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.
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Schoy, Al-Bir. p. 81 (No. 14); cf. also below p.776.
MC T(UV ypappii v, meaning “rigorous” methods (cf. below p.771 n. 1).
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).
Cf., e.g., ed. Heiberg I, p. 317, 22f., et passim.
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.
VII, 3 (1-leib. II, p. 30,18).
Cf. Rome, CA I, p. 569.
For plane trigonometry cf. above p.26.
Contrast: “sphaera obliqua” denotes geographical latitudes different from zero; the corresponding rising times are called “oblique ascensions.”
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).
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.
Almagest II, 3 Heiberg, p. 95, 6 to 13.
Cf. p.142.
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.
VI, 11 Heiberg, p. 543, 24f. and plate at the end of Vol. I.
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. 144
Cf. our Pl. I.
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.
Cf. above p.32.
We ignore here the change of the solar longitude between sunrise and sunset.
Above p.36(1).
Alm. VII, 3 Heib. II, p.33, 3ff. For the longitude cf. below p.60.
III, 1, 61 and VIII, 8, 3 (Nobbe, p. 151, 26 and p. 205, 7f.).
Almagest II, 9.
From Alm. II, 11.
One can express this also in the form that α′ is reckoned from the winter solstitial point 0°, because α′(0)=0°.
For an example cf. below p. 979.
Cf. above I A 4, 3.
Cf., e.g., the tables in Alm. II, 13 (below p. 50ff.).
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.
Cf. above I A 3, I.
E.g. VIII 16, 3-14 (Nobbe, p. 221-223); several numbers are garbled.
Probably written sometime between A.D. 500 and 600.
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.
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.
Cf. below p. 50.
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.
Instead of the above found angle m, =32;10° for Rhodes and 1i0° the table gives only 122;7-90= 32;7°.
The values for 2n2 are rounded to the nearest degree while 23;51° is taken for e.
Cf. below p.51.
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.
The Greek text has no technical term for “zenith distance” but says simply “arc.” 2 Above p. 40.
Cf. below p. 529.
Cf. also below p. 294.
For an apparent confirmation of this constant of precession from the motion of the apsidal line of Mercury cf. below p. 160.
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, ....
Tannery, AA, p. 163.
I, 10 (p. 47, 3 Heib.) and III, 1 (p. 209, 13 ff. Heib.).
Cf. IVB2,1.
Cf. p. 361.
Cf. H Intr. 5.
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.
Cf. Neugebauer [1962, 2], p. 267. Actually the motion is slightly faster than precession.
Below p.73 ff.
Below p. 173 ff.
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.
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.
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.
Heiberg II, p. 33, 3 ff. = Manitius II, p. 28, 14 ff.
For the solution of this problem, assuming 2=x23° given, cf. above p. 41.
Correctly about 1;10” = 17;30°. Thus Ptolemy’s error is only 2;30° (not 4° as Manitius II, p. 27 note a) says).
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).
Cf. Neugebauer [1938], p. 22. Cf. below p. 848.
Cf. p. 563, n. 3.
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.
Cf. above p.62.
Below p. 67.
Cf. above p. 59.
Almagest IV, 6; cf. below p. 77. The dates are −720 March 19, −719 March 8 and September 1 respectively.
Cf. p. 63.
As always K = 2 − II 5;30 = 2 − 65;30°.
Ptolemy, Opera II, p. 162, 23-163, 6 ed. Heiberg. Cf. also below p. 984f.
Cf. Neugebauer [1958], p. 97ff.
Flamsteed’s treatise “De inaequalitate dierum solarium” (London 1672) is supposedly the first modern treatment of the subject; cf. Wolf, Handbuch 2, p. 261.
Cf. p. 63.
Cf. above p. 62.
Because of the equivalence theorem (p. 57) we need not distinguish between an eccenter-and an epicycle-model.
Above p. 55.
Below p. 79.
Below p. 81.
Above p. 55.
Above p. 57f.
Cf. below p. 546.
Cf. p. 481 f.
Below p. 310.
It seems possible that the relation (5), quoted below p. 310, is the result of these observations. Cf. p. 310.
Cf. below p.78.
Below p. 76ff.
