cicero, De natura deorum II 34/35, 88 (Loeb, p. 206–209).Google Scholar
  396. 7.
    I do not see how the daily (mean) motions of sun and moon can be combined with the planetary retrogradations (even ignoring latitudes) in one spherical model. D. Price [ 1974 ], p. 57f. tries to explain Archimedes’ planetarium by means of gearings of a type he had discovered in the Antikythera mechanism that represents lunar motions. But even these intricate devices cannot produce more than the mean motions of the outer planets. Hence the most characteristic features of planetary motions, stations and retrogradations, are omitted and the inner planets must be ignored altogether.Google Scholar
  397. 2.
    Cf. for Canopus (modern: α Carinae) above p. 576, n. 3.Google Scholar
  398. 3.
    Cf. below p. 671. The same data in Pliny, NH II, 178 (Budé II, p. 78 ).Google Scholar
  399. 4.
    Letronne (Oeuvres choisies, ser. 2, Vol. 1, p. 263) argues against the authenticity of the whole story but seems to have found no followers.Google Scholar
  400. 5.
    Actually Rhodes is about 4;50° north, 1;50° west of Alexandria, Syene about 7;10° south, 3° east (cf. Fig. 17).Google Scholar
  401. 6.
    Actually for Rhodes 90−φ=54° while the declination of Canopus is about −52;30°. Hence the star culminates at Rhodes at an altitude of about 1 1/2° which leads to a visibility of about 2 1/2b (Drabkin [1943], p.510, n.5).Google Scholar
  402. 7.
    Strabo, Geogr. II 5, 7 (Loeb I, p. 436/437); also Heron, Dioptra 35 (Opera III, p. 302, 12–17); Geminus (Manitius, p. 166, 2), etc.Google Scholar
  403. 9.
    Strabo, Geogr. II 5, 24 (Loeb I, p. 482/483). Cf. also Pliny NH V, 132 (Jan-Mayhoff I, p. 418, 2) where he ascribes to Eratosthenes the estimate of 469 miles, i.e. ≈ 3750 st.Google Scholar
  404. 10.
    It is, of course, nonsense when Strabo says that this distance was determined by means of sun dials; these instruments can only furnish angles, never absolute distances.Google Scholar
  405. 11.
    Strabo, Geogr. II 2, 2 (Loeb I, p. 364/365). The estimate (6) is also mentioned by St. Basil (Hexaemeron IX, Sources Chrétiennes, p. 483) who correctly observes that Moses was not concerned with the shape or the size of the earth, a fact held against the sciences.Google Scholar
  406. 12.
    Cf. below (9). Perhaps an obscure remark by Pliny (NH II 247, Budé II, p. 111, p. 266) can be taken as an indication that Hipparchus also did not accept (4) unreservedly: he is said to have added “a little less than 26000 stades” to the 252000. [Note: ce=252000+25500=277500st. leads with the Babylonian approximation π ≈ 3;7,30 to exactly re = 44 400 st.].Google Scholar
  407. 14.
    Jan-Mayhoff I, p. 227 ad line 3. The other variants (584, 583, 588, 573 miles) make no sense (NH V, 132 I.c. p. 418, 2 and var.).Google Scholar
  408. 1.
    Cf. above p. 635 (4) and below p. 667.Google Scholar
  409. 2.
    Cleomedes II, 1 (Ziegler, p. 146, 18–25); cf. also II, 3 (Ziegler, p. 178, 12).Google Scholar
  410. 3.
    Cf. also Fig. 15 in the Eudoxus Papyrus (below p. 1453, Pl. VII col. X II ).Google Scholar
  411. 5.
    Cf. above p.650(35). Note that Archimedes made Rm=5544130 stades (above p.649(2)), using obviously the same type of stades as Apollonius and Posidonius.Google Scholar
  412. 7.
    De natura deorum II, 103 (Loeb, p. 221).Google Scholar
  413. 1.
    The famous paper by Hultsch [1897] on “Poseidonius über die Grösse und Entfernung der Sonne” is a collection of implausible hypotheses which are not worth discussing.Google Scholar
  414. 2.
    Cleomedes II, 1 (Ziegler, p. 144, 22–146, 16).Google Scholar
  415. 3.
    Also Ziegler, p. 140, 7–9; cf. also below p. 726: n. 14. Cf. also Pliny, N.H. II, 182 (Budé Il, p. 80).Google Scholar
  416. 5.
    Pliny, NH II, 21 (Budé II, p. 37 and p. 172–175).Google Scholar
  417. 6.
    Actually only from the upper limit of the atmosphere (the region of winds and clouds) which adds 40 stades to the radius of the earth.Google Scholar
  418. 3.
    Distances expressed in lunar diameters are mentioned, e.g., in the Almagest (IX, 7 and IX, 10 in observations from the Era Dionysius (−264 and −261), in X, 1 from Ptolemy and Theon, A.D. 140 and 127). Also P. Lond. 130 for A.D. 81 (Neugebauer-Van Hoesen, Gr. Hor., p. 26).Google Scholar
  419. 4.
    E.g. Aristarchus, according to Archimedes, Sandreckoner (Opera II, p. 222, 6–8); cf. also above IV B 3, I E and p. 592.Google Scholar
  420. 7.
    In the second part of P. Oslo 73 or Proclus, Hypotyp. IV, 73–75; etc.Google Scholar
  421. 8.
    Alm. V, 14 (Manitius I, p. 305); Tetrabiblos II, 2 (Boer, p. 110/111; Loeb, p. 231); also Proclus, Hypotyposis IV, 80–86.Google Scholar
  422. 9.
    Diels; VS(5), p. 68, 6–8; cf. also Heath, Aristarchus, p. 21/22.Google Scholar
  423. 14.
    De facie 935 D, Loeb, Moralia XII, p. 142/143.Google Scholar
  424. 15.
    Simplicius (≈ A.D. 530) in his Commentary to Aristotles’ De caelo, ed. Heiberg, p. 504, 25–505,19; French transi. Duhem SM I, p.401. Cf. also Proclus, Hypot. IV 98f. (Manitius, p. 130/131). In the same context we read that Polemarchus (the contemporary of Eudoxus and Callippus, cf. below p. 676) considered the variation of the apparent lunar diameter negligible and ignored it on purpose because he preferred the theory of homocentric spheres (Heiberg l.c. p. 505, 21–23; Duhem I.c. p. 402).Google Scholar
  425. 16.
    Cf. Alm. VI, 7 (Manitius I, p. 374, 28; 376, 31–377, 1); same in Cleomedes II, 3 (Ziegler, p. 172, 25 ).Google Scholar
  426. 17.
    Cleomedes II, 3 (ed. Ziegler, p. 172, 22–27).Google Scholar
  427. 18.
    Comm. Ar., Manitius, p. 90, 10.Google Scholar
  428. 19.
    Archimedes, Sandreckoner; cf. Lejeune [1947] (above p. 647).Google Scholar
  429. 20.
    Cf. above p. 644 (1). I do not understand a sentence in Plutarch (Moralia, Loeb XIV, p. 64/65) in which he ascribes to Archimedes the discovery of a certain ratio of the solar diameter to the circumference.Google Scholar
  430. 21.
    Alm. IV, 9 (Manitius I, p. 237, 8); also Pappus, Coll. VI, 37 (Hultsch, p. 556, 14ff., trsl. Heath, Aristarchus, p. 413).Google Scholar
  431. 22.
    De nuptiis VIII, 860 (ed. Dick, p. 452, 13 f.); cf. also below p. 664.Google Scholar
  432. 2.
    De animae procr., Moralia 1028B (ed. Hubert, p. 183, 17–24 ).Google Scholar
  433. 4.
    The same intellectual level is still present in Hegel’s “Dissertatio philosophica de Orbitis Pianetarum,” accepted (1801) “pro licentia docendi” at the University of Jena (Hegel, Sämtliche Werke I, Stuttgart 1927, p. 1–29; German transi.: Philos. Bibl., Leipzig 1928, p. 347–402).Google Scholar
  434. 6.
    Pliny, NH II 83 (Budé II, p. 36, p. 172–175); also Plutarch, De facie, Loeb XII, p. 75.Google Scholar
  435. 7.
    He has it from Sulpicius Gallus (≈160 B.C. — cf. below p. 666, n. 8). For the subsequent speculation about harmonies cf., e.g., van der Waerden, RE Suppl. 10 col. 857–859.Google Scholar
  436. 9.
    Cf. the variants in Lydus, De mensibus (ed. Wuensch, p. 54, 7–10) and Diels, Dox., p. 362, 25–363, 4. Tannery (Mém. Sci. I, p. 391 f.) suggested drastic changes of all numbers in order to obtain reasonable results.Google Scholar
  437. 10.
    Exactly the ratio 6 would require Rm=680000 st.Google Scholar
  438. 17.
    This is the meaning of the first sentence on p. 44 (misinterpreted in note (a) as referring to the lunar eccentricity); cf. for the terminology, e.g., Archimedes Opera II, p. 218, 18.Google Scholar
  439. 18.
    Cf. above I E 5, 4: even the mean distances are greater: 67;20 re or 77 re; the maximum is 83 e.Google Scholar
  440. 21.
    Book I, 20 (ed. Eyssenhardt, p. 564–570; trsl. Stahl, p. 168–174).Google Scholar
  441. 22.
    Cf. the corresponding waterclock “observation” above p. 658. The coefficient c/216 is mentioned once before in Book I, 16 (Eyssenhardt, p. 550, 29–32; trsl. Stahl, p. 154).Google Scholar
  442. 3.
    Cf. below IV C 1, 3 B the “Eudoxus Papyrus”, Tannery HAA, p. 290, No. 33.Google Scholar
  443. 5.
    Cleomedes II, 4 (Ziegler, p. 190, 4); a similar theory also in the Āryabhatīya IV, 47 (trsl. Clark, p. 81).Google Scholar
  444. 6.
    Sandreckoner, Archimedes Opera II, p. 220, 20f. (trsl. Ver Eecke, p. 356); also Lasserre, p. 17, D 13.Google Scholar
  445. 11.
    The same argument, in a slightly improved form (sidereal rotations instead of synodic) is also found with Posidonius (cf. above p. 656). Cf. for this whole type of arguing Aristotle, De caelo II, 10 (Loeb, p. 196/199).Google Scholar
  446. 12.
    Macrobius, Comm. XX, 9, Eyssenhardt, p. 565, 25f.; trsl. Stahl, p. 170.Google Scholar
  447. 13.
    Cramer, Anecd. Gr. I, p. 373, 27–30; the author is not Dionysius as assumed by Cramer (cf. CCAG 8, 3, p. 10, F. 103; CCAG 8, 4, p. 5, F. 192v; CCAG 7, p. 45, F. 75v).Google Scholar
  448. 14.
    Also Joannes Damascenus (x 700), Expositio fidei 21 (ed. Kotter, Patristische Texte u. Stud. 12, 1973, p. 60, 164= Migne PG 94, 895C) quotes “the Holy Fathers” for the same opinion.Google Scholar
  449. 15.
    The same numbers also in Proclus, Hypot. IV, 101 (Manitius, p. 132/133).Google Scholar
  450. 17.
    Diets, Dox. p. 63 n. 2 = Maass, Comm. Ar. rel., p. 445, 18–22.Google Scholar
  451. 18.
    Isidore, Nat. rer., ed. Fontaine, p. 333, 31. Isidore himself talks only very cautiously about the sizes of the luminaries (Etym. III, 47, 48; Nat. rer. XVI).Google Scholar
  452. 23.
    De nuptiis VIII 859 (Dick, p. 452); cf. also below p. 668 and p. 964.Google Scholar
  453. 4.
    P. Carlsberg 31 (probably second century A.D.); cf. Neugebauer-Parker, EAT III, p. 241–243, Pl. 79 A (not B). The character of this text was not understood in the edition because the scribe had consistently replaced “month” by “year”. The proper explanation was found by A. Aaboe [ 1972 ], p. 111f.Google Scholar
  454. 6.
    Heron, Dioptra 35; Opera III, p. 302/303, Sect. 22; cf. also Rome [1931, 1].Google Scholar
  455. 7.
