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The Almagest and its Direct Predecessors

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

Abstract

The Chaps. 10 and 11 of Book I of the Almagest contain the ancient theory of plane trigonometry and the resulting tables. The basic function, however, is not the sine function but its equivalent, the length of the chord subtended by the given angle in the unit circle. We shall use the notation crd α if the radius of the unit circle is 1, but Crd α if the radius R of the circle is the sexagesimal unit R = 1,0 = 60. Ptolemy uses Crd α exclusively but we shall frequently replace it by crd α which only implies a shift in the sexagesimal place value.

References

  1. 1.
    Schoy, Al-Bir. p. 81 (No. 14); cf. also below p.776.Google Scholar
  2. 2.
    MC T(UV ypappii v, meaning “rigorous” methods (cf. below p.771 n. 1).Google Scholar
  3. 3.
    If one could construct crd(1/2°), one could find, by virtue of the preceding steps, the chord for 1°, thus for 2°, 4°, 4+6=10°, hence also for 20° and finally for 40° which is the side of the regular 9-gon. But Gauss has shown (Disquis. arithm. § 365, Werke I, p. 461) that the construction of a regular n-gon by ruler and compass is only possible when n is a prime number of the form 2+1 (k an integer).Google Scholar
  4. 5.
    Cf., e.g., ed. Heiberg I, p. 317, 22f., et passim.Google Scholar
  5. 1.
    Two additional formulae (or their counterparts for chords), namely cos α = cos α/sin β and cos α = cot α cot β which express the sides by means of the angles never occur in ancient spherical trigonometry, although it was known that a spherical triangle is determined by its angles (Menelaos I, 18; Krause, p. 138). The equivalent of (1), (4) and cos a = cos α/sin β is proved by Copernicus (De revol. I, 14 Theorems 3 and 4) but he has still no formula in which a tangent occurs.Google Scholar
  6. 2.
    VII, 3 (1-leib. II, p. 30,18).Google Scholar
  7. 3.
    Cf. Rome, CA I, p. 569.Google Scholar
  8. 4.
    For plane trigonometry cf. above p.26.Google Scholar
  9. 5.
    Contrast: “sphaera obliqua” denotes geographical latitudes different from zero; the corresponding rising times are called “oblique ascensions.”Google Scholar
  10. 6.
    The equivalence with (5) of (1) plus (4) requires the use of the relation cos a cos S =cos A for which see p. 26 (4).Google Scholar
  11. 1.
    This omission has misled Manitius in his translation of the Almagest (edition of 1912) and then Vogt (in his Griech. Kal. 4, p. 44) who blamed Ptolemy for an essential error, instead of Manitius.Google Scholar
  12. 5.
    Almagest II, 3 Heiberg, p. 95, 6 to 13.Google Scholar
  13. 6.
    Cf. p.142.Google Scholar
  14. 7.
    Ptolemy does not indicate his method of computation but it is easy to reconstruct it. As soon as the maximum rising amplitude r10 is known the ortive amplitude /j(2) = EH of any point of longitude A on the ecliptic (cf. Fig. 30) is given by sin q = sin A sin r2. This follows immediately from p. 30 (5) in the triangle EHV, since sin /sin E = sin d/sin (90+(p); hence sin φ = sin .i sin E/cos cp and with (6) the formula sin φ = sin 110 sin A.Google Scholar
  15. 8.
    VI, 11 Heiberg, p. 543, 24f. and plate at the end of Vol. I.Google Scholar
  16. 9.
    The same numbers are found also tabulated in the ordinary fashion; cf., e.g., Manitius I, p. 454 or Vat. gr. 208 fol. 122 and Vat. gr. 1594 fol. 144Google Scholar
  17. 10.
    Cf. our Pl. I.Google Scholar
  18. 11.
    For the modifications and far reaching influences of this doctrine cf. the article “ Paranatellonta” by W. Gundel in RE 18, 3 (1949), col. 1241-1275.Google Scholar
  19. 12.
    Cf. above p.32.Google Scholar
  20. 13.
    We ignore here the change of the solar longitude between sunrise and sunset.Google Scholar
  21. 14.
    Above p.36(1).Google Scholar
  22. 15.
    Alm. VII, 3 Heib. II, p.33, 3ff. For the longitude cf. below p.60.Google Scholar
  23. 16.
    III, 1, 61 and VIII, 8, 3 (Nobbe, p. 151, 26 and p. 205, 7f.).Google Scholar
  24. 17.
    Almagest II, 9.Google Scholar
  25. 18.
    From Alm. II, 11.Google Scholar
  26. 19.
    One can express this also in the form that α′ is reckoned from the winter solstitial point 0°, because α′(0)=0°.Google Scholar
  27. 20.
    For an example cf. below p. 979.Google Scholar
  28. 1.
    Cf. above I A 4, 3.Google Scholar
  29. 2.
    Cf., e.g., the tables in Alm. II, 13 (below p. 50ff.).Google Scholar
  30. 3.
    Cf. above I A 4, 1. A similar list, relating M and φ, is given in the “Geography” (I 23; Mik, Ptol. Erdkunde, p. 65 f.), all values of cp being rounded to the nearest multiple of 0;5°. The boundaries are φ = −16;25° and φ = +63°; cf. below p. 935.Google Scholar
  31. 4.
    Cf. above I A 3, I.Google Scholar
  32. 5.
    E.g. VIII 16, 3-14 (Nobbe, p. 221-223); several numbers are garbled.Google Scholar
  33. 6.
    Probably written sometime between A.D. 500 and 600.Google Scholar
  34. 10.
    Converted to sexagesimal fractions in Manitius translation and in our Table 2. In No. 8 so = 30;50 is the correct value, found in MS D, whereas Heiberg and Manitius accepted the obviously wrong version 36;50. In No. 11 so = 43 1/2 1/3 = 43;50 is taken from Heiberg p. 109,9 whereas Ptolemy in Alm. II, 5 (Heiberg, p. 100, 15) had found (correctly) 43;36.Google Scholar
  35. The correct computation for so = 43;36 and of s, and s2 for rp = 36° is given in Alm. II, 5 (cf. above p. 24, No. 1) and agrees for s, and s2 with the values in Alm. II, 6, No.11.Google Scholar
  36. 1.
    Cf. below p. 50.Google Scholar
  37. At sphaera recta not only the meridian but also the horizon contains the north pole N; therefore both circles are perpendicular to the equator. Cf. Fig. 39 which depicts this situation in stereographic projection.Google Scholar
  38. 2.
    Instead of the above found angle m, =32;10° for Rhodes and 1i0° the table gives only 122;7-90= 32;7°.Google Scholar
  39. 1.
    The values for 2n2 are rounded to the nearest degree while 23;51° is taken for e.Google Scholar
  40. 2.
    Cf. below p.51.Google Scholar
  41. 3.
    For clima I Op <6) this rule has to be modified to α + β = 2γ ± 180 since Ptolemy counts angles in such a fashion as to avoid negative values. Cf. also below p. 992.Google Scholar
  42. 4.
    The Greek text has no technical term for “zenith distance” but says simply “arc.” 2 Above p. 40.Google Scholar
  43. 1.
    Cf. below p. 529.Google Scholar
  44. 2.
    Cf. also below p. 294.Google Scholar
  45. 3.
    For an apparent confirmation of this constant of precession from the motion of the apsidal line of Mercury cf. below p. 160.Google Scholar
  46. 1.
    The sidereal mean motion would be smaller since one tropical year would correspond only to a progress of 360-1/100°. One finds in this way 0;59,8,11,27, ....Google Scholar
  47. 2.
    Tannery, AA, p. 163.Google Scholar
  48. 3.
    I, 10 (p. 47, 3 Heib.) and III, 1 (p. 209, 13 ff. Heib.).Google Scholar
  49. 1.
    Cf. IVB2,1.Google Scholar
  50. 2.
    Cf. p. 361.Google Scholar
  51. 3.
    Cf. H Intr. 5.Google Scholar
  52. 4.
    The inverse influence is practically excluded since it would mean the transformation of the arithmetical methods into a simple geometrical argument; but this is not feasible in a simple fashion.Google Scholar
  53. 1.
    Cf. Neugebauer [1962, 2], p. 267. Actually the motion is slightly faster than precession.Google Scholar
  54. 2.
    Below p.73 ff.Google Scholar
  55. 3.
    Below p. 173 ff.Google Scholar
  56. 4.
    The values given here are the ones used by Ptolemy in this computations. The tables of mean motions (Aim. III, 2) would give 93;8,33 and 91;10,16, respectively.Google Scholar
  57. Accurate computation with these tables results, however, in dK = 211;25,43°. Cf. p. 63 where it is shown that Ptolemy’s result is exact if one includes the equation of time.Google Scholar
  58. 3.
    It is important to realize that the “mean sun” in ancient terminology is not the same as the “mean sun” in modern astronomy. The latter moves in the equator and coincides with the true sun at 2=0. 3 The same result can be obtained from dK+x°=320;12+265;15=225;27.Google Scholar
  59. 4.
    Heiberg II, p. 33, 3 ff. = Manitius II, p. 28, 14 ff.Google Scholar
  60. 5.
    For the solution of this problem, assuming 2=x23° given, cf. above p. 41.Google Scholar
  61. 6.
    Correctly about 1;10” = 17;30°. Thus Ptolemy’s error is only 2;30° (not 4° as Manitius II, p. 27 note a) says).Google Scholar
  62. 7.
    Halma III, p. 34 and Halma I, p. 38 where 24° = 1 1/2 1/10” appear as approximation of 24;10° = 60;30-36;20 (correcting Halma’s errors by means of Vat. gr. 208, fol. 52 and Vat. gr. 1291) (Honigmann, SK, p. 197, 94 and 198, 168).Google Scholar
  63. 8.
    Cf. Neugebauer [1938], p. 22. Cf. below p. 848.Google Scholar
  64. 1.
    Cf. p. 563, n. 3.Google Scholar
  65. 2.
    I do not know where this term originated; it is found neither in Ptolemy nor in Theon. The Islamic term is “equation of day” (e.g. Battâni, Nallino II, p. 61) and similar in Byzantine tables (ópOoociç rry”ç 4jzèpaç) and in Latin works (Toledan Tables, verbatim from the Arabic: equationes dierum cum noctibus suis). Cf. also Wolf, Hdb. d. Astr. II, No. 494.Google Scholar
  66. 1.
    Cf. above p.62.Google Scholar
  67. 2.
    Below p. 67.Google Scholar
  68. 3.
    Cf. above p. 59.Google Scholar
  69. 4.
    Almagest IV, 6; cf. below p. 77. The dates are −720 March 19, −719 March 8 and September 1 respectively.Google Scholar
  70. 5.
    Cf. p. 63.Google Scholar
  71. 1.
    As always K = 2 − II 5;30 = 2 − 65;30°.Google Scholar
  72. 2.
    Ptolemy, Opera II, p. 162, 23-163, 6 ed. Heiberg. Cf. also below p. 984f.Google Scholar
  73. 3.
    Cf. Neugebauer [1958], p. 97ff.Google Scholar
  74. 4.
    Flamsteed’s treatise “De inaequalitate dierum solarium” (London 1672) is supposedly the first modern treatment of the subject; cf. Wolf, Handbuch 2, p. 261.Google Scholar
  75. 5.
    Cf. p. 63.Google Scholar
  76. 6.
    Cf. above p. 62.Google Scholar
  77. 7.
    Because of the equivalence theorem (p. 57) we need not distinguish between an eccenter-and an epicycle-model.Google Scholar
  78. 1.
    Above p. 55.Google Scholar
  79. 2.
    Below p. 79.Google Scholar
  80. 3.
    Below p. 81.Google Scholar
  81. 4.
    Above p. 55.Google Scholar
  82. 1.
    Above p. 57f.Google Scholar
  83. 2.
    Cf. below p. 546.Google Scholar
  84. 3.
    Cf. p. 481 f.Google Scholar
  85. 4.
    Below p. 310.Google Scholar
  86. 6.
    It seems possible that the relation (5), quoted below p. 310, is the result of these observations. Cf. p. 310.Google Scholar
  87. 8.
    Cf. below p.78.Google Scholar
  88. 1.
    Below p. 76ff.Google Scholar
  89. 2.
    Cf. also below p.77.Google Scholar
  90. 3.
    The interior of the triangle would have the signature − − −(or + + + since we are dealing in fact with the projective plane).Google Scholar
  91. 4.
    Fig. 68 does not give the positions of P1, P2, P3 as required in the case of the eclipses I, II, III (for which cf. Figs. 66 and 67).Google Scholar
  92. 1.
    Cf. above p. 74.Google Scholar
  93. 2.
    As usual these computations contain many small inaccuracies such that r=5;13 would be the nearest common solution.Google Scholar
  94. 3.
    Below p. 79.Google Scholar
  95. 4.
    Below p. 79.Google Scholar
  96. 5.
    That is “Marduk gave an heir”; biblical distortion: Merodach-baladan.Google Scholar
  97. 6.
    Accurate value: 57 minutes.Google Scholar
  98. 7.
    Above p. 75f.Google Scholar
  99. 8.
    From P. V. Neugebauer, Kanon d. Mondf.Google Scholar
  100. 9.
    Cf. P. V. Neugebauer, Astr. Chron. II, p. 128.Google Scholar
  101. 1.
    For the method cf. above p. 76 (2).Google Scholar
  102. 2.
    Since dt is close to an integer number of years the influence of the equation of time can be ignored.Google Scholar
  103. 3.
    Actually the tables give d.I.=123;22,33 and da=103;35,23.Google Scholar
  104. 4.
    Above p.63.Google Scholar
  105. 1.
    490 Apr. 25.Google Scholar
  106. 2.
    A.D. 125 Apr. 5.Google Scholar
  107. 3.
    Modern values: 1.7” and 2.0”, respectively.Google Scholar
  108. 4.
    If we assume, e.g., a nodal motion of −0;3,10,40°” we find from multiplication by the time interval It =1,2,23,29° a motion of about −15° (mod. 360). This suffices to exclude opposite nodes for two lunar positions of nearly the same longitude (cf. the dates given in notes 1 and 2). Cf. also below p. 82 n. 4.Google Scholar
  109. 5.
    Ptolemy’s corrections for the equation of time are slightly inaccurate because of the use of unit fractions of hours but the effect on the anomalies is negligible.Google Scholar
  110. 6.
    Drawn to scale.Google Scholar
  111. 1.
    This is the same eclipse used before as No. II for the determination of r (cf. p. 74, p. 77, and Fig. 70, p. 1228). The julian date is − 719 March 8.Google Scholar
  112. 2.
    501 Nov. 19/20. Compare Fig. 75 with Fig. 70 II, p. 1228.Google Scholar
  113. 3.
    Actually only 1.5” and 2.1” respectively.Google Scholar
  114. 4.
    To show this one has to remark that the nodal motion during dt=218 310d amounts to nearly 90°. At the eclipse (C) the moon, and thus one node, was near tip 15 (cf. above p. 77). Consequently, this node was at the eclipse (D) near .015. But the moon was at (D) near 4f 23 thus near the opposite node.Google Scholar
  115. 5.
    Actually one finds 160;3,9.Google Scholar
  116. 6.
    Cf. above p. 77.Google Scholar
  117. 7.
    For the more primitive method used by Hipparchus cf. below p. 313 f.Google Scholar
  118. 8.
    As we have shown on p. 64, the equation of time has only a negligible effect.Google Scholar
  119. 1.
    Solar eclipses remain outside of these discussions because they depend also on geographical elements.Google Scholar
  120. 1.
    Observational data which supposedly confirm this round value for the extremal latitude of the moon are mentioned only later by Ptolemy (Alm. V, 12; cf. below p.101).Google Scholar
  121. 2.
    Cf., e.g., Almagest IV, 6 (Man. I, p. 218 f.) or V, 2 (Man. !, p. 260ff.).Google Scholar
  122. 3.
    Below I C 7.Google Scholar
  123. 4.
    Alm. IV, 6 (Man. I, p. 219).Google Scholar
  124. 5.
    The error thus committed reaches only about 0;6° for w between about 30° and 60°.Google Scholar
  125. 6.
    Above p. 30.Google Scholar
  126. 1.
    Cf. above p. 80 and Fig. 72.Google Scholar
  127. 2.
    Cf. below p.1109f.Google Scholar
  128. 3.