Cf. also below p.77.
The interior of the triangle would have the signature − − −(or + + + since we are dealing in fact with the projective plane).
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).
Cf. above p. 74.
As usual these computations contain many small inaccuracies such that r=5;13 would be the nearest common solution.
Below p. 79.
Below p. 79.
That is “Marduk gave an heir”; biblical distortion: Merodach-baladan.
Accurate value: 57 minutes.
Above p. 75f.
From P. V. Neugebauer, Kanon d. Mondf.
Cf. P. V. Neugebauer, Astr. Chron. II, p. 128.
For the method cf. above p. 76 (2).
Since dt is close to an integer number of years the influence of the equation of time can be ignored.
Actually the tables give d.I.=123;22,33 and da=103;35,23.
Above p.63.
490 Apr. 25.
A.D. 125 Apr. 5.
Modern values: 1.7” and 2.0”, respectively.
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.
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.
Drawn to scale.
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.
501 Nov. 19/20. Compare Fig. 75 with Fig. 70 II, p. 1228.
Actually only 1.5” and 2.1” respectively.
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.
Actually one finds 160;3,9.
Cf. above p. 77.
For the more primitive method used by Hipparchus cf. below p. 313 f.
As we have shown on p. 64, the equation of time has only a negligible effect.
Solar eclipses remain outside of these discussions because they depend also on geographical elements.
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).
Cf., e.g., Almagest IV, 6 (Man. I, p. 218 f.) or V, 2 (Man. !, p. 260ff.).
Below I C 7.
Alm. IV, 6 (Man. I, p. 219).
The error thus committed reaches only about 0;6° for w between about 30° and 60°.
Above p. 30.
Cf. above p. 80 and Fig. 72.
Cf. below p.1109f.
Below p. 88ff. For a comparison with modern theory cf. p. 1108.
Below p. 155.
Cf. below p. 88.
Below, p. 91ff.
Cf. p. 80.
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.
A.D. 139 Febr.9.
Cf. above p. 50.
For some textual difficulties cf. below p. 92. -127 Aug. 5.
Above p.76.
In the Canobic Inscription (below p. 903) the parameters r and e are renormed such that R—e obtains the value 60.
Thirteenth and fourteenth century; cf. Roberts [1957].
Copernicus, De Revol. IV, 3, IV, 8, IV, 9. Cf. also Neugebauer [1968, 2].
The same term also occurs in the theory of eclipses (below p. 141) but with totally different meaning.
Angles are drawn nearly to scale but the eccentricity, and particularly the radius of the epicycle, are exaggerated.
126 May 2 and July 7, respectively.
Cf. below p. 92.
Including the equation of time.
Above p. 87.
Above p. 87 and p. 89.
Above p. 89, No. II.
Above p. 48 ff.
From p(H)=a(M)+90; cf. p.42.
Above pp. 87, 89.
Above p. 89.
Aim. III, 13 for the angle between ecliptic and altitude circle, Alm. V, 18 for parallax.
For the Hipparchian theory of parallax cf. below I E 5, 3.
Ptolemy’s rounded values (A(=1IL 10°, 1;30” west) would give an angle of 83°.
Ideler, Astron. Beob., p. 217 and Chron., p. 345. Also Ginzel, Hdb. II, p. 410.
Heiberg, p. 363, 18f.
Cf. p. 80.
Cf. above p. 88.
Cf. p.80.
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°.
Cf. p.83.
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°.
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.
A.D. 98 Jan. 14.
Cf. p. 84.
Alm. VII, 3 (Heib. II, p. 33, 19 ).
Ptolemy 2;10° as Heiberg and Manitius interpret the text (cf. below p. 117, note 7).
Above p.84.
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.
The maximum of cQ=5;1 (cf. p.80), of co = 2;23 (cf. p.59).
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.
Cf. below I E 5, 3 and I E 5, 4 B.
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).
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.
Below p.104.
Below p.115.
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°.
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).
A.D. 135 Oct. 1.
Accurate computation with the tables of chords (Alm. II, 11) leads to EP=39;49,31rex39;50re (modern tables: 39;49,48).
