A History of Ancient Mathematical Astronomy pp 19-343 | Cite as

# The Almagest and its Direct Predecessors

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## 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.Schoy, Al-Bir. p. 81 (No. 14); cf. also below p.776.Google Scholar
- 2.MC T(UV ypappii v, meaning “rigorous” methods (cf. below p.771 n. 1).Google Scholar
- 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
- 5.Cf., e.g., ed. Heiberg I, p. 317, 22f., et passim.Google Scholar
- 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
- 2.VII, 3 (1-leib. II, p. 30,18).Google Scholar
- 3.Cf. Rome, CA I, p. 569.Google Scholar
- 4.For plane trigonometry cf. above p.26.Google Scholar
- 5.Contrast: “sphaera obliqua” denotes geographical latitudes different from zero; the corresponding rising times are called “oblique ascensions.”Google Scholar
- 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
- 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
- 5.Almagest II, 3 Heiberg, p. 95, 6 to 13.Google Scholar
- 6.Cf. p.142.Google Scholar
- 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 - 8.VI, 11 Heiberg, p. 543, 24f. and plate at the end of Vol. I.Google Scholar
- 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
- 10.Cf. our Pl. I.Google Scholar
- 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
- 12.Cf. above p.32.Google Scholar
- 13.We ignore here the change of the solar longitude between sunrise and sunset.Google Scholar
- 14.Above p.36(1).Google Scholar
- 15.Alm. VII, 3 Heib. II, p.33, 3ff. For the longitude cf. below p.60.Google Scholar
- 16.III, 1, 61 and VIII, 8, 3 (Nobbe, p. 151, 26 and p. 205, 7f.).Google Scholar
- 17.Almagest II, 9.Google Scholar
- 18.From Alm. II, 11.Google Scholar
- 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 - 20.For an example cf. below p. 979.Google Scholar
- 1.Cf. above I A 4, 3.Google Scholar
- 2.Cf., e.g., the tables in Alm. II, 13 (below p. 50ff.).Google Scholar
- 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 - 4.Cf. above I A 3, I.Google Scholar
- 5.E.g. VIII 16, 3-14 (Nobbe, p. 221-223); several numbers are garbled.Google Scholar
- 6.Probably written sometime between A.D. 500 and 600.Google Scholar
- 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
- 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
- 1.Cf. below p. 50.Google Scholar
- 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
- 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
- 1.The values for 2n2 are rounded to the nearest degree while 23;51° is taken for e.Google Scholar
- 2.Cf. below p.51.Google Scholar
- 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
- 4.The Greek text has no technical term for “zenith distance” but says simply “arc.” 2 Above p. 40.Google Scholar
- 1.Cf. below p. 529.Google Scholar
- 2.Cf. also below p. 294.Google Scholar
- 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
- 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
- 2.Tannery, AA, p. 163.Google Scholar
- 3.I, 10 (p. 47, 3 Heib.) and III, 1 (p. 209, 13 ff. Heib.).Google Scholar
- 1.Cf. IVB2,1.Google Scholar
- 2.Cf. p. 361.Google Scholar
- 3.Cf. H Intr. 5.Google Scholar
- 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
- 1.Cf. Neugebauer [1962, 2], p. 267. Actually the motion is slightly faster than precession.Google Scholar
- 2.Below p.73 ff.Google Scholar
- 3.Below p. 173 ff.Google Scholar
- 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
- 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
- 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
- 4.Heiberg II, p. 33, 3 ff. = Manitius II, p. 28, 14 ff.Google Scholar
- 5.For the solution of this problem, assuming 2=x23° given, cf. above p. 41.Google Scholar
- 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
- 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
- 8.Cf. Neugebauer [1938], p. 22. Cf. below p. 848.Google Scholar
- 1.Cf. p. 563, n. 3.Google Scholar
- 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
- 1.Cf. above p.62.Google Scholar
- 2.Below p. 67.Google Scholar
- 3.Cf. above p. 59.Google Scholar
- 4.Almagest IV, 6; cf. below p. 77. The dates are −720 March 19, −719 March 8 and September 1 respectively.Google Scholar
- 5.Cf. p. 63.Google Scholar
- 1.As always K = 2 − II 5;30 = 2 − 65;30°.Google Scholar
- 2.Ptolemy, Opera II, p. 162, 23-163, 6 ed. Heiberg. Cf. also below p. 984f.Google Scholar
- 3.Cf. Neugebauer [1958], p. 97ff.Google Scholar
- 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
- 5.Cf. p. 63.Google Scholar
- 6.Cf. above p. 62.Google Scholar
- 7.Because of the equivalence theorem (p. 57) we need not distinguish between an eccenter-and an epicycle-model.Google Scholar
- 1.Above p. 55.Google Scholar
- 2.Below p. 79.Google Scholar
- 3.Below p. 81.Google Scholar
- 4.Above p. 55.Google Scholar
- 1.Above p. 57f.Google Scholar
- 2.Cf. below p. 546.Google Scholar
- 3.Cf. p. 481 f.Google Scholar
- 4.Below p. 310.Google Scholar
- 6.It seems possible that the relation (5), quoted below p. 310, is the result of these observations. Cf. p. 310.Google Scholar
- 8.Cf. below p.78.Google Scholar
- 1.Below p. 76ff.Google Scholar
- 2.Cf. also below p.77.Google Scholar
- 3.The interior of the triangle would have the signature − − −(or + + + since we are dealing in fact with the projective plane).Google Scholar
- 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
- 1.Cf. above p. 74.Google Scholar
- 2.As usual these computations contain many small inaccuracies such that r=5;13 would be the nearest common solution.Google Scholar
- 3.Below p. 79.Google Scholar
- 4.Below p. 79.Google Scholar
- 5.That is “Marduk gave an heir”; biblical distortion: Merodach-baladan.Google Scholar
- 6.Accurate value: 57 minutes.Google Scholar
- 7.Above p. 75f.Google Scholar
- 8.From P. V. Neugebauer, Kanon d. Mondf.Google Scholar
- 9.Cf. P. V. Neugebauer, Astr. Chron. II, p. 128.Google Scholar
- 1.For the method cf. above p. 76 (2).Google Scholar
- 2.Since dt is close to an integer number of years the influence of the equation of time can be ignored.Google Scholar
- 3.Actually the tables give d.I.=123;22,33 and da=103;35,23.Google Scholar
- 4.Above p.63.Google Scholar
- 1.490 Apr. 25.Google Scholar
- 2.A.D. 125 Apr. 5.Google Scholar
- 3.Modern values: 1.7” and 2.0”, respectively.Google Scholar
- 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
- 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
- 6.Drawn to scale.Google Scholar
- 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
- 2.501 Nov. 19/20. Compare Fig. 75 with Fig. 70 II, p. 1228.Google Scholar
- 3.Actually only 1.5” and 2.1” respectively.Google Scholar