    Below p. 88ff. For a comparison with modern theory cf. p. 1108.Google Scholar
  129. 4.
    Below p. 155.Google Scholar
  130. 5.
    Cf. below p. 88.Google Scholar
  131. 1.
    Below, p. 91ff.Google Scholar
  132. 2.
    Cf. p. 80.Google Scholar
  133. 3.
    In his discussion of two observations (below p. 87) Ptolemy does not make use of this criterium but simply computes the epicyclic anomaly a for the given dates and finds them near ±90°. I do not see the practical advantage of the formulation (1) over the direct computation of a.Google Scholar
  134. 4.
    A.D. 139 Febr.9.Google Scholar
  135. 5.
    Cf. above p. 50.Google Scholar
  136. 6.
    For some textual difficulties cf. below p. 92. -127 Aug. 5.Google Scholar
  137. 8.
    Above p.76.Google Scholar
  138. 9.
    In the Canobic Inscription (below p. 903) the parameters r and e are renormed such that R—e obtains the value 60.Google Scholar
  139. 10.
    Thirteenth and fourteenth century; cf. Roberts [1957].Google Scholar
  140. 11.
    Copernicus, De Revol. IV, 3, IV, 8, IV, 9. Cf. also Neugebauer [1968, 2].Google Scholar
  141. 12.
    The same term also occurs in the theory of eclipses (below p. 141) but with totally different meaning.Google Scholar
  142. 13.
    Angles are drawn nearly to scale but the eccentricity, and particularly the radius of the epicycle, are exaggerated.Google Scholar
  143. 14-.
    126 May 2 and July 7, respectively.Google Scholar
  144. 15.
    Cf. below p. 92.Google Scholar
  145. 16.
    Including the equation of time.Google Scholar
  146. 17.
    Above p. 87.Google Scholar
  147. 18.
    Above p. 87 and p. 89.Google Scholar
  148. 19.
    Above p. 89, No. II.Google Scholar
  149. 20.
    Above p. 48 ff.Google Scholar
  150. 21.
    From p(H)=a(M)+90; cf. p.42.Google Scholar
  151. 22.
    Above pp. 87, 89.Google Scholar
  152. 23.
    Above p. 89.Google Scholar
  153. 24.
    Aim. III, 13 for the angle between ecliptic and altitude circle, Alm. V, 18 for parallax.Google Scholar
  154. 25.
    For the Hipparchian theory of parallax cf. below I E 5, 3.Google Scholar
  155. 26.
    Ptolemy’s rounded values (A(=1IL 10°, 1;30” west) would give an angle of 83°.Google Scholar
  156. 28.
    Ideler, Astron. Beob., p. 217 and Chron., p. 345. Also Ginzel, Hdb. II, p. 410.Google Scholar
  157. 29.
    Heiberg, p. 363, 18f.Google Scholar
  158. 3.
    Cf. p. 80.Google Scholar
  159. 4.
    Cf. above p. 88.Google Scholar
  160. 5.
  161. 6.
    Above p. 87 f. It is, however, not quite correct to subtract the first maximum equation from the second because they do not belong to the same value of a. At the syzygies the maximum occurs near a =96°, at the quadratures near 102°.Google Scholar
  162. 7.
  163. 8.
    The differences for the tabulated values c6 (Alm. V, 8) show several constant stretches. This indicates linear interpolation between values accurately computed for greater intervals than 3° or 6°.Google Scholar
  164. 9.
    It is again only approximately correct to deal with the maximum equations as if they belonged to the same epicyclic anomaly, independent of the elongation.Google Scholar
  165. 10.
    A.D. 98 Jan. 14.Google Scholar
  166. 11.
    Cf. p. 84.Google Scholar
  167. 12.
    Alm. VII, 3 (Heib. II, p. 33, 19 ).Google Scholar
  168. 13.
    Ptolemy 2;10° as Heiberg and Manitius interpret the text (cf. below p. 117, note 7).Google Scholar
  169. 14.
    Above p.84.Google Scholar
  170. 15.
    The seventh day, marked by H, corresponds to Hipparchus’ observation of May 2 (above p. 89). Our computation includes the equation of time, using Ptolemy’s approximation —0;206 for the whole interval. Actually it would change from —0;216 in No. 1 to —0;246 in No. 16.Google Scholar
  171. 1.
    The maximum of cQ=5;1 (cf. p.80), of co = 2;23 (cf. p.59).Google Scholar
  172. 2.
    Here, as well as in the next case, Ptolemy computes all corrections ab ovo from the given parameters of the model. Actually the tables in V, 8 give the same result.Google Scholar
  173. 1.
    Cf. below I E 5, 3 and I E 5, 4 B.Google Scholar
  174. 2.
    The basic assumption made by Eratosthenes that Alexandria and Syene lie on the same meridian does not agree with Ptolemy’s Geography IV, 15,15 where Syene is placed 1/10 of one hour to the east of Alexandria, i.e. 1;30° in longitude (in agreement with Geogr. IV, 5,9 and IV, 5, 73).Google Scholar
  175. 3.
    Fig. 93 does not pretend to reconstruct the technical details of the instrument. This has been attempted by A. Rome [1927], certainly successfully in the main elements. The fact that the use of an arm AC of the same length as AB restricts the instrument to zenith distances <60° might have to do, according to Rome, with an intentional avoidance of refraction. It remains difficult, however, to see how this instrument could have produced results of greater accuracy than direct readings on a quadrant.Google Scholar
  176. 4.
    Below p.104.Google Scholar
  177. 5.
    Below p.115.Google Scholar
  178. 1.
    30;58° corresponds exactly to an equinoctial noon shadow 5:3, mentioned by Vitruvius (Arch. IX, 7). In the Geography Ptolemy gives cp = 31° (IV, 5, 9, p. 251, Nobbe); the same value is found in the Handy Tables (Halma I, p.119). “Lower Egypt” has cp=30;22° (Aim. II, 8). Actually Alexandria is at cp=31;13°.Google Scholar
  179. 2.
    A maximal southern latitude would place the moon at a zenith distance of about 60°, thus at the limit of Ptolemy’s instrument and perhaps too near to the horizon to ignore refraction (cf. above p.100, note 3).Google Scholar
  180. 3.
    A.D. 135 Oct. 1.Google Scholar
  181. 4.
    Accurate computation with the tables of chords (Alm. II, 11) leads to EP=39;49,31rex39;50re (modern tables: 39;49,48).Google Scholar
  182. 5.
    P. V. Neugebauer, Astron. Chron. I, p. 72f.Google Scholar
  183. 6.
    For p1 one finds about —0;3,36° whereas Ptolemy assumes p,= O.Google Scholar
  184. This condition simplifies the problem insofar as the angle is directly known under which the center of the disk crosses the horizon, namely 90—cp. The diameter is then given by At cos qp when At is the time required for the rising or setting of the whole disk (obviously an extremely ill-defined quantity).Google Scholar
  185. 1.
    A popular (and obviously meaningless) version of this procedure is the story that it takes the sun a 1/720th part of one day to cross the horizon, from which one concludes that the solar diameter is 1/2°. Cf., e.g., P. Oslo 73 (for the literature cf. Neugebauer [ 1962 ], No.24); also Hultsch [1899], p. 193 (but misleading hypotheses).Google Scholar
  186. 2.
    The instrumental problems are discussed by Hultsch [1897], [1899], [1900]; by Rome, Pappus Comm., p.87ff.; by Lejeune, Euclide et Ptol., pp. 131, 151.Google Scholar
  187. 3.
    Below p.106.Google Scholar
  188. 4.
    Proclus, Hypotyp. I, 19 Manitius, p. 10, 18) mentions the observation of annular eclipses by “earlier” astronomers. In IV, 98 (Manitius, p. 130, 18) Sosigenes (teacher of Alexander of Aphrodisias, thus before A.D. 200) is said to have observed one. The only eclipse possible is the one of A.D. 164 Sept. 4, annular for Greece, cf. Ginzel, Spez. Kanon pl. X I.Google Scholar
  189. 5.
    Accurate computation, however, gives only 80;35, mainly because the equation of time amounts to —0;20h instead of Ptolemy’s —0;15h. Pappus in his commentary (Rome p. 102, 7) accepts Ptolemy’s number without checking.Google Scholar
  190. 10.
    Cf., e.g., the list in Houzeau, Vade-mecum, p.404f.Google Scholar
  191. 11.
    Ed. Rome, p. 100, 10 to 103, 11 and p. 184, 1 to 187, 7. Cf. also above p.104, note 5. The value 0;41 for b„ is not expressly given by Pappus but results from repeating his computations for case II.Google Scholar
  192. 12.
    From P. V. Neugebauer, Kanon d. Mondf. The eclipse I was considered invisible by Kepler (Werke 5, p. 270f.) “luna enim sub terra fuit.” The cause of Kepler’s error lies in the insufficiently known geographical longitudes; he assumes, e.g., that Alexandria lies 2h to the east of Hven, instead of actually only 1h (e.g. Werke 3, p. 419, 3). The commentary in Werke 5, p. 453/4 is wrong.Google Scholar
  193. 9.
    Rome p. 107, 10ff.Google Scholar
  194. 10.
    Above p. 58.Google Scholar
  195. Hypotyposis, ed. Manitius, p. 222, 4 and p. 224, 13. Comm. Tim. ed. Diehl III, p. 62, 30 trsl. Festugière IV, p. 86 (also Hypotyposis ed. Manitius, p. 131, 1). Both works of Proclus also contain errors: in the Hypotyposis (Manitius, p. 222, 3) 1210 re is incorrectly called maximum, instead of mean, distance; and in the Comm. Tim. (ed. Diehl III, p. 62, 30 and p. 63, 12) he gives 1076 re as minimum distance, obviously invented to fit the approximate computation given in the commentary. Thâbit ben Qurra omits this last number, although he otherwise follows Proclus in all the preceding steps, including the use of 1260 re for the maximum distance of the sun (Thâbit b. Qurra, ed. Carmody, p. 137, De hiis, Nos. 43-45). For Proclus cf. below p. 920.Google Scholar
  196. 12.
    De revol. I, 10 (Gesamtausg. II, pp. 22, 27f.); also 64;10 re as maximum distance of the moon (I.c. p. 22, 25 f.).Google Scholar
  197. 13.
    De revol. IV, 21 (Gesamtausg. II, p. 257).Google Scholar
  198. 14.
    Cf. above p. 104, note 4.Google Scholar
  199. 1.
    Epitome IV, 1, IV (Werke 7, p. 279).Google Scholar
  200. 2.
    Cf. below p.148 f.Google Scholar
  201. 3.
    This, incidentally, implies that the solar parallax is also considered to be smaller than directly observable; indeed, its values are only computed from the distances found by the Hipparchian method (above p.109). Cf. also Hipparchus’ assumptions about the solar parallax (below I E 5,4 B).Google Scholar
  202. 4.
    For this part of the work, preserved only in Arabic, cf. below p. 918.Google Scholar
  203. 5.
    Ptolem., Opera II, p. 118 (Heiberg).Google Scholar
  204. 6.
    Cf. below V B 7, 6.Google Scholar
  205. 7.
    Above p. 109, (1).Google Scholar
  206. 8.
    Above p. 112.Google Scholar
  207. 9.
    This whole procedure has nothing to do with the Eudoxan-Aristotelian concentric spheres since it is based solely on Ptolemy’s model of Mercury and Venus and the parameters given in the Almagest.Google Scholar
  208. 1.
    Above p. 50.Google Scholar
  209. 2.
    Cf. above p. 104, (a).Google Scholar
  210. 3.
    Cf. above p. 110.Google Scholar
  211. 4.
    Above p. 93 ff.Google Scholar
  212. 5.
    This concerns an observation made by Menelaos in Rome (A.D. 98 Jan. 14). Cf. above p. 96 for Ai and p. 43, for M. Explicit examples of parallax computations are rare; two are found in Pappus, Comm., ed. Rome, p. 115, 6 to 117, 12 and p. 125, 15 to 126, 10.Google Scholar
  213. 3.
    Pappus, Comm., ed. Rome, p. 166, 16ff.Google Scholar
  214. 4.
    Above p.115.Google Scholar
  215. 5.
    Cf. p. 96.Google Scholar
  216. 6.
    Alm. VII, 3 (Man. II, p. 29).Google Scholar
  217. 7.
    Heiberg’s edition (II, p.33, 19-21) gives for β 2 and 6−, for β′ 1 and 3— or (in MS D) 1 These numbers must be interpreted as 2;6° and 1;3°, respectively, not as 2 1/6=2;10° and 1 1/3=1;20° (as in Manitius II, p.29) because this would give only p6= −0;50°.Google Scholar
  218. 8.
    Ptolemy assigns to β Sco in A.D. 137 the longitude 116;20. Thus he assumed for the observation 40 years earlier a longitude of 115;55. The actual coordinates in A.D. 100 are 116;46 and + 1;15°. The latitude of the moon was in fact about 0;7° greater than computed by Ptolemy. These two corrections bring the star near the center of the moon.Google Scholar
  219. 9.
    Cf. Alm. VII, 3 (Manitius II, p. 28). Cf. above p. 79, p. 82, and p. 60.Google Scholar
  220. 2.
    This relation was probably known in Egypt long before hellenistic times; cf., p. 563.Google Scholar
  221. 3.
    For syzygies the second inequality is zero.Google Scholar
  222. 1.
    Cf. above p.104, (1) and (2); p. 105, (3) and (4); p. 109, (2).Google Scholar
  223. 2.
    Above p. 80.Google Scholar
  224. 3.
    Above p. 59.Google Scholar
  225. 4.
    Cf. our discussion of these statements below p.127.Google Scholar
  226. 5.
    Pappus, Comm., ed. Rome p. 194-197.Google Scholar
  227. 6.
    For a proof of this theorem and further discussion cf. Neugebauer, Al-Khwar., p. 122f.Google Scholar
  228. 7.
    He finds p,=0;28,43 (instead of Ptolemy’s 0;30) and 0;15,30 (for Ptolemy’s 0;15). Correct would be 0;28,43,56 and 0;15,3(!), respectively.Google Scholar
  229. 8.
    This includes the rounding errors mentioned on p.127.Google Scholar
  230. 1.
    For Ptolemy’s much refined investigation of consecutive syzygies cf. below p.133, F.Google Scholar
  231. 2.
    Example: Oppolzer, Canon, Nos. 2271 and 2272 (A.D. 265 Oct. 12 and 266 March 8).Google Scholar
  232. 3.
    We ignore here, as always in these discussions, the influence of geographical longitude which can exclude eclipses because of the time of the day.Google Scholar
  233. 4.
    Again supplemented in Pappus’ Commentary (ed. Rome, p. 226 to 231).Google Scholar
  234. 5.
    Of course, as always, lunar minus solar parallax.Google Scholar
  235. 6.
    Example: Oppolzer, Canon, Nos. 5356 and 5357 (A.D. 1049 March 6 and August 1); cf. also p.133, n. 7.Google Scholar
  236. 7.
    Example: Oppolzer, Canon, Nos. 4678 and 4681 (A.D. 752 Jan. 21 and August 14). Between these two eclipses, however. occur two more eclipses (Febr. 20 and July 15) such that we have a sequence of four eclipses with intervals 1 month, 5 months, 1 month, respectively.Google Scholar
  237. 8.
    Examples passim in Oppolzer, Canon; e.g. Nos. 6201 and 6202, or 6501 and 6502. Cf. also p. 133, n. 7.Google Scholar
  238. 1.
    Cf. below p. 140.Google Scholar
  239. 2.
    Exactly the same numerical values are found as coefficients of interpolation in the table of parallaxes (Alm. V, 18) column 7 though associated with a/2 instead of a; cf. above p. 113/114.Google Scholar
  240. 6.
    Using the values from (1), p.125 but with small roundings in the results.Google Scholar
  241. 9.
    For solar eclipses one has to replace s by re.Google Scholar
  242. 10.
    Duration” is here always meant in the sense of the elongations tabulated in the Almagest.Google Scholar
  243. 1.
    Alm. VI, 7 Heib., p. 512, 8.Google Scholar
  244. 2.
    One finds 6;23 and 15;36, respectively. For the first value Ptolemy gives incorrectly 6;10, an error discussed by Pappus (Rome CA I, p. 261 ff.).Google Scholar
  245. 3.
    Ptolemy approximates rz by 3;8,30, a value which he motivates (Heib., p. 513, 2-5) as mean value between the Archimedean approximations 3 1/7 (=3;8,34,…) and 3 10/71 ( =3;8,27, …).Google Scholar
  246. 1.