P. V. Neugebauer, Astron. Chron. I, p. 72f.
For p1 one finds about —0;3,36° whereas Ptolemy assumes p,= O.
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).
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).
The instrumental problems are discussed by Hultsch [1897], [1899], [1900]; by Rome, Pappus Comm., p.87ff.; by Lejeune, Euclide et Ptol., pp. 131, 151.
Below p.106.
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.
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.
Cf., e.g., the list in Houzeau, Vade-mecum, p.404f.
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.
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.
Rome p. 107, 10ff.
Above p. 58.
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.
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.).
De revol. IV, 21 (Gesamtausg. II, p. 257).
Cf. above p. 104, note 4.
Epitome IV, 1, IV (Werke 7, p. 279).
Cf. below p.148 f.
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).
For this part of the work, preserved only in Arabic, cf. below p. 918.
Ptolem., Opera II, p. 118 (Heiberg).
Cf. below V B 7, 6.
Above p. 109, (1).
Above p. 112.
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.
Above p. 50.
Cf. above p. 104, (a).
Cf. above p. 110.
Above p. 93 ff.
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.
Pappus, Comm., ed. Rome, p. 166, 16ff.
Above p.115.
Cf. p. 96.
Alm. VII, 3 (Man. II, p. 29).
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°.
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.
Cf. Alm. VII, 3 (Manitius II, p. 28). Cf. above p. 79, p. 82, and p. 60.
This relation was probably known in Egypt long before hellenistic times; cf., p. 563.
For syzygies the second inequality is zero.
Cf. above p.104, (1) and (2); p. 105, (3) and (4); p. 109, (2).
Above p. 80.
Above p. 59.
Cf. our discussion of these statements below p.127.
Pappus, Comm., ed. Rome p. 194-197.
For a proof of this theorem and further discussion cf. Neugebauer, Al-Khwar., p. 122f.
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.
This includes the rounding errors mentioned on p.127.
For Ptolemy’s much refined investigation of consecutive syzygies cf. below p.133, F.
Example: Oppolzer, Canon, Nos. 2271 and 2272 (A.D. 265 Oct. 12 and 266 March 8).
We ignore here, as always in these discussions, the influence of geographical longitude which can exclude eclipses because of the time of the day.
Again supplemented in Pappus’ Commentary (ed. Rome, p. 226 to 231).
Of course, as always, lunar minus solar parallax.
Example: Oppolzer, Canon, Nos. 5356 and 5357 (A.D. 1049 March 6 and August 1); cf. also p.133, n. 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.
Examples passim in Oppolzer, Canon; e.g. Nos. 6201 and 6202, or 6501 and 6502. Cf. also p. 133, n. 7.
Cf. below p. 140.
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.
Using the values from (1), p.125 but with small roundings in the results.
For solar eclipses one has to replace s by re.
Duration” is here always meant in the sense of the elongations tabulated in the Almagest.
Alm. VI, 7 Heib., p. 512, 8.
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.).
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, …).
Unfortunately from the viewpoint of our historical interests, Ptolemy completely ignored the problems of first and last visibility of the moon.
Above p.88.
E.g. Tetrabiblos II, 10 (p. 92, 2, 13 Boll-Boer).
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).
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.
E.g. to Pappus; cf. his Commentary to Alm. VI ed. Rome, p. 309. Cf. p. 997 f.
Cf. Roberts [ 1957 ]; Neugebauer [1968, 2].
Cf. below V C 4, 5 B and 5 C.
Cf. p. 386ff.
Cf. Appendix VI B 7, 2.
Cf. below p. 208 and p. 212.
Cf. below p. 1101.
Computed for A.D. 100 from Almagest XI, 11 using Ptolemy’s constant of precession.
Case (1); for the elliptic approximation of the deferent of Mercury cf. below p. 168.
Below p. 227ff.
Above p. 109ff.
Cf. p.111.
For more historical details cf. below p.270f.
As we shall see (below p. 421) the Babylonian planetary computations make use of precisely this fact.
Cf. below I D 3, 1.
In the lunar theory also the apsidal line is movable, but proceeds with the difference velocity (cf. above p. 68).
Cf. above p. 58.
Details to be discussed later; cf. below p.207.