- 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
- 5.Actually one finds 160;3,9.Google Scholar
- 6.Cf. above p. 77.Google Scholar
- 7.For the more primitive method used by Hipparchus cf. below p. 313 f.Google Scholar
- 8.As we have shown on p. 64, the equation of time has only a negligible effect.Google Scholar
- 1.Solar eclipses remain outside of these discussions because they depend also on geographical elements.Google Scholar
- 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
- 2.Cf., e.g., Almagest IV, 6 (Man. I, p. 218 f.) or V, 2 (Man. !, p. 260ff.).Google Scholar
- 3.Below I C 7.Google Scholar
- 4.Alm. IV, 6 (Man. I, p. 219).Google Scholar
- 5.The error thus committed reaches only about 0;6° for w between about 30° and 60°.Google Scholar
- 6.Above p. 30.Google Scholar
- 1.Cf. above p. 80 and Fig. 72.Google Scholar
- 2.Cf. below p.1109f.Google Scholar
- 3.Below p. 88ff. For a comparison with modern theory cf. p. 1108.Google Scholar
- 4.Below p. 155.Google Scholar
- 5.Cf. below p. 88.Google Scholar
- 1.Below, p. 91ff.Google Scholar
- 2.Cf. p. 80.Google Scholar
- 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
- 4.A.D. 139 Febr.9.Google Scholar
- 5.Cf. above p. 50.Google Scholar
- 6.For some textual difficulties cf. below p. 92. -127 Aug. 5.Google Scholar
- 8.Above p.76.Google Scholar
- 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
- 10.Thirteenth and fourteenth century; cf. Roberts [1957].Google Scholar
- 11.Copernicus, De Revol. IV, 3, IV, 8, IV, 9. Cf. also Neugebauer [1968, 2].Google Scholar
- 12.The same term also occurs in the theory of eclipses (below p. 141) but with totally different meaning.Google Scholar
- 13.Angles are drawn nearly to scale but the eccentricity, and particularly the radius of the epicycle, are exaggerated.Google Scholar
- 14-.126 May 2 and July 7, respectively.Google Scholar
- 15.Cf. below p. 92.Google Scholar
- 16.Including the equation of time.Google Scholar
- 17.Above p. 87.Google Scholar
- 18.Above p. 87 and p. 89.Google Scholar
- 19.Above p. 89, No. II.Google Scholar
- 20.Above p. 48 ff.Google Scholar
- 21.From p(H)=a(M)+90; cf. p.42.Google Scholar
- 22.Above pp. 87, 89.Google Scholar
- 23.Above p. 89.Google Scholar
- 24.Aim. III, 13 for the angle between ecliptic and altitude circle, Alm. V, 18 for parallax.Google Scholar
- 25.For the Hipparchian theory of parallax cf. below I E 5, 3.Google Scholar
- 26.Ptolemy’s rounded values (A(=1IL 10°, 1;30” west) would give an angle of 83°.Google Scholar
- 28.Ideler, Astron. Beob., p. 217 and Chron., p. 345. Also Ginzel, Hdb. II, p. 410.Google Scholar
- 29.Heiberg, p. 363, 18f.Google Scholar
- 3.Cf. p. 80.Google Scholar
- 4.Cf. above p. 88.Google Scholar
- 5.Cf. p.80.Google Scholar
- 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
- 7.Cf. p.83.Google Scholar
- 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
- 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
- 10.A.D. 98 Jan. 14.Google Scholar
- 11.Cf. p. 84.Google Scholar
- 12.Alm. VII, 3 (Heib. II, p. 33, 19 ).Google Scholar
- 13.Ptolemy 2;10° as Heiberg and Manitius interpret the text (cf. below p. 117, note 7).Google Scholar
- 14.Above p.84.Google Scholar
- 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
- 1.The maximum of cQ=5;1 (cf. p.80), of co = 2;23 (cf. p.59).Google Scholar
- 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
- 1.Cf. below I E 5, 3 and I E 5, 4 B.Google Scholar
- 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
- 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
- 4.Below p.104.Google Scholar
- 5.Below p.115.Google Scholar
- 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
- 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
- 3.A.D. 135 Oct. 1.Google Scholar
- 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
- 5.P. V. Neugebauer, Astron. Chron. I, p. 72f.Google Scholar
- 6.For p1 one finds about —0;3,36° whereas Ptolemy assumes p,= O.Google Scholar
- 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
- 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
- 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
- 3.Below p.106.Google Scholar
- 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
- 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
- 10.Cf., e.g., the list in Houzeau, Vade-mecum, p.404f.Google Scholar
- 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
- 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
- 9.Rome p. 107, 10ff.Google Scholar
- 10.Above p. 58.Google Scholar
- 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
- 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
- 13.De revol. IV, 21 (Gesamtausg. II, p. 257).Google Scholar
- 14.Cf. above p. 104, note 4.Google Scholar
- 1.Epitome IV, 1, IV (Werke 7, p. 279).Google Scholar
- 2.Cf. below p.148 f.Google Scholar
- 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
- 4.For this part of the work, preserved only in Arabic, cf. below p. 918.Google Scholar
- 5.Ptolem., Opera II, p. 118 (Heiberg).Google Scholar
- 6.Cf. below V B 7, 6.Google Scholar
- 7.Above p. 109, (1).Google Scholar
- 8.Above p. 112.Google Scholar
- 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
- 1.Above p. 50.Google Scholar
- 2.Cf. above p. 104, (a).Google Scholar
- 3.Cf. above p. 110.Google Scholar
- 4.Above p. 93 ff.Google Scholar
- 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
- 3.Pappus, Comm., ed. Rome, p. 166, 16ff.Google Scholar
- 4.Above p.115.Google Scholar
- 5.Cf. p. 96.Google Scholar
- 6.Alm. VII, 3 (Man. II, p. 29).Google Scholar
- 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
- 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
- 9.Cf. Alm. VII, 3 (Manitius II, p. 28). Cf. above p. 79, p. 82, and p. 60.Google Scholar
- 2.This relation was probably known in Egypt long before hellenistic times; cf., p. 563.Google Scholar
- 3.For syzygies the second inequality is zero.Google Scholar
- 1.Cf. above p.104, (1) and (2); p. 105, (3) and (4); p. 109, (2).Google Scholar
- 2.Above p. 80.Google Scholar
- 3.Above p. 59.Google Scholar
- 4.Cf. our discussion of these statements below p.127.Google Scholar
- 5.Pappus, Comm., ed. Rome p. 194-197.Google Scholar
- 6.For a proof of this theorem and further discussion cf. Neugebauer, Al-Khwar., p. 122f.Google Scholar
- 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
- 8.This includes the rounding errors mentioned on p.127.Google Scholar
- 1.For Ptolemy’s much refined investigation of consecutive syzygies cf. below p.133, F.Google Scholar
- 2.Example: Oppolzer, Canon, Nos. 2271 and 2272 (A.D. 265 Oct. 12 and 266 March 8).Google Scholar
- 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