    Plutarch, De facie (Loeb, Moralia XII, p. 131) citing parameters also known from Babylonian astronomy; cf. above p. 321.Google Scholar
  456. 8.
    E.g. the “prediction” by Helicon of the (annular) eclipse of — 360 May 12 (Ginzel, Kanon, p. 183, No. 15) or of the lunar eclipse of —167 June 21 by Sulpicius Gallus (Pliny, NH II 53; Budé II, p. 23 f. and p. 142f.; Ginzel, Kanon, p. 190ff., No. 27).Google Scholar
  457. 9.
    Dio Cassius, Roman History (completed about A.D. 230) LX, 26 (Loeb VII, p. 432–435).Google Scholar
  458. 12.
    Cicero, Dedivinatione II 6, 17 (Loeb, p. 388/389), written 44 B.C.Google Scholar
  459. 13.
    Quaest. nat. VII, I11, 3 (written between A.D. 62 and 65); cf also above p. 572, n. 4. Tannery, Mém. Sci. 111, p. 353 substituted “Chaldeans” for the implausible “Egyptians”.Google Scholar
  460. 14.
    Catullus, Carmina 66, 3 (around 50 B.C.).Google Scholar
  461. 16.
    Maass, Comm. Ar. rel., p. 47, 13f.; cf. also above p.321.Google Scholar
  462. 19.
    Stobaeus, ed. Wachsmuth, p. 221, 6; cf. also above p. 574.Google Scholar
  463. 20.
    Diogenes Laertius VII 146 (Loeb II, p. 248/251).Google Scholar
  464. 22.
    Hipparchus, Ar. Comm., p. 90, 10–12, ed. Manitius. Note that he says nothing about solar eclipses.Google Scholar
  465. 23.
    Cf. above p. 635 f. and below p. 689; also Cleomedes (from Posidonius): above p. 654; cf. also below p. 963.Google Scholar
  466. 24.
    Plutarch, De facie 923 B (Loeb, Moralia XII, p. 57, noted) and De anim. procr. 1028 D (ed. Hubert, p. 184, 12 ).Google Scholar
  467. 25.
    Isagoge XI, 7 (Manitius, p. 134/135); cf. above p.593.Google Scholar
  468. 28.
    Strabo, Geogr. 11, 12 (Loeb I, p. 24/25).Google Scholar
  469. 30.
    Pliny, NH II, 180 (Loeb I, p. 312/313; Budé II. p. 79 and p. 234, n. 5) gives the 2nd hour of night for Arbela, moon-rise (i.e. sun-set) in Sicily. Ptolemy, however, in Geogr.I,4 (Nobbe, p. 11, 19f.; Miik, p. 21) reports the 5th hour of night for Arbela, the 2nd for Carthage. Cf. also Ginzel, Kanon, p. 184f. (No. 18) and P. V. Neugebauer, Kanon d. Mondf., p. 42. Cleomedes I, 8 (Ziegler, p. 76, 8–16) gives 45 as difference for eclipses seen in “Persia and Spain”.Google Scholar
  470. 31.
    All data are reduced to Arbela local time: for Syracuse Δt =1;55h, for Carthage 2;15h.Google Scholar
  471. 32.
    Pliny, NH II, 180; cf. Ginzel, Kanon, p. 201f. (No. 39).Google Scholar
  472. 33.
    Cf. above p. 327f. and for the identification of the eclipse p. 316, n. 9.Google Scholar
  473. 34.
    Martianus Capella, De nuptiis VII, p. 859 f. (Dick, p. 452 f.).Google Scholar
  474. 35.
    Cf. above p. 663f., and below p. 964.Google Scholar
  475. 36.
    Cf. below p. 688 and p. 689, n. 21.Google Scholar
  476. 38.
    Proclus, Hypot. I, 19 (Manitius, p. 10/11).Google Scholar
  477. 39.
    Proclus, Hypot. IV, 99 (Manitius, p. 130/131). For Ptolemy’s parameters cf. above p. 106.Google Scholar
  478. 40.
    Cleomedes II, 4 (Ziegler, p. 190, 17–24).Google Scholar
  479. 41.
    Simplicius, Comm. Arist., Vol. III, ed. Heiberg, p. 505, 1–9; trans]. Duhem, SM I, p. 401.Google Scholar
  480. 42.
    Above I B 6, 7; for later modifications cf. below p. 997.Google Scholar
  481. 43.
    Cf., e.g., Schaumberger, Erg., p. 246–249 or Parker, Vienna Pap.Google Scholar
  482. 6.
    E.g. Marc. gr. 325 fol. 21v, 1–14 (unpublished). Here the argument ω′=345;12° is described as “in the 6th step and about 1/60 [thus 0;15° for the 0;121, northerly and ascending” (cf. Fig. 21).Google Scholar
  483. 7.
    Cf., e.g., below p. 671, or P. Carlsberg 9 which starts its enumeration of zodiacal signs with Leo (Neugebauer-Parker, EAT III, p. 223 ).Google Scholar
  484. 9.
    Halma HT I, p. 144/145; cf. below p. 979, n. 3. The steps are not indicated, however, in the corresponding tables of Vat. gr. 1291 (fol. 44r, 45v, 46r).Google Scholar
  485. 10.
    Similar commentaries, e.g. by Stephanus of Alexandria (: A.D. 620) demonstrate the continuity of the tradition (cf. Usener, K1. Schr. III, p. 296, Chap. 22, unpublished); cf. below p. 1049 ).Google Scholar
  486. 11.
    Nallino, Batt. I, p. 13, 22–26; II, p. 58, p. 221.Google Scholar
  487. 12.
    E.g. Vat. gr. 1059 fol. 112r (unpublished). A unique V-shaped diagram is found in Vat. gr. 1291 fol. 47v (originally the last page) but with incorrect legend (latitudes of Mercury instead of solar declinations; the MS was written about A.D. 820; cf. below p. 969f.).Google Scholar
  488. 13.
    Cf. above IV A 3, 3. Cf. also Ptolemy, Tetrab. II, 12 Loeb, p. 209 and p. 213; ed. Boll-Boer, p. 99, 3, p. 100, 8.Google Scholar
  489. 15.
    CCAG 7 p. 128, 12–24 (from Antiochus, ≈A.D.200); CCAG 5, 1, p. 198, 8f. (Anonymus of 379); CCAG 8, 1, p. 243 a, 12 f. and p. 243 b, 23f. (from Rhetorius); CCAG 9, 1, p. 180,19 (cod. 16, note 12); etc.Google Scholar
  490. 16.
    Cf. for the planetary latitudes in the Handy Tables below p. 1016. The customary names for the four quadrants of planetary latitudes are also found in Cleomedes 1,4 (Ziegler, p. 34, 23–36, 1) but no steps or winds.Google Scholar
  491. 19.
    Neugebauer-Van Hoesen, Gr. Hor., p. 154 (cf. also p. 188 f. concerning the question of authorship of Julianus of Laodicaea or Eutocius).Google Scholar
  492. 22.
    Also the use of χηλαί, i.e. “the Claws” (of Scorpio), for Libra conforms to an early date of Diogenes’ source. It is a funny coincidence that Yonge in 1895 (in Bohn’s Classical Library) mistook χηλαί for “Cancer” and that the same translation appears in the German translations of 1921, 1955, 1967, in the English translation (Loeb) of 1925 and 1950, in the French translation of 1933, and in the Italian translation of 1962. The older Latin translations give simply chelae.Google Scholar
  493. 23.
    Cf. above p. 652; also Geminus V, 25 (Manitius, p. 52, 4).Google Scholar
  494. 26.
    Both operate almost exclusively with fractions of whole quadrants, not of sections of a quadrant; cf. below p. 772f.Google Scholar
  495. 27.
    Steps of 15° are commonly used in Indian astronomy, e.g., just as in Greek astronomy, for solar declinations (Kh.-Kh. I 29. Sengupta, p. 31 for ε=24°); also for the equation of center of the sun (Kh: Kh. 116, Sengupta, p.19 for 2;14° as maximum equation). For the tables of sines cf., e.g., Āryabhatiya 110 (≈A.D. 500) or the “modern” Sūrya-Siddhānta II 15–17 (12th cent.). In the Middle Ages in Europe these tables are known as “kardaga”; cf., e.g., Millas-Vallicrosa, Est. Azar., p. 44 (in six steps to the quadrant) and Goldstein, Ibn al-Muthann5, p. 196/197. The same in Kh.-Kh. I 30 or IX 8 (Sengupta, p. 32 and p. 142) for R= 150.Google Scholar
  496. 30.
    Fol. 47v; cf. below V C 4, 1 D 2, p. 978.Google Scholar
  497. 31.
    Terminology: βορρά/νοτία/άνάβασις/κατάβασις. Cf. also Tihon [1973]. p. 103.Google Scholar
  498. 32.
    In the part published by A. Tihon [1973]; I am using here Marc. gr. 314 fol. 222r (unpublished).Google Scholar
  499. 33.
    I correct trivial scribal errors.Google Scholar
  500. 34.
    Marc. gr. 314 fol. 218v, et al. (cf. Tihon [1973], p. 52).Google Scholar
  501. 36.
    Cf. above p. 358 and below p. 1069.Google Scholar
  502. 37.
    Alm. VI, 8 for m = 12 digits.Google Scholar
  503. 38.
    Marc. gr. 314 fol. 222r, 222v, 223v; also Tihon [1973], p. 70, note 1 (counting from the solar apogee).Google Scholar
  504. 2.
    For references cf. above p. 573. It also has been noted above (p. 599, n. 10) that Rehm tried to distinguish two astronomers named Eudoxus.Google Scholar
  505. 2.
    Berlin 1966; cf. also the review Toomer [1968].Google Scholar
  506. 4.
    E.g. Hipparchus, Ar. Comm., Manitius, p. 9.Google Scholar
  507. 5.
    Collected in Lasserre as Fragm. No. 146–267. Cf. also above p. 588 and below p.929.Google Scholar
  508. 7.
    Mentioned by Diogenes Laertius VIII 89 (Lasserre, Fragm. T 7, p. 5, 17–20; Loeb, Diog. L. II, p. 402/403). The title has inspired a long sequence of learned (often funny) conjectures.Google Scholar
  509. 8.
    This title is taken from a sentence by Simplicius (Comm. in Arist. De caelo, ed. Heiberg, p. 494, 12; Lasserre Fragm., p. 69, 10 ).Google Scholar
  510. 10.
    The often repeated stories about Eudoxus learning astronomy from Egyptian priests, or about his observatories in Egypt, are not worth refuting. Among the “observatories”, shown to Strabo by his guides three centuries later, is mentioned (Strabo, Geogr. XVII 1, 30; Leob VIII, p. 84/85) “Kerkesoura in Lybia” which is Kerkeosiris in the Faiyiim, near Tebtunis (cf. RE 11, 1 col. 291), some 120 km from Heliopolis.Google Scholar
  511. 11.
    Diogenes Laertius VIII 87 (Lasserre, Fragm. T 7; Loeb II, p. 402/403). This does not agree too well with the familiar story that it was at Plato’s suggestion that Eudoxus undertook to explain planetary motions by means of uniform rotations (Simplicius, Comm., ed. Heiberg, p. 488, 18–24; p. 492, 31–493, 5; also Schiaparelli, Scritti II, p.95f.; [1877], p. 182).Google Scholar
  512. 12.
    What we know about the school of Eudoxus can all be found in Böckh, Sonnenkr., p. 150–159.Google Scholar
  513. 14.
    Theon Smyrn., ed. Hiller, p. 201, 25; Martin, p. 332 and p. 58f.; Dupuis, p. 326/327.Google Scholar
  514. 16.
    Cf. above p. 623; also Heath, Aristarchus, p. 212.Google Scholar
  515. 18.
    Cf., e.g., Heath, Euclid I, p. 137; II, p. 112; II, p. 365 (et passim).Google Scholar
  516. 1.
    Greek text: Metaphysics Λ(=XI), 8 (Opera, ed. Bekker, II, p. 1073b, 17–1074 a, 14). Italian trsl.: Schiaparelli, Scritti II, p.94f. (German: [1877], p. 180f.); English trsl.: Heath, Aristarchus, p. 194f.; p. 212; p. 217.Google Scholar
  517. 2.