Cf. above p. 54.
In the so-called “Goal-year-texts”; cf. below p. 351 and p. 554.
Cf. for the outer planets below p.180f. and p. 182; for the inner planets p. 157 and p.167f.
Cf. above p. 54.
Cf. above p. 55.
Cf. for the outer planets below p.182, for Venus p.157, for Mercury p. 167.
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.
Cf. below p.155; p. 171.
Identical with the mean longitude of the sun at epoch (cf. p.60).
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.
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.
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.
Ptolemy’s own roundings would give as final result e=1;16,23 and r=43;10,18.
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.
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).
Below p. 172 ff.
Above p. 151. Z Above p. 153.
For the sake of clarity the eccentricity is exaggerated in these figures but the angles at O are drawn essentially correctly.
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.
Cf. p. 151.
The approximate period relation leads to about 0;36,58°/d.
Above p. 60, (5).
Cf. also Fig. 16 (p.1375).
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).
Cf. below II A 5, I C and IIA7,6.
Cf. Neugebauer [1968, 21.
Tetrabiblos IV, 3 (Robbins, p. 381).
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).
Cf. p. 161 and IC 3, 5.
Not with Nisan (April) as in the Mesopotamian version of the Seleucid era which is used in the cuneiform texts.
Cf for details p.1066.
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.
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.
Above p.85.
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).
The velocity of this motion is, of course, equal to the mean motion of the sun. Above p.159.
Below p. 168 f.
Cf. above p. 159 f.
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.
Heiberg, p. 288, 20/289, 1.
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.
Using Tuckerman, Tables.
According to Schoch, Planetentafeln. Cf. above p. 160.
Cf. above p. 159.
Cf. p.151.
In checking this division I find for the last three digits only 58,39,48.
Hartner [1955], p. 109 to 117.
Obviously the Cartesian coordinates of K are ML=(R + e) cos a, and LK =(R — e) sin ~i, respectively.
Cf. below p.389; also above p. 150f., (1) and (2).
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.
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.
Cf. below V A 1, 4.
For the corresponding term “equant” cf. below p. 1102.
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.
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.
Cf. above p. 57 ff.
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.
Above p. 73 ff.
Computed by Mr. E. S. Ginsberg.
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.
Cf. above p.177, (6) and (7), p. 179.
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.
The determination of OC is required by the absence of the tangent function as well as by the following steps.
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.
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.
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.
Alm. XI, 10 (Heib. II, p. 429).-For the Handy Tables cf. below p.1002, (1). Cf. below p. 185.
Cf. above p.182, n. 15.
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).
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.
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.
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.
Above p. 95.
For Saturn: Manitius II, pp. 281, 7 and 283, 14; for the remaining planets shortened to “maximum/minimum distance.” Cf. also below p. 193.
Below p. 195 and p. 197ff.
Cf. above p. 146.
Ptolemy’s procedure differs from the one given here only in so far as he first finds PT and then p.
Cf. above p.192 and note I there.
Cf. p. 183 f.
Cf. p. 192.
Above p. 193. For the numerical details cf. below p. 199 (Table 18). Cf. above p. 193.
Cf. above p. 183 and below p. 204.
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 ).
Described in the case of maximum distance of Mars in Alm. XII, 6 (Man. II, p. 301f.).
Above p. 193, values of b in the table.
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.
From (2b), p.194.
Above p. 193.
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.
For Ptolemy’s procedure cf. p.165 and Fig.149.
Above p.193.
It is probably only accidental that Ptolemy’s correction at maximum distance (0;24) is 3 times the correction at minimum distance (0;8).
Cf. above p.164, (1) and Table 14 (p.169).
Above pp. 193 and 195.
From Table 18 (p. 199), slightly rounded.
Cf. Tables 17 and 19 (pp. 197 and 201).
Cf. (8), p.195.
From Table 17 (p. 197), rounded.
Cf. Tables 17 and 20 (pp.197 and 201).
Cf. (9), p. 196.
From (1).
He mentions only some special cases, e.g. for k=30° (Manitius II, p. 258).
Cf. below p. 232.
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.
Cf. Fig. 207, p. 1272, moving C into A or B.