- 4.Again supplemented in Pappus’ Commentary (ed. Rome, p. 226 to 231).Google Scholar
- 5.Of course, as always, lunar minus solar parallax.Google Scholar
- 6.Example: Oppolzer, Canon, Nos. 5356 and 5357 (A.D. 1049 March 6 and August 1); cf. also p.133, n. 7.Google Scholar
- 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
- 8.Examples passim in Oppolzer, Canon; e.g. Nos. 6201 and 6202, or 6501 and 6502. Cf. also p. 133, n. 7.Google Scholar
- 1.Cf. below p. 140.Google Scholar
- 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
- 6.Using the values from (1), p.125 but with small roundings in the results.Google Scholar
- 9.For solar eclipses one has to replace s by re.Google Scholar
- 10.Duration” is here always meant in the sense of the elongations tabulated in the Almagest.Google Scholar
- 1.Alm. VI, 7 Heib., p. 512, 8.Google Scholar
- 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
- 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
- 1.Unfortunately from the viewpoint of our historical interests, Ptolemy completely ignored the problems of first and last visibility of the moon.Google Scholar
- 2.Above p.88.Google Scholar
- 3.E.g. Tetrabiblos II, 10 (p. 92, 2, 13 Boll-Boer).Google Scholar
- 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
- 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
- 6.E.g. to Pappus; cf. his Commentary to Alm. VI ed. Rome, p. 309. Cf. p. 997 f.Google Scholar
- 1.Cf. Roberts [ 1957 ]; Neugebauer [1968, 2].Google Scholar
- 2.Cf. below V C 4, 5 B and 5 C.Google Scholar
- 3.Cf. p. 386ff.Google Scholar
- 5.Cf. Appendix VI B 7, 2.Google Scholar
- 6.Cf. below p. 208 and p. 212.Google Scholar
- 7.Cf. below p. 1101.Google Scholar
- 8.Computed for A.D. 100 from Almagest XI, 11 using Ptolemy’s constant of precession.Google Scholar
- 9.Case (1); for the elliptic approximation of the deferent of Mercury cf. below p. 168.Google Scholar
- 10.Below p. 227ff.Google Scholar
- 11.Above p. 109ff.Google Scholar
- 12.Cf. p.111.Google Scholar
- 13.For more historical details cf. below p.270f.Google Scholar
- 14.As we shall see (below p. 421) the Babylonian planetary computations make use of precisely this fact.Google Scholar
- 15.Cf. below I D 3, 1.Google Scholar
- 16.In the lunar theory also the apsidal line is movable, but proceeds with the difference velocity (cf. above p. 68).Google Scholar
- 17.Cf. above p. 58.Google Scholar
- 18.Details to be discussed later; cf. below p.207.Google Scholar
- 19.Cf. above p. 54.Google Scholar
- 20.In the so-called “Goal-year-texts”; cf. below p. 351 and p. 554.Google Scholar
- 21.Cf. for the outer planets below p.180f. and p. 182; for the inner planets p. 157 and p.167f.Google Scholar
- 22.Cf. above p. 54.Google Scholar
- 23.Cf. above p. 55.Google Scholar
- 24.Cf. for the outer planets below p.182, for Venus p.157, for Mercury p. 167.Google Scholar
- 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
- 26.Cf. below p.155; p. 171.Google Scholar
- 27.Identical with the mean longitude of the sun at epoch (cf. p.60).Google Scholar
- 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
- 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
- 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
- 4.Ptolemy’s own roundings would give as final result e=1;16,23 and r=43;10,18.Google Scholar
- 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
- 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
- 1.Below p. 172 ff.Google Scholar
- 2.Above p. 151. Z Above p. 153.Google Scholar
- 3.For the sake of clarity the eccentricity is exaggerated in these figures but the angles at O are drawn essentially correctly.Google Scholar
- 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
- 5.Cf. p. 151.Google Scholar
- 6.The approximate period relation leads to about 0;36,58°/d.Google Scholar
- 7.Above p. 60, (5).Google Scholar
- 8.Cf. also Fig. 16 (p.1375).Google Scholar
- 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
- 2.Cf. below II A 5, I C and IIA7,6.Google Scholar
- 3.Cf. Neugebauer [1968, 21.Google Scholar
- 4.Tetrabiblos IV, 3 (Robbins, p. 381).Google Scholar
- 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
- 6.Cf. p. 161 and IC 3, 5.Google Scholar
- 7.Not with Nisan (April) as in the Mesopotamian version of the Seleucid era which is used in the cuneiform texts.Google Scholar
- 8.Cf for details p.1066.Google Scholar
- 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
- 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
- 2.Above p.85.Google Scholar
- 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
- 1.The velocity of this motion is, of course, equal to the mean motion of the sun. Above p.159.Google Scholar
- 2.Below p. 168 f.Google Scholar
- 1.Cf. above p. 159 f.Google Scholar
- 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
- 3.Heiberg, p. 288, 20/289, 1.Google Scholar
- 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
- 5.Using Tuckerman, Tables.Google Scholar
- 6.According to Schoch, Planetentafeln. Cf. above p. 160.Google Scholar
- 8.Cf. above p. 159.Google Scholar
- 9.Cf. p.151.Google Scholar
- 10.In checking this division I find for the last three digits only 58,39,48.Google Scholar
- 1.Hartner [1955], p. 109 to 117.Google Scholar
- 2.Obviously the Cartesian coordinates of K are ML=(R + e) cos a, and LK =(R — e) sin ~i, respectively.Google Scholar
- 1.Cf. below p.389; also above p. 150f., (1) and (2).Google Scholar
- 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
- 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
- 2.Cf. below V A 1, 4.Google Scholar
- 3.For the corresponding term “equant” cf. below p. 1102.Google Scholar
- 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
- 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
- 2.Cf. above p. 57 ff.Google Scholar
- 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
- 4.Above p. 73 ff.Google Scholar
- 5.Computed by Mr. E. S. Ginsberg.Google Scholar
- 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
- 9.Cf. above p.177, (6) and (7), p. 179.Google Scholar
- 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
- 11.The determination of OC is required by the absence of the tangent function as well as by the following steps.Google Scholar
- 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
- 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
- 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
- 3.Alm. XI, 10 (Heib. II, p. 429).-For the Handy Tables cf. below p.1002, (1). Cf. below p. 185.Google Scholar
- 1.Cf. above p.182, n. 15.Google Scholar
- 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
- 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