    Greek text: Simplicius. Comm. in Arist. De caelo, ed. Heiberg, p. 493–507. Italian trsl.: Schiaparelli, Scritti II, p. 95–112 (German: [1877], p. 182–198); English trsl. of the major passages: Heath, Aristarchus, p. 201 f.; p. 213; p. 221–223. Greek text and German trsl. of what is considered to be an Eudoxan “fragment” proper: Lasserre, p. 67–74.Google Scholar
  518. 3.
    For the corresponding theory for sun and moon cf. above p. 624 f. and p. 627.Google Scholar
  519. 4.
    Schiaparelli, Scritti II, p. 3–112; German translation (with some changes): Schiaparelli [ 1877 ]. For earlier work on the homocentric spheres cf. Heath, Aristarchus, p. 194.Google Scholar
  520. 5.
    For the modern discussion of the mathematical properties of the hippopede cf. Schiaparelli, Scritti II, p. 53–55 ([1877], p. 145–147) notes. Also G. Loria, Curve sghembe speciali algebriche e transcendenti (Bologna 1925 ), Vol. I, p. 199–201.Google Scholar
  521. 6.
    Because of obvious symmetries we may restrict α to the first quadrant.Google Scholar
  522. 7.
    It is not necessary to know that the curve AC is an ellipse. In order to determine the position of Pl we first consider in the horizontal plane a rotation by the angle a from A to P0. Then we tilt this plane about OA until the axis OX reaches the position OΞ (cf. Fig. 25; inclinations in these figures are not drawn to scale) and B comes to B1. As a result B1 is projected into C and P0 moves into P1′ such that P0P1′G is perpendicular to OA.Google Scholar
  523. 9.
    According to Diogenes Laertius VIII 86 (Tannery HAA, p. 295; Loeb, Diog. L. II, p. 400/401) on the authority of Callimachus, librarian of the Museum in Alexandria in the third century B.C.Google Scholar
  524. 11.
    Cf. the examples shown in Figs. 228, 233; in Fig. 35; in V C 4, 5 B 2, Figs. 115 and 116.Google Scholar
  525. 12.
    Simplicius, Comm., ed. Heiberg, p. 497, 4; Schiaparelli, Scritti II, p. 100, [1877], p. 186.Google Scholar
  526. 14.
    Simplicius, Comm., ed. Heiberg, p. 497, 5; cf. above note 12. Also Heath, Aristarchus, p. 202.Google Scholar
  527. 15.
    Schiaparelli, Scritti II, p. 70ff., p. 74ff.; [1877], p. 159ff.; cf. below p. 683.Google Scholar
  528. 2.
    Simplicius, Comm., ed. Heiberg, p. 496, 6–9; Schiaparelli, Scritti II, p. 99; [1877], p. 185.Google Scholar
  529. 3.
    Lasserre in his translation (p. 72) gives 103 days by mistake.Google Scholar
  530. 4.
    Examples for this reckoning are collected in Heath, Aristarchus, p. 285f.Google Scholar
  531. 5.
    Simplicius, p.495, 26–28; Schiaparelli II, p. 98/99; [1877], p. 185. The same parameters are also found, e.g., in the “Eudoxus Papyrus” (cf. below p. 688).Google Scholar
  532. 6.
    Cf. e.g., the tables in Alm. IX, 4.Google Scholar
  533. 7.
    Cf. for Ptolemy’s estimates at mean distance above p. 193. These values also agree with averages obtainable from Babylonian schemes (cf. ACT II, p. 315, p. 312, p. 305, respectively).Google Scholar
  534. 8.
    Even crude observations of Mars during one or two decades should have given a greater value, e.g. Δλ≈6,50°, hence about γ≈1,50° in (15).Google Scholar
  535. 9.
    This was stated, for Mars and Venus, from the very beginning by Schiaparelli (Scritti II, p. 70–72; [1877], p.159–161).Google Scholar
  536. 10.
    Schiaparelli. Scritti II, p. 73f.; [1877], p. 162. Similar curves are also shown in Hargreave [1970], p. 342–345 but the numerical discussions are without interest for the historical problem.Google Scholar
  537. 1.
    Simplicius had it from Sosigenes (2nd cent. A.D., the teacher of Alexander of Aphrodisias, not Caesar’s contemporary) who had it from Eudemus’ “History of Astronomy” (4th cent. B.C.; cf. Simplicius, Comm. ed. Heiberg, p.488, 18–21; Lasserre Fragm., p.67 (F. 121); Schiaparelli, Scritti II, p.95; [1877], p. 182). For the complicated relations between these sources (including Aristotle) cf. Schramm, Haitham. p. 32–63. The difficulties are increased by a gap in the text of Simplicius (cf. Schiaparelli, Scritti II, p. 101; [1877], p. 187).Google Scholar
  538. 2.
    Cf. for sun and moon above p. 625 and p. 627, respectively.Google Scholar
  539. 3.
    Schiaparelli, Scritti II, p. 79–82; [1877], p. 167–169. For his model he assumes the greatest permissible inclination between the two axes XX′ and ΞΞ′, i.e. γ = 90°. The planet P, however, is not located on the equator of the sphere with axis ΞΞ′ but on the equator of the new innermost sphere whose axis ZZ′ is inclined to ΞΞ′ and rotates about it with twice the angular velocity of XX′ with respect to ΞΞ′. The motion starts when XZΞP (in this order) are located on the same great circle (which then becomes the line of symmetry of the curve and represents the ecliptic).Google Scholar
  540. 7.
    Arist., Metaphys. Λ (Opera II, Bekker, p. 1073b, 38–1074a,14); also Simplicius, Comm. ed. Heiberg, p. 497f; Schiaparelli, Scritti II, p. 101–112; [1877], p. 187–198.Google Scholar
  541. 8.
    For some ancient difficulties with the count of spheres cf. Simplicius, p. 503,10–504,3; Schiaparelli. Scritti II, p. 107f.; [ 1877 ], p. 193f.Google Scholar
  542. 1.
    With this find is connected the famous story of the Arabs burning papyri for the pleasant smell of the smoke (Not. et Extr. 18, 2, p. 6, note 1).Google Scholar
  543. 2.
    The only color used is some reddish brown, indicated by shading in the edition, appearing dark on our photograph PI. VII, p. 1453.Google Scholar
  544. 3.
    Blass, p. 4f. (Nos. 27 and 50 in Tannery’s count). Suidas (ed. Adler, II, p. 445 = Lasserre T 8) says that Eudoxus had composed a poem called “Astronomy”.Google Scholar
  545. 4.
    A second hand wrote in the space between these lines “Work, you men, in order not to work” and “Oracles of Serapis” and “Oracles of Hermes”. The words “Oracles of Serapis” are also found at the end of the preceding column (XXIII) and inside the zodiac in col. XXIV.Google Scholar
  546. 8.
    Adopting Tannery’s sectioning of the text one has the following parallels: No. 1/No. 53; No.7/Nos. 34, 35, 40; No. 9/No. 41; Nos. 26, 27/Nos. 42, 50: No. 39/No. 44.Google Scholar
  547. 1.
    Tannery No. 27 and No. 50. We have seen that these sections belong to the earliest form of the treatise; cf. above p. 686, note 3.Google Scholar
  548. 2.
    Tannery (p. 286) replaces 3 by 7 months but the text has words, not alphabetic numerals; nor is the result, 579 days, attested elsewhere.Google Scholar
  549. 4.
    This parapegma does not agree with what has been reconstructed as Eudoxus’ parapegma; cf. Rehm, RE Par. col. 1322, 37–1323, 51.Google Scholar
  550. 8.
    The sun is said to remain in each sign 30 days and 5 hours, supposedly as a result of the division by 12 of 365. This implies that one “hour” (ωρα) is taken as 1/12 of one day. Similarly in No. 37 (Tannery HAA, p. 290; Lasserre F 128) 1/2 zodiacal sign is equated to 1/2 hour. Since Letronne (Journal des Savants 1839, p. 585–587; Blass, p. 8; Boll, Sphaera, p. 313, etc.) this has been considered as evidence for the use of the Babylonian “double-hours” (danna). To me this seems very implausible on general historical grounds. I would consider a simple arithmetical error of the redactor of our text much more likely; in particular, since a section from the original poetic version (No. 3, on the variation of the length of daylight) uses ωρα for the equinoctial hours.Google Scholar
  551. 9.
    For the moon 2 1/4 days per sign (Tannery No. 41 and No. 9).Google Scholar
  552. 12.
    Cf. Simplicius, Comm. p. 505, 3–23 ed. Heiberg; Schiaparelli, Scritti II, p. 109f., [1877], p. 195f. ‘3 For the problem of relative size and distances of the luminaries cf. above IV B 3.Google Scholar
  553. 15.
    This fits in with our observation of duplications in the contents (cf. above p. 687).Google Scholar
  554. 17.
    Cf. above p. 686. I have numbered all figures consecutively; some are still better preserved on the facsimile. Nos. 11 to 19 are shown on PI. VII; this plate is based on photographs which could not be exactly matched to the text in col. XIII.Google Scholar
  555. 18.
    As shown on p. 600 we have probably a reference here to the solar longitude in the month Thoth, using the Eudoxan norm for the zodiacal signs.Google Scholar
  556. 21.
    No. 49. Another representation of an annular eclipse (why 3 concentric circles?) is found in Fig. 26, col. XVIII.Google Scholar
  557. 22.
    Neither text nor figures refer to astrological doctrines.Google Scholar
  558. 1.
    Cf. Neugebauer-Parker, EAT III, p. 175ff. (cf. also the index).Google Scholar
  559. 3.
    Discovered by Kugler; cf. for (2): SSB I, p. 9. p. 11; for (3): p. 40. Conveniently listed in Boll [ 1913 ], p. 342.Google Scholar
  560. 4.
    Boll-Boer, p. 47, 17ff.; Robbins I, 21 (Loeb, p. 98–101).Google Scholar
  561. 5.
    We shall meet similar sequences, e.g., in the hellenistic shadow tables; cf. below p.739 (5).Google Scholar
  562. 6.
    Tetrabiblos, ed. Boll-Boer, p. 47 top; Loeb, p. 97.Google Scholar
  563. 7.
    Boll-Boer, p. 51/52. The list of the Loeb edition (p. 107) has to be modified on the basis of the apparatus in Boll-Boer.Google Scholar
  564. 8.
    Οί περί μετέωρα δεινοί (Achilles, Isag. 16; Maass, Comm. Ar. rel., p. 42, 25).Google Scholar
  565. 9.
    Planetary Hypotheses,” below V B 7, 3.Google Scholar
  566. 10.
    Cicero, De divin. II, 43 (Loeb, p. 474/5). In De nat. deorum II, 53 (Loeb, p. 174/5), however, he follows the order (6c), p. 692.Google Scholar
  567. 13.
    Plutarch, De animae procr. 31 (Moralia VI, 1, ed. Hubert, p. 183, 20–25 ).Google Scholar
  568. 15.
    Cf., e.g., Varâhamihira, Pc: Sk., Neugebauer-Pingree, Vol. II, p. 13f., Vol. I, p. 121 (XIII, 39); cf., however, Vol. II, p. 109. Two different derivations of the sequence (5) were given by Cassius Dio (≈A.D.200), one based on musical intervals, the second on the rulers of the hours (Roman History 37, 18 and 19; Loeb III, p. 128–131 ).Google Scholar
  569. 23.
    Stobaeus, Ecl., ed. Wachsmuth I, p. 185, 14–19.Google Scholar
  570. 23a.
    For references see W. Bousset, Jüdisch-Christlicher Schulbetrieb in Alexandria and Rom, p. 31 ff. (Göttingen 1915 ).Google Scholar
  571. 24.
    Theon Smyrn. XV, ed. Hiller, p. 142, 7ff.; Dupuis, p. 232/233. Similarly Chalcidius, Chap. 73, ed. Wrobel, p. 141, 2ff.Google Scholar
  572. 25.
    Achilles, Isag. 16 (Maass, Comm. Ar. rel., p. 42, 30–43, 2 ).Google Scholar
  573. 26.
    Maass, Comm. Ar. rel., p. 43, 28.Google Scholar
  574. 27.
