# The Almagest and its Direct Predecessors

• Otto Neugebauer
Chapter
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.
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.
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.
20. 13.
We ignore here the change of the solar longitude between sunrise and sunset.Google Scholar
21. 14.
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.
25. 18.
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.
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.
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.
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.
48. 3.
I, 10 (p. 47, 3 Heib.) and III, 1 (p. 209, 13 ff. Heib.).Google Scholar
49. 1.
50. 2.
51. 3.
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.
55. 3.
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.
67. 2.
68. 3.
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.
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.
76. 6.
77. 7.
Because of the equivalence theorem (p. 57) we need not distinguish between an eccenter-and an epicycle-model.Google Scholar
78. 1.
79. 2.
80. 3.
81. 4.
82. 1.
83. 2.
84. 3.
85. 4.
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.
88. 1.
89. 2.
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.
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.
95. 4.
96. 5.
97. 6.
98. 7.
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.
105. 1.
106. 2.
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.
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.
116. 6.
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.
123. 4.
Alm. IV, 6 (Man. I, p. 219).Google Scholar
124. 5.
125. 6.
126. 1.
Cf. above p. 80 and Fig. 72.Google Scholar
127. 2.
128. 3.
Below p. 88ff. For a comparison with modern theory cf. p. 1108.Google Scholar
129. 4.
130. 5.
131. 1.
132. 2.
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.
135. 5.
136. 6.
For some textual difficulties cf. below p. 92. -127 Aug. 5.Google Scholar
137. 8.
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.
145. 16.
Including the equation of time.Google Scholar
146. 17.
147. 18.
Above p. 87 and p. 89.Google Scholar
148. 19.
Above p. 89, No. II.Google Scholar
149. 20.
150. 21.
151. 22.
152. 23.
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.
158. 3.
159. 4.
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.
166. 11.
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.
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.
177. 5.
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.
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.
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.
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.
194. 10.
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.
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.
206. 8.
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.
209. 2.
Cf. above p. 104, (a).Google Scholar
210. 3.
211. 4.
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.
215. 5.
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.
224. 3.
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.
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.
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.
255. 5.
Cf. Appendix VI B 7, 2.Google Scholar
256. 6.
Cf. below p. 208 and p. 212.Google Scholar
257. 7.
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.
261. 11.
262. 12.
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.
268. 18.
Details to be discussed later; cf. below p.207.Google Scholar
269. 19.
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.
273. 23.
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.
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.
289. 6.
290. 7.
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.
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.
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.
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.
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.
311. 6.
According to Schoch, Planetentafeln. Cf. above p. 160.Google Scholar
312. 8.
313. 9.
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.
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.
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.
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.
348. 5.
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.
356. 6.
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.
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.
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.
369. 8.
370. 1.
He mentions only some special cases, e.g. for k=30° (Manitius II, p. 258).Google Scholar
371. 2.
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.
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.
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.
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.
402. 11.
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.
406. 5.
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.
410. 9.
According to p. 153, A=It25.Google Scholar
411. 12.
Following the rules given p. 222 ff.Google Scholar
412. 13.
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.
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.
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.
429. 3.
430. 4.
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.
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.
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.
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.
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.
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.
462. 20.
Ptolemy, Opera II, p. 4 Heiberg.Google Scholar
463. 21.
Plan. Hyp. I; cf. Goldstein [ 1967 ], p. 8.Google Scholar
464. 22.
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.
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.
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.
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.
500. 4.
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.
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.
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.
520. 21.
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.
527. 6.
528. 1.
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.
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.
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.
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.
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.
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.
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.
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.
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.
585. 10.
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.
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.
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.
600. 25.
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.
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.
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.
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.
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.
624. 11.
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.
633. 5.
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.
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.
649. 21.
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.
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.
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.
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.
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.
666. 9.
667. 10.
Ginzel, Hdb. II, p. 391. Similar already Ideler, Chron. I, p. 353.Google Scholar
668. 11.
669. 12.
E.g. Archons, p. 414: “Therefore we may assume that the authorities … called in a specialist, namely, Hipparchos.”Google Scholar
670. 1.
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.
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.
684. 8.
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.
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.
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.
705. 18.
Given in Ar. Comm., p. 244-271 (Manitius).Google Scholar
706. 19.
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.
719. 3.
720. 4.
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.
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.
737. 7.
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.
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.
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.
757. 1.
758. 2.
Manitius I, p. 245 f; cf. also Man. I, p. 212, 25.Google Scholar
759. 3.
760. 4.
A denominator 4;45 would have been slightly better because 1,0/4;45Google Scholar
761. 5.
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.
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.
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.
785. 1.
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.
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.
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.
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.
811. 3.
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.
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.
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.
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.
829. 10.
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.
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.
842. 11.
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.
860. 12.
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.
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.
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.
875. 27.
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.
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.
903. 3.
904. 4.
I E 3, 1, p. 299; VI B 1, 6.Google Scholar
905. 5.
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.
909. 9.
I E 2, 1 C, p. 284f.Google Scholar
910. 10.
911. 11.
912. 12.
I E 2,2 A, p. 294; I E 2, 2C, p. 298.Google Scholar
913. 13.
914. 14.
915. 15.
Cf. below IV C 3, 8.Google Scholar
916. 16.
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