- 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
- 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
- 6.Above p. 95.Google Scholar
- 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
- 2.Below p. 195 and p. 197ff.Google Scholar
- 3.Cf. above p. 146.Google Scholar
- 4.Ptolemy’s procedure differs from the one given here only in so far as he first finds PT and then p.Google Scholar
- 1.Cf. above p.192 and note I there.Google Scholar
- 2.Cf. p. 183 f.Google Scholar
- 5.Cf. p. 192.Google Scholar
- 6.Above p. 193. For the numerical details cf. below p. 199 (Table 18). Cf. above p. 193.Google Scholar
- 8.Cf. above p. 183 and below p. 204.Google Scholar
- 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
- 2.Described in the case of maximum distance of Mars in Alm. XII, 6 (Man. II, p. 301f.).Google Scholar
- 3.Above p. 193, values of b in the table.Google Scholar
- 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
- 5.From (2b), p.194.Google Scholar
- 6.Above p. 193.Google Scholar
- 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
- 8.For Ptolemy’s procedure cf. p.165 and Fig.149.Google Scholar
- 9.Above p.193.Google Scholar
- 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
- 11.Cf. above p.164, (1) and Table 14 (p.169).Google Scholar
- 1.Above pp. 193 and 195.Google Scholar
- 2.From Table 18 (p. 199), slightly rounded.Google Scholar
- 3.Cf. Tables 17 and 19 (pp. 197 and 201).Google Scholar
- 4.Cf. (8), p.195.Google Scholar
- 5.From Table 17 (p. 197), rounded.Google Scholar
- 6.Cf. Tables 17 and 20 (pp.197 and 201).Google Scholar
- 7.Cf. (9), p. 196.Google Scholar
- 8.From (1).Google Scholar
- 1.He mentions only some special cases, e.g. for k=30° (Manitius II, p. 258).Google Scholar
- 2.Cf. below p. 232.Google Scholar
- 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
- 1.Cf. Fig. 207, p. 1272, moving C into A or B.Google Scholar
- 3.In some cases the result is 0;0,12 less than expected.Google Scholar
- 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
- 5.One finds a cubic equation for the sine of the angle under which the eccentricity e= 3 is seen from C.Google Scholar
- 6.Cf. Figs. 188 and 189 (p. 1265 f.).Google Scholar
- 1.Cf. above p. 148.Google Scholar
- 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
- 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
- 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
- 5.Cf. above p. 151 (4).Google Scholar
- 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
- 7.Cf. p. 151 (4).Google Scholar
- 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
- 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
- 3.AA =1f 25 (cf. p.153).Google Scholar
- 4.Ptolemy needs two steps here, having no tables for tan a. 3 115;30 (cf. p. 58).Google Scholar
- 5.Below p. 241.Google Scholar
- 8.Above p. 159 and Fig.144, p. 1252.Google Scholar
- 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
- 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
- 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
- 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
- 3.Cf., e.g., Strabo, Geogr. II, 5, 39 (Loeb I, p. 511 ).Google Scholar
- 4.Cf. below p. 367; cf., however, below p. 249, note 12.Google Scholar
- 5.Cf. for this value below p. 236.Google Scholar
- 6.Accurately 0;46,57,23 and 0;37,21,3, respectively. Cf. above p. 218.Google Scholar
- 7.Cf. below p. 237 f.Google Scholar
- Ptol. Opera II, p. 153, 15.Google Scholar
- 10.Cf. below p.1017 (2).Google Scholar
- 11.Cf. below p. 535.Google Scholar
- 1.Cf. Table 3, p. 47 and Fig. 41, p. 1218.Google Scholar
- 2.Cf. below pp. 239 and 241.Google Scholar
- 3.Cf. below p. 245ff.Google Scholar
- 5.Cf. p. 219, (7).Google Scholar
- 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
- 7.Cf. above p. 208 and Fig. 213, p. 1275.Google Scholar
- 8.Above p. 180.Google Scholar
- 9.According to p. 153, A=It25.Google Scholar
- 12.Following the rules given p. 222 ff.Google Scholar
- 13.Above p. 159.Google Scholar
- 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
- 1.I,17; I, 22; VII, 18. Translation Manitius, pp. 11, 13, 221, respectively.Google Scholar
- 2.Cf. IC7,2B.Google Scholar
- 3.Cf. Fig. 218 (p. 1278 ).Google Scholar
- 4.Above p.226 (2), and p. 215 (2).Google Scholar
- 5.Cf. above p. 154 (8).Google Scholar
- 6.E.g. Vat. gr. 1291 fol. 89”.Google Scholar
- 7.This was recognized by A. Aaboe [1960], p. 20.Google Scholar
- 8.Below p.403.Google Scholar
- 9.Cf. Fig. 218 (p. 1278 ).Google Scholar
- 10.According to the tables for maximum elongation, Alm. XII, 10. Cf. also Fig. 238 (p. 1288 ).Google Scholar
- 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
- 12.Above pp. 236 and 240.Google Scholar
- 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
- 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
- 2.Cf. below p. 256f.Google Scholar
- 3.Cf. below p. 261.Google Scholar
- 4.Cf. also below p.260f.Google Scholar
- 5.Alm. XIII, 9 (Manitius II, p. 393).Google Scholar
- 6.IBi can reach about 5;30° for Mercury and almost 10° for Venus.Google Scholar
- Cf., e.g., p. 235 for all planets, p. 240 (2) for Venus, p. 241 (1) for Mercury.Google Scholar
- 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
- 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
- 10.Aaboe [1960], p. 7.Google Scholar
- 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
- 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
- 13.Above p. 236.Google Scholar
- 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
- 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
- 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
- 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
- 8.Cf. p. 253 (2).Google Scholar
- 9.For the significance of a negative elongation at r cf. above p.241.Google Scholar
- 10.For F in up +0;6, for Sl +0;4 and —0;11, respectively.Google Scholar
- 1.Cf. above p. 241 and note 11 there.Google Scholar
- 3.The omitted cases are marked by a xGoogle Scholar
- 4.Above p.254; below p. 259.Google Scholar
- 9.Cf. below p.1024.Google Scholar
- 10.Halma, H.T. III, p. 30 to 32.Google Scholar
- 11.Ptolemy, Opera I, 2, Heiberg, p. 606/607.Google Scholar
- 12.Cf., e.g., Cumont [1934].Google Scholar
- 13.CCAG 5, 4, p. 228, 15 to 19; cf. also below V A 3, 2.Google Scholar
- 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
- 15.CCAG 7, p.214 to 224.Google Scholar
- 16.Cf. above p. 235 (1).Google Scholar
- 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
- 17.Cf. also Nallino, Batt. II, p. 255 to 268.Google Scholar
- 18.Cf. Kennedy-Agha [1960], p. 138, Fig. 2.Google Scholar
- 19.Above p. 244 ff.Google Scholar