    Unfortunately from the viewpoint of our historical interests, Ptolemy completely ignored the problems of first and last visibility of the moon.Google Scholar
  247. 2.
    Above p.88.Google Scholar
  248. 3.
    E.g. Tetrabiblos II, 10 (p. 92, 2, 13 Boll-Boer).Google Scholar
  249. 4.
    E.g. Tetrabiblos II, 13, 14 (pp. 99, 3f.; 100, 8f.; 102, 3 Boll-Boer). Cf. also the “wind” toward which points the latitudinal component of the lunar parallax (below p. 999, n. 29).Google Scholar
  250. 5.
    Heiberg I, pp. 512, 9; 536, 21; 537, 8; 545, 3, 4. In fact this holds for all the above-mentioned appendices: cf for area digits Heiberg I, p. 512, 9, for heliacal phenomena II, p. 204, 7.Google Scholar
  251. 6.
    E.g. to Pappus; cf. his Commentary to Alm. VI ed. Rome, p. 309. Cf. p. 997 f.Google Scholar
  252. 1.
    Cf. Roberts [ 1957 ]; Neugebauer [1968, 2].Google Scholar
  253. 2.
    Cf. below V C 4, 5 B and 5 C.Google Scholar
  254. 3.
    Cf. p. 386ff.Google Scholar
  255. 5.
    Cf. Appendix VI B 7, 2.Google Scholar
  256. 6.
    Cf. below p. 208 and p. 212.Google Scholar
  257. 7.
    Cf. below p. 1101.Google Scholar
  258. 8.
    Computed for A.D. 100 from Almagest XI, 11 using Ptolemy’s constant of precession.Google Scholar
  259. 9.
    Case (1); for the elliptic approximation of the deferent of Mercury cf. below p. 168.Google Scholar
  260. 10.
    Below p. 227ff.Google Scholar
  261. 11.
    Above p. 109ff.Google Scholar
  262. 12.
    Cf. p.111.Google Scholar
  263. 13.
    For more historical details cf. below p.270f.Google Scholar
  264. 14.
    As we shall see (below p. 421) the Babylonian planetary computations make use of precisely this fact.Google Scholar
  265. 15.
    Cf. below I D 3, 1.Google Scholar
  266. 16.
    In the lunar theory also the apsidal line is movable, but proceeds with the difference velocity (cf. above p. 68).Google Scholar
  267. 17.
    Cf. above p. 58.Google Scholar
  268. 18.
    Details to be discussed later; cf. below p.207.Google Scholar
  269. 19.
    Cf. above p. 54.Google Scholar
  270. 20.
    In the so-called “Goal-year-texts”; cf. below p. 351 and p. 554.Google Scholar
  271. 21.
    Cf. for the outer planets below p.180f. and p. 182; for the inner planets p. 157 and p.167f.Google Scholar
  272. 22.
    Cf. above p. 54.Google Scholar
  273. 23.
    Cf. above p. 55.Google Scholar
  274. 24.
    Cf. for the outer planets below p.182, for Venus p.157, for Mercury p. 167.Google Scholar
  275. 25.
    This is confirmed by the fact that, e.g., the quotient (6) is not accurate since one would obtain... 41,33,... instead of... 41,43,40. Furthermore in this division 0;0,12d is disregarded in the denominator. Hence the accurate result would be only... 39,46,... instead of... 41, 43, 40.Google Scholar
  276. 26.
    Cf. below p.155; p. 171.Google Scholar
  277. 27.
    Identical with the mean longitude of the sun at epoch (cf. p.60).Google Scholar
  278. 1.
    A.D. 136. The date of this observation of Venus in “maximum elongation” as evening star is only 37 days earlier than the one used just before: Hadrian 21 Tybi 2/3 (A.D. 136 Nov. 18; cf. Fig. 136) and Mechir 9/10 (Dec. 25; cf. Fig. 135), respectively. Obviously Ptolemy uses here the term “maximum elongation” only in a vague sense. In fact the actual maximum elongation occurs about midway between the two dates as Table 13 shows. Ptolemy had to select different dates in order to obtain elongations symmetric to Theon’s observations. Incidentally: the latter also were made somewhat later than the accurate moment of the greatest elongation.Google Scholar
  279. 2.
    In the case of Mercury, however, the same procedure has its advantages; cf below p. 161. The same geometrical problem occurs once more in the derivation of (9), p. 155. The absence of algebraic notations obscures such parallelisms.Google Scholar
  280. 3.
    The resulting extremal geocentric distances of Venus are M=R+e+r=104;25, m=R —e—r=15;35 hence M/m x 6;42.Google Scholar
  281. 4.
    Ptolemy’s own roundings would give as final result e=1;16,23 and r=43;10,18.Google Scholar
  282. 5.
    Again it is only Brahe and Kepler who returned to Ptolemy’s attitude and required that the distances OM and ME should be determined empirically. In applying this principle to the orbits of Mars and of the earth Kepler went far beyond Ptolemy. Cf. Kepler, Werke 3, Astronomia Nova, Chaps. 16 and 23.Google Scholar
  283. 6.
    Cf. below p. 161. In the final presentation of the theory in the Almagest Mercury precedes Venus (to be followed by Mars, Jupiter, Saturn). For the chronology of the observations cf. Fig. 16 (p. 1375).Google Scholar
  284. 1.
    Below p. 172 ff.Google Scholar
  285. 2.
    Above p. 151. Z Above p. 153.Google Scholar
  286. 3.
    For the sake of clarity the eccentricity is exaggerated in these figures but the angles at O are drawn essentially correctly.Google Scholar
  287. 4.
    We have for the distance of P from 11: 61=m.20;55—np4;10=76;45 and 52=1.25-1,6;30=18;30.Google Scholar
  288. 5.
    Cf. p. 151.Google Scholar
  289. 6.
    The approximate period relation leads to about 0;36,58°/d.Google Scholar
  290. 7.
    Above p. 60, (5).Google Scholar
  291. 8.
    Cf. also Fig. 16 (p.1375).Google Scholar
  292. 1.
    Leverrier: “Nulle planète n’a demandé aux astronomes plus de soins et de peines que Mercure, et ne leur a donné en recompense tant d’inquiétudes, tant de contrariétés” (Annales de l’observ. de Paris 5, p. 1, quoted by Tisserand [1880], p. 35).Google Scholar
  293. 2.
    Cf. below II A 5, I C and IIA7,6.Google Scholar
  294. 3.
    Cf. Neugebauer [1968, 21.Google Scholar
  295. 4.
    Tetrabiblos IV, 3 (Robbins, p. 381).Google Scholar
  296. 5.
    These observations by Ptolemy belong to the years A.D. 132, 134, 135, 138, and 141; cf. above p.158 and Fig. 16 (p.1375).Google Scholar
  297. 6.
    Cf. p. 161 and IC 3, 5.Google Scholar
  298. 7.
    Not with Nisan (April) as in the Mesopotamian version of the Seleucid era which is used in the cuneiform texts.Google Scholar
  299. 8.
    Cf for details p.1066.Google Scholar
  300. 9.
    Kepler seems to have been the first to recognize an independent displacement of the planetary apsides (Houzeau, Vade-Mecum, p. 384). Real insight came, of course, only with Newton.Google Scholar
  301. 1.
    Using Ptolemy’s own tables of chords accurately one finds 39;10,10, 99;13,20, and 10;23,20 respectively, very close to the values obtainable by modern tables.Google Scholar
  302. 2.
    Above p.85.Google Scholar
  303. Gerard of Cremona gave the correct reading in his translation of the Almagest (Venice 1515, p. 106) while George of Trapezunt had missed it (ed. 1451, p. 101). Copernicus, De revol. V, 27 (Gesamtausgabe, p. 343, 10) adopted the correct date (as noted by Menzzer note 444, only to be again overlooked in the Gesamtausgabe).Google Scholar
  304. 1.
    The velocity of this motion is, of course, equal to the mean motion of the sun. Above p.159.Google Scholar
  305. 2.
    Below p. 168 f.Google Scholar
  306. 1.
    Cf. above p. 159 f.Google Scholar
  307. 2.
    Obtained from the catalogue of stars in Alm. VIII, 1 by subtracting 4° of longitude for precession during 400 years (instead of 5;30°). This is one of the cases which illustrate the intricate interplay between constant of precession and values of specific parameters, here mean motion in anomaly.Google Scholar
  308. 3.
    Heiberg, p. 288, 20/289, 1.Google Scholar
  309. 4.
    Copernicus, De revol. V, 29 (Gesamtausg., p. 346) simply inverted the data: 2 lunar diameters east, 1 north. But the continuation of the text excludes this remedy, even if it were not too drastic.Google Scholar
  310. 5.
    Using Tuckerman, Tables.Google Scholar
  311. 6.
    According to Schoch, Planetentafeln. Cf. above p. 160.Google Scholar
  312. 8.
    Cf. above p. 159.Google Scholar
  313. 9.
    Cf. p.151.Google Scholar
  314. 10.
    In checking this division I find for the last three digits only 58,39,48.Google Scholar
  315. 1.
    Hartner [1955], p. 109 to 117.Google Scholar
  316. 2.
    Obviously the Cartesian coordinates of K are ML=(R + e) cos a, and LK =(R — e) sin ~i, respectively.Google Scholar
  317. 1.
    Cf. below p.389; also above p. 150f., (1) and (2).Google Scholar
  318. a The small circles give the position of Saturn in 20-day intervals, for Jupiter in 10-day steps. The degrees of latitude are represented in units twice as large as the longitudes.Google Scholar
  319. b The planet is visible from F to S2, invisible from Q to F. The graph shows clearly the motion of an epicycle along an inclined deferent and the return to a loop of similar shape and position after 30 years and after 12 years, respectively. This periodicity would be still more outspoken after a period of 59 years.Google Scholar
  320. 2.
    Cf. below V A 1, 4.Google Scholar
  321. 3.
    For the corresponding term “equant” cf. below p. 1102.Google Scholar
  322. The observations are specified as made at extremal distances of the epicycle, i.e. when C is a point of the straight line OE. In this case the determination of the eccentricity OM which accounts for the observed retrograde arc causes no difficulty (whereas a general position of C leads to a rather complicated computation). By reason of symmetry it is clear that also M must lie on OE. This is no proof, however, that OME are always on one line but I do not know whether such a proof has ever been attempted.Google Scholar
  323. Hill [ 1900 ] has shown that the problem can be made definite by requiring that the center M of the circle through C„ C2, C, is the midpoint of OE. In this form the problem leads to an algebraic equation of the 8th degree and one of its 6 real roots corresponds to Ptolemy’s solution which he obtained by an iteration process; cf. below p. 178, note 6. I owe the reference to Hill’s paper to Mr. Stephen Gross.Google Scholar
  324. 2.
    Cf. above p. 57 ff.Google Scholar
  325. 3.
    Determinations of the solar eccentricity from differently located observations occur in Islamic astronomy (e.g. Birúni, Chronol., p. 167), also by Copernicus (De revol. III, 16 = Gesamtausg. II, p. 190f.), and Brahe (Progymn. I=Opera II, p. 19ff.); cf. Neugebauer [1962, 2], p.274f.Google Scholar
  326. 4.
    Above p. 73 ff.Google Scholar
  327. 5.
    Computed by Mr. E. S. Ginsberg.Google Scholar
  328. 6.
    Hill [1900] has shown, for the case of Mars, that the modern solution agrees excellently with Ptolemy’s results, i.e. with the third approximation. Hill finds e=60.0.1000026 (instead of Ptolemy’s 60.0.1) and for the apogee êv 25;29,33.01 (instead of Ptolemy’s 25;30; cf. below p. 179, (6)). But the above given fourth approximation shows slightly larger deviations, a fact that underlines the accidental character of purely numerical comparisons.Google Scholar
  329. 9.
    Cf. above p.177, (6) and (7), p. 179.Google Scholar
  330. 10.
    Actually the tables of the Almagest (IX, 4) would give 1;22° and 1;33°,respectively. This is one of the many cases where unnecessarily inaccurate data are used for a computation of seemingly higher accuracy. For the final result the present deviations are without effect.Google Scholar
  331. 11.
    The determination of OC is required by the absence of the tangent function as well as by the following steps.Google Scholar
  332. 12.
    Ptolemy does not determine corrections for the mean motion A in longitude since, for an outer planet, A is the difference Ao —â of the known mean motion A0 of the sun and of the mean motion â in anomaly of the planet.Google Scholar
  333. From a strictly logical viewpoint this procedure appears to be circular since Ptolemy determined the parameters of his model by means of angles S, and SZ (cf. above p. 174) which require the knowledge of A; hence â is no longer free. In fact, however, no high accuracy of A is required for S, and 83; hence it is legitimate to determine in a second step â as accurately as possible and then correct A accordingly such that d+A=Ao is exactly satisfied, as is the case in the tables of Alm. IX, 4.Google Scholar
  334. 13.
    The position of A is known under the assumption, made by Ptolemy for all planets (cf. above p. 160 and below p. 182), that the apsidal line participates with all fixed stars in the motion of precession.Google Scholar
  335. 3.
    Alm. XI, 10 (Heib. II, p. 429).-For the Handy Tables cf. below p.1002, (1). Cf. below p. 185.Google Scholar
  336. 1.
    Cf. above p.182, n. 15.Google Scholar
  337. 2.
    Ptolemy gives the list of the maxima of O in Alm. XI, 10 (Heib. II, p.433, 15-19). Out of the 15 values 8 differ from the values obtainable from the tables XI, 11 by 1, 2, or 3 minutes. In the case of Mars and Venus even the value of c6 itself is differently given in the text (41;10 and 46;0) and in the tables (41;9 and 45;57).Google Scholar
  338. 3.
    For a=93 one finds only 0=5;52 which would mean that there existed one maximum at 90 and a second at 96. It follows, however, from the differences of c6(a) that c6(90) should be 6;9 or 6;10 but not 6;12. On the basis of this correction one finds only one maximum at =96.Google Scholar
  339. 4.
    In the few cases where Ptolemy’s values differ from the value in our table one has to use Ptolemy’s values if one wishes to recompute the table in the Almagest; cf. note 2.Google Scholar
  340. 5.
    I recomputed a sequence of values of c8(ic) for Mercury from C=120 to is=180 in steps of 12°. The results of these rather longish computations of Bo deviate only once by as much as 0;1° from Ptolemy’s values. The deviations from c8 reach in one case 0;0,17 and are otherwise 0;0,4, 0;0,2, and 0;0,1. It seems clear that Ptolemy had computed at least one more digit than the tabulated values show.Google Scholar
  341. 6.
    Above p. 95.Google Scholar
  342. 1.
    For Saturn: Manitius II, pp. 281, 7 and 283, 14; for the remaining planets shortened to “maximum/minimum distance.” Cf. also below p. 193.Google Scholar
  343. 2.
    Below p. 195 and p. 197ff.Google Scholar
  344. 3.
    Cf. above p. 146.Google Scholar
  345. 4.
    Ptolemy’s procedure differs from the one given here only in so far as he first finds PT and then p.Google Scholar
  346. 1.
    Cf. above p.192 and note I there.Google Scholar
  347. 2.
    Cf. p. 183 f.Google Scholar
  348. 5.
    Cf. p. 192.Google Scholar
  349. 6.
    Above p. 193. For the numerical details cf. below p. 199 (Table 18). Cf. above p. 193.Google Scholar
  350. 8.
    Cf. above p. 183 and below p. 204.Google Scholar
  351. 1.
    We consider here only longitudes. The real motion in longitude and latitude need not produce stationary points at all; cf., e.g., the orbits shown in Fig. 228 (p. 1283 ).Google Scholar
  352. 2.
    Described in the case of maximum distance of Mars in Alm. XII, 6 (Man. II, p. 301f.).Google Scholar
  353. 3.
    Above p. 193, values of b in the table.Google Scholar
  354. 4.
    Assuming, of course, the values of c3 and c4 as found in Alm. XI, 11. These values themselves, however, show the effects of irregular roundings and interpolations. Cf. also p. 200, n. 7.Google Scholar
  355. 5.
    From (2b), p.194.Google Scholar
  356. 6.
    Above p. 193.Google Scholar
  357. 7.