In some cases the result is 0;0,12 less than expected.
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.
One finds a cubic equation for the sine of the angle under which the eccentricity e= 3 is seen from C.
Cf. Figs. 188 and 189 (p. 1265 f.).
Cf. above p. 148.
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.
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 ].
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.
Cf. above p. 151 (4).
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.
Cf. p. 151 (4).
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.
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).
AA =1f 25 (cf. p.153).
Ptolemy needs two steps here, having no tables for tan a. 3 115;30 (cf. p. 58).
Below p. 241.
Above p. 159 and Fig.144, p. 1252.
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.
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.
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]).
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).
Cf., e.g., Strabo, Geogr. II, 5, 39 (Loeb I, p. 511 ).
Cf. below p. 367; cf., however, below p. 249, note 12.
Cf. for this value below p. 236.
Accurately 0;46,57,23 and 0;37,21,3, respectively. Cf. above p. 218.
Cf. below p. 237 f.
Ptol. Opera II, p. 153, 15.
Cf. below p.1017 (2).
Cf. below p. 535.
Cf. Table 3, p. 47 and Fig. 41, p. 1218.
Cf. below pp. 239 and 241.
Cf. below p. 245ff.
Cf. p. 219, (7).
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.
Cf. above p. 208 and Fig. 213, p. 1275.
Above p. 180.
According to p. 153, A=It25.
Following the rules given p. 222 ff.
Above p. 159.
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.
I,17; I, 22; VII, 18. Translation Manitius, pp. 11, 13, 221, respectively.
Cf. IC7,2B.
Cf. Fig. 218 (p. 1278 ).
Above p.226 (2), and p. 215 (2).
Cf. above p. 154 (8).
E.g. Vat. gr. 1291 fol. 89”.
This was recognized by A. Aaboe [1960], p. 20.
Below p.403.
Cf. Fig. 218 (p. 1278 ).
According to the tables for maximum elongation, Alm. XII, 10. Cf. also Fig. 238 (p. 1288 ).
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.
Above pp. 236 and 240.
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.
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.
Cf. below p. 256f.
Cf. below p. 261.
Cf. also below p.260f.
Alm. XIII, 9 (Manitius II, p. 393).
IBi can reach about 5;30° for Mercury and almost 10° for Venus.
Cf., e.g., p. 235 for all planets, p. 240 (2) for Venus, p. 241 (1) for Mercury.
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.
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.
Aaboe [1960], p. 7.
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 ).
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).
Above p. 236.
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.
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.
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.
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.
Cf. p. 253 (2).
For the significance of a negative elongation at r cf. above p.241.
For F in up +0;6, for Sl +0;4 and —0;11, respectively.
Cf. above p. 241 and note 11 there.
The omitted cases are marked by a x
Above p.254; below p. 259.
Cf. below p.1024.
Halma, H.T. III, p. 30 to 32.
Ptolemy, Opera I, 2, Heiberg, p. 606/607.
Cf., e.g., Cumont [1934].
CCAG 5, 4, p. 228, 15 to 19; cf. also below V A 3, 2.
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.
CCAG 7, p.214 to 224.
Cf. above p. 235 (1).
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.
Cf. also Nallino, Batt. II, p. 255 to 268.
Cf. Kennedy-Agha [1960], p. 138, Fig. 2.
Above p. 244 ff.
Ptolemy, Opera II, p. 4 Heiberg.
Plan. Hyp. I; cf. Goldstein [ 1967 ], p. 8.
Cf above p. 257.
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).
Kepler, Astronomia Nova (Werke III), Chaps. 59 and 60.
A careful discussion of these biographical data has been given by G.J.Toomer in the Dictionary of Scientific Biography I (1970), p.179f.
Cf for this earlier phase below IV B 3, 4.
Apollonius, Opera II (ed. Heiberg), p. 139 frgm. 60. Cf. also below pp. 650 and 655.
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.
Vettius Valens, Anthol., ed. Kroll, p. 354, 4-7; Cumont [1910], p.16I. Cf. also below p. 602.
On Sudines and Kidenas cf. below p. 611; on the norm with 8° below IV A4, 2A.
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.
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.