- 20.Ptolemy, Opera II, p. 4 Heiberg.Google Scholar
- 21.Plan. Hyp. I; cf. Goldstein [ 1967 ], p. 8.Google Scholar
- 22.Cf above p. 257.Google Scholar
- 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
- 2.Kepler, Astronomia Nova (Werke III), Chaps. 59 and 60.Google Scholar
- 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
- 4.Cf for this earlier phase below IV B 3, 4.Google Scholar
- 5.Apollonius, Opera II (ed. Heiberg), p. 139 frgm. 60. Cf. also below pp. 650 and 655.Google Scholar
- 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
- 7.Vettius Valens, Anthol., ed. Kroll, p. 354, 4-7; Cumont [1910], p.16I. Cf. also below p. 602.Google Scholar
- 8.On Sudines and Kidenas cf. below p. 611; on the norm with 8° below IV A4, 2A.Google Scholar
- 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
- 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
- 2.Cf. above Fig. 51, p. 1220.Google Scholar
- 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
- 4.Apollonius, Opera I, p. 402-413; ed. Heiberg; trsl. Ver Eecke, p. 249-255.Google Scholar
- 1.Alm. III, 3 (Manitius I, p. 162).Google Scholar
- 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
- 2.Cf. above IB4,1.Google Scholar
- 3.Cf. for his results below p. 315.Google Scholar
- 4.Alm. IV, 6, Manitius I, p. 223.Google Scholar
- 5.Rome CA III, p. 1053-1056.Google Scholar
- 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
- 1.Cf. above I C 6.Google Scholar
- 2.Cf. Fig. 195, p. 1268.Google Scholar
- 3.Cf., e.g., Fig. 134, p. 1248.Google Scholar
- 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
- 5.Alm. XII, 1 (Manitius II, p. 277). Cf. also above p.191.Google Scholar
- 6.Cf. above p. 264 f.Google Scholar
- 7.Alm. XII,1 (Manitius II, p. 272 f.).Google Scholar
- 1.Cf. below p. 643.Google Scholar
- 2.Cf. above I B 3, 4 A and p. 267.Google Scholar
- 3.Cf. Fig. 68, p. 1227 (and similarly Fig. 268, p. 1303 ).Google Scholar
- 5.Aaboe [1963], p. 8 f.Google Scholar
- 6.ACT, No. 801, Sections 4 and 5 for Saturn, No. 810, Sections 3 and 4 for Jupiter. Cf. below p. 832.Google Scholar
- 1.Almagest VII, 5/VIII, 1.Google Scholar
- 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
- 3.Cf. below IE6,3.Google Scholar
- 4.Cf. below p.823.Google Scholar
- 5.Suidas, ed. Adler II, p. 657, No. 521; cf. also the preceding note 2.Google Scholar
- 6.Ptolemy, Opera II, p. 67, 10 and 16 to 18 (ed. Heiberg). Cf below p. 928.Google Scholar
- 7.Cf. Table 28, below p. 276.Google Scholar
- 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
- a Maass, Aratea, p. 121.Google Scholar
- 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
- 9.Cf. below p. 301.Google Scholar
- 10.Hipparchus, Arat. Comm. ed. Manitius, p. 184/5; cf. also p. 292, note 3.Google Scholar
- 11.The inscription of Keskinto, e.g., shows that other astronomers had worked at Rhodes (cf. below p. 698).Google Scholar
- 12.Alm. VII, 2 and 3 (Manitius II, p. 15, 9 and 20, 21).Google Scholar
- 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
- 13.Below p. 281.Google Scholar
- 14.Vogt [1925], col. 25. Cf. below p.284.Google Scholar
- 15.Hipparchus, Arat. Comm., Manitius, p. 182 to 270.Google Scholar
- 16.Hipparchus, Arat. Comm., Manitius, p. 270 to 280. Cf. also below p. 279, note 22.Google Scholar
- 17.Alm. VII, 3 (Manitius II, p. 18 to 20).Google Scholar
- 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
- 19.Delambre HAA I, p. XXII to XXIV.Google Scholar
- 20.Fotheringham [1918], p. 408.Google Scholar
- 21.Cf. below p. 284.Google Scholar
- 22.Alm. III, 1 (Manitius I, p. 133, 32).Google Scholar
- 1.Ptolemy, Opera II, p. 1 to 67. Cf. also below V B 8,1 B.Google Scholar
- 2.Cf. below p. 301, n. 1.Google Scholar
- 3.Cf. below p. 301, n. 2.Google Scholar
- 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
- 5.Below p. 283.Google Scholar
- 6.Below pp. 285 ff.Google Scholar
- 1.Cf. below p. 368.Google Scholar
- 2.Cf., e.g., IV A 4, 2 A and 2 B.Google Scholar
- 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
- 4.Aratus Comm., p. 132, 7.Google Scholar
- 5.E.g. Aratus Comm., p. 48, 5 to 7.Google Scholar
- 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
- 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
- 8.Aratus Comm., p. 89.Google Scholar
- 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
- 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
- 11.Arat. Comm., p. 68, 20ff. ed. Manitius. Cf. also Vogt [1925], col. 29.Google Scholar
- 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
- 13.Aratus Comm., p. 8 to 182, ed. Manitius.Google Scholar
- 14.ÉRÉXE1 or KEITa,, respectively.Google Scholar
- 15.Aratus Comm., p. 182 to 280.Google Scholar
- 16.This result agrees with the tables in Alm. II, 8.Google Scholar
- 17.Cf. for this problem below p. 868 f.Google Scholar
- 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
- 19.niimuç, (Arai. Comm., p. 272, 1); n,jyvri ov Slâarriµa (p. 190, 10).Google Scholar
- 20.ryµtnryXtov (Aral. Comm., p. 186, 11; 190, 8, 26 etc.); Siio µépry znryyewç (p. 254, 11, 25; 268, 8, etc.).Google Scholar
- 21.Cf. below p. 591; also p. 304.Google Scholar
- 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
- 23.Manitius, p. 150, 2f.; cf. below p. 299.Google Scholar
- 24.Almagest VII, 3 (Manitius II, p. 18 to 20).Google Scholar
- 25.Cf. below p. 284.Google Scholar
- 26.Cf. below p. 283, note 13.Google Scholar
- 27.Cf. below p. 286. With a few exceptions all these stars are near the ecliptic.Google Scholar
- 28.Cf. below p. 934.Google Scholar
- 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
- 2.Copernicus, De revol. II, 14 (Gesamtausgabe II, p. 102, Thorn, p. 115).Google Scholar
- 3.Cf., e.g., the accusations of dishonesty in HAA I, p. XXXI. Cf. also Vogt [1925], col. 33.Google Scholar
- 4.No. 5354-55 (1925).Google Scholar
- 5.In section C, below p. 284.Google Scholar
- 6.Hipparchus, Arat. Comm., p. 186 to 270 ed. Manitius.Google Scholar
- 7.Almagest I, 12 (Manitius, p. 44).Google Scholar
- 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
- 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
- 10.I.e. the “polar longitude” p of E. Cf. above p. 279.Google Scholar
- 11.Alm. VII, 3 (Manitius II, p. 18 to 20); Geography I, 7, 4 (ed. Nobbe, p. 15, 6 ).Google Scholar
- 12.Strabo, Geogr. II, 5, 41 (ed. Meineke, p. 181, 21 to 25).Google Scholar
- 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
- 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
- 15.Cf. above p. 275.Google Scholar
- 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
- 17.Cf. for details Vogt [1925], col. 23 to 26.Google Scholar
- 18.Vogt [1925], col. 23: Hipparchus —0.06° (±0.065°), Ptolemy +0.01° (±0.03°).Google Scholar