    For the counting of positions in the sequence (4) beginning with Saturn, cf., e.g. Pseudo-Plutarch, De plac. II, 16 (Diels, Dox., p. 344, 17–345, 3); here (4) is ascribed to Plato (!).Google Scholar
  575. 31.
    Cf. above, p. 646; this opinion is ascribed to Empedocles (5th cent.) by Pseudo-Plutarch, De Plac. II 1, 4 (Diets, Dox., p. 328, 1–3 ).Google Scholar
  576. 35.
    De animae procr. 1028 C and D (Moralia VI, 1, ed. Hubert, p. 184, 2–17).Google Scholar
  577. 37.
    Mentioned before (p. 663); similarities with Hipparchian ratios (p.326 (3) and (7)) are probably accidental.Google Scholar
  578. 1.
    E.g. with Philolaus (second half of the 5th cent. B.C.); Diels. Dox., p. 378, 6–9.Google Scholar
  579. 2.
    Translated, e.g., in Heath, Aristarchus, p. 165.Google Scholar
  580. 3.
    From Heraclea in Pontus. He was nicknamed “Pompicus” by the Athenians; for some moderns he is “the Paracelsus of Antiquity” or “un des romanciers les plus lus” (Bidez-Cumont, Mages I, p. 14). At the death of Plato (347 B.C.) he was about 40 years of age.Google Scholar
  581. 4.
    The main sources are two short passages, one in Pseudo-Plutarch and one in Proclus, and a more substantial discussion by Simplicius in his Commentary to Aristotle’s De caelo (Ps.-Plut.: Diels, Dox., p. 378, 10–15; trsl. Heath, Aristarchus, p. 251. Proclus, Comm. Tim., ed. Diehl III, p. 137; trsl. Heath, p. 255, Festugière IV, p. 176. Simplicius, ed. Heiberg, p. 541, 29; trsl. Heath, p. 255).Google Scholar
  582. 5.
    Chalcidius, Comm. Tim., Chap. 110 (ed. Wrobel, p. 176, 22–25; trsl. Heath, Aristarchus, p. 256).Google Scholar
  583. 6.
    Theodosius, De diebus et noct. I 9, I 10, etc. (ed. Fecht); cf. below IV D 3, 3 B.Google Scholar
  584. 7.
    Archit. IX 1, 6 (Budé, p. 11; p. 89–92).Google Scholar
  585. 8.
    Theon Sm., Chap. 32f. (Dupuis, p. 300–303). The epicycles are represented by “solid spheres” which carry the planet, rolling inside of “hollow” spheres that correspond to the deferents. The radii of the epicycles increase from the sun to Mercury and to Venus.Google Scholar
  586. 9.
    Since the sun retains its solid sphere one must assume that the orbits of Mercury and Venus have their center in the mean sun.Google Scholar
  587. 10.
    Macrobius, Comm. I. 19, ed. Willis II, p. 73f.; trsl. Stahl, p. 162–164; Heath Aristarchus, p.258f.; summary in Dreyer, Plan. Syst., p. 129.Google Scholar
  588. 11.
    Martianus Capella VIII 857; ed. Dick, p. 450, trsl. Heath, Aristarchus, p. 256.Google Scholar
  589. 12.
    Initiated by Schiaparelli (1873); Scritti I, p. 361 ff.; II, p. 113 ff. Actually this overlooks a fundamental aim in the work of Copernicus, that is to show that all technical details of the Ptolemaic numerical procedures can also be explained heliocentrically under the severe restriction to uniform circular motions.Google Scholar
  590. 13.
    Cf., e.g., the simile of the ants on the potter’s wheel or of the traveler on a boat: Vitruvius, Arch. IX 1, 15 (Budé, p. 15, p. 111); Geminus, Isag. XII, 18 (Manitius, p. 140, 23–142, 12); Achilles, Isag. 20 (Maass, Comm. Ar. rel., p. 48, 16–24); Anon. (Maass, p. 97, 33f.); Cleomedes I, 3 (Ziegler, p. 30, 8–15). Cf. also Aryabhatiya IV, 8 (Clark, p. 64); Bar Hebraeus (13th cent.), L’asc. I, 6 (trsl. Nau, p. 10).Google Scholar
  591. 14.
    Van der Waerden [1944] and [1951, 2] (p. 69, Fig. 5).Google Scholar
  592. 15.
    Pannekoek [ 1952 ]. Twenty years later van der Waerden ([1970, 1], p. 51) treats his theory of the earth’s motion as an established fact.Google Scholar
  593. 16.
    Chalcidius, Chap. 111 (ed. Wrobel, p. 178, 5–8 ).Google Scholar
  594. 17.
    Cf. below p. 804 for the tradition of this parameter (Theon of Smyrna-Chalcidius-Cleomedes-Martianus Capella).Google Scholar
  595. 23.
    English translation by Heath, Aristarchus, p. 302. Cf. also above p. 646.Google Scholar
  596. 24.
    Plutarch, De facie (Loeb XII, p. 54/55; Heath, Arist., p. 304); also Pseudo-Plutarch, De Plac. II, 24 (Diels, Dox.; Heath, Arist., p. 305).Google Scholar
  597. 25.
    Sextus Emp., Adv. Math. X (= Against Phys. II) 174 (Loeb III, p. 298/9; Heath, Arist., p. 305).Google Scholar
  598. 26.
    Simplicius, Comm. Arist. De caelo (ed. Heiberg, p. 444, 34; trsl. Heath, Arist., p. 254).Google Scholar
  599. 27.
    Plutarch, Plat. Quaest. 1006 C (ed. Hubert, Moralia VI, 1, p. 129, 21–25).Google Scholar
  600. 28.
    Heath, Aristarchus, p. 305.Google Scholar
  601. 29.
    On Seleucus (ca. 170 B.C.) cf. above p.610f.Google Scholar
  602. 31.
    Van der Waerden [1970, 1], p. 7, p. 51. Needless to say such tables would be useless for a terrestrial observer.Google Scholar
  603. 33.
    That Seleucus considered the cosmos to be infinite is also known from Pseudo-Plutarch (Diels, Dox., p. 328, 4–6); in Stobaeus’ version not only Seleucus but also Heraclides Ponticus are credited with this opinion.Google Scholar
  604. 34.
    Plutarch, De facie 11 and 14 (Loeb XII, p. 77–79; p. 89–91).Google Scholar
  605. 36.
    Erected under Augustus; the hour lines on the pavement were designed by the “mathematician” Novius Facundus.Google Scholar
  606. 1.
    Ancient Lartos, about 7 km west of Lindos (cf. the map on PI. I in IG XII, 1). Note that the text does not come from a systematic excavation (as has been occasionally asserted).Google Scholar
  607. 2.
    The text seems to have been lost in the Museum for many years (cf. Hiller v. Gaertringen [1942], p. 165), only a squeeze was reproduced in Herz [1894], p. 1144 (upside down) and in Tannery, Mém. Sci. 15, p. 119. After the second World War Prof. Derek Price of Yale University obtained a new squeeze from Berlin. The fragment measures about 76 by 28 cm.Google Scholar
  608. 7.
    Hipparchus, Ar. Comm., Introduction et passim; cf. also above p. 278.Google Scholar
  609. 9.
    Cf. however, the “stadium” of 1/2° in Manilius III 282ff. (below p.719) and the “solar-cubit” in P. Oslo 73 (above p.592).Google Scholar
  610. 11.
    Cf. for this terminology Tannery, Mém. Sci. 15, p. 182E For the three preceding terms cf., e.g., below p. 933.Google Scholar
  611. 12.
    Tannery, Mém. Sci. 2, p. 509 suggests restoring 918543 for the above mentioned number 10n for Mercury’s phases.Google Scholar
  612. 13.
    The initial 40 is doubtful.Google Scholar
  613. 14.
    The final 10 is doubtful.Google Scholar
  614. 15.
    Cf. below p.704 (22).Google Scholar
  615. 16.
    Cf. above p.150 (1), p.170(1), and p. 389 (7) or p. 420 (2).Google Scholar
  616. 17.
    Cf., e.g., above p.151 (2).Google Scholar
  617. 18.
    From an epigraphic viewpoint the change from 57 to 50 (or rather from 570 to 500) is unpleasant but seems unavoidable.Google Scholar
  618. 19.
    Cf. for these concepts above p. 377.Google Scholar
  619. 20.
    Cf. above p.388f.Google Scholar
  620. 21.
    Cf., e.g., ACT II, p. 283 or above p.426.Google Scholar
  621. 22.
    The latter without emendations; perhaps one should read 15436 for Mars and 2456 for Jupiter.Google Scholar
  622. 23.
    Herz [1894], p. 1142.Google Scholar
  623. 24.
    Cf. below p.704.Google Scholar
  624. 25.
    Decreasing L by 20 would lead in (13) to P≈10;59 for Jupiter, to ≈28;57 for Saturn, i.e. to values hardly permissible.Google Scholar
  625. 26.
    Evidence for planetary models which assume an incorrect sense of rotation on the epicycle will be discussed in V A 1, 4.Google Scholar
  626. 28.
    Prof. Toomer reminded me of this parallelism; cf. also Toomer [ 1965 ], p. 62, Fig. V.Google Scholar
  627. 29.
    For further details of the Indian procedure cf., e.g., Neugebauer-Pingree, Varah. Pc: Sk. II, p. 101 and Fig. 59 there.Google Scholar
  628. 2.
    Cf. Neugebauer-Parker, EAT I, p. 119f. for the difficulties connected with the unexpected ratio M: m = 3:1 and for its possible connection with the origin of the 24-division of the day which appears for the first time in this period.Google Scholar
  629. 1.
    In ancient mathematical geography the name of this area is “(Mouth of the) Borysthenes” (i.e. Dniepr); cf., e.g., below p.725 (1).Google Scholar
  630. 2.
    We have textual evidence for the use of water clocks since Old-Babylonian times; cf. Neugebauer [1947, 2].Google Scholar
  631. 5.
    Cf. for the texts Neugebauer [1947, 2], p. 41.Google Scholar
  632. 7.
    Written by Christians since it contains, e.g., a discussion between Gregory of Nazianz and Basil.Google Scholar
  633. 12.
    Published CCAG 8, 4, p. 232. The treatise was perhaps compiled by Balbillus who had great influence on Nero and Vespasianus. (Serious objections against the commonly accepted conjecture that Balbillus was the son of Thrasyllus, the astrological advisor of Claudius, were raised by Gagé, Basiléia, p. 76ff.).Google Scholar
  634. 13.
    Pañcasiddhāntikā XII 5; cf. Neugebauer-Pingree, Varāham., Pc. Sk. I, p. 105, II, p. 83. For the Paitāmahasiddhānta cf. I, p. 10.Google Scholar
  635. 14.
    Cf. above p. 706 and Fig. 38.Google Scholar
  636. 15.
    Mich. Pap. III, p. 77f. (XII, 11–48), p. 104f., p. 114f.Google Scholar
  637. 16.
    Published by E. Maass, Anal. Erat., p. 141–146 (from Par. gr. 2426; cf. CCAG 8, 3, p. 61. cod. 46, F. T ). Cf. also above p. 332, n. 15.Google Scholar
  638. 17.
    E.g. March = Dystros = Phamenoth. The text was compiled for a city in Asia with a harbour; cf. CCAG 9, 1, p. 128–137.Google Scholar
  639. 18.
    E.g. CCAG 11, 1, p. 33/34, F. 71 (also Revilla, CataL I, p. 298, No. 7).Google Scholar
  640. 19.
    E.g. CCAG 8, 3, p. 123–125, p. 168f. (ascribed to the Prophet David).Google Scholar
  641. a Monumenta Germ. hist., Scriptores rerum Merov. I, 2, p. 405. I owe this reference to D. Pingree.Google Scholar
  642. 20.
    Acta Sanctorum, Propylaeum ad Acta Sanctorum Novembris. Hippolytus Delehaye, Synaxarium Ecclesiae Constantinopolitanae e Codice Sirmondiano, nunc Berolinensi. Bruxelles, Soc. Bolland., 1902 ( Greek).Google Scholar
  643. 21.
    Sauget [1967] (Syriac) and Patrol. Or. 10, p. 347–353 (Arabic).Google Scholar
  644. a Garitte [1964].Google Scholar
  645. 22.