Cf. above Fig. 51, p. 1220.
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 ).
Apollonius, Opera I, p. 402-413; ed. Heiberg; trsl. Ver Eecke, p. 249-255.
Alm. III, 3 (Manitius I, p. 162).
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.
Cf. above IB4,1.
Cf. for his results below p. 315.
Alm. IV, 6, Manitius I, p. 223.
Rome CA III, p. 1053-1056.
For the sake of greater clarity the points on the circle in Fig. 268 have been spaced more conveniently than in Fig. 267.
Cf. above I C 6.
Cf. Fig. 195, p. 1268.
Cf., e.g., Fig. 134, p. 1248.
Cf. for these problems and the role of Eudoxus: Hasse-Scholz, Die Grundlagenkrisis der Griechischen Mathematik, Charlottenburg 1928 (Pan Bücherei, Philosophie No. 3).
Alm. XII, 1 (Manitius II, p. 277). Cf. also above p.191.
Cf. above p. 264 f.
Alm. XII,1 (Manitius II, p. 272 f.).
Cf. below p. 643.
Cf. above I B 3, 4 A and p. 267.
Cf. Fig. 68, p. 1227 (and similarly Fig. 268, p. 1303 ).
Aaboe [1963], p. 8 f.
ACT, No. 801, Sections 4 and 5 for Saturn, No. 810, Sections 3 and 4 for Jupiter. Cf. below p. 832.
Almagest VII, 5/VIII, 1.
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).
Cf. below IE6,3.
Cf. below p.823.
Suidas, ed. Adler II, p. 657, No. 521; cf. also the preceding note 2.
Ptolemy, Opera II, p. 67, 10 and 16 to 18 (ed. Heiberg). Cf below p. 928.
Cf. Table 28, below p. 276.
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).
a Maass, Aratea, p. 121.
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).
Cf. below p. 301.
Hipparchus, Arat. Comm. ed. Manitius, p. 184/5; cf. also p. 292, note 3.
The inscription of Keskinto, e.g., shows that other astronomers had worked at Rhodes (cf. below p. 698).
Alm. VII, 2 and 3 (Manitius II, p. 15, 9 and 20, 21).
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.
Below p. 281.
Vogt [1925], col. 25. Cf. below p.284.
Hipparchus, Arat. Comm., Manitius, p. 182 to 270.
Hipparchus, Arat. Comm., Manitius, p. 270 to 280. Cf. also below p. 279, note 22.
Alm. VII, 3 (Manitius II, p. 18 to 20).
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.
Delambre HAA I, p. XXII to XXIV.
Fotheringham [1918], p. 408.
Cf. below p. 284.
Alm. III, 1 (Manitius I, p. 133, 32).
Ptolemy, Opera II, p. 1 to 67. Cf. also below V B 8,1 B.
Cf. below p. 301, n. 1.
Cf. below p. 301, n. 2.
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 (?).”
Below p. 283.
Below pp. 285 ff.
Cf. below p. 368.
Cf., e.g., IV A 4, 2 A and 2 B.
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.”
Aratus Comm., p. 132, 7.
E.g. Aratus Comm., p. 48, 5 to 7.
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.”
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.
Aratus Comm., p. 89.
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).
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).
Arat. Comm., p. 68, 20ff. ed. Manitius. Cf. also Vogt [1925], col. 29.
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).
Aratus Comm., p. 8 to 182, ed. Manitius.
ÉRÉXE1 or KEITa,, respectively.
Aratus Comm., p. 182 to 280.
This result agrees with the tables in Alm. II, 8.
Cf. for this problem below p. 868 f.
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).
niimuç, (Arai. Comm., p. 272, 1); n,jyvri ov Slâarriµa (p. 190, 10).
ryµtnryXtov (Aral. Comm., p. 186, 11; 190, 8, 26 etc.); Siio µépry znryyewç (p. 254, 11, 25; 268, 8, etc.).
Cf. below p. 591; also p. 304.
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.
Manitius, p. 150, 2f.; cf. below p. 299.
Almagest VII, 3 (Manitius II, p. 18 to 20).
Cf. below p. 284.
Cf. below p. 283, note 13.