    One would not only obtain Ptolemy’s result but also smoother differences for c3+c4 if one could replace c3(24)+c4(24)=4;16 by 4;15. Unfortunately the Handy Tables confirm the value 4;16.Google Scholar
  358. 8.
    For Ptolemy’s procedure cf. p.165 and Fig.149.Google Scholar
  359. 9.
    Above p.193.Google Scholar
  360. 10.
    It is probably only accidental that Ptolemy’s correction at maximum distance (0;24) is 3 times the correction at minimum distance (0;8).Google Scholar
  361. 11.
    Cf. above p.164, (1) and Table 14 (p.169).Google Scholar
  362. 1.
    Above pp. 193 and 195.Google Scholar
  363. 2.
    From Table 18 (p. 199), slightly rounded.Google Scholar
  364. 3.
    Cf. Tables 17 and 19 (pp. 197 and 201).Google Scholar
  365. 4.
    Cf. (8), p.195.Google Scholar
  366. 5.
    From Table 17 (p. 197), rounded.Google Scholar
  367. 6.
    Cf. Tables 17 and 20 (pp.197 and 201).Google Scholar
  368. 7.
    Cf. (9), p. 196.Google Scholar
  369. 8.
  370. 1.
    He mentions only some special cases, e.g. for k=30° (Manitius II, p. 258).Google Scholar
  371. 2.
    Cf. below p. 232.Google Scholar
  372. 3.
    Since, for a short interval of time, a increases like and k, i.e. proportional with time, the function a(k) will be very nearly a linear function. Cf. below the vertical graphs in Figs. 204 and 205.Google Scholar
  373. 1.
    Cf. Fig. 207, p. 1272, moving C into A or B.Google Scholar
  374. 3.
    In some cases the result is 0;0,12 less than expected.Google Scholar
  375. 4.
    In this figure, as in all our similar graphs, the scale of the latitude is twice the scale of the longitudes. The modern positions are taken from the Tuckerman Tables.Google Scholar
  376. 5.
    One finds a cubic equation for the sine of the angle under which the eccentricity e= 3 is seen from C.Google Scholar
  377. 6.
    Cf. Figs. 188 and 189 (p. 1265 f.).Google Scholar
  378. 1.
    Cf. above p. 148.Google Scholar
  379. 2.
    Proclus, Hypotyp. V, 121T. (Manitius, p. 142 f .) repeats and elaborates (in part incorrectly) Ptolemy’s arguments but he does not attempt a numerical confirmation.Google Scholar
  380. 3.
    Kepler, Opera 7, p. 592-594; also Werke 4, p.429-433 and for the theory Halley [1691]. Ptolemy himself eventually found in the brightness of the sun the true cause for the impossibility of a naked eye observation of transits of Mercury and Venus (Planetary Hypotheses, Goldstein [1967], p. 6 ). On medieval reports on alleged transits cf. Goldstein [ 1969 ].Google Scholar
  381. 4.
    Fig. 234 shows only the cases from K0=0 to 3. For K0 =180 one has a mirrored arrangement with slightly greater equations for the sun and the planet.Google Scholar
  382. 5.
    Cf. above p. 151 (4).Google Scholar
  383. 6.
    The alternative of one single transit occurs at a nearly central position of the path across the sun. For this case one obtains as maximum duration of a transit, using Ptolemy’s parameters, about 7;40h which is only about 0;15” too short.Google Scholar
  384. 7.
    Cf. p. 151 (4).Google Scholar
  385. 8.
    The details of the arrangement of Mercury’s transits are much more intricate than for Venus. The actual intervals are 3 1/2 and 7 years or 6, 9 1/2, and 13 years. The maximum durations of about 8h at apogee, 6” at perigee, easily follow from Ptolemy’s parameters.Google Scholar
  386. 2.
    It should be noted that we are no longer considering “elongations” with respect to the mean sun as was the case in I C 2, 1 (p. 153) and I C 3,1 (p. 160).Google Scholar
  387. 3.
    AA =1f 25 (cf. p.153).Google Scholar
  388. 4.
    Ptolemy needs two steps here, having no tables for tan a. 3 115;30 (cf. p. 58).Google Scholar
  389. 5.
    Below p. 241.Google Scholar
  390. 8.
    Above p. 159 and Fig.144, p. 1252.Google Scholar
  391. 9.
    Cf. above p.165 f. and Table 14 (p. 169). In principle one could reconstruct the table for p(k) from the relation p=e sin k/sin ri with, known from Alm. XI, 11 as c3(k)+c4(k). Unfortunately the roundings in the tabulated values of c3 and c4 have a great influence on the small values of sin ry and hence produce large errors for p, much too large for the accuracy required in our present problem.Google Scholar
  392. 10.
    The entry 19;14 for Mercury as evening star in Capricorn is incorrect and should be 18;54. Halma H.T. III, p. 32 gives 18;14. Fig. 238 shows the correct value.Google Scholar
  393. 1.
    I do not know where this term originated. Nallino, Battani II, p. 256 considers it to be of Arabic origin. Petavius says “arcus ille, qui fulsionis, vel visionis vulgo nuncupatur” (De doctrina temporum III, Var. Diss. Lib. I, cap. III, p.5 [Verona 1736]). Neither Brahe nor Kepler seem to use the term, though Regiomontanus knows it (Epitoma in Almag., XIII, propos. 23 [1496]).Google Scholar
  394. 2.
    As we shall see later Ptolemy in the Handy Tables determined planetary phases for each of the seven climata (cf below p. 257 and V C 4, 5 C) and phases for the fixed stars for the five climata II to VI (cf. below V B 8, 1).Google Scholar
  395. 3.
    Cf., e.g., Strabo, Geogr. II, 5, 39 (Loeb I, p. 511 ).Google Scholar
  396. 4.
    Cf. below p. 367; cf., however, below p. 249, note 12.Google Scholar
  397. 5.
    Cf. for this value below p. 236.Google Scholar
  398. 6.
    Accurately 0;46,57,23 and 0;37,21,3, respectively. Cf. above p. 218.Google Scholar
  399. 7.
    Cf. below p. 237 f.Google Scholar
  400. Ptol. Opera II, p. 153, 15.Google Scholar
  401. 10.
    Cf. below p.1017 (2).Google Scholar
  402. 11.
    Cf. below p. 535.Google Scholar
  403. 1.
    Cf. Table 3, p. 47 and Fig. 41, p. 1218.Google Scholar
  404. 2.
    Cf. below pp. 239 and 241.Google Scholar
  405. 3.
    Cf. below p. 245ff.Google Scholar
  406. 5.
    Cf. p. 219, (7).Google Scholar
  407. 6.
    Fig. 241 is drawn to scale, representing the situation at first visibility of the planet. At last visibility the point C and the direction 00 would have a mirrored position with respect to the line OP. Cf. also below note 10.Google Scholar
  408. 7.
    Cf. above p. 208 and Fig. 213, p. 1275.Google Scholar
  409. 8.
    Above p. 180.Google Scholar
  410. 9.
    According to p. 153, A=It25.Google Scholar
  411. 12.
    Following the rules given p. 222 ff.Google Scholar
  412. 13.
    Above p. 159.Google Scholar
  413. 14.
    By constructing a figure to scale, similar to Figs. 241 and 242, one finds that actually a z 39°. The influence of this correction on the value for β is negligible.Google Scholar
  414. 1.
    I,17; I, 22; VII, 18. Translation Manitius, pp. 11, 13, 221, respectively.Google Scholar
  415. 2.
    Cf. IC7,2B.Google Scholar
  416. 3.
    Cf. Fig. 218 (p. 1278 ).Google Scholar
  417. 4.
    Above p.226 (2), and p. 215 (2).Google Scholar
  418. 5.
    Cf. above p. 154 (8).Google Scholar
  419. 6.
    E.g. Vat. gr. 1291 fol. 89”.Google Scholar
  420. 7.
    This was recognized by A. Aaboe [1960], p. 20.Google Scholar
  421. 8.
    Below p.403.Google Scholar
  422. 9.
    Cf. Fig. 218 (p. 1278 ).Google Scholar
  423. 10.
    According to the tables for maximum elongation, Alm. XII, 10. Cf. also Fig. 238 (p. 1288 ).Google Scholar
  424. 11.
    The computation of these latitudes causes no difficulties since the basic parameters, K° and a (cf. p. 223) are readily available as Fig. 248 shows. In the first case, P1 in M0°, one has —rc01=A11 —20=20;58-20l and a, =90+ d 1, x 111. In the second case, P2 in if 0°, u° 2 =180+ 20+422 x 222 and a2 =180+90 —11.12 x 248. Computing with these elements one finds from the tables Alm. XIII, 5 β, x — 3;1 and β2x - 3;7 in agreement with Ptolemy’s rounded values.Google Scholar
  425. 12.
    Above pp. 236 and 240.Google Scholar
  426. 13.
    Above p.235 (4 b). In the Handy Tables (cf below p. 257) the value of h for Mercury is increased to 12°, raising the lower limits of visibility given in (3) to — 25;33 and + 25;48, respectively.Google Scholar
  427. 1.
    For some corrections that must be made in the text as accepted by Heiberg (II, p. 606 f. z Manitius II, p. 394) cf. p. 248, note 9 and 11, p. 252, note 2, p. 256, note 2. According to our norm (p. 240) the elongations A A for the evening phenomena (IF, and 0) are reckoned negative.Google Scholar
  428. 2.
    Cf. below p. 256f.Google Scholar
  429. 3.
    Cf. below p. 261.Google Scholar
  430. 4.
    Cf. also below p.260f.Google Scholar
  431. 5.
    Alm. XIII, 9 (Manitius II, p. 393).Google Scholar
  432. 6.
    IBi can reach about 5;30° for Mercury and almost 10° for Venus.Google Scholar
  433. Cf., e.g., p. 235 for all planets, p. 240 (2) for Venus, p. 241 (1) for Mercury.Google Scholar
  434. 8.
    In Fig. 245, p. 1291 A is associated with the rising point of the ecliptic, not with the planet. 6 These are, of course, not coordinated phases Q and T but two independent cases.Google Scholar
  435. 9.
    The MSS give for d Ar(8) either 20;8 or 20;16 but d dn(lfl.) = 20;19. The first value, though accepted by Heiberg and Manitius, cannot be correct since it would mean that v = 34;50 instead of Ptolemÿ s 34;30. For 20;16 one finds v=34;34 but v=34;28 for 20;19 which I therefore use for Table 24, p. 243.Google Scholar
  436. 10.
    Aaboe [1960], p. 7.Google Scholar
  437. 11.
    The first entry of Saturn d Ar = 23;1 is definitely wrong as the computation shows (cf. below Table 25, p.251). Obviously one has to accept the variant 23;30 given by Halma H.T. III, p. 30; cf. also the MSS D and K. An emendation 23;[2]1 would agree better with computation and also with Jupiter (sin v =0;32,37,58, hence v = 32; 57 ).Google Scholar
  438. 12.
    This fits also very well Ptolemy’s geographical data for Phoenicia; cf. above p. 44, Table 2, No. 10 (q, = 33;18) and Geogr. V 15, 5 (ed. Nobbe, p. 58). Babylon, however, is given a latitude of 35° (Geogr. V 20, 6 ed. Nobbe, p. 78).Google Scholar
  439. 13.
    Above p. 236.Google Scholar
  440. 1.
    For the computation of βo we need c3(0) and c4(0) not given in the tables (cf. above p. 246, note 7). I found c3(0)=0;8 (=c3(6)) and c4(0)=0;3.Google Scholar
  441. 2.
    The best of the attested values for F in if is 20;16. It should be the same as for Q in IL which is 20;19 and which shows better agreement; cf. also above p. 248, note 9.Google Scholar
  442. 6.
    The maximum deviations are: for E once +0;9°, for 3 twice +0;7°. The arithmetical mean of the deviations is +0;1° for 8, zero for E.Google Scholar
  443. 7.
    The rounded values (±6;20) for the latitudes at Q and r in X and up used by Ptolemy in explaining the greatly variable duration of invisibility of Venus (above p.239) are not accurate enough for the present purpose. Indeed the tables require the latitudes β1z6;18 βm6;29 in X and β,. —6;30 in r:, —6;20 in tlp.Google Scholar
  444. 8.
    Cf. p. 253 (2).Google Scholar
  445. 9.
    For the significance of a negative elongation at r cf. above p.241.Google Scholar
  446. 10.
    For F in up +0;6, for Sl +0;4 and —0;11, respectively.Google Scholar
  447. 1.
    Cf. above p. 241 and note 11 there.Google Scholar
  448. 3.
    The omitted cases are marked by a xGoogle Scholar
  449. 4.
    Above p.254; below p. 259.Google Scholar
  450. 9.
    Cf. below p.1024.Google Scholar
  451. 10.
    Halma, H.T. III, p. 30 to 32.Google Scholar
  452. 11.
    Ptolemy, Opera I, 2, Heiberg, p. 606/607.Google Scholar
  453. 12.
    Cf., e.g., Cumont [1934].Google Scholar
  454. 13.
    CCAG 5, 4, p. 228, 15 to 19; cf. also below V A 3, 2.Google Scholar
  455. 14.
    In CCAG 8, 4, p. 180, 19 and 29 the longitudes of Aldebaran and of Antares are given, respectively as Taurus and Scorpio 16;20°, i.e. 3;40° greater than in the Almagest (VII, 5/VIII, 1; cf. also p.980). Hence the epoch is A.D. 138+366=504 (Cumont [ 1918 ], p.43). We also have horoscopes in the works of Rhetorius which confirm this date; cf. Neugebauer-Van Hoesen, Gr. Hor., p. 187f.Google Scholar
  456. 15.
    CCAG 7, p.214 to 224.Google Scholar
  457. 16.
    Cf. above p. 235 (1).Google Scholar
  458. 16a.
    One version, Monac. 287 and Vat gr. 208, is published CCAG 7, p. 119ff. and Neugebauer [ 1958, 2]. A slightly different version comes from a group of notes to the Handy Tables, published by Tihon [1973], No.XIV. Cf. below pp. 1053f.Google Scholar
  459. 17.
    Cf. also Nallino, Batt. II, p. 255 to 268.Google Scholar
  460. 18.
    Cf. Kennedy-Agha [1960], p. 138, Fig. 2.Google Scholar
  461. 19.
    Above p. 244 ff.Google Scholar
  462. 20.
    Ptolemy, Opera II, p. 4 Heiberg.Google Scholar
  463. 21.
    Plan. Hyp. I; cf. Goldstein [ 1967 ], p. 8.Google Scholar
  464. 22.
    Cf above p. 257.Google Scholar
  465. 1.
    No printed edition of the Arabic text exists; Halley’s edition (Oxford 1710) gives only a Latin translation. An epigram on the Conic Sections from the Byzantine period is found in the Greek Anthology (Loeb III, p. 323, No. 578).Google Scholar
  466. 2.
    Kepler, Astronomia Nova (Werke III), Chaps. 59 and 60.Google Scholar
  467. 3.
    A careful discussion of these biographical data has been given by G.J.Toomer in the Dictionary of Scientific Biography I (1970), p.179f.Google Scholar
  468. 4.
    Cf for this earlier phase below IV B 3, 4.Google Scholar
  469. 5.
    Apollonius, Opera II (ed. Heiberg), p. 139 frgm. 60. Cf. also below pp. 650 and 655.Google Scholar
  470. 6.
    Opera II, p. 139 frgm. 61 or Photius, ed. Henry, vol. III, p. 66 (Collection Budé). The connection of the letter s with the moon probably originated in the coordination of the seven vowels of the Greek alphabet with the seven planets; cf. the restoration of P. Ryl. 63 in Neugebauer-Van Hoesen [1964], p. 64, No. 131 and Dornseiff, Alph., p. 43.Google Scholar
  471. 7.
    Vettius Valens, Anthol., ed. Kroll, p. 354, 4-7; Cumont [1910], p.16I. Cf. also below p. 602.Google Scholar
  472. 8.
    On Sudines and Kidenas cf. below p. 611; on the norm with 8° below IV A4, 2A.Google Scholar
  473. 9.
    Cf. Cumont [1910], p. 163, n. 2; also Kroll, RE Suppl. V, col. 45 (No. 114) and Honigmann in Mich. Pap. III, p. 310. The date of the Myndian is extremely insecure, based on a huge web of very tenuous arguments.Google Scholar
  474. 1.