- 19.I.e. around —138 and +48, respectively.Google Scholar
- 20.Cf. below p. 294.Google Scholar
- 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
- 2.Biot [1843], p. 61.Google Scholar
- 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
- 4.First edition p. 563, 4th edition (1851), p.474. Also referred to by Humboldt, Kosmos III (1850), p. 221.Google Scholar
- 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
- 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
- 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
- 8.From CCAG 9, 1, p. 189 (ignoring variants).Google Scholar
- 9.Rehm [1899], Boll [1901].Google Scholar
- 10.Boll [1901], p. 192ff.Google Scholar
- 11.Almagest VIII, 1 (Manitius II, p. 64).Google Scholar
- 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
- 13.Heiberg II, p. 37, 15: noLlagry.Google Scholar
- 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
- 15.Maass, Comm. Ar. rel., p. 128 (No. 12). The same formulation also in CCAG 8, 4, p. 94, fol. 10.Google Scholar
- 16.HAA I, p. 290.Google Scholar
- 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
- 18.The one exceptional case (Gundel HT, p. 25, 8 and p. 152, No. 63) gives 30 minutes beyond integer degrees.Google Scholar
- 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
- 20.Gundel, HT, p. 127ff.Google Scholar
- 21.y Canc; Almagest: 2=0910;20, β=2;40.Google Scholar
- 22.We also known that the Babylonian division of seasons took the summer solstice as the point of departure (Neugebauer [1948]).Google Scholar
- 23.Cf. below p. 309 f.Google Scholar
- 24.Cf. above p. 279.Google Scholar
- 25.aPlyccrtaizoi.Google Scholar
- 26.Alm. VII, 1, Manitius II, p. 5 to 8.Google Scholar
- 27.Cf. above p. 277, note 4.Google Scholar
- 28.Cf. above p. 283.Google Scholar
- 29.Alm. VII, 4 (Manitius II, p. 31f.).Google Scholar
- 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
- 31.Below p. 577.Google Scholar
- 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
- 33.Rehm [1899], p. 265.Google Scholar
- 34.Hipparchus, Comm. Arat., p. 184, 23 ed. Manitius.Google Scholar
- 35.Alm. VII, 3 (Manitius II, p. 26 and p. 28, respectively).Google Scholar
- 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
- 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
- 1.The division in sentences is mine, made for easier reference.Google Scholar
- 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
- 2.For a comparison with modern standards cf. Peters-Knobel, Ptol. Cat., p. 120f. and the literature quoted there.Google Scholar
- 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
- 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
- 5.Hipparchus, Comm. Ar., p. 238, 31.Google Scholar
- 6.E.g. p. 42 to 45.Google Scholar
- 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
- 8.De signis”; for the title of the “Catalogue of Stars” cf. above p. 277, note 4.Google Scholar
- 9.Only twice in the Commentary to Aratus; once more in a quotation from Attalus.Google Scholar
- 10.NH II, 95.Google Scholar
- 11.Cf above p. 285.Google Scholar
- 12.Maass, Comm. Ar. rel., p. 137.Google Scholar
- 13.In the Almagest three stars of the Pleiades are considered to be of the 5th magnitude, only one of the 4th.Google Scholar
- 14.Changed by Housman arbitrarily to sextumque which makes no sense.Google Scholar
- 15.Manilius V, 710-717 ed. Housman (V, p. 89-91), 711-719 ed. Breiter (I, p. 148, II, p. 178f.).Google Scholar
- 1.From observations made by Timocharis in Alexandria in the years —294/-282 (Alm. VII, 3, Manitius II, p. 22 to 27).Google Scholar
- 2.Alm. VII, 2 (Heib. II, p. 12, 21): nepì Tfjç itETa7rT(6aECOç ran TpoiriKCUV Kai ia11/4Eprvthv a11nwiojv.Google Scholar
- 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
- 4.Cf. below p. 308.Google Scholar
- 5.Cf. below p. 296.Google Scholar
- 6.Alm. III, 1 (Manitius, p. 145) from the work on the length of the year.Google Scholar
- 7.Alm. III, 1 (Manitius, p. 132).Google Scholar
- 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
- 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
- 10.Alm. VII, 2 (Manitius II, p. 15).Google Scholar
- 11.Preserved only in Arabic; published by Walzer [ 1935 ]. Cf. also Neugebauer [1949, 1].Google Scholar
- 12.Anthol., ed. Kroll, p. 353,12f.; cf. below p.601.Google Scholar
- 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
- 14.Alm. VII, 1 (Manitius II, p. 4).Google Scholar
- 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
- 16.Alm. III, 1 (Manitius, p. 143ff.).Google Scholar
- 17.The diagonals in Fig. 283 represent intervals of exactly 365 1/4°.Google Scholar
- 18.Alm. III, 1 (Manitius, p. 135).Google Scholar
- 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
- 20.Below p. 295.Google Scholar
- 21.Cf. Alm. VII 2.Google Scholar
- 22.Alm. III, 1 (Manitius, p. 137f.).Google Scholar
- 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
- 24.Alm. IV, 11, Manitius I, p. 252.Google Scholar
- 26.Cf. below p. 318.Google Scholar
- 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
- 28.Alm. VII, 1 (Manitius II, p.4).Google Scholar
- 29.Cf. above p. 280, p. 287.Google Scholar
- 30.Above p. 294, note 15.Google Scholar
- 1.ed. Jahn, Proleg., p.V.Google Scholar
- 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
- 3.Cf. below p. 624.Google Scholar
- 4.Cf. below II Intr. 3, 1.Google Scholar
- 5.The resulting length of the synodic month would be 29;31,51,3,49, … days.Google Scholar
- 6.Cf. above p. 293.Google Scholar
- 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
- 8.Cf. below p. 310.Google Scholar
- 9.Cf. note 5.Google Scholar
- 10.Ginzel, Hdb. II, p. 391. Similar already Ideler, Chron. I, p. 353.Google Scholar
- 11.Dinsmoor, Archons, p. 410-423.Google Scholar
- 12.E.g. Archons, p. 414: “Therefore we may assume that the authorities … called in a specialist, namely, Hipparchos.”Google Scholar
- 1.Above p. 293.Google Scholar
- 2.Alm. VII, 2 (Man. II, p. 15).Google Scholar
- 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
- 4.Alm. IV, 2 (Man. I, p. 196); cf. below p. 310 (5).Google Scholar
- 5.Above p. 295.Google Scholar
- 6.Cf. for the details below IV B 2, 3.Google Scholar
- 7.Cf. above Table 28, p. 276.Google Scholar
- 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
- 1.Cf. also Toomer [1973], p. 19/20.Google Scholar
- 2.Theon indeed admires the conciseness of Ptolemy’s derivations (ed. Rome CA II, p. 451).Google Scholar
- 3.Aratus Comm., Manitius, p. 150, 2; cf. above p. 279.Google Scholar
- 4.Cf. below IV B 5.Google Scholar
- 5.Burgess, Sûr. Siddh., p. 64; cf. also Nallino, Scritti V, p. 220f.Google Scholar