Cf. below p. 286. With a few exceptions all these stars are near the ecliptic.
Cf. below p. 934.
Remarked by Dreyer [1918], p. 348/9. Cf. Brahe, Opera II, p. 151, 10f.; p. 281, 11ff.; III, p. 335, 31 ff.
Copernicus, De revol. II, 14 (Gesamtausgabe II, p. 102, Thorn, p. 115).
Cf., e.g., the accusations of dishonesty in HAA I, p. XXXI. Cf. also Vogt [1925], col. 33.
No. 5354-55 (1925).
In section C, below p. 284.
Hipparchus, Arat. Comm., p. 186 to 270 ed. Manitius.
Almagest I, 12 (Manitius, p. 44).
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.
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.
I.e. the “polar longitude” p of E. Cf. above p. 279.
Alm. VII, 3 (Manitius II, p. 18 to 20); Geography I, 7, 4 (ed. Nobbe, p. 15, 6 ).
Strabo, Geogr. II, 5, 41 (ed. Meineke, p. 181, 21 to 25).
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.
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.
Cf. above p. 275.
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.
Cf. for details Vogt [1925], col. 23 to 26.
Vogt [1925], col. 23: Hipparchus —0.06° (±0.065°), Ptolemy +0.01° (±0.03°).
I.e. around —138 and +48, respectively.
Cf. below p. 294.
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.).
Biot [1843], p. 61.
This date is generally accepted by Chinese scholars; cf. Yoke [1962], p. 145, No.41 or Hsi [ 1958 ], p. 114, No. 6.
First edition p. 563, 4th edition (1851), p.474. Also referred to by Humboldt, Kosmos III (1850), p. 221.
The Chinese sources seem not to exclude the possibility that the “Nova of —133” was only a comet. Cf. Needham SCC III, p. 425f.
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 …”
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 ].
From CCAG 9, 1, p. 189 (ignoring variants).
Rehm [1899], Boll [1901].
Boll [1901], p. 192ff.
Almagest VIII, 1 (Manitius II, p. 64).
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.
Heiberg II, p. 37, 15: noLlagry.
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.
Maass, Comm. Ar. rel., p. 128 (No. 12). The same formulation also in CCAG 8, 4, p. 94, fol. 10.
HAA I, p. 290.
Gundel, HT. His assumption (p. 135, p. 142, note 1) that the original number of stars must have been 72 seems to me unfounded.
The one exceptional case (Gundel HT, p. 25, 8 and p. 152, No. 63) gives 30 minutes beyond integer degrees.
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.
Gundel, HT, p. 127ff.
y Canc; Almagest: 2=0910;20, β=2;40.
We also known that the Babylonian division of seasons took the summer solstice as the point of departure (Neugebauer [1948]).
Cf. below p. 309 f.
Cf. above p. 279.
aPlyccrtaizoi.
Alm. VII, 1, Manitius II, p. 5 to 8.
Cf. above p. 277, note 4.
Cf. above p. 283.
Alm. VII, 4 (Manitius II, p. 31f.).
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).
Below p. 577.
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.
Rehm [1899], p. 265.
Hipparchus, Comm. Arat., p. 184, 23 ed. Manitius.
Alm. VII, 3 (Manitius II, p. 26 and p. 28, respectively).
Nallino, Battâni I, p. 124, p. 292; II, p. 269ff.; Dreyer [1917], [1918]; Vogt [1925], col. 37f. Cf. also Knobel [ 1877 ], p. 3f.
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.
The division in sentences is mine, made for easier reference.
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.
For a comparison with modern standards cf. Peters-Knobel, Ptol. Cat., p. 120f. and the literature quoted there.
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.
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.
Hipparchus, Comm. Ar., p. 238, 31.
E.g. p. 42 to 45.
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.”
De signis”; for the title of the “Catalogue of Stars” cf. above p. 277, note 4.
Only twice in the Commentary to Aratus; once more in a quotation from Attalus.
NH II, 95.
Cf above p. 285.
Maass, Comm. Ar. rel., p. 137.
In the Almagest three stars of the Pleiades are considered to be of the 5th magnitude, only one of the 4th.
Changed by Housman arbitrarily to sextumque which makes no sense.