    Petri [ 1964 ] Table, p. 283 (without realizing its schematic character).Google Scholar
  646. 23.
    Cf. below p. 739.Google Scholar
  647. 23a.
    Porphyry. Introd. 193: cf. CCAG 5, 4, p. 209, 1–7.Google Scholar
  648. b Cf. below p.725 (1).Google Scholar
  649. 24.
    Cf. above p. 581, notes 7 to 10.Google Scholar
  650. 25.
    Cf. below p. 731.Google Scholar
  651. 26.
    Strabo, Geogr. II 5, 40 (Loeb I, p. 513) says that M =15h belongs to an area south of Rome but north of Naples. Geminus, Isag. VI 8 (Manitius, p. 70, 16) says that M =15h means “around (περί) Rome.” Also the “Calendarium Colotianum” (1st cent. A.D.) and “Vallense” (Degrassi, Inscr., p. 284–287, Pl. 81–86) assume M =15h. Pliny, NH II 186 (Loeb I, p. 319) associates “Italy” with M =15h. Hyginus, Astron. IV (ed. Bunte, p. 100f.) says that where he lives M:m = 5:3 and he therefore divides the day in 8 parts. John of Damascus, in the 8th century, declares, without any geographical specification, that M =15h, m=9h (Expositio fidei 21, ed. Kotter, Patristische Texte u. Stud. 12, 1973, p. 57, 69f. and 84f. = Migne PG 94 col. 889–892).Google Scholar
  652. 3.
    These were correctly determined, e.g., in the Almagest; cf. above I A 4, 1.Google Scholar
  653. 5.
    For M= 12t we have, of course, ρ= 30° for both systems. At M =17h one finds ρ1= 9;10°, ρ6= 50;50° for System A, ρ1=11;15°, ρ6=48;45° for System B.Google Scholar
  654. 5.
    The time of Hypsicles, about 150 to 120 B.C., is suggested by his preface to his treatise commonly known as “Book XIV” of the “Elements” of Euclid where he refers to Apollonius (cf., e.g., Heath, Euclid III, p. 512); cf. also Huxley [1963], p. 102f.Google Scholar
  655. 6.
    Cf. ed. De Falco-Krause-Neugebauer, Hypsikles (1966).Google Scholar
  656. 7.
    Cf., e.g., Neugebauer MKT III, p. 76–80; cf also, p. 83 s.v. Reihen.Google Scholar
  657. 8.
    Book XIV of the Elements, mentioned above note 5; cf., e.g., Heath. GM I, p. 419–421.Google Scholar
  658. 9.
    Opera I, p. 470, 17–472, 22 ed. Tannery; also Heath, Dioph., p. 125 f.: p. 252 f.Google Scholar
  659. 10.
    Vettius Valens, Anthol. (p. 157, 14f., ed. Kroll) remarks that “the king” (i.e. Nechepso) — in contrast to Hypsicles? — gave the rising times only for the first clima (i.e. for Alexandria). What follows in this section are horoscopes which make use of System A for Alexandria and for Babylon (cf. Neugebauer-Van Hoesen, Greek Horosc. Nos. L 82 and L 102 IV a and IV b). The reference to Nechepso is, of course, historically valueless.Google Scholar
  660. 11.
    Cf. RE 14,1 col. 1116, 12–1117, 10.Google Scholar
  661. 12.
    Astron. III, 394: “mihi debeat artem.”Google Scholar
  662. 13.
    For System B cf. below p.721.Google Scholar
  663. 14.
    Astron. III, 247–293; ed. Housman III, p.22–26 and commentary p. XIII-XX.Google Scholar
  664. 15.
    Cf., e.g., above p. 368 (1).Google Scholar
  665. 16.
    Cf. below p.719.Google Scholar
  666. 17.
    Cf. Hypsikles, ed. De Falco-Krause-Neugebauer, p. 16, p. 48 f.Google Scholar
  667. 18.
    CCAG 1, p. 163, 4–14.Google Scholar
  668. 19.
    For the date of Rhetorius cf above p.258, n. 14.Google Scholar
  669. 20.
    Ed. Kroll, p. 23f.Google Scholar
  670. 21.
    Ed. Kroll, p. 304, 4, 9, 18.Google Scholar
  671. 22.
    Above p. 597, n. 31.Google Scholar
  672. 23.
    Above p.718 (11). Manilius III 275, 279, 291, 418, 437; ed. Housman III, p. 24ff., p.41, p. 44; commentary p. XIV.Google Scholar
  673. 24.
    Cf. above p.Google Scholar
  674. 25.
    Since 0;30°=0;0,5d and since 1 beru =0;5d (the “double-hour” of the older literature, e.g., Bilfinger) one could say that the stadia are the “minutes” of the béru. But in Babylonian astronomy the units below the béru are the u3, i.e. the degrees (1/30 béru) and their sexagesimal parts, and not units of 1/60 béru.Google Scholar
  675. 26.
    Tetrab. I, 21 Boll-Boer, p. 46, 10–14=I, 20 Robbins, p. 94/95.Google Scholar
  676. 27.
    CCAG V, 4, p. 211, § 41 (194 Wolf).Google Scholar
  677. 28.
    Ed. Boer, p. 3, 4–8, 2; p. 10, 17–11, 3; p. 81, 13–19.Google Scholar
  678. 29.
    Firmicus, Math. II, 11 (ed. Kroll-Skutsch I, p. 53–55 ).Google Scholar
  679. 30.
    In fact the above p.718 (11) mentioned values for Babylon (in degrees).Google Scholar
  680. Cf. Ryssel [1893], p. 47f. Syriac text and Latin translation in Patrologia Syriaca, Vol. I, part 2 (Paris 1907), p. 513, No. 8.Google Scholar
  681. 1.
    Cf. for Paulus (z A.D. 375) below V C 2, 4 B; for Chap. 30 (which is not found in all MSS) cf. ed. Boer, p. 81 f.Google Scholar
  682. 4.
    Tetrab. I, 21 Boll-Boer, p. 46, 7–9; = I, 20 Robbins, p. 20.Google Scholar
  683. 5.
    Pliny NH VII 49, 160 (Jan-Mayhoff II, p. 55, 19–56, 6; Loeb II, p. 612–615 but unreliable in the translation); Censorinus, De die nat. 17, 4 (ed. Hultsch, p. 31, 15–24). The best presentation of these passages is given by Honigmann in Mich. Pap. III, p. 307–311.Google Scholar
  684. 6.
    One has for Babylon M = 3,36°, d = 4°; for Alexandria M = 3,30°, d = 3:20°. For System B one would find κ = 110 for Babylon, κ = 114 for Alexandria.Google Scholar
  685. 7.
    Geminus, Isag. VI, 8 (Manitius, p. 70, 16f.); or above p. 581.Google Scholar
  686. 8.
    There is no proof, however, for Honigmann’s conjecture (Mich. Pap. III, p. 316) to see in Epigenes the “inventor” of the “astrological climata”.Google Scholar
  687. 5.
    Cf., e.g., Geminus, Isag., ed. Manitius, p. 70, 16; cf. also above p.Google Scholar
  688. 6.
    In Book III, 458–462; cf. also Housman, Manilius III, p. XIX f.Google Scholar
  689. 8.
    Mich. Pap. III; text: XI, 38-XII, 11 (p. 76/77); transl.: p. 114; 301–321 and Neugebauer [1942, 2], p. 255–257.Google Scholar
  690. 10.
    It follows from (7) or (10), p.713f., respectively, that also d and progression if the M form such a sequence.Google Scholar
  691. a Patrol. Or. 10, p. 59–87 (menologium from Aleppo); slightly garbled versions: p.93–97 (from Scete in the Wâdi Natrûn, Lower Egypt), p. 102–107 (Antioch), p. 127–151 (Aleppo).Google Scholar
  692. 17.
    De nuptiis VIII, 878 (ed. Dick, p. 463,10–15).Google Scholar
  693. 18.
    Gerbert, Opera, ed. Bubnov, p. 39.Google Scholar
  694. 19.
    Martianus, De nupt. VIII, 844f. (ed. Dick, p. 444, 1–445, 13 ).Google Scholar
  695. 20.
    Cf. CCAG 12, P. 216, P. 223–228. On Gergis cf. also Ruska, Turba, P. 26 (No. 27) and p. 56f.Google Scholar
  696. 1.
    Both terms are used interchangeably, e.g., by Geminus (cf. the index in Manitius, p. 307 f., p. 325) or by Vettius Valens (Anthol., ed. Kroll, p. 317, 1/2, p. 343, 8/17, etc.).Google Scholar
  697. 2.
    E.g. in the Latin version of Ptolemy’s Analemma (Opera II, p. 217, 17).Google Scholar
  698. 3.
    In geometry κλίμα can denote the length of the generating line of a cone or the edge of a pyramid (e.g. Heron, Stereom. 14 and 30, Opera V, p. 12, 15–20 and p. 28, a4/b 5).Google Scholar
  699. 5.
    An entirely different meaning of κλίμα is associated with the four cardinal directions, East, West, etc. (e.g. Heron, Geom., Opera IV, p. 176, 18f. or Isidorus, Etym. III 42, 1; XIII 1, 3) or with the four principal winds (Cramer, Anecd. gr. Par. I, p. 369, 3f.).Google Scholar
  700. 7.
    Geogr. I, 23, ed. Nobbe, p. 45–47; trsl. Mžik, p. 65 f. The same spacing is also found in the “Diagnosis” (cf. Diller [1943], p. 44, verso 12-p. 46).Google Scholar
  701. 9.
    Angles between ecliptic and meridian: Alm. II, 13 (cf. p.50f.); diagram for ortive amplitudes: Alm. VI, I1 (cf. Fig. 32 below p. 1216), with explicit reference to the “seven climata” (ed. Heiberg, p. 538, 25/539, 1); tables for oblique ascensions in the “Handy Tables” (cf. V C 4, 2 A); tables for parallaxes, ibid. (cf. p. 990); values for φ in the nomogram of the “Analemma” (cf. Fig. 35, p. 1382 ).Google Scholar
  702. 12.
    Vettius Valens, Anthol., ed. Kroll, p. 24, 13–21 for seven climata, p. 157, 14 for Alexandria=No. I, p. 157, 22 for Babylon = No. II. Cf. also Honigmann SK, p.42f. and Neugebauer-Van Hoesen, Greek Horosc. L82, L102, IVa and b; also p. 184.Google Scholar
  703. 13.
    This unfortunate classification was introduced by Honigmann (SK p. 3 et passim). When Paulus Alex. in his astrological treatise calls the clima of Alexandria the “third” (ed. Boer, p. 3, 5 and p. 10, 18), thus following (1), then Honigmann simply speaks about an “unpassende Reminiszenz.”Google Scholar
  704. 14.
    Honigmann, SK, p. 11/12. The width of a zone of practically constant conditions is occasionally specified to be 400 stades (Geminus, Isagoge, ed. Manitius, p. 62/65, p. 170/171). Cleomedes 1, 10 (ed. Ziegler, p. 98, 4f.) says that the gnomon near Syene casts no noon-shadow at the summer solstice in an area of 300 stades diameter. Pliny NH II 182 (Jan-Mayhoff I, p.197, 5–7; Loeb I, p. 315) considers 300 to 500 stades the limits for practically equal shadow lengths, whereas Posidonius takes 400 stades as within observational accuracy in the determination of latitudes (Strabo, Geogr. II 1, 35, Loeb I, p. 330/1; Budé I 2, p. 44 ).Google Scholar
  705. 18.
    Cf. above p. 721; also below p. 733, n. 28 for a possible connection with Eudoxus, i.e. evidence from the fourth century B.C.Google Scholar
  706. 19.
    According to Strabo, Geogr. II 5, 36–42 (Loeb I, p. 509–517) Hipparchus singled out 10 parallels between M= 13h and 17h for which AM is either 1/2h or 1/4h, and once, at the end, 1h. For Ptolemy see above p. 725.Google Scholar
  707. 20.
    Entirely unfounded is the association of some Old-Babylonian mythology with this hellenistic invention (Honigmann, SK, p. 8).Google Scholar
  708. 21.