    CCAG 5, 1, p. 204, 16; 5, 2, p. 128, 16 and note 1; CCAG 1, p. 80, 8 and p. 113, note 1. 1 Cf. below p. 658, n. 15.Google Scholar
  475. 2.
    Cf. above Fig. 51, p. 1220.Google Scholar
  476. 3.
    E.g. in Copernicus, De revol. III, 15. Theon of Smyrna (2nd cent. A.D.) says that Hipparchus considered it worth the attention of mathematicians to investigate the cause of so greatly different explanations of the phenomena. Theon gives the impression that Adrastus (around A.D. 100) first proved the mathematical equivalence (ed. Hiller, p. 166, 6-12; Dupuis, p. 268/269). This only goes to show that even an ancient author may have an incorrect view of the chronological sequence of events. ° Alm. XII, 1 (Manitius II, pp. 268, 1 and 272, 18 ).Google Scholar
  477. 4.
    Apollonius, Opera I, p. 402-413; ed. Heiberg; trsl. Ver Eecke, p. 249-255.Google Scholar
  478. 1.
    Alm. III, 3 (Manitius I, p. 162).Google Scholar
  479. 2.
    Manitius II, p. 270. The relation (2 a) motivates the term “reciprocal radii” since e =1/R for r=1. Above I B 3, 4 A.Google Scholar
  480. 2.
    Cf. above IB4,1.Google Scholar
  481. 3.
    Cf. for his results below p. 315.Google Scholar
  482. 4.
    Alm. IV, 6, Manitius I, p. 223.Google Scholar
  483. 5.
    Rome CA III, p. 1053-1056.Google Scholar
  484. 10.
    For the sake of greater clarity the points on the circle in Fig. 268 have been spaced more conveniently than in Fig. 267.Google Scholar
  485. 1.
    Cf. above I C 6.Google Scholar
  486. 2.
    Cf. Fig. 195, p. 1268.Google Scholar
  487. 3.
    Cf., e.g., Fig. 134, p. 1248.Google Scholar
  488. 4.
    Cf. for these problems and the role of Eudoxus: Hasse-Scholz, Die Grundlagenkrisis der Griechischen Mathematik, Charlottenburg 1928 (Pan Bücherei, Philosophie No. 3).Google Scholar
  489. 5.
    Alm. XII, 1 (Manitius II, p. 277). Cf. also above p.191.Google Scholar
  490. 6.
    Cf. above p. 264 f.Google Scholar
  491. 7.
    Alm. XII,1 (Manitius II, p. 272 f.).Google Scholar
  492. 1.
    Cf. below p. 643.Google Scholar
  493. 2.
    Cf. above I B 3, 4 A and p. 267.Google Scholar
  494. 3.
    Cf. Fig. 68, p. 1227 (and similarly Fig. 268, p. 1303 ).Google Scholar
  495. 5.
    Aaboe [1963], p. 8 f.Google Scholar
  496. 6.
    ACT, No. 801, Sections 4 and 5 for Saturn, No. 810, Sections 3 and 4 for Jupiter. Cf. below p. 832.Google Scholar
  497. 1.
    Almagest VII, 5/VIII, 1.Google Scholar
  498. 2.
    For example the note on Hipparchus by Suidas (ed. Adler II, p. 657, No. 521) gives his time as “under the consuls” which is not only meaningless but also contradicts Suidas’ way of dating (cf. Rohde, Ki Schr. I, p. 134, no. 1). Aelian, De natura animalium VII, 8 (ed. Herscher, Didot, p. 119, 20 or Teubner I, p. 175, 2) puts an anecdote about Hipparchus under “Neron the Tyrant”. Following Herscher this is usually emended to “Hieron the Tyrant” (in order to find at least some motivation for the error).Google Scholar
  499. 3.
    Cf. below IE6,3.Google Scholar
  500. 4.
    Cf. below p.823.Google Scholar
  501. 5.
    Suidas, ed. Adler II, p. 657, No. 521; cf. also the preceding note 2.Google Scholar
  502. 6.
    Ptolemy, Opera II, p. 67, 10 and 16 to 18 (ed. Heiberg). Cf below p. 928.Google Scholar
  503. 7.
    Cf. Table 28, below p. 276.Google Scholar
  504. 8.
    E.g. Zeitschr. f. Numismatik 9 (1882), p. 127f. Coins with the picture of Hipparchus are known from the reigns of Antoninus (138 to 161), Commodus (180 to 192), Marinus (217), Alexander Severus (222 to 235), Gallus (251 to 253).Google Scholar
  505. a Maass, Aratea, p. 121.Google Scholar
  506. b Geogr. 14, 2,13 (Loeb VI, p. 279/281. Hipparchus is mentioned, of course, among the learned men of Bithynia (Geogr. 12, 4, 9; Loeb V, p. 467).Google Scholar
  507. 9.
    Cf. below p. 301.Google Scholar
  508. 10.
    Hipparchus, Arat. Comm. ed. Manitius, p. 184/5; cf. also p. 292, note 3.Google Scholar
  509. 11.
    The inscription of Keskinto, e.g., shows that other astronomers had worked at Rhodes (cf. below p. 698).Google Scholar
  510. 12.
    Alm. VII, 2 and 3 (Manitius II, p. 15, 9 and 20, 21).Google Scholar
  511. a A marginal note to the Royal Canon of the “Handy Tables” (in a version of the 9th cent.) assigns the lifetime of Hipparchus to the reign of Euergetes II, equated with the years 179 to 207 of the era Philip (i.e. —145/4 to —117/6); cf. Monumenta 13, 3, p. 451, 9.Google Scholar
  512. 13.
    Below p. 281.Google Scholar
  513. 14.
    Vogt [1925], col. 25. Cf. below p.284.Google Scholar
  514. 15.
    Hipparchus, Arat. Comm., Manitius, p. 182 to 270.Google Scholar
  515. 16.
    Hipparchus, Arat. Comm., Manitius, p. 270 to 280. Cf. also below p. 279, note 22.Google Scholar
  516. 17.
    Alm. VII, 3 (Manitius II, p. 18 to 20).Google Scholar
  517. 18.
    Rome [1937], p. 217 quotes a passage by Theon (Comm. Alm. III,1 ed. Rome, p. 817, 11 f.) in which he refers to the equinox observations as made by Hipparchus. But Theon’s source is obviously only the Almagest and hence not an unambiguous new witness.Google Scholar
  518. 19.
    Delambre HAA I, p. XXII to XXIV.Google Scholar
  519. 20.
    Fotheringham [1918], p. 408.Google Scholar
  520. 21.
    Cf. below p. 284.Google Scholar
  521. 22.
    Alm. III, 1 (Manitius I, p. 133, 32).Google Scholar
  522. 1.
    Ptolemy, Opera II, p. 1 to 67. Cf. also below V B 8,1 B.Google Scholar
  523. 2.
    Cf. below p. 301, n. 1.Google Scholar
  524. 3.
    Cf. below p. 301, n. 2.Google Scholar
  525. 4.
    The Greek title is not certain; cf. Rehm in RE 8, 2, col. 1670, 58. Ptolemy, Alm. VII, 1 (Heiberg, p. 3, 9) quotes “On the fixed stars”, Suidas (ed. Adler II, p. 657) “On the arrangement of the stars and the Catasterism (?).”Google Scholar
  526. 5.
    Below p. 283.Google Scholar
  527. 6.
    Below pp. 285 ff.Google Scholar
  528. 1.
    Cf. below p. 368.Google Scholar
  529. 2.
    Cf., e.g., IV A 4, 2 A and 2 B.Google Scholar
  530. 3.
    E.g. Aratus Comm. ed. Manitius, p. 48, 8; 128, 25; 132,10, etc., e.g. p. 56, 15: “18° of Pisces, or, as Eudoxus divides the zodiacal circle, at 3° of Aries.”Google Scholar
  531. 4.
    Aratus Comm., p. 132, 7.Google Scholar
  532. 5.
    E.g. Aratus Comm., p. 48, 5 to 7.Google Scholar
  533. 6.
    E.g. Aratus Comm., p. 98, 19: “the bright star in the middle of the body of Perseus lies 40° to the north of the equator.”Google Scholar
  534. 7.
    E.g. Aratus Comm., p. 82, 24: “Arcturus is 59° distant from the northern pole while the bright star in the middle of the Altar is 46° distant from the southern pole.” Cf. also below p. 283.Google Scholar
  535. 8.
    Aratus Comm., p. 89.Google Scholar
  536. 9.
    This is motivated by variations in the positions of the solstices, supposedly observed by Eudoxus, according to a passage in his “Enoptron” quoted by Hipparchus (Aratus Comm., p. 88, 19).Google Scholar
  537. 10.
    Aratus Comm., p. 98, 21. Similarly p. 98, 2; 102, 9; 120, 16; 150, 26, etc. Similarly, for the summer tropic “1/2 and 1/12 of one zodiacal sign” (i.e. 17;30°) below the horizon (quoted by Strabo, Geogr. II 5, 42; Loeb I, p. 514/5).Google Scholar
  538. 11.
    Arat. Comm., p. 68, 20ff. ed. Manitius. Cf. also Vogt [1925], col. 29.Google Scholar
  539. 12.
    This has been done by Manitius, p. 288 f of his edition, but ignored in his translation. For a clear formulation cf. Vogt [1925], col. 27 to 29. Cf. also below (p.596, n. 19).Google Scholar
  540. 13.
    Aratus Comm., p. 8 to 182, ed. Manitius.Google Scholar
  541. 14.
    ÉRÉXE1 or KEITa,, respectively.Google Scholar
  542. 15.
    Aratus Comm., p. 182 to 280.Google Scholar
  543. 16.
    This result agrees with the tables in Alm. II, 8.Google Scholar
  544. 17.
    Cf. for this problem below p. 868 f.Google Scholar
  545. 18.
    Cf. below p.1081 and Fig. 12 there. The corresponding second coordinate, the “polar latitude” 6, seems not to be attested as such in the writings of Hipparchus (cf. the statistics of coordinates given given below p. 283). Instead he seems to prefer to define the position of a star by its declination or by its distance from the pole (cf., e.g., below I E 2,1 C I).Google Scholar
  546. 19.
    niimuç, (Arai. Comm., p. 272, 1); n,jyvri ov Slâarriµa (p. 190, 10).Google Scholar
  547. 20.
    ryµtnryXtov (Aral. Comm., p. 186, 11; 190, 8, 26 etc.); Siio µépry znryyewç (p. 254, 11, 25; 268, 8, etc.).Google Scholar
  548. 21.
    Cf. below p. 591; also p. 304.Google Scholar
  549. 22.
    Manitius, p.270 to 281. For the high accuracy of these hour-circles cf. Schjellerup [1881], p. 38f. The solstitial meridian is correct for the year —140. Cf. also above p. 276.Google Scholar
  550. 23.
    Manitius, p. 150, 2f.; cf. below p. 299.Google Scholar
  551. 24.
    Almagest VII, 3 (Manitius II, p. 18 to 20).Google Scholar
  552. 25.
    Cf. below p. 284.Google Scholar
  553. 26.
    Cf. below p. 283, note 13.Google Scholar
  554. 27.
    Cf. below p. 286. With a few exceptions all these stars are near the ecliptic.Google Scholar
  555. 28.
    Cf. below p. 934.Google Scholar
  556. 1.
    Remarked by Dreyer [1918], p. 348/9. Cf. Brahe, Opera II, p. 151, 10f.; p. 281, 11ff.; III, p. 335, 31 ff.Google Scholar
  557. 2.
    Copernicus, De revol. II, 14 (Gesamtausgabe II, p. 102, Thorn, p. 115).Google Scholar
  558. 3.
    Cf., e.g., the accusations of dishonesty in HAA I, p. XXXI. Cf. also Vogt [1925], col. 33.Google Scholar
  559. 4.
    No. 5354-55 (1925).Google Scholar
  560. 5.
    In section C, below p. 284.Google Scholar
  561. 6.
    Hipparchus, Arat. Comm., p. 186 to 270 ed. Manitius.Google Scholar
  562. 7.
    Almagest I, 12 (Manitius, p. 44).Google Scholar
  563. 8.
    Hipparchus, Arat. Comm., p. 184, 1: longest day=141/2 hours. This is the equivalent of cp=36° according to Arat. Comm., p. 72, 23f.Google Scholar
  564. 9.
    From the numerical examples given by Vogt one sees that the discrepancies between the alternative possibilities are usually very small or zero, reaching only in a few cases 10 or 15 minutes.Google Scholar
  565. 10.
    I.e. the “polar longitude” p of E. Cf. above p. 279.Google Scholar
  566. 11.
    Alm. VII, 3 (Manitius II, p. 18 to 20); Geography I, 7, 4 (ed. Nobbe, p. 15, 6 ).Google Scholar
  567. 12.
    Strabo, Geogr. II, 5, 41 (ed. Meineke, p. 181, 21 to 25).Google Scholar
  568. 13.
    These two exceptions (Alm. VII, 2 Manitius II, p. 12, 26-28 and p. 15, 1-3) are positions of Regulus and Spica, observed by Hipparchus in connection with the problem of determining the constant of precession. In Alm. VII, 3 (Man. II, p. 16, 27-17, 3) the permanency of the latitude of Spica (β= —2) is stressed, quoting Hipparchus.Google Scholar
  569. 14.
    In 17 cases Vogt had to replace the identifications by Manitius in his translation of the catalogue of stars in the Almagest by the identifications given in Peters-Knobel, Catal.Google Scholar
  570. 15.
    Cf. above p. 275.Google Scholar
  571. 16.
    Taken from Vogt [1925], Tables III and IV where the deviations for the single stars are arranged in decreasing order of Al A and d β.Google Scholar
  572. 17.
    Cf. for details Vogt [1925], col. 23 to 26.Google Scholar
  573. 18.
    Vogt [1925], col. 23: Hipparchus —0.06° (±0.065°), Ptolemy +0.01° (±0.03°).Google Scholar
  574. 19.
    I.e. around —138 and +48, respectively.Google Scholar
  575. 20.
    Cf. below p. 294.Google Scholar
  576. 1.
    Pliny NH II, 95 (Ian-Mayhoff I, p. 159, 12-14; trsl. Loeb Class. Libr. I, p. 239; trsl. Collect. Budé II, p. 41, p. 180f.).Google Scholar
  577. 2.
    Biot [1843], p. 61.Google Scholar
  578. 3.
    This date is generally accepted by Chinese scholars; cf. Yoke [1962], p. 145, No.41 or Hsi [ 1958 ], p. 114, No. 6.Google Scholar
  579. 4.
    First edition p. 563, 4th edition (1851), p.474. Also referred to by Humboldt, Kosmos III (1850), p. 221.Google Scholar
  580. 5.
    The Chinese sources seem not to exclude the possibility that the “Nova of —133” was only a comet. Cf. Needham SCC III, p. 425f.Google Scholar
  581. 6.
    Variant: uel aliam. The text is probably corrupt. Ph. H. Külb in Balss, Ant. Astr., p. 140/141 deletes whatever is found between stellam and in aevo; on the other hand he inserts die between qua and fulsit. Consequently he translates “… entdeckte auch einen neuen, zu seiner Zeit entstandenen Stern und wurde durch dessen Bewegung an dem Tage selbst, an dem er zum Leuchten kam, zu dem Zweifel veranlaβt …”Google Scholar
  582. 7.
    First discovery: Maass, Aratea (1892), p. 377; republished in Maass, Comm. Ar. rel., p. 134. Most recent edition by Weinstock, CCAG 9, 1, p. 189f. I count a total of 9 Greek and 2 Latin MSS. Cf. also Rehm [1899] and Boll [ 1901 ].Google Scholar
  583. 8.
    From CCAG 9, 1, p. 189 (ignoring variants).Google Scholar
  584. 9.
    Rehm [1899], Boll [1901].Google Scholar
  585. 10.
    Boll [1901], p. 192ff.Google Scholar
  586. 11.
    Almagest VIII, 1 (Manitius II, p. 64).Google Scholar
  587. 12.
    Hipparchus, Arat. Comm., p. 186 to 271 ed. Manitius. Also the number of stars within the single constellations (using the index Manitius, p. 364 to 372) shows the expected relation: in 7 cases the numbers are equal, in 2 cases the Commentary to Aratus has 2 more stars than the excerpts, in one case one more. For the remaining 35 constellations the totals in the excerpts exceed the number of stars mentioned in the Commentary; the latter was not intended to enumerate all stars in each constellation.Google Scholar
  588. 13.