- 6.Cf. above p. 278.Google Scholar
- 8.Cf. below p. 315.Google Scholar
- a Cf. above p. 140, n. 3.Google Scholar
- Sb Cf. the diagram Toomer [1973], p. 19, Table II.Google Scholar
- o Cf. above p. 23.Google Scholar
- 9.Metrica I, 17-25, Heron, Opera III, ed. Schöne, p. 46, 23-64, 31.Google Scholar
- 10.Opera III, p. 58, 19; p.62, 17-18. This need not to be understood as an exact title.Google Scholar
- 11.This is a conclusion first clearly established by A. Rome [1933].Google Scholar
- 12.Opera III, p. 66, 6-68, 5.Google Scholar
- 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
- 5.Cf. below p.304f.Google Scholar
- 6.VB3,7B.Google Scholar
- 7.Cf., e.g., Varâhamihira, Paficasiddhântikâ IV, 41 to 44; Neugebauer-Pingree, II, p. 41 - 44.Google Scholar
- 9.Presumably the work on simultaneous risings (cf. above p. 301).Google Scholar
- 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
- 11.This is the “day-radius” of Indian astronomy.Google Scholar
- 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
- 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
- 14.Ar. Comm., p. 96, 11.Google Scholar
- 15.Ar Comm., p.98, 20f.; also Theon Smyrn., p. 202, 19ff. (Hiller); Dupuis, p. 327.Google Scholar
- 16.Cf. above p. 301, note 2.Google Scholar
- 17.Ar. Comm., p. 182-185.Google Scholar
- 18.Given in Ar. Comm., p. 244-271 (Manitius).Google Scholar
- 19.Ar. Comm., p. 124/5.Google Scholar
- 20.This is again a term of Indian astronomy (cf., e.g., Pc.-Sk. IV, 27 and 28).Google Scholar
- 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
- 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
- 28.Cf. above Table 2, p. 44 (from Alm. II, 6).Google Scholar
- 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
- 31.Cf. below IV D 3.Google Scholar
- 32.Eratosthenes, however, can probably be excluded since his distancesdeviate from the Hipparchian (cf. Fig. 291, p. 1313 ).Google Scholar
- 33.Cf. also below p. 1014.Google Scholar
- 34.Cf., e.g., the treatise by Hypsicles (below IV D 1, 2 A) who is about contemporary with Hipparchus.Google Scholar
- 35.E.g. column J of the lunar theory of System B. Cf. below II B 3, 5 B.Google Scholar
- 1.Vettius Valens, Anthol. IX, 11 (ed. Kroll, p. 354, 4-6); also above p. 263.Google Scholar
- 2.Cf. above p. 58.Google Scholar
- 3.Rome [1950], p. 214f.Google Scholar
- 4.Cf. above p.55.Google Scholar
- 5.Cf. below p.971, n. 21.Google Scholar
- 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
- 7.Cf. Fig. 286 as compared with Fig. 53 (p. 1221 ).Google Scholar
- 8.Nallino, Batt. I, p. 43 f.Google Scholar
- 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
- 10.Alm. III, 4 (Manitius I, p. 166); cf. Fig. 53, p. 1221.Google Scholar
- 11.Alm. III, 4 (Manitius I, p. 170); also Theon of Smyrna (Dupuis, p. 218/219), etc.Google Scholar
- 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
- 13.Cf. above p. 297 (2) or below p. 310 (5).Google Scholar
- 14.Cf. also above p. 55. note 1.Google Scholar
- 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
- 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
- 3.From Suidas, ed. Adler II, p. 657, 27 f.Google Scholar
- 4.Cf. below I E 5, 2 A.Google Scholar
- 5.Kugler, Mondrechnung (1900), p. 111; cf. p.348 ff.Google Scholar
- 6.Cf. below p. 310.Google Scholar
- 7.Cf. above p. 271.Google Scholar
- 8.Alm. V, 2 and V, 3; cf. above p. 84 and p. 89.Google Scholar
- 1.Manitus I, p. 197f.; cf above p. 69 (1) to (3).Google Scholar
- 2.Cf. below p. 483 (3); p.478(2 c); p. 523 (2c) or ACT, p. 75 (20).Google Scholar
- 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
- 5.Alm. IV, 2 (Manitius I, p. 195).Google Scholar
- 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
- 8.Copernicus, De Revol. IV, 4, silently correcting Ptolemy’s value (Gesamtausg., p. 215, 31 f).Google Scholar
- 9.ACT I, p. 272.Google Scholar
- 10.This was pointed out by A. Aaboe [1955].Google Scholar
- 11.Cf. below p. 378 (15h). p. 396 (5b), and p. 496 (20)Google Scholar
- 12.Cf., e.g., above p. 125 f.Google Scholar
- 3.Cf. above p. 306.Google Scholar
- 4.The proper understanding of this passage in relation to Ptolemy’s method is due to Olaf Schmidt [1937].Google Scholar
- 5.We shall discuss these parameters later on (below p.325).Google Scholar
- 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
- 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
- 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
- 10.Alm. IV, 9 (Manitius I, p. 238-241).Google Scholar
- 11.Cf. above p. 81.Google Scholar
- 1.Cf. above p. 84.Google Scholar
- 2.Manitius I, p. 245 f; cf. also Man. I, p. 212, 25.Google Scholar
- 3.Below p. 318.Google Scholar
- 4.A denominator 4;45 would have been slightly better because 1,0/4;45Google Scholar
- 5.Cf. above p.84.Google Scholar
- 6.In Alm. IV, 11; cf. below p.316.Google Scholar
- 8.Rome, CA I, p. 68; translation Toomer [1967], p. 147.Google Scholar
- 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
- 10.Cf. below I E 5, 4 B.Google Scholar
- 12.These observations antedate Hipparchus’ lifetime.Google Scholar
- 13.For a Babylonian record of this eclipse cf. Schaumberger, Erg., p. 368, note 1.Google Scholar
- 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
- 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
- 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
- 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
- 19.Manitius I, p. 247, 5.Google Scholar
- 20.Taking the equation of time into consideration does not help matters.Google Scholar
- 21.Toomer [1973], p.9-16. Cf. also above p. 299f.Google Scholar
- 22.Above p.74f. and Figs. 65 to 67 there.Google Scholar
- 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
- 2.Cf., e.g., Rome [1950].Google Scholar
- 3.Below II B 6 and II B 7.Google Scholar
- 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
- 5.Aim. III, (Man. I, p. 183, 5 ).Google Scholar
- 6.Cf. above p. 73 ff.Google Scholar
- 7.Cf. e.g., above p.72; p. 77.Google Scholar
- 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
- 9.Below p. 322.Google Scholar
- 1.Above p. 310 (3).Google Scholar
- 2.Published in CCAG 8, 2, p. 126 to 134.Google Scholar
- 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
- 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
- 5.CCAG 8, 2, p. 126, 21 to 28.Google Scholar
- 7.NH II, 57 (Budé II, p. 25 f.).Google Scholar
- 7.Above IB6,4.Google Scholar
- 9.Below p. 523 (2 c).Google Scholar
- 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
- 11.Maass, Comm. Ar. rel., p. 47, 13; also below p. 666.Google Scholar
- 1.Cf. below I E 5, 4 B.Google Scholar
- 2.Cf. for this method, e.g., above p. 295.Google Scholar