    Eusebius, Praep. evang. VI 10 (278) [164], ed. Dindorf, Opera I, p. 321, 3–6.Google Scholar
  709. 22.
    Perhaps there exists some relation to the geographical version of the doctrine of the “Heptomades”; cf. Boll, Lebensalter, p. 137ff.; also Kranz [1938], p. 139.Google Scholar
  710. 2.
    Cf., e.g., Millás-Vallicrosa, Est. Azar., p. 64 and p. 67.Google Scholar
  711. 3.
    As we have seen the “System” in combination with M determines all parameters needed for the computation of a table of oblique ascensions (cf. p. 713).Google Scholar
  712. 4.
    Note the remark “since there exist seven climata” (ed. Kroll, p. 24, 13 f.).Google Scholar
  713. 12.
    Math. II, 11, ed. Kroll-Skutsch I, p. 53–55. For details of the reconstruction cf. Neugebauer [1942, 2], p. 258 f.Google Scholar
  714. 13.
    Pliny, NH VI, 211–218 (Jan-Mayhoff I, p. 517–521; Loeb II, p.4 4–501). Cf. also below p. 747.Google Scholar
  715. 14.
    Above p. 725 (1). Cf., however, for the special role of M=15h above p. 711f.Google Scholar
  716. 15.
    Since Pliny refers to Nigidius Figulus (1st cent. B.C.) in connection with the value M=15;12h for Rome it has been assumed that Nigidius was Pliny’s direct source for the whole selection of “climata” (cf., e.g., Honigmann, SK, p. 31, p. 45, etc). Honigmann then appointed Serapion gnomonicus, supposedly a pupil of Hipparchus, as “ältesten Urheber” of the whole scheme. I see no gain in this web of hypotheses. Incidentally, M=15;12h is also in Ptolemy’s Geography (VIII 8, 3 Nobbe, p. 205, 7 f.) the value given for Rome.Google Scholar
  717. 17.
    For the details cf. Neugebauer [1942, 2], p. 256f.Google Scholar
  718. 18.
    One more name is only partially preserved, perhaps A[sia] or A[rmenia].Google Scholar
  719. 19.
    Cf., e.g., Strabo, Geogr. II 2, 2 (Loeb I, p. 363); I 2, 24 (Loeb I, p. 111ff.); etc.Google Scholar
  720. 23.
    Above p. 723; cf. also for Manilius, p. 722.Google Scholar
  721. 25.
    Cf. above p. 723 (1). For earlier Syriac sources on rising times cf above p. 720. See also Bar Hebraeus, L’asc. II, 3, ed. Nau, p. 143–157; on climata 1.c. II, 1, 7–9, p. 125–129 and Candel. p. 583–590.Google Scholar
  722. 28.
    Hipparchus tells us (Ar. Comm., Manitius, p. 29) that Eudoxus applied in the “Phenomena” the ratio M:m = 12: 7, presumably for a certain region in Greece, after having used 5:3 (i.e. M =15s) in the “Enoptron” for Greece in general (Manitius, p. 23; cf. also above p. 581, n. 8). The ratio M:m = 12:7 can hardly be correct, however, because it cannot be expressed in units of hours or degrees (M ≈ 15;9,28,...h = 3,47;22,... o). It is tempting to emend the ratio to 11:7 which belongs as climate III b to the Babylon sequence, associated with “Athens” (cf. above Table 2, p. 730) or with “Rhodes” (above p. 730 (2)). If one accepts this emendation we would have here the earliest evidence for the arithmetical climata, indeed from prehellenistic times.Google Scholar
  723. 3.
    Geminus, Intr. V, 45–48 (Manitius, p. 58, 18–60, 13); XVI, 6–12 (Manitius, p. 164, 22–168, 20 ).Google Scholar
  724. 7.
    Archit. IX 7 (ed. Krohn, p. 216, 6 f.; Loeb II, p. 250/251; Budé, p. 27).Google Scholar
  725. 8.
    Ed. Hiller, p. 151, 16–18 (Martin, p.214/215; Dupuis, p.246/247); p.199,7f. (Martin p. 324/325; Dupuis p. 320/321); p. 202, 12; p. 203, 10–14, etc.Google Scholar
  726. 9.
    Cf. Ver Eecke, Dioph., Introd., p. XI, n. 2; also Heron, Opera IV, p. 168, 10–12=Theon Smyrn., ed. Hiller, p. 199, 7.Google Scholar
  727. 10.
    Comm. Euclid., ed. Friedlein, p. 269, 13–18 (trsl. Ver Eecke, p. 231); Hypotyposis III, 28 (ed. Manitius, p. 54, 1–5), VI, 13 (Manitius, p. 206, 6–8). Obviously wrong is a reconstruction by Mugler of a corrupt passage in Proclus, Comm. Rep., trsl. Festugière II, p. 152, n. 1 which leads to ε=20°(!).Google Scholar
  728. 11.
    About” 24°: Hipparchus, Comm. Ar. I 10, 2 (Manitius, p. 96, 21); Plutarch, Moralia 590 F (Loeb VII, p. 464/465); Ptolemy, Planisph. (Opera II, p. 259, 13) or Geogr. VII 6, 7 (Neugebauer [1959, 1], p. 23, n. 6). Also without such specification ε=24° is common: e.g. Anonymus, Maass, Comm. Ar. rel., p. 132,1 (≈ A.D. 500) or Anon., Logica et Quadr. (11th cent.), Heiberg, p. 104, 21 f.; Bar Hebraeus (13th cent.), East. II 1, 1; II 2, 3f. etc. (Nau, p. 113, p. 134ff.) beside the accurate value 23;55 in II 1, 9 (Nau, p. 128).Google Scholar
  729. 12.
    Strabo II 5, 7 (Berger, Geogr. Fr. Hipp., p. 111 f., II B, 15 and II B, 23; Loeb I, p. 436/437f.). Cf. also above p. 590.Google Scholar
  730. 13.
    Opera I, 1, p. 67, 17–68, 6. Theon’s commentary (Rome CA II, p. 528/529) is, as usual, only a paraphrase of Ptolemy’s text.Google Scholar
  731. 14.
    By arguments which are bound to lead to absurd results Diller [1934] tried to show that Hipparchus assumed ε=23;40° “although this fact has disappeared entirely from the tradition and is not attested by any ancient author.” Diller first computes latitudes φ from distances given for parallels of longest daylight M (i.e. he converts rounded numbers of stadia into accurate degree values) and then operates with a formula of modern spherical trigonometry to find ε from φ and M. This then is taken seriously to establish a deviation of 0;10°.Google Scholar
  732. 15.
    Berger, Geogr. Fr. Erat., p. 131 tries to show that (5) does not belong to Eratosthenes. His arguments are much too pedantic in view of Ptolemy’s clear text (cf., e.g., Thalamas, Erat., p. 121 f.). In any case there remains the problem of explaining the origin of the peculiar fraction 11/83 in (5).Google Scholar
  733. 18.
    It should not have been difficult to obtain in hellenistic Egypt a fair estimate of the reduction of sailing time to account for the huge bend of the Nile in the region of Tentyra-Diospolis Parva.Google Scholar
  734. 19.
    A summary of the literature supporting this view is found in Prell [1959].Google Scholar
  735. 21.
    For the little we know about Hipparchus’ methods in mathematical geography cf. above I E 6, 3.Google Scholar
  736. 23.
    Indeed between M=12 1/2h and 16h are about 40°=48000st.Google Scholar
  737. 26.
    Cf. for details Neugebauer [1975], based on Marc. gr. 314 fol. 222r and several parallels.Google Scholar
  738. 30.
    Alm. II, 8 has for clima VII only tp =48° (in MS D expressly 48;0). This may be a residue of the simple arithmetical pattern.Google Scholar
  739. 3.
    A Babylonian shadow table belongs to the second tablet of the “series” mulApin (cf. above p. 598), published by Weidner [1912], p. 198f. The length s of the shadow (measured in cubits) and the time t after sunrise (measured in time degrees) are related through the formula s · t = c with c = 1,0 at the summer solstice, c=1,15 at the equinoxes, c=1,30 at the winter solstice; cf. for details above p. 544. The Greek scheme, described below p. 738 (1), is obviously unrelated to this Babylonian approach.Google Scholar
  740. 4.
    Cf. above p.581, notes 7 to 10. Only in text (q) do we find M:m=14:10.Google Scholar
  741. 5.
    Ambros. C 37 sup. fol. 137r-139r (unpublished); cf. CCAG 3, p. 7, cod. 11.Google Scholar
  742. 6.
    This text also operates with M:m = 15:9 but refers it correctly to a latitude φ = 41° (cf. Almagest II, 8 where φ=40;56° corresponds to M =15h). For a gnomon of 7 feet cf. also below p. 744.Google Scholar
  743. 8.
    Texts (a), (b), (d, V), (I), (m); cf also the Ethiopic texts V1 and V2 below p. 742.Google Scholar
  744. 9.
    This is certainly the case with Bar Hebraeus (13th cent.) because Macedonia is mentioned in this context; cf. below p. 744.Google Scholar
  745. 12.
    Generally identified as Philip of Opus; cf. RE 19, 2 col. 2351, 67–2352.5 [v. Fritz]. For his association with Euctemon see, e.g., Ptolemy, Phaseis, ed. Heiberg, p.17, 15; 18.5 etc. Rehm says in Parap., p. 99, n. 3 (and similarly in RE Par. col. 1346, 13–41): “So ist das Parapegma des Philippos tatsächlich sicher nichts anderes als eine noch dazu sehr wenig selbständige Bearbeitung des euktemonischen,” referring to Griech. Kal. III [1913], p. 36 where he represents in a diagram Euctemon and Philip as independent (!) sources of Ptolemy’s Phaseis.Google Scholar
  746. 13.
    Ptolemy, Phaseis, p. 67, 5, ed. Heiberg. Also Hipparchus, Comm. Ar.. Manitius, p. 29, 13–18.Google Scholar
  747. 15.
    Also for meteorological data a schematic transfer to Alexandria from Greece has been established; cf. Hellmann [1916], p.332–241.Google Scholar
  748. 16.
    The same error we noticed above (p. 724) with Gerbert (10th cent.) for the length of daylight.Google Scholar
  749. 18.
    Cf. Schissel [1936], p. 115–117 (who did, however, not grasp the simple background).Google Scholar
  750. 1.
    All three texts are edited and translated in Bouriant [1898]; an added astronomical commentary by Ventre Bey is without value.Google Scholar
  751. 2.
    This selection is probably determined by the hours of prayer.Google Scholar
  752. 3.
    Maskaram = Thoth ≈ September.Google Scholar
  753. 4.
    For details cf. Neugebauer [1964], p. 62–67, p. 69. All our texts (as Ethiopic texts in general) are of very recent date (last three centuries).Google Scholar
  754. 6.
    Found, e.g., in Berlin, Eth. 84 fol. 41” and in Vindob. Aeth. 6 fol. 32” II, 16–33’ I, 16.Google Scholar
  755. 8.
    Chaine, Chron., p. 251. Actually our manuscript was written in 1768 (cf. Sobhy [ 1942 ], p. 169 ).Google Scholar
  756. 9.
    Sobhy [1942], p. 187, Arabic text [1943], p.250.Google Scholar
  757. 10.
    The increment of 6 feet corresponds exactly to the 3rd hour after noon.Google Scholar
  758. 19.
    Beyond some wrongly placed numbers in the first two columns all numbers for 3h (and thus for 9h) are one unit too low.Google Scholar
  759. 21.
    I.e. a moral and social code. Our text is Chap. XXI in a supplementary section; cf. West, Pahl. T., p. 397–400; also Kotwal [1969], p. 86–89. The arithmetical pattern makes it very easy to correct the few doubtful passages in the text.Google Scholar
  760. 22.
    The text uses “parts” as smaller units such that 1 foot =12 parts.Google Scholar
  761. 24.
    Cf. p. 741 (L, A), later than the 9th century. The codex was probably written in the monastery of Flavigny (on the Moselle, south of Nancy) in the 9th century but our table is “manu recentiore.”Google Scholar
  762. 25.
    In this table also the hours are paired: “hora 1 et 11,” etc.Google Scholar
  763. 26.