    Heiberg II, p. 37, 15: noLlagry.Google Scholar
  589. 14.
    RE 6, 2, col. 2417. 1. Another suggestion was made by Dreyer [1917], p. 529 note (counting the external groups of stars as constellations, resulting in a total of 70) which seems to me less plausible.Google Scholar
  590. 15.
    Maass, Comm. Ar. rel., p. 128 (No. 12). The same formulation also in CCAG 8, 4, p. 94, fol. 10.Google Scholar
  591. 16.
    HAA I, p. 290.Google Scholar
  592. 17.
    Gundel, HT. His assumption (p. 135, p. 142, note 1) that the original number of stars must have been 72 seems to me unfounded.Google Scholar
  593. 18.
    The one exceptional case (Gundel HT, p. 25, 8 and p. 152, No. 63) gives 30 minutes beyond integer degrees.Google Scholar
  594. 19.
    From about 130 to 60 B.C.; cf. for details my Exact Sciences (2), p. 68f.; also below p. 287, n. 30. Gundel based his hypothesis of partly pre-Hipparchian origin on the comparison of rounded with not rounded numbers and dealing with the resulting differences as if they were exact.Google Scholar
  595. 20.
    Gundel, HT, p. 127ff.Google Scholar
  596. 21.
    y Canc; Almagest: 2=0910;20, β=2;40.Google Scholar
  597. 22.
    We also known that the Babylonian division of seasons took the summer solstice as the point of departure (Neugebauer [1948]).Google Scholar
  598. 23.
    Cf. below p. 309 f.Google Scholar
  599. 24.
    Cf. above p. 279.Google Scholar
  600. 25.
    aPlyccrtaizoi.Google Scholar
  601. 26.
    Alm. VII, 1, Manitius II, p. 5 to 8.Google Scholar
  602. 27.
    Cf. above p. 277, note 4.Google Scholar
  603. 28.
    Cf. above p. 283.Google Scholar
  604. 29.
    Alm. VII, 4 (Manitius II, p. 31f.).Google Scholar
  605. 30.
    Alm. VII, 1 (Manitius II, p. 4). The longitude of Spica, observed by Timocharis as UP 22;20 (in —293) and as 1722;30 (in —282) and referred to by Hipparchus as “about 1722 in the time of Timocharis” (Alm. VII, 3 and VII, 2, respectively) appears also with IIp22 in Gundel’s Hermes Trismegistos (p. 149, No. 15).Google Scholar
  606. 31.
    Below p. 577.Google Scholar
  607. 32.
    First remarked by Boehme [1887], p. 298. Cf. for the text Maass, Comm. Ar. rel., p. 183, 186, 189. Translation and commentary below p. 288 ff.Google Scholar
  608. 33.
    Rehm [1899], p. 265.Google Scholar
  609. 34.
    Hipparchus, Comm. Arat., p. 184, 23 ed. Manitius.Google Scholar
  610. 35.
    Alm. VII, 3 (Manitius II, p. 26 and p. 28, respectively).Google Scholar
  611. 36.
    Nallino, Battâni I, p. 124, p. 292; II, p. 269ff.; Dreyer [1917], [1918]; Vogt [1925], col. 37f. Cf. also Knobel [ 1877 ], p. 3f.Google Scholar
  612. 37.
    Nallino, Battâni II, p. 144 to 186. Much shorter lists of fixed star positions are found in earlier zijes, e.g. in the Mumtahan zij (about A.D. 830) for 24 stars. Cf. Kennedy, Survey, p. 146.Google Scholar
  613. 1.
    The division in sentences is mine, made for easier reference.Google Scholar
  614. 2.
    For the determination of stellar coordinates I have used a and S whenever given in P. V. Neugebauer, This is the case for 154 stars among 1008. For the distribution of the different magnitudes see the summaries at the end of each section (Manitius 11, pp. 43, 45, 64) and the total at the end of the catalogue.Google Scholar
  615. 2.
    For a comparison with modern standards cf. Peters-Knobel, Ptol. Cat., p. 120f. and the literature quoted there.Google Scholar
  616. 3.
    Manitius, p. 293f. in his edition of the Commentary to Aratus, gave a list of all occurrences of these terms in relation to the individual stars. He came to the conclusion that “bright” (a.aµinpoi) are the stars of the first three magnitudes. This result is, however, not too well founded because the term “bright” (and, repeatedly, for the same star “very bright”) occurs about 5 times as frequently than all four remaining terms together. What is really made evident, it seems to me, by Manitius’ statistics is the absence of an accurate terminology.Google Scholar
  617. 4.
    Gundel HT, p. 133 and p. 134. Only one star (3 Enid.) is called “magnitudinis primae” (in agreement with the Almagest). In the Commentary to Aratus this star is called “very bright.” Also Servius (around A.D. 400) denotes in his Commentary to Vergil’s Georgics I,137 (ed. Thilo, p. 164) a terminology as Hipparchian which is similar to the one in the Hermes Trismegistus. Cf. note 7.Google Scholar
  618. 5.
    Hipparchus, Comm. Ar., p. 238, 31.Google Scholar
  619. 6.
    E.g. p. 42 to 45.Google Scholar
  620. Servius, Comm. in Verg. Georg. I, 137 (ed. Thilo, p. 164): “nam Hipparchus scripsit de signis et commemoravit etiam, unumquodque signum quot claras,quot secundae Iucis, quot obscuras stellas habeat.”Google Scholar
  621. 8.
    De signis”; for the title of the “Catalogue of Stars” cf. above p. 277, note 4.Google Scholar
  622. 9.
    Only twice in the Commentary to Aratus; once more in a quotation from Attalus.Google Scholar
  623. 10.
    NH II, 95.Google Scholar
  624. 11.
    Cf above p. 285.Google Scholar
  625. 12.
    Maass, Comm. Ar. rel., p. 137.Google Scholar
  626. 13.
    In the Almagest three stars of the Pleiades are considered to be of the 5th magnitude, only one of the 4th.Google Scholar
  627. 14.
    Changed by Housman arbitrarily to sextumque which makes no sense.Google Scholar
  628. 15.
    Manilius V, 710-717 ed. Housman (V, p. 89-91), 711-719 ed. Breiter (I, p. 148, II, p. 178f.).Google Scholar
  629. 1.
    From observations made by Timocharis in Alexandria in the years —294/-282 (Alm. VII, 3, Manitius II, p. 22 to 27).Google Scholar
  630. 2.
    Alm. VII, 2 (Heib. II, p. 12, 21): nepì Tfjç itETa7rT(6aECOç ran TpoiriKCUV Kai ia11/4Eprvthv a11nwiojv.Google Scholar
  631. 3.
    According to a quotation from Hipparchus (Alm. III, 1, Heib., p. 207, 20) the title was nspì Tor) év,auaíou xpóvou βi13,1íov ìiv. Ptolemy quotes it as 7repi év,avaiov µsyé9ovç (Alm. III, 1, Heib. I, p. 206, 24; VII, 2 and 3, Heib. II, p. 15, 18 and 17, 21). From the last quoted references we know that this work was written when Hipparchus was aware of the existence of precession.Google Scholar
  632. 4.
    Cf. below p. 308.Google Scholar
  633. 5.
    Cf. below p. 296.Google Scholar
  634. 6.
    Alm. III, 1 (Manitius, p. 145) from the work on the length of the year.Google Scholar
  635. 7.
    Alm. III, 1 (Manitius, p. 132).Google Scholar
  636. 8.
    Astrological computations, e.g., are commonly based on sidereal coordinates. Cf., e.g., for the second century A.D., Vettius Valens (Neugebauer-Van Hoesen, Gr. Hor., p. 172, p. 180). The astrologers of the 5th century use, in general, more sophisticated astronomical methods and hence adopt with the tables of Ptolemy and Theon tropical coordinates.Google Scholar
  637. 9.
    Alm. III, 1 (Manitius, p. 146). The same value is also found in the Romaka Siddhânta (Panca-Siddhântikâ I,15 and VIII, 1; Neugebauer-Pingree I, p. 31, p. 85; II, p. 11, p. 59 ).Google Scholar
  638. 10.
    Alm. VII, 2 (Manitius II, p. 15).Google Scholar
  639. 11.
    Preserved only in Arabic; published by Walzer [ 1935 ]. Cf. also Neugebauer [1949, 1].Google Scholar
  640. 12.
    Anthol., ed. Kroll, p. 353,12f.; cf. below p.601.Google Scholar
  641. 13.
    The value 365 1/41/144 is, however, not attested in cuneiform sources. This is not very significant since we know only little about the Babylonian solar year. Cf. below II B 8.Google Scholar
  642. 14.
    Alm. VII, 1 (Manitius II, p. 4).Google Scholar
  643. 15.
    Cf., e.g., his doubts whether or not the poles of the ecliptic are really the center of motion (Alm. VII, 3 Manitius II, p. 17).Google Scholar
  644. 16.
    Alm. III, 1 (Manitius, p. 143ff.).Google Scholar
  645. 17.
    The diagonals in Fig. 283 represent intervals of exactly 365 1/4°.Google Scholar
  646. 18.
    Alm. III, 1 (Manitius, p. 135).Google Scholar
  647. 19.
    Cf., e.g., the statement about the variations found with the permanently mounted ring in Alexandria (Alm. III, 1 Manitius, p. 134, 1 to 8).Google Scholar
  648. 20.
    Below p. 295.Google Scholar
  649. 21.
    Cf. Alm. VII 2.Google Scholar
  650. 22.
    Alm. III, 1 (Manitius, p. 137f.).Google Scholar
  651. 23.
    The vernal equinoxes are the Nos. 1 and 2 listed in Table 29, p. 294 (cf. also Nos. 5 and 10 in Table 28, p. 276). The lunar eclipses are both total ( —145 Apr. 21 and —134 March 21). An additional detail is given by Theon in his commentary to this passage (cf. Rome CA III, p. 828, note (2)) where he tells us that (at least at the second of the two lunar eclipses) a star emerged from occultation by the moon at the moment of first contact (determined by Rome as h Virg).Google Scholar
  652. 24.
    Alm. IV, 11, Manitius I, p. 252.Google Scholar
  653. 26.
    Cf. below p. 318.Google Scholar
  654. 27.
    Ptolemy’s corrected data lead to very satisfactory dates for the solstices in — 382/381 (from R and S) and for the equinoxes in —200/199 (from U and V).Google Scholar
  655. 28.
    Alm. VII, 1 (Manitius II, p.4).Google Scholar
  656. 29.
    Cf. above p. 280, p. 287.Google Scholar
  657. 30.
    Above p. 294, note 15.Google Scholar
  658. 1.
    ed. Jahn, Proleg., p.V.Google Scholar
  659. 2.
    De die natali 18,9, ed. Hultsch, p. 38, 18f. One may assume that we have here a fragment from the work “On intercalary months and days” mentioned above p. 293.Google Scholar
  660. 3.
    Cf. below p. 624.Google Scholar
  661. 4.
    Cf. below II Intr. 3, 1.Google Scholar
  662. 5.
    The resulting length of the synodic month would be 29;31,51,3,49, … days.Google Scholar
  663. 6.
    Cf. above p. 293.Google Scholar
  664. 7.
    The insight that the Hipparchian cycle is motivated by his estimate for the length of the tropical year is due to Ideler, Chron. I, p. 352; also Ginzel, Hdb. II, p. 390.Google Scholar
  665. 8.
    Cf. below p. 310.Google Scholar
  666. 9.
    Cf. note 5.Google Scholar
  667. 10.
    Ginzel, Hdb. II, p. 391. Similar already Ideler, Chron. I, p. 353.Google Scholar
  668. 11.
    Dinsmoor, Archons, p. 410-423.Google Scholar
  669. 12.
    E.g. Archons, p. 414: “Therefore we may assume that the authorities … called in a specialist, namely, Hipparchos.”Google Scholar
  670. 1.
    Above p. 293.Google Scholar
  671. 2.
    Alm. VII, 2 (Man. II, p. 15).Google Scholar
  672. 3.
    The following is due to Viggo M. Petersen [ 1966 ]. Mr. boomer drew my attention to the fact that exactly the same conclusions had been reached by L. Am. Sédillot in 1840 (cf. his Matériaux... des sciences mathématiques chez les grecs et les orientaux, Paris 1845, p. 11-14). Van der Waerden [1970,2] accepting Petersen’s result as of absolute numerical accuracy expanded its consequences to all related parameters, in my opinion much too rigorously.Google Scholar
  673. 4.
    Alm. IV, 2 (Man. I, p. 196); cf. below p. 310 (5).Google Scholar
  674. 5.
    Above p. 295.Google Scholar
  675. 6.
    Cf. for the details below IV B 2, 3.Google Scholar
  676. 7.
    Cf. above Table 28, p. 276.Google Scholar
  677. 8.
    E.g. Rehm in R.E. 8, 2, col. 1669, 11 ff. who then discusses the “Lebensperiode” into which this work must have fallen. He takes from Theon’s commentary to Alm. I, 10 (ed. Rome CA II, p. 451, 4f.) as title Ilepi tnç npaypateiaç tiuv e69e16v βiβ lía iβ′. But Rome [1933], p. 178 has pointed out that the sentence in question does not contain a book title but has to be rendered as “a study on the chords was also made by Hipparchus in 12 books and so by Menelaos in 6.Google Scholar
  678. 1.
    Cf. also Toomer [1973], p. 19/20.Google Scholar
  679. 2.
    Theon indeed admires the conciseness of Ptolemy’s derivations (ed. Rome CA II, p. 451).Google Scholar
  680. 3.
    Aratus Comm., Manitius, p. 150, 2; cf. above p. 279.Google Scholar
  681. 4.
    Cf. below IV B 5.Google Scholar
  682. 5.
    Burgess, Sûr. Siddh., p. 64; cf. also Nallino, Scritti V, p. 220f.Google Scholar
  683. 6.
    Cf. above p. 278.Google Scholar
  684. 8.
    Cf. below p. 315.Google Scholar
  685. a Cf. above p. 140, n. 3.Google Scholar
  686. Sb Cf. the diagram Toomer [1973], p. 19, Table II.Google Scholar
  687. o Cf. above p. 23.Google Scholar
  688. 9.
    Metrica I, 17-25, Heron, Opera III, ed. Schöne, p. 46, 23-64, 31.Google Scholar
  689. 10.
    Opera III, p. 58, 19; p.62, 17-18. This need not to be understood as an exact title.Google Scholar
  690. 11.
    This is a conclusion first clearly established by A. Rome [1933].Google Scholar
  691. 12.
    Opera III, p. 66, 6-68, 5.Google Scholar
  692. 4.
    Above I A 2, 1. Cf. also Pappus’ remarks concerning Menelaus (Hultsch, p. 600, 25-602, 1; Ver Eecke II, p. 459).Google Scholar
  693. 5.
    Cf. below p.304f.Google Scholar
  694. 6.
  695. 7.
    Cf., e.g., Varâhamihira, Paficasiddhântikâ IV, 41 to 44; Neugebauer-Pingree, II, p. 41 - 44.Google Scholar
  696. 9.
    Presumably the work on simultaneous risings (cf. above p. 301).Google Scholar
  697. 10.
    Ar. Comm., p. 150, 1-3. Manitius’ translation (p. 151) and commentary (p. 297f.) are incorrect since he assumes that 1/20 means 3 minutes instead 1/20 of 15°. This error furthermore forces him to assume as underlying geographical latitude q,=36;29° (instead of simply 36°) in order to obtain agreement with modern computation, as if this were of any interest.Google Scholar
  698. 11.
    This is the “day-radius” of Indian astronomy.Google Scholar
  699. 12.
    This is the equivalent of the Indian limit of 3;45° in a table of sines; cf., e.g., Pc.-Sk. IV, 1.Google Scholar
  700. 13.
    The respective values are: crd 2cp=1,10;19,26 crd(180-2(p)=1,37;1,55 crd 2b=55;4,31 crd (180 —2 S) =1,46;33,20 thus crd 2 n = 44;57,3.Google Scholar
  701. 14.
    Ar. Comm., p. 96, 11.Google Scholar
  702. 15.
    Ar Comm., p.98, 20f.; also Theon Smyrn., p. 202, 19ff. (Hiller); Dupuis, p. 327.Google Scholar
  703. 16.
    Cf. above p. 301, note 2.Google Scholar
  704. 17.
    Ar. Comm., p. 182-185.Google Scholar
  705. 18.