- 3.Aim. V, 19, Heiberg I, p. 450, 1 and 4 (rzapaaLlcocrncâ).Google Scholar
- 4.Rome CA I, p. 150, 20-155, 27.Google Scholar
- 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
- 6.Above p. 90.Google Scholar
- 7.Cf. above I A 5, 5.Google Scholar
- 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
- 9.Cf. above p. 304 ff.Google Scholar
- 10.Alm. V, 19, Manitius I, p. 329, 24 - 29.Google Scholar
- 11.Cf. below p. 324.Google Scholar
- 12.Actually Ptolemy’s own methods are just as crude (cf. above I B 5, 6).Google Scholar
- 13.Cf. the apparatus to Alm. II, 13 in Heiberg, p. 181, 28.Google Scholar
- 14.Rome, CA I, p. 152, note (2); similarly p. 168, note (1).Google Scholar
- 1.Alm. V, 8 gives for AL =30 the latitude 2;30°. Thus the latitude at L should be greater. Hultsch [ 1900 ].Google Scholar
- 2.Swerdlow [1969].Google Scholar
- 3.Swerdlow [1969], p. 297/298.Google Scholar
- 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
- 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
- 6.Dupuis, p. 318/319; ed. Hiller, p. 197, 9.Google Scholar
- 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
- 9.Cf below p. 962.Google Scholar
- 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
- 2.Cf. for this eclipse (of —189 March 14) above p. 316, n. 9.Google Scholar
- 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
- 1.Cf above p.296.Google Scholar
- 2.Alm. IX, 3 (Man. II, p. 99). Cf. above I C 1, 4; also Neugebauer [ 1956 ], p. 295.Google Scholar
- 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
- 4.Ptolemy, Alm. IX, 2, Man. II, p. 96.Google Scholar
- 5.Cf. below p. 823 f.Google Scholar
- 6.Below V A 1.Google Scholar
- 7.Goldstein [1967]; cf. below V B 7.Google Scholar
- 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
- 9.Cf. above p. 327.Google Scholar
- 10.Cf. above p. 291.Google Scholar
- 11.Wessely [ 1900 ]; Neugebauer [1962, 3], p. 40, col. II, 7-10. Cf. for this text also below p. 737 (n).Google Scholar
- 12.Cf. below p.693.Google Scholar
- 1.Cf. above p. 306 and below p. 823.Google Scholar
- 2.NH II, 95, Loeb I, p. 239, Budé II, p. 41.Google Scholar
- 3.Continued in the passage discussed above p. 285.Google Scholar
- 4.AG, p. 543/4; also Pfeiffer, Sterngl., p. 115.Google Scholar
- 5.Cumont [1909], p. 268; similar Eg. astrol., p. 156, etc.Google Scholar
- 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
- 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
- 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
- 9.Mentioned by Ptolemy in his “Analemma”; cf. below p.1380 and Figs. 26 and 27.Google Scholar
- 10.Engelbrecht, p. 47, 20.Google Scholar
- 11.Engelbrecht, p. 60, 30.Google Scholar
- 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
- 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
- 14.Cf. above p.283; p. 287.Google Scholar
- 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
- 1.Even a hemispherical cupola is by no means an a priori concept; cf. below p. 577.Google Scholar
- 2.Cf. above I E 2, 1 A.Google Scholar
- 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
- 1.Cf. below p.934; also above p.280.Google Scholar
- 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
- 2.Honigmann, SK (widely accepted); Dicks [1955] opposing Honigmann.Google Scholar
- 3.Reinhardt (cf., e.g., Honigmann, SK, p. 8f.).Google Scholar
- 5.Cf., e.g. below IV D 1, 3.Google Scholar
- 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
- 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
- 9.Cf. above p. 305, n. 27.Google Scholar
- 10.It is, of course, absurd to give latitudes to seconds (rounded!), as, e.g., in the Loeb translation.Google Scholar
- 11.Cf. above p. 305.Google Scholar
- 12.Cf. below p. 653.Google Scholar
- 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
- 14.Cf. below p. 746, n. 3.Google Scholar
- 15.Alm. VI, 11 (Heiberg I, p. 538/539). Introduction to the Handy Tables (Opera II, ed. Heiberg, p. 174, 17 ).Google Scholar
- 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
- 17.Above p. 43 f. and Table 2.Google Scholar
- 18.Ptolemy, Opera II ed. Heiberg, p. 4, 3-20; below p. 928.Google Scholar
- 19.Above p. 304.Google Scholar
- 20.Cf. p. 1313, Fig. 291.Google Scholar
- 21.Strabo II 5, 35 (Loeb I, p. 507; Budé I, 2, p. 119).Google Scholar
- 22.Above p. 290.Google Scholar
- 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
- 24.Aratus Comm., p. 82, 24 f. Actually S x 31;17° in —125.Google Scholar
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- 27.Cf. above p. 335.Google Scholar
- 28.Strabo II 5, 38 (Loeb I, p. 511, Budé I, 2, p. 120).Google Scholar
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- 32.Ptolemy, Geogr. I 4, 2 (Nobbe, p. 11; Milk, p.21).Google Scholar
- 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
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- 4.Strabo, Geogr. 14, 1 (Loeb I, p. 233; Budé I, 1, p. 167); cf. also below p.652.Google Scholar
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- 2.Alm. III, 1 Heiberg, p. 207, 18.Google Scholar
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- 4.Cf. belowVB8,1B.Google Scholar
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- 6.Comm. in Arist., Vol. VII, p. 264, 25-266, 29. Discussed, e.g., by Duhem, SM I, p. 386, p. 394.Google Scholar
- 7.Diels, Dox., p. 404 (also Diels VS15) I, p. 226, 25 or Galen, Opera XIX, p. 307 ed. Kühn).Google Scholar
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- 1.Above I E 2, 1 B.Google Scholar
- 2.Above p. 283.Google Scholar
- 3.VB3,7B.Google Scholar
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- 5.IE2,IC, p.287; IE2,1C1.Google Scholar
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- 10.IE2,2A.Google Scholar
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- 12.I E 2,2 A, p. 294; I E 2, 2C, p. 298.Google Scholar
- 13.IE4.Google Scholar
- 14.IE5,1C, p.317.Google Scholar
- 15.Cf. below IV C 3, 8.Google Scholar
- 16.IE5,1C, p.315.Google Scholar
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- 19.IE5,1A.Google Scholar
- 20.I E 6, 1, p. 329. Also the use of Babylonian units (“cubit” of 2°) points in the same direction (cf.Google Scholar
- 21.E 2, 1 A, p. 279; I E 3, 2, p. 304).Google Scholar
- 21.I E 3, 2, p. 306.Google Scholar
- 33.I E 6, 1, p. 330; also I C 8, 5, p. 261.Google Scholar
- 34.TB 5, 4 A, p. 109.Google Scholar
- 35.Copernicus found 1179 re ( De revol. IV, 19 ).Google Scholar
- 36.IE 5,4 A.Google Scholar
- 37.Swerdlow [1969], against Hultsch [1900].Google Scholar
- 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
- 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

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