    Above p. 741 (LF); cf. also the calendar of Ibn al-Bann (above p. 743).Google Scholar
  764. 29.
    Exeter Cathedral and a Cotton MS (LD); probably also in the missal of Jumièges (west of Rouen) of 1150 (LG). A similar, also slightly corrupt pattern, is found in the manuscript (L, I). The hourly increments are said to be 1, 2, 2, 5, 10 feet, i.e. 2 and 5 instead of 3 and 4. The longest shadows run with the difference 2 between 19 and 29 instead of between 21 (i.e. u = 11 and 31 (i.e. U = 11).Google Scholar
  765. 31.
    West of Koblenz, north of Trier; cf. above p. 741 (LC).Google Scholar
  766. 1.
    Hipparchus, Ar. Comm. Manitius, p. 27, p. 35, and p. 29. Cf. for the last mentioned ratio above p. 733, n.28.Google Scholar
  767. 2.
    Vitruvius, Archit. IX, 7 ed. Krohn, p. 215, 8 f.; Loeb II, p 248/249. Similar data in the same section (for Rhodes, Tarentum, and Alexandria) are not transmitted securely enough to be usable as independent evidence.Google Scholar
  768. 3.
    Strabo, Geogr. II 5, 41; Loeb I, p.513. Of little interest, because isolated, is another remark by Strabo (II 5, 38; Loeb I, p. 511) according to which Hipparchus (if he is meant) assumed for Carthage g:s0=11:7. This would lead to φ: 32;28°, almost 4 1/2° less than in reality.Google Scholar
  769. 5.
    The correct determination of φ from M or vice versa suggests the use of analemma methods by Hipparchus (cf. above I E 3, 2).Google Scholar
  770. 6.
    Hipparchus, Ar. Comm. I, III 10, Manitius, p. 28/29.Google Scholar
  771. 7.
    Above p. 739, p. 743 and p. 744. Cf. also p. 929.Google Scholar
  772. 8.
    Hypsikles, Anaph., ed. De Falco-Krause-Neugebauer, p. 48.Google Scholar
  773. 11.
    Pliny, NH VI, 211–218 (Jan-Mayhoff I, p. 517–521; Loeb II, p.494–501).Google Scholar
  774. 12.
    Readings and emendations vary between 100:77 (≈ 1;17,55), 105:77 (=15:11 ≈ 1;21,4,51), and 100:74 (= 50: 37 ≈ 1;21,49). Using a sequence of constant second difference 0;0,35 for all six climata from la to III b one finds for IIIa the value g:s0=1;22,50=1,46;18,10:1,17≈106:77 and a corresponding φ ≈ 35;55°.Google Scholar
  775. 13.
    Pliny NH II, 182 and VI, 218 (Jan-MayhoffI, p. 197, 11–13; p. 521, 18f. Loeb I, p. 317; II, p. 501).Google Scholar
  776. 14.
    The ratio 9:8 is also found in Vitruvius, Arch. IX 7,1 (ed. Krohn, p. 215, 6f., Budé, p. 26, Loeb II, p. 249).Google Scholar
  777. 1.
    Rotating” sphere conveys a better idea of the meaning of this title than “moving” sphere.Google Scholar
  778. 2.
    From “De Habitationibus,” as usually quoted.Google Scholar
  779. 3.
    For the general background of these treatises cf. Tropfke GEM V, p. 118–121 and Heath, GM I, p. 348–353; II, p. 245–252.Google Scholar
  780. 4.
    Euclid, Phaen. 1.Google Scholar
  781. 5.
    Theodosius, Dieb. II, 19; a similar discussion of irrational quantities is found in Theod., Sph. III, 9 (ed. Heiberg, p. 146, 10) and in Pappus, Coll. VI, 8 (ed. Hultsch, p. 484).Google Scholar
  782. 6.
    The number of preserved copies is relatively high, e.g. some 40 manuscripts of Autolycus (none older than about A.D. 900, i.e. not less than 12 centuries of transmission).Google Scholar
  783. 7.
    Cf. below IV D 3, 6.Google Scholar
  784. 8.
    At the end of the 14th century Ibn Khaldûn still names the Spherics of Theodosius as preparatory for the study of Menelaus (trsl. Rosenthal III, p. 131).Google Scholar
  785. 13.
    Cf. for details Fecht, Theod., Introduction, and Ver Eecke, Theodosius, Introduction.Google Scholar
  786. 14.
    Strabo, Geogr. XII 4, 9 (Loeb V, p. 466/467).Google Scholar
  787. 15.
    Theorem 10; of course Hipparchus’ name is not mentioned in the text.Google Scholar
  788. 16.
    Cf. below p. 757.Google Scholar
  789. 17.
    E.g. still in Czwalina’s translation of the Spherics (1931) who overlooked Heiberg’s “Tripolites deleatur ubique” (“Corrigenda”, p. XVI).Google Scholar
  790. 18.
    Suidas, ed. Adler I, 2, p. 693; cf. also Konrat Ziegler in RE VA (1934) col. 1931.Google Scholar
  791. 19.
    Date suggested by Vogt [1912].Google Scholar
  792. 20.
    On the coast of Asia Minor, opposite Lesbos.Google Scholar
  793. 21.
    For the chronological details cf. the introduction in Mogenet, Autol.. p. 5–19.Google Scholar
  794. 22.
    The beginning of a roll is always particularly exposed.Google Scholar
  795. 23.
    Nokk [1850]; Heiberg, Stud. Eukl. (1882); Hultsch [1886] etc.; Björnbo [ 1902 ]; Heath, GM I, p. 348–352 (1921); Mogenet [1947].Google Scholar
  796. 24.
    For the underlying argument cf. below p. 761.Google Scholar
  797. 25.
    Above p. 748; for the testimonia in general see Euclid, Opera VIII, ed. Menge-Heiberg, p. XXXII-XXXIV.Google Scholar
  798. 29.
    In Book I only mentioned at the end of the proof of Theorem 10, though used implicitly long before.Google Scholar
  799. 30.
    Such a “revised edition” could go back to the author himself; cf., e.g., the preface of Apollonius to Book I of his “Conics.”Google Scholar
  800. 1.
    Occasionally explicit references to figures are found in the texts, e.g. in Euclid, Phaen. 12b (p. 74, 1, ed. Menge), Autolycus, Rot. Sph. 2 (p. 199, 16, ed. Mogenet), Theodosius, Dieb. I, 1 (p. 58, 5/6) and Hab., Scholion 13 (p. 46, 17, ed. Fecht). Cf., however, below n. 3.Google Scholar
  801. 2.
    Cf., e.g., the contrast between the ancient figure Mogenet, Autol., p. 207 and its absurd counterpart in Hultsch’s edition, p. 31.Google Scholar
  802. 3.
    Normally figures are inserted in spaces left free in the text (and often remaining blank) but this is not the case in Vat. gr. 191. Hence it seems possible that the figures on the margins are later additions.Google Scholar
  803. 4.
    For another drastic case of total disrespect for the importance of diagrams for the understanding of a text (Heron) cf. Neugebauer [1938, 2] II.Google Scholar
  804. 5.
    Euclid, Phaen. 2 and 3 from Vat. gr. 204 fol. 61v/62v (for this manuscript cf. Mogenet, Autol., p. 145 and p. 187).Google Scholar
  805. 7.
    This has been first observed and widely utilized by A. Rome (CA I, p. 141 note).Google Scholar
  806. 8.
    Heinrich Schäfer, Von ägyptischer Kunst, besonders der Zeichenkunst, Leipzig, Hinrichs, 1919; this first edition is the best one (in my opinion).Google Scholar
  807. 9.
    Cf. also the figures in the “Eudoxus Papyrus,” above p. 689.Google Scholar
  808. 10.
    Cf., e.g., Rome CA II and his notes to the figures.Google Scholar
  809. 11.
    E.g. Euclid, Phaen., p. 118, 19/20 (ed. Menge).Google Scholar
  810. 12.
    Theodosius, Dieb. II, 19: EΓΔZ instead of EΓZΔ (p. 154, 12, ed. Fecht).Google Scholar
  811. 15.
    E.g. Theodosius, Dieb. Lemma II, 10 has the letters T and Y (and the corresponding arc) misplaced (ed. Fecht, p. 122).Google Scholar
  812. 16.
    Th’eodosius, Dieb. II 10 and 11: the arc representing the winter tropic is missing.Google Scholar
  813. 17.
    I do not know what kind of drawing instruments were used that could produce with high accuracy constructions of great complexity (cf., e.g., the astrolabic figures 1 to 8 in Delatte AA II).Google Scholar
  814. 18.
    Cf. below Figs. 52a and 52b; or Euclid, Phaen. 14, Versions (a) and (b) (ed. Menge, p. 86/87).Google Scholar
  815. 19.
    Cf. above p.749. Also the references to Callippus and to Meton and Euctemon look strange in this context.Google Scholar
  816. 20.
    It also may be noted that the concluding theorems to Theodosius, Hab. (10 to 12), do not fit the rest of the treatise very well; cf. below p. 757.Google Scholar
  817. 1.
    For a summary cf. Mogenet, Autol., p. 160; text: ibid. note (3).Google Scholar
  818. 2.
    Examples: Euclid, Phaen. 12 (ed. Menge, p. 72, 6–74, 1); Theodosius, Sph. II, 11 and 12; III, 2 and 3, etc.Google Scholar
  819. 3.
    Angles” subtending an arc never occur; arcs are either “similar” (öμοιος) or “equal” if located on the same circle (e.g. Theodosius, Dieb. 9, p. 122, 11 f., ed. Fecht). Otherwise arcs are “greater than” or “less than” similar (e.g. Euclid, Phaen., p. 48, 4; p. 66 (b), 8; Autol., Rot. Sph., p. 34, 2).Google Scholar
  820. 5.
    Elements XI, Definitions 14 to 17 (Opera IV, p. 4, 21–6, 3; trsl. Heath, Vol. III, p. 261 ).Google Scholar
  821. 6.
    Heron, Definitions 76 to 81 (Opera IV, p. 52–55).Google Scholar
  822. 7.
    Cf. also Mogenet [1947].Google Scholar
  823. 8.
    Theorem 12, which states that both these circles are great circles. Also Theorem 11 does not mention the ecliptic.Google Scholar
  824. 9.
    Tannery, HAA, p.287f., No.21.Google Scholar
  825. 10.
    Autolycus, Rot. Sph. 7; Euclid, Phaen. 3.Google Scholar
  826. 11.
    Euclid, Phaen. 1 (p. 12, 9, ed. Menge).Google Scholar
  827. 12.
    Euclid, Phaen. Introd. (p. 6, 5, ed. Menge); cf. for this topic also above p. 576f.Google Scholar
  828. 13.
    A reference “as shown in the Optics” (Phaen., p. 2, 8, ed. Menge) seems to have no basis in the extant works on optics.Google Scholar
  829. 14.
    References to “Theodosius” and phrases like “he says,” etc. (ed. Fecht, p. 54, 11 etc.).Google Scholar
  830. 15.
    The definition of axis and poles appears only in one of the Greek manuscripts; cf. Mogenet, Autol., p. 195, 9–11 appar.Google Scholar
  831. 16.
    Tannery, HAA, p. 287 f., Nos. 15 to 21.Google Scholar
  832. 1.
    Of course no value for the obliquity of the ecliptic is ever mentioned in any of our treatises.Google Scholar
  833. 2.
    Cf. below IV D 3, 4; also V A 3.Google Scholar
  834. 3.
    Also the author of scholion No. 146 to Autolycus, Ris. Set. II, 1 (Mogenet, p. 273) begins the “day” when the sun is still 15° distant from the horizon.Google Scholar
  835. 4.
    Cf. above p. 58 (1).Google Scholar
  836. 5.
    Here, as always in such general estimates, “month” means 30 days; cf., e.g., Geminus, !sag. VI, 5 (Manitius, p. 70, 7).Google Scholar
  837. 6.
    Theodosius, Dieb. I,1 (Fecht, p. 56, 26f.; also p. 54, 2 ).Google Scholar
  838. 5.
    Quoted by Hipparchus, Comm. Ar.; cf. ed. Manitius, index p. 310.Google Scholar
  839. 1.