    Given in Ar. Comm., p. 244-271 (Manitius).Google Scholar
  706. 19.
    Ar. Comm., p. 124/5.Google Scholar
  707. 20.
    This is again a term of Indian astronomy (cf., e.g., Pc.-Sk. IV, 27 and 28).Google Scholar
  708. 21.
    Cf., e.g., above I A 4, 3. According to Varâhamihira’s introduction to the Brhat-Samhitâ the determination of the day-radius and of the ascensional differences belongs to the topics which must be mastered by the astrologer (cf. Kern, Verspr. Geschr. I, p. 175 ).Google Scholar
  709. 27.
    Strictly speaking we know only for certain that Hipparchus followed Eratosthenes in assuming that the circumference c of the earth measures 252000 stades (Strabo, Geogr. II, 5, 7 and II, 5, 34), but we are not sure whether it was Hipparchus who first introduced the division of c into 360 degrees or Eratosthenes (who perhaps remained at a strictly sexagesimal division of the circle; cf. below p. 590).Google Scholar
  710. 28.
    Cf. above Table 2, p. 44 (from Alm. II, 6).Google Scholar
  711. 29.
    This also holds for the intermediate values at 1/4h and 1/2”, not listed in Table 31. 3° Cf. above p. 304.Google Scholar
  712. 31.
    Cf. below IV D 3.Google Scholar
  713. 32.
    Eratosthenes, however, can probably be excluded since his distancesdeviate from the Hipparchian (cf. Fig. 291, p. 1313 ).Google Scholar
  714. 33.
    Cf. also below p. 1014.Google Scholar
  715. 34.
    Cf., e.g., the treatise by Hypsicles (below IV D 1, 2 A) who is about contemporary with Hipparchus.Google Scholar
  716. 35.
    E.g. column J of the lunar theory of System B. Cf. below II B 3, 5 B.Google Scholar
  717. 1.
    Vettius Valens, Anthol. IX, 11 (ed. Kroll, p. 354, 4-6); also above p. 263.Google Scholar
  718. 2.
    Cf. above p. 58.Google Scholar
  719. 3.
    Rome [1950], p. 214f.Google Scholar
  720. 4.
    Cf. above p.55.Google Scholar
  721. 5.
    Cf. below p.971, n. 21.Google Scholar
  722. 6.
    ed. Hiller p. 188, 15 (trsl. Dupuis, p. 305); what Theon has to say otherwise about Hipparchus and the equivalence of eccenters and epicycles does not inspire confidence (Hiller, p. 166, 6; p. 185,17; trsl. Dupuis, p. 269, p. 299). Cf also above p.264, n. 3.Google Scholar
  723. 7.
    Cf. Fig. 286 as compared with Fig. 53 (p. 1221 ).Google Scholar
  724. 8.
    Nallino, Batt. I, p. 43 f.Google Scholar
  725. 9.
    Cf., e.g., Geminus VI, 28-33 (Manitius, p. 78/81); also Aaboe-Price [1964], p. 6-10. The use of the octants by Thabit b. Qurra is motivated by the difficulty of accurately observing the solstices (cf. Neugebauer [1962, 2], p. 274/5).Google Scholar
  726. 10.
    Alm. III, 4 (Manitius I, p. 166); cf. Fig. 53, p. 1221.Google Scholar
  727. 11.
    Alm. III, 4 (Manitius I, p. 170); also Theon of Smyrna (Dupuis, p. 218/219), etc.Google Scholar
  728. 12.
    The parameters in Alm. III, 4 also determine the two remaining seasons, because the mean motion a3 is given by 90—(be +62)=86;51° and thus C4=88;49°. This, then, gives for the corresponding seasons s3 x 88 1/8e and s4 z 90 118e, with a total of 365 1/4e for the year. Again it is impossible to distinguish between Ptolemy’s and Hipparchus’ parameters on the basis of these round numbers.Google Scholar
  729. 13.
    Cf. above p. 297 (2) or below p. 310 (5).Google Scholar
  730. 14.
    Cf. also above p. 55. note 1.Google Scholar
  731. 1.
    Cf. below p. 339, n. 10 and Galen’s commentary to Hippocrates’ “On epidemics” (Galen, Opera XVII, 1 ed. Kühn, p. 23); cf. also above p.296.Google Scholar
  732. 2.
    Alm. III, 1 (Heiberg, p. 207, 7/8; Manitius, p. 145). No title of a work by Hipparchus on the length of the synodic month is ever mentioned in the extant sources; Rehm’s Hepì pryviaíov xpóvou (RE 8, 2, col. 1670, 20) is a pure conjecture. For the Arabic tradition cf. Walzer [1935], p. 347 (110/75), based on Galen’s treatise “On seven-month children”; cf. above p. 293.Google Scholar
  733. 3.
    From Suidas, ed. Adler II, p. 657, 27 f.Google Scholar
  734. 4.
    Cf. below I E 5, 2 A.Google Scholar
  735. 5.
    Kugler, Mondrechnung (1900), p. 111; cf. p.348 ff.Google Scholar
  736. 6.
    Cf. below p. 310.Google Scholar
  737. 7.
    Cf. above p. 271.Google Scholar
  738. 8.
    Alm. V, 2 and V, 3; cf. above p. 84 and p. 89.Google Scholar
  739. 1.
    Manitus I, p. 197f.; cf above p. 69 (1) to (3).Google Scholar
  740. 2.
    Cf. below p. 483 (3); p.478(2 c); p. 523 (2c) or ACT, p. 75 (20).Google Scholar
  741. 3.
    Galen in his treatise “On Seven-Month Children” ascribes this value to Hipparchus in a form which is equivalent to saying “1 syn. m. = 29;31,50,8d and a little.” Cf. for the details Neugebauer [1949].Google Scholar
  742. 5.
    Alm. IV, 2 (Manitius I, p. 195).Google Scholar
  743. 6.
    A fragment from an anonymous commentary (probably from the third century A.D.; cf. below p. 321, note 3) gives consistently, but wrongly, 235 instead of 239 for the number of anomalistic months (CCAG 8, 2, p. 127, lines 12, 16, 17).Google Scholar
  744. 8.
    Copernicus, De Revol. IV, 4, silently correcting Ptolemy’s value (Gesamtausg., p. 215, 31 f).Google Scholar
  745. 9.
    ACT I, p. 272.Google Scholar
  746. 10.
    This was pointed out by A. Aaboe [1955].Google Scholar
  747. 11.
    Cf. below p. 378 (15h). p. 396 (5b), and p. 496 (20)Google Scholar
  748. 12.
    Cf., e.g., above p. 125 f.Google Scholar
  749. 3.
    Cf. above p. 306.Google Scholar
  750. 4.
    The proper understanding of this passage in relation to Ptolemy’s method is due to Olaf Schmidt [1937].Google Scholar
  751. 5.
    We shall discuss these parameters later on (below p.325).Google Scholar
  752. 6.
    Ptolem. Opera II, p. 153, 18-20. The ratio (6) is also used in Tamil eclipse computations; cf. Neugebauer [1952], p. 272 (3).Google Scholar
  753. 7.
    Both eclipses have been discussed before: the earlier one belongs to a triple (recorded in Babylon) used for the determination of the radius of the lunar epicycle (above p. 77), the later one is one of a pair that served to find the apparent diameter of the moon (above p.104).Google Scholar
  754. 8.
    Actually the magnitudes are 1.5 and 2.8, respectively, (P. V. Neugebauer, Kanon d. Mondf.). Alm. VI, 9 (Manitius I, p. 394 - 396 ).Google Scholar
  755. 10.
    Alm. IV, 9 (Manitius I, p. 238-241).Google Scholar
  756. 11.
    Cf. above p. 81.Google Scholar
  757. 1.
    Cf. above p. 84.Google Scholar
  758. 2.
    Manitius I, p. 245 f; cf. also Man. I, p. 212, 25.Google Scholar
  759. 3.
    Below p. 318.Google Scholar
  760. 4.
    A denominator 4;45 would have been slightly better because 1,0/4;45Google Scholar
  761. 5.
    Cf. above p.84.Google Scholar
  762. 6.
    In Alm. IV, 11; cf. below p.316.Google Scholar
  763. 8.
    Rome, CA I, p. 68; translation Toomer [1967], p. 147.Google Scholar
  764. 9.
    This eclipse is also discussed by Cleomedes (II, 3 ed. Ziegler, p. 172, 20-174, 15) who is about a generation younger than Pappus (cf. below V C 2, 5 A and p. 963). Hultsch ([1900], p. 198 f.) suggested its identification with the eclipse of —128 Nov. 20, P. V. Neugebauer (Astron. Chron. I, p. 132 and p. 113) with the “Agathocles” eclipse of — 309 Aug. 15. In analyzing Hipparchus’ procedure in his determination of the effect of a measurable or not measurable solar parallax on the moon’s distance G. Toomer has shown [1974, 2] that only the eclipse of —189 March 14 satisfies the conditions imposed by Hipparchus’ method.Google Scholar
  765. 10.
    Cf. below I E 5, 4 B.Google Scholar
  766. 12.
    These observations antedate Hipparchus’ lifetime.Google Scholar
  767. 13.
    For a Babylonian record of this eclipse cf. Schaumberger, Erg., p. 368, note 1.Google Scholar
  768. 14.
    Fig. 288 is drawn to scale with Hipparchus’ values for e and r. Black dots denote mean positions, white dots true positions of the moon; O=observer, M =center of eccenter.Google Scholar
  769. 16.
    The method for finding r is described in Alm. IV, 6 (cf. above I B 3, 4 A). For the eccenter model Prolemy refers briefly to the arrangement obtainable by a transformation with reciprocal radii, known since Apollonius (cf. above p. 265).Google Scholar
  770. 17.
    Cf. above p. 76. The corrections mentioned in note 15 have practically no influence on the final results; the eccentricity, e.g. changes only from 5;16,22 to 5;15,33.Google Scholar
  771. 18.
    Ptolemy says (Manitius I, p. 246, 7f.) that the corresponding maximum equation amounts to 5;49° and Theon repeats this figure (Rome, CA III, p. 1084, 6). In fact, however, 1/2 arc crd 2 e = 5;59°.Google Scholar
  772. 19.
    Manitius I, p. 247, 5.Google Scholar
  773. 20.
    Taking the equation of time into consideration does not help matters.Google Scholar
  774. 21.
    Toomer [1973], p.9-16. Cf. also above p. 299f.Google Scholar
  775. 22.
    Above p.74f. and Figs. 65 to 67 there.Google Scholar
  776. 1.
    Pliny NH II, 53 (Ian-Mayhoff 1, p. 143; Budé II, p. 24; Loeb I, p. 203). About five centuries later Lydus improved on this story by mentioning only solar eclipses (Lydus, De ost., p. 15, 2f ed. Wachsmuth).Google Scholar
  777. 2.
    Cf., e.g., Rome [1950].Google Scholar
  778. 3.
    Below II B 6 and II B 7.Google Scholar
  779. 4.
    D. R. Dicks, in his Hipparchus, p. 51 (H), seems to think that the passage becomes more acceptable by using a variant reading which he translates “Hipparchus foretold the course of both the sun and moon for hundreds of years.” In fact this makes even less sense. For mean positions one needs for all times nothing but a few tables (e.g. Alm. VI, 3). Hence one must assume true positions. But 600 years contain almost 15000 syzygies which Hipparchus should have undertaken to compute with no useful purpose at all.Google Scholar
  780. 5.
    Aim. III, (Man. I, p. 183, 5 ).Google Scholar
  781. 6.
    Cf. above p. 73 ff.Google Scholar
  782. 7.
    Cf. e.g., above p.72; p. 77.Google Scholar
  783. 8.
    E.g. Manitius in Hipparchus, Aratus Comm., p. 286. Rehm (RE 8,2, col. 1668, 63-1669, 1) rightly objected against constructing a title of a treatise from this note in which Achilles names four astronomers as occupied with the same topic (cf. below p. 666).Google Scholar
  784. 9.
    Below p. 322.Google Scholar
  785. 1.
    Above p. 310 (3).Google Scholar
  786. 2.
    Published in CCAG 8, 2, p. 126 to 134.Google Scholar
  787. 3.
    Rome [1931, 2] made it plausible that this commentary was not written before A. D. 213; cf. also Rome [1931, 1], p. 97, note 2. The terminology shows parallels with Proclus’ Hypotyposis (e.g. the use of tinxncóç).Google Scholar
  788. 4.
    Strictly speaking one should say that intervals between lunar eclipses are always of the form 5 m+6 n months, where m and n are non-negative integers.Google Scholar
  789. 5.
    CCAG 8, 2, p. 126, 21 to 28.Google Scholar
  790. 7.
    NH II, 57 (Budé II, p. 25 f.).Google Scholar
  791. 7.
    Above IB6,4.Google Scholar
  792. 9.
    Below p. 523 (2 c).Google Scholar
  793. 10.
    Below p.549; cf. also Neugebauer [1973, 3], p. 248ff. or Aaboe [1972], p. 114. The emphasis on the 5-month intervals is a common feature in Babylonian eclipse texts.Google Scholar
  794. 11.
    Maass, Comm. Ar. rel., p. 47, 13; also below p. 666.Google Scholar
  795. 1.
    Cf. below I E 5, 4 B.Google Scholar
  796. 2.
    Cf. for this method, e.g., above p. 295.Google Scholar
  797. 3.
    Aim. V, 19, Heiberg I, p. 450, 1 and 4 (rzapaaLlcocrncâ).Google Scholar
  798. 4.
    Rome CA I, p. 150, 20-155, 27.Google Scholar
  799. 5.
    Alm. V, 5 (Manitius I, p. 271, 6-8), above p. 89(1). A second observation (-126 July 7) is of no interest for our present problem because the moon is so near to the highest point of the ecliptic that obviously px~0.Google Scholar
  800. 6.
    Above p. 90.Google Scholar
  801. 7.
    Cf. above I A 5, 5.Google Scholar
  802. 8.
    In principle one could reduce the problem to plane trigonometry by means of stereographic projection. This, however, would require the knowledge of conformality of this mapping, a property unknown in antiquity (cf. below p. 860).Google Scholar
  803. 9.
    Cf. above p. 304 ff.Google Scholar
  804. 10.
    Alm. V, 19, Manitius I, p. 329, 24 - 29.Google Scholar
  805. 11.
    Cf. below p. 324.Google Scholar
  806. 12.
    Actually Ptolemy’s own methods are just as crude (cf. above I B 5, 6).Google Scholar
  807. 13.
    Cf. the apparatus to Alm. II, 13 in Heiberg, p. 181, 28.Google Scholar
  808. 14.
    Rome, CA I, p. 152, note (2); similarly p. 168, note (1).Google Scholar
  809. 1.
    Alm. V, 8 gives for AL =30 the latitude 2;30°. Thus the latitude at L should be greater. Hultsch [ 1900 ].Google Scholar
  810. 2.
    Swerdlow [1969].Google Scholar
  811. 3.
    Swerdlow [1969], p. 297/298.Google Scholar
  812. 4.
    Cf. above p. 313. The parameters (1) are also mentioned by Pappus, Coll. VI (the passage in question is translated in Heath, Arist., p. 412). The ratio u/rQ=2;30 occurs also in Tamil eclipse computations; cf. Neugebauer [1952], p. 272 (3).Google Scholar
  813. 5.
    Cf. above p. 109 and Fig. 98. Calling now EM = Rm, ES=RS, a=r„ we have MC=2—r0, hence from (5) AC= MC— rm=2—rm—r„ and from (6) Rm=(1— AC) Rs=(rm+ro-1)Rs which is our present relation (5a).Google Scholar
  814. 6.
    Dupuis, p. 318/319; ed. Hiller, p. 197, 9.Google Scholar
  815. 7.
    The same ratios are also mentioned by Chalcidius (4th cent.) and Proclus (5th cent.); Chalcidius gives as title of Hipparchus’ work “De secessibus atque intervallis solis et Iunae” (ed. Mullach, Fragm. II, p.202b, ch. 90; ed. Wrobel, p. 161, ch. 91); Proclus, Hypot., ed. Manitius, p. 133. In CCAG 7, p. 20 n. 1 one finds a passage quoted which erroneously assumes Vs=1880 Vm.Google Scholar
  816. 9.
    Cf below p. 962.Google Scholar
  817. 1.