    άσύμπτωτοι; the restriction to semicircles is, of course, essential since complete great-circles would always intersect on the sphere.Google Scholar
  840. 3.
    Theodosius, Sph. ed. Heiberg, p. 67.Google Scholar
  841. 4.
    The theorem for sphaera recta that corresponds to II, 13 is formulated in II, 10.Google Scholar
  842. 2.
    Definitions are given in the introduction to Euclid, Phaen. (p. 10, 3–10, ed. Menge), in the introduction to Theodosius, Dieb. (p. 54, 7–16, Fecht), and in the scholia Nos. 106 and 114 to Euclid, Phaen. (p. 147 and 150, ed. Menge). In spite of variants in the formulations these definitions are obviously derived from a common source.Google Scholar
  843. 5.
    The sun at the midpoint of the arc AB (cf. Fig. 55) is 15° distant from the horizon; CD is the do-decatemorion diametrically opposite to AB. While AB interchanges one hemisphere CD interchanges the other. The arc CD is visible almost all night.Google Scholar
  844. 3.
    Autolycus, Ris. Set. II, 6. Expressly formulated in II, 1 and then consistently applied in II, 3 and II, 9 to 18; implicitly used in I, 2 and 3; casually mentioned in the proof of I, 10, a theorem that is parallel to II, 15.Google Scholar
  845. 4.
    Cf. for the notation below VI B 5, 2 or VI D 3, 4.Google Scholar
  846. 11.
    Using the south pole as center of projection. The figures in the text are much more primitive than in the Rot. Sph. or in Euclid and Theodosius and show no trend toward stereographic projection.Google Scholar
  847. 12.
    Theorem II,9 is also related to this group but at least the proof seems to be corrupt since it intro-duces the meridian which has no connection with the phases.Google Scholar
  848. 15.
    The translations by Czwalina and by Bruin-Vondjidis repeatedly err by adding the word star where only the phases are meant.Google Scholar
  849. 1.
    Cf. for the continued interest in these quantities above p. 38.Google Scholar
  850. 2.
    The proof more or less assumes what should be demonstrated. Scholion 81 (ed. Menge, p. 143) rightly refers for the necessary lemma to Theodosius, Sph. III, 7.Google Scholar
  851. 3.
    Below p. 765.Google Scholar
  852. 4.
    Above IV D 3, 3 D.Google Scholar
  853. 5.
    Similar discussions are also found in Theodosius, Dieb., Lemma II, 10 and in its applications in II, 10to14.Google Scholar
  854. 6.
    Cf. below p. 766f.Google Scholar
  855. 7.
    Scholion 7 (ed. Fecht, p. 156), but calling it Phaen. 18 instead of 14.Google Scholar
  856. 8.
    Dieb. II, 5 and 6.Google Scholar
  857. 9.
    Dieb. I, 11.Google Scholar
  858. 10.
    Mid-day” is the exact midpoint between sunrise and sunset and not identical with “noon.”Google Scholar
  859. 11.
    Dieb. II, 10.Google Scholar
  860. 12.
    Cf. above I B 2.Google Scholar
  861. 13.
    Cf. above p. 749, n. 5.Google Scholar
  862. 14.
    Or, to use the terminology of our treatises: with with the radius of the greatest always visible circle.Google Scholar
  863. 15.
    Its assertion concerns the monotonic decrease of as function of the distance from the summer solstice (cf. above p. 765).Google Scholar
  864. 16.
    It may be noted that (4) accidentally covers the space of the five climata from Syene to Mid-Pontus to which Ptolemy restricts his discussion in the “Phaseis” (cf. above p. 726).Google Scholar
  865. 17.
    Cf., e.g., above p.43f.Google Scholar
  866. 18.
    Euclid, Phaen. 2; Theodosius, Hab. 3 and 5.Google Scholar
  867. 1.
    Pappus, ed. Hultsch II, p. 474, 10, 6f., and 12f., respectively.Google Scholar
  868. 2.
    Book VI, Sect. 27 (Hultsch, p. 518–524).Google Scholar
  869. 3.
    It suffices to refer to Mogenet’s discussion (Autol., p. 167–170) of Pappus’ attitude toward Autolycus.Google Scholar
  870. 4.
    Hultsch, p. 536, 8–540, 25.Google Scholar
  871. 5.
    Hultsch, p. 540, 26–546, 2.Google Scholar
  872. 6.
    Translated by Heath, Arist., p. 412–414.Google Scholar
  873. 7.
    Cf. also above p. 640.Google Scholar
  874. 8.
    Cf. Hultsch, apparatus to p. 568, 12.Google Scholar
  875. 9.
    Sect. 55, Hultsch, p. 600, 9–11; German translation in Björnbo, Menelaos, p. 70.Google Scholar
  876. 10.
    Cf. above p. 301.Google Scholar
  877. 11.
    Hultsch (p. 622, 19–27) has maltreated the text because he was unfamiliar with the geographical terminology; cf. Honigmann SK, p. 80f.Google Scholar
  878. 13.
    Cf. Hultsch, p. 475, n. 1.Google Scholar
  879. 14.
    As suggested by Mogenet [1956].Google Scholar
  880. 16.
    D. Pingree, Gnomon 40 (1968), p. 15 f. has enumerated the passages which are commonly invoked to support the hypothesis of the existence of a collection under the name “Little Astronomy.” Among these sources I consider the following ones entirely unrelated to our problem: Theon’s commentary on the “little astrolabe” (mentioned by Suidas) is well-known to be concerned only with the “planisphaerium” (cf. below V B 3, 7 F); Cassiodorus’ distinction between Ptolemy’s “minor” and “major” astronomy concerns only Ptolemy and has nothing to do with a collection; Philoponus discusses specifically the methodology of Theodosius, Autolycus, and Euclid from the viewpoint of philosophical classification, without any reference to a larger “collection” (cf. Mogenet, Autol., p. 160; cf. also above p. 755).Google Scholar
  881. Boll [1916], p. 72 states that a work of Ptolemy is referred to as μικρòς άστρομούμενος by the “Anonymous of 379” (CCAG 5, 1, p. 197, 23 and p.205, 18 — the latter repeated by “Palchus” CCAG 1, p. 81, 2) and thus adds the “Phaseis” to the well-known “collection.” In fact, however, the word μικρός is not in the text, being Boll’s own arbitrary addition.Google Scholar
  882. 17.
    Steinschneider [ 1865 ]; Pingree, Gnomon 40, p. 16.Google Scholar
  883. 18.
    Cf. for this date above IV A 3, 1.Google Scholar
  884. 19.
    In astrological parlance such signs are called “seeing each other”; cf., e.g., Bouché-Leclercq, AG, p. 159–162; also P. Mich. 149, XII 27 (Mich. Pap. III, p. 104).Google Scholar
  885. 20.
    Isag. II, 27–45; VI, 44–50; VII, 18–31.Google Scholar
  886. 21.
    This has been observed, long ago, by Smyly (Hibeh Papyri I, p. 141).Google Scholar
  887. 22.
    Cf. above IV D 2, 1 A.Google Scholar
  888. 23.
    This does not imply the assumption of observational improvements.Google Scholar
  889. 24.
    Isag. VII, 32–34.Google Scholar
  890. 25.
    For the trigonometrically computed oblique ascensions (Alm. II, 8) one must use the smaller intervals of 10° in order to obtain proper results by simple linear interpolation.Google Scholar
  891. 26.
    Isag. XIII.Google Scholar
  892. 27.
    Cf., e.g., above p. 761f. and Fig. 56 (p.1368).Google Scholar
  893. 28.
    Alm. VIII, 4.Google Scholar
  894. 1.
    It was P. Luckey ([1927], p. 29–31) who first understood that τιά τών γραμμών “by rigorous methods,” in contrast to merely numerical results. Luckey started from the passages in Ptolemy’s “Analemma,” adding occurrences in the Almagest (which can easily be multiplied). Chronologically the earliest occurrence of this expression for “rigorous,” known to me, is found in Hipparchus’ commentary to Aratus (ed. Manitius, p. 150, 14–17). For its continued use can be quoted (without any claim to completeness): Pappus, Comm. to Alm. VI, ed. Rome, p. 171, 16–17 (≈A.D. 320).Google Scholar
  895. Theon, Comm. to Alm. I, ed. Rome, p.451, 11–12 (≈A.D. 370); in the Great Commentary to the Handy Tables he uses the comparative (γραμμικώτερον) which can only mean “more accurate” (quoted Tihon [1971] I, p. VII, note 1).Google Scholar
  896. Basil of Caesarea, Homily III 57B, ed. Giet, Sources Chrétiennes [1950], p. 198, ridiculing “their proofs... as exact and artificial nonsense”; Basil died 379.Google Scholar
  897. Proclus, Comm. to Plato’s Rep., ed. Kroll II, p.27, 16–17 (≈450).Google Scholar
  898. Heliodorus”, Comm. to Paulus Alex., ed. Boer, p.92, apparatus (≈560).Google Scholar
  899. Theodoros Metochites, Logos 14, 35, ed. Ševčenko, Métoch., p. 263, 32–33 (≈1300).Google Scholar
  900. A Latin equivalent is found in Pliny, NH II, 63 (Jan-Mayhoff I, p. 147, 3 f.): ratione circini semper indubitata.Google Scholar
  901. 2.
    This does not mean that “trigonometric” problems are absent from Babylonian mathematics. On the contrary, we know of typical trigonometric topics (“chord” and “arrow”) in Old-Babylonian mathematical texts (cf., e.g. Neugebauer, MKT I, p. 180) but we find no trigonometry in connection with astronomical problems. Also the general approach is not the same as in Greek trigonometry: in the cuneiform texts a numerical answer is sought to geometrically formulated examples whereas Greek trigonometry uses general geometrical theorems to obtain specific numerical results.Google Scholar
  902. 3.
    Cf. above I A 1.Google Scholar
  903. 4.
    Cf. above I E 3, 1.Google Scholar
  904. 5.
    Cf. above p. 302; p. 299. Also “four-sixtieths of a great circle” (i.e. 24°) for the obliquity of the ecliptic (Strabo Geogr. II 5, 43, Loeb I, p. 520/521).Google Scholar
  905. 6.
    Archimedes, Sand-Reckoner, Opera II, p. 226, 19–20; p. 228, 12 and 18; etc. Aristarchus, Heath, p. 352, 11–12; p. 380, 16–17; etc. Once, p. 376, 22, a right angle is divided in 60 parts but only to show that the ratio 1/4 (R): 1/30 (R)= 15: 2.Google Scholar
  906. 7.
    Aristarchus, Heath, p. 366, 6; p. 368, 9–10.Google Scholar
  907. 8.
    Aristarchus, Heath, p. 366, 2–3; p. 380, 17; etc.Google Scholar
  908. 9.
    Aristarchus, Heath, p. 352, 14–15; p. 364, 21–366, 2; etc.Google Scholar
  909. 10.
    Cf. for this treatise above IV B 3, 1.Google Scholar
  910. 11.
    Cf above p. 645 (8 b).Google Scholar
  911. 12.
    Archimedes, Opera II, p. 332, 3–10; trsl. Ver Eecke, p. 361.Google Scholar
  912. 13.
    Equivalent to (1) since c′=a/ sin α′, etc.Google Scholar
  913. 14.
    Cf. above I E 3, 1.Google Scholar
  914. 15.
    Cf. above p. 640 and Fig. 11 there; the present angle α′ is the angle α in Fig. 11.Google Scholar
  915. 18.
    Actually cos 1° ≈ 0;59,59,27 while 89/90 = 0;59,20≈cos 8;30°.Google Scholar
  916. 19.
    Alm. I, 10; cf. above p. 24.Google Scholar
  917. 20.
    Cf. above p.23 (C) and (A).Google Scholar
  918. 21.
    In the “Book on the determination of the chords in a circle (Boilot [1955], p. 197 No. 64; translated in Suter [1910]) and in a chapter of the Qānūn (Boilot [1955], p. 211 f., No. 104; summary in Schoy, Bir.). Cf. also Toomer [1973], p. 20–23.Google Scholar
  919. 22.
    Cf. for details Tropfke [1928].Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1975

Authors and Affiliations

  • Otto Neugebauer
    • 1
  1. 1.Brown UniversityProvidenceUSA

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