    This assumption is made, of course, only as a preliminary simplification of the mathematical discussion. Pappus in his “Collections” (VI, 37 ed. Hultsch, p. 554, 21 f.; p. 556, 6-10; translated in Heath, Arist., p.413) says that Hipparchus and Ptolemy considered the earth’s size negligible only with respect to the sphere of the fixed stars. Proclus (Hypot., p. 112, 15 f. ed. Manitius) ascribes the “leadership” in this question to Hipparchus; cf. also Hypot., p. 228, 19 f.Google Scholar
  818. 2.
    Cf. for this eclipse (of —189 March 14) above p. 316, n. 9.Google Scholar
  819. 3.
    For the epicycle radii which result from (1) and (2) cf. above p. 316 (5). Swerdlow [1969], p. 299. For parallax cf. above I B 5.Google Scholar
  820. 1.
    Cf above p.296.Google Scholar
  821. 2.
    Alm. IX, 3 (Man. II, p. 99). Cf. above I C 1, 4; also Neugebauer [ 1956 ], p. 295.Google Scholar
  822. 3.
    Perhaps to this group belongs his interest in the distance of Mercury from Spica (Alm. IX, 7, Manitius II, p. 134, 29; above p. 159).Google Scholar
  823. 4.
    Ptolemy, Alm. IX, 2, Man. II, p. 96.Google Scholar
  824. 5.
    Cf. below p. 823 f.Google Scholar
  825. 6.
    Below V A 1.Google Scholar
  826. 7.
    Goldstein [1967]; cf. below V B 7.Google Scholar
  827. 8.
    Goldstein [1967], p.8. This is also assumed by Bar Hebraeus (L’asc. II, ch. 7, Nau, p. 194f.) who quotes from the k. al-manshúrât (cf. Goldstein, p. 4, n. 8). Caution is nevertheless necessary since the values for the moon are based on the Ptolemaic model, not on the simple Hipparchian.Google Scholar
  828. 9.
    Cf. above p. 327.Google Scholar
  829. 10.
    Cf. above p. 291.Google Scholar
  830. 11.
    Wessely [ 1900 ]; Neugebauer [1962, 3], p. 40, col. II, 7-10. Cf. for this text also below p. 737 (n).Google Scholar
  831. 12.
    Cf. below p.693.Google Scholar
  832. 1.
    Cf. above p. 306 and below p. 823.Google Scholar
  833. 2.
    NH II, 95, Loeb I, p. 239, Budé II, p. 41.Google Scholar
  834. 3.
    Continued in the passage discussed above p. 285.Google Scholar
  835. 4.
    AG, p. 543/4; also Pfeiffer, Sterngl., p. 115.Google Scholar
  836. 5.
    Cumont [1909], p. 268; similar Eg. astrol., p. 156, etc.Google Scholar
  837. 6.
    Similar Boll, KI. Schr., p. 5, n. 1 (1908); Rehm in RE 8, 2, col. 1680, 29ff. (1913); Gundel HT, p. 303/4 (1936) who makes Hipparchus responsible for the astrology of his “pupil” Serapion (a relationship which is very doubtful: cf. Neugebauer [1958, 1], p. I11, note 39). A naive blunder is committed by Dicks, Hipp., p. 3, who did not realize that the “testimony K” from CCAG 5, 1, p. 205 (also in CCAG 1, p. 80) is taken from Ptolemy’s “Phaseis” (Heiberg, p. 67) and has nothing to do with astrology.Google Scholar
  838. 7.
    Firmicus Maternus, Mathesis II, Praef (ed. Kroll-Skutsch I, p. 40, 8 ff.; p. 41, 5 f.) I see no reason for considering “antiscia” a book title (Rehm, RE 8, 2, col. 1668, 32).”Google Scholar
  839. 8.
    The theory of “antiscia” is described by Firmicus in 11, 29 (Kroll-Skutsch, p. 77-85); cf. also Vettius Valens III, 7 (ed. Kroll, p. 142, 28) from Critodemus (1st B.C.?). For discussion cf. Bouché-Leclercq AG, p. 161 f.; p. 275, note 2.Google Scholar
  840. 9.
    Mentioned by Ptolemy in his “Analemma”; cf. below p.1380 and Figs. 26 and 27.Google Scholar
  841. 10.
    Engelbrecht, p. 47, 20.Google Scholar
  842. 11.
    Engelbrecht, p. 60, 30.Google Scholar
  843. 12.
    These passages from Hephaistio are conveniently tabulated in a monograph by Karl Trüdinger, Studien zur Geschichte der griechisch-römischen Ethnographie (Basel 1918), p. 84.Google Scholar
  844. 13.
    As Rehm has pointed out (RE 8, 2, col. 1680 ) Hephaistio’s text has a close parallel in Vettius Valens I, 2. Since both texts mention Corinth as existent (Engelbrecht, p. 63, 4; Kroll, p. 11, 27) and speak of the “domain of Carthage” instead of Roman “Africa” (Engelbrecht, p. 61, 6; Kroll, p. 7, 23 but in different context), a situation before 146 B.C. is assumed.Google Scholar
  845. 14.
    Cf. above p.283; p. 287.Google Scholar
  846. 15.
    The text exists in several versions; the two best ones were edited by Maass, Anal. Erat., p. 141-149. The longer version is based on CCAG 8, 3, p. 61, Cod. 46, F. 9”, the shorter one is recorded in CCAG 2, p. 1, Cod. I, F. 221”; 3, p. 10, Cod. 12, F. 188; 4, p. 23, Cod. 7, F. 88”; 9, 2, p. 3, Cod. 38, F. 9. Abridged versions are CCAG 9, 2, p. 6, Cod. 39, F. 101 and CCAG 9, 2, p. 62, Cod. 65, F. 154; 11, 1, p. 6, Cod. 1, F. 126; p. 122, Cod. 14, F. 394”.Google Scholar
  847. 1.
    Even a hemispherical cupola is by no means an a priori concept; cf. below p. 577.Google Scholar
  848. 2.
    Cf. above I E 2, 1 A.Google Scholar
  849. 3.
    That there was still room for systematic improvements in the time of Ptolemy is shown by his introduction of new coordinates in the “Analemma” (cf. below V B 2, 5).Google Scholar
  850. 1.
    Cf. below p.934; also above p.280.Google Scholar
  851. Mainly in Book II, 1 of his “Geography.” Cf also the sharp division of topics in Ptolemy’s Geography: Book I mathematical theory, lists of localities and their coordinates in the remaining books.Google Scholar
  852. 2.
    Honigmann, SK (widely accepted); Dicks [1955] opposing Honigmann.Google Scholar
  853. 3.
    Reinhardt (cf., e.g., Honigmann, SK, p. 8f.).Google Scholar
  854. 5.
    Cf., e.g. below IV D 1, 3.Google Scholar
  855. 6.
    Strabo, Geogr. I 1,12 (Loeb I, p. 23; Budé I, 1, p. 74); cf. also II 5, 34 (Loeb I, p. 503; Budé I, 2, p. 117 ).Google Scholar
  856. 8.
    The boundaries are Meroe (13h) and Borysthenes (166). I think Honigmann is right (SK, p. 13) when he takes this as indicating that Eratosthenes was familiar with the seven half-hour zones.Google Scholar
  857. 9.
    Cf. above p. 305, n. 27.Google Scholar
  858. 10.
    It is, of course, absurd to give latitudes to seconds (rounded!), as, e.g., in the Loeb translation.Google Scholar
  859. 11.
    Cf. above p. 305.Google Scholar
  860. 12.
    Cf. below p. 653.Google Scholar
  861. 13.
    Cf., e.g., the estimate in Strabo II 5, 7 or XVII 3, 1 (Loeb I, p. 439; VIII, p. 157). For the Cinnamon-producing country cf. below p.335.Google Scholar
  862. 14.
    Cf. below p. 746, n. 3.Google Scholar
  863. 15.
    Alm. VI, 11 (Heiberg I, p. 538/539). Introduction to the Handy Tables (Opera II, ed. Heiberg, p. 174, 17 ).Google Scholar
  864. 16.
    Angles between meridian and ecliptic (Alm. II, 13; cf. above p. 50); Analemma (below p. 853 and p. 854). Similarly in the Handy Tables: oblique ascensions and parallaxes (below p. 978).Google Scholar
  865. 17.
    Above p. 43 f. and Table 2.Google Scholar
  866. 18.
    Ptolemy, Opera II ed. Heiberg, p. 4, 3-20; below p. 928.Google Scholar
  867. 19.
    Above p. 304.Google Scholar
  868. 20.
    Cf. p. 1313, Fig. 291.Google Scholar
  869. 21.
    Strabo II 5, 35 (Loeb I, p. 507; Budé I, 2, p. 119).Google Scholar
  870. 22.
    Above p. 290.Google Scholar
  871. 23.
    AIm.II, 6 gives M =12;45” for 1p =12;30°. The 8800” from the equator would correspond to cp =12;34°.Google Scholar
  872. 24.
    Aratus Comm., p. 82, 24 f. Actually S x 31;17° in —125.Google Scholar
  873. 25.
    E.g. Manitius in Ar. Comm., p. 301, n. 28. Cf., however, Schjellerup [1881], p. 30 about q Cas.Google Scholar
  874. 26.
    Manitius, p. 120, 18.Google Scholar
  875. 27.
    Cf. above p. 335.Google Scholar
  876. 28.
    Strabo II 5, 38 (Loeb I, p. 511, Budé I, 2, p. 120).Google Scholar
  877. 29.
    Mich. Pap. III, p. 316.Google Scholar
  878. 30.
    Strabo II 5, 41 (Loeb I, p. 514, Budé I, 2, p. 122).Google Scholar
  879. 31.
    Strabo II 5, 34 (Loeb I, p. 503, Budé I, 2, p. 117).Google Scholar
  880. 32.
    Ptolemy, Geogr. I 4, 2 (Nobbe, p. 11; Milk, p.21).Google Scholar
  881. 33.
    Aratus Comm. II 4, 3 (Manitius, p. 184/185). The above quoted passage in Strabo (above note 31) is only a clumsy paraphrase of Hipparchus’ words to which Strabo added of his own (Air) Si)” from the equator to the north pole.”Google Scholar
  882. 1.
    Strabo, Geogr. I 1, 12 (Loeb I, p. 25; Budé I, 1, p. 74); cf also below p. 667.Google Scholar
  883. 2.
    Strabo, Geogr. II, 5, 7 (Loeb I, p. 439; Budé I, 2, p. 86); cf. also below p. 590.Google Scholar
  884. 3.
    Cf. below V B 4, 1.Google Scholar
  885. 4.
    Strabo, Geogr. 14, 1 (Loeb I, p. 233; Budé I, 1, p. 167); cf. also below p.652.Google Scholar
  886. 6.
    Loeb I, p. 202, note b.Google Scholar
  887. 1.
    Budé II, p. 24. Honigmann, SK, p. 72/73, note 3 obscures the situation with learned irrelevancies. Rehm in RE 8, 2, col. 1666, 46 and 1671, 25, following Heiberg in Ptol. Opera II, Index, p. 276 s.v. Avay paupq.Google Scholar
  888. 2.
    Alm. III, 1 Heiberg, p. 207, 18.Google Scholar
  889. 3.
    Halma I, p. 164, Manitius I, p. 145, 23.Google Scholar
  890. 4.
    Cf. belowVB8,1B.Google Scholar
  891. 5.
    Cf. Maass, Comm. Ar. rel., p. 330 and Anal. Erat., p. 45-49; p. 139. Cf. also Gudeman in RE 3A, 2, col. 1879fGoogle Scholar
  892. 6.
    Comm. in Arist., Vol. VII, p. 264, 25-266, 29. Discussed, e.g., by Duhem, SM I, p. 386, p. 394.Google Scholar
  893. 7.
    Diels, Dox., p. 404 (also Diels VS15) I, p. 226, 25 or Galen, Opera XIX, p. 307 ed. Kühn).Google Scholar
  894. 8.
    Plutarch, Moralia 732 F (Loeb IX, p. 196/7) and 1047 C, D. Cf. Biermann-Mau, J. of Symbolic Logic 23 (1958), p. 129-132; also Rome, Annales de la Soc. Sci. de Bruxelles, Sér. A 50 (1930), Mém., p. 101.Google Scholar
  895. 9.
    Cf. also above p. 293.Google Scholar
  896. 10.
    In one of his commentaries to Hippocrates (edited in the Corp. Med. Gr. V 9, 2, p. 333, 12-334, 14 =ed. Kühn XVII, 2, p. 240), commenting on a remark by Hippocrates that neither the year nor the lunar month amounts to an integer number of days, Galen says that in particular the question of the length of the interval between consecutive conjunctions of sun and moon requires long discussions and that “Hipparchus wrote a whole book” about it, “such as our work On the length of the year” (I1epì rob éviavaíou Xpóvou aúyypapps gpétapov). The same title, however, is quoted by Hipparchus himself for one of his writings (Alm. III, 1 Heiberg, p. 207, 20; cf above p. 292, n. 3) and Galen in De crisibus III (ed. Kühn IX, p.907, 14-16) again mentions only Hipparchus as having written “one whole book” on the length of the lunar month. All this casts doubts on Galen’s authorship of a book on the length of the year and suggests a corruption of the text in the commentary on Hippocrates. Unfortunately the problem has again been obscured by a conjecture of Bergsträsser; cf. the subsequent note.Google Scholar
  897. 12.
    Arab. Übers. aus dem Griech., p.226 (from ZDMG 50, 1896, p. 350).Google Scholar
  898. 13.
    Chronogr. IV, trsl. Budge, p. 29.Google Scholar
  899. 14.
    Cf., e.g., Diels VS(5) I, p.421, No.5.Google Scholar
  900. 15.
    Cf. his preface to De revolutionibus (Opera II, p. 3 and p. 30 note, for Copernicus’ Latin translation which he deleted from the printed edition).Google Scholar
  901. 1.
    Above I E 2, 1 B.Google Scholar
  902. 2.
    Above p. 283.Google Scholar
  903. 3.
  904. 4.
    I E 3, 1, p. 299; VI B 1, 6.Google Scholar
  905. 5.
    IE2,IC, p.287; IE2,1C1.Google Scholar
  906. 6.
    E.g. the use of “zodiacal signs” for arcs in any direction (cf. I E 2, 1 A, p. 278).Google Scholar
  907. 7.
    I E 2, 1 B, p. 283, n.13.Google Scholar
  908. 8.
    IE2,ID; IE6,1, p.330.Google Scholar
  909. 9.
    I E 2, 1 C, p. 284f.Google Scholar
  910. 10.
  911. 11.
    IE 2, 2C, p.298.Google Scholar
  912. 12.
    I E 2,2 A, p. 294; I E 2, 2C, p. 298.Google Scholar
  913. 13.
  914. 14.
    IE5,1C, p.317.Google Scholar
  915. 15.
    Cf. below IV C 3, 8.Google Scholar
  916. 16.
    IE5,1C, p.315.Google Scholar
  917. 17.
    I E 5,1 C, p. 315 f.Google Scholar
  918. 18.
    I E 5,2 B and I B 6,4, p. 129.Google Scholar
  919. 19.
  920. 20.
    I E 6, 1, p. 329. Also the use of Babylonian units (“cubit” of 2°) points in the same direction (cf.Google Scholar
  921. 21.
    E 2, 1 A, p. 279; I E 3, 2, p. 304).Google Scholar
  922. 21.
    I E 3, 2, p. 306.Google Scholar
  923. 33.
    I E 6, 1, p. 330; also I C 8, 5, p. 261.Google Scholar
  924. 34.
    TB 5, 4 A, p. 109.Google Scholar
  925. 35.
    Copernicus found 1179 re ( De revol. IV, 19 ).Google Scholar
  926. 36.
  927. 37.
    Swerdlow [1969], against Hultsch [1900].Google Scholar
  928. 38.
    Acceptance of the meridian Alexandria-Rhodes (cf. below p. 939) or the equivalence of 700 stades and 1° on the meridian (I E 6, 3 A).Google Scholar
  929. 39.
    Birrtni, Tandid (trsl. Ali), Chaps. V to XXII and Kennedy’s commentary. In contrast the nonsense about the “hellenische Geist,” approvingly cited by Kubitschek, RE 10, 2, col. 2058, 43 - 49.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1975

Authors and Affiliations

  • Otto Neugebauer
    • 1
  1. 1.Brown UniversityProvidenceUSA

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