Manilius V, 710-717 ed. Housman (V, p. 89-91), 711-719 ed. Breiter (I, p. 148, II, p. 178f.).
From observations made by Timocharis in Alexandria in the years —294/-282 (Alm. VII, 3, Manitius II, p. 22 to 27).
Alm. VII, 2 (Heib. II, p. 12, 21): nepì Tfjç itETa7rT(6aECOç ran TpoiriKCUV Kai ia11/4Eprvthv a11nwiojv.
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.
Cf. below p. 308.
Cf. below p. 296.
Alm. III, 1 (Manitius, p. 145) from the work on the length of the year.
Alm. III, 1 (Manitius, p. 132).
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.
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 ).
Alm. VII, 2 (Manitius II, p. 15).
Preserved only in Arabic; published by Walzer [ 1935 ]. Cf. also Neugebauer [1949, 1].
Anthol., ed. Kroll, p. 353,12f.; cf. below p.601.
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.
Alm. VII, 1 (Manitius II, p. 4).
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).
Alm. III, 1 (Manitius, p. 143ff.).
The diagonals in Fig. 283 represent intervals of exactly 365 1/4°.
Alm. III, 1 (Manitius, p. 135).
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).
Below p. 295.
Cf. Alm. VII 2.
Alm. III, 1 (Manitius, p. 137f.).
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).
Alm. IV, 11, Manitius I, p. 252.
Cf. below p. 318.
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).
Alm. VII, 1 (Manitius II, p.4).
Cf. above p. 280, p. 287.
Above p. 294, note 15.
ed. Jahn, Proleg., p.V.
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.
Cf. below p. 624.
Cf. below II Intr. 3, 1.
The resulting length of the synodic month would be 29;31,51,3,49, … days.
Cf. above p. 293.
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.
Cf. below p. 310.
Cf. note 5.
Ginzel, Hdb. II, p. 391. Similar already Ideler, Chron. I, p. 353.
Dinsmoor, Archons, p. 410-423.
E.g. Archons, p. 414: “Therefore we may assume that the authorities … called in a specialist, namely, Hipparchos.”
Above p. 293.
Alm. VII, 2 (Man. II, p. 15).
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.
Alm. IV, 2 (Man. I, p. 196); cf. below p. 310 (5).
Above p. 295.
Cf. for the details below IV B 2, 3.
Cf. above Table 28, p. 276.
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.
Cf. also Toomer [1973], p. 19/20.
Theon indeed admires the conciseness of Ptolemy’s derivations (ed. Rome CA II, p. 451).
Aratus Comm., Manitius, p. 150, 2; cf. above p. 279.
Cf. below IV B 5.
Burgess, Sûr. Siddh., p. 64; cf. also Nallino, Scritti V, p. 220f.
Cf. above p. 278.
Cf. below p. 315.
a Cf. above p. 140, n. 3.
Sb Cf. the diagram Toomer [1973], p. 19, Table II.
o Cf. above p. 23.
Metrica I, 17-25, Heron, Opera III, ed. Schöne, p. 46, 23-64, 31.
Opera III, p. 58, 19; p.62, 17-18. This need not to be understood as an exact title.
This is a conclusion first clearly established by A. Rome [1933].
Opera III, p. 66, 6-68, 5.
Above I A 2, 1. Cf. also Pappus’ remarks concerning Menelaus (Hultsch, p. 600, 25-602, 1; Ver Eecke II, p. 459).
Cf. below p.304f.
VB3,7B.
Cf., e.g., Varâhamihira, Paficasiddhântikâ IV, 41 to 44; Neugebauer-Pingree, II, p. 41 - 44.
Presumably the work on simultaneous risings (cf. above p. 301).
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.
This is the “day-radius” of Indian astronomy.
This is the equivalent of the Indian limit of 3;45° in a table of sines; cf., e.g., Pc.-Sk. IV, 1.
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.
Ar. Comm., p. 96, 11.
Ar Comm., p.98, 20f.; also Theon Smyrn., p. 202, 19ff. (Hiller); Dupuis, p. 327.
Cf. above p. 301, note 2.
Ar. Comm., p. 182-185.
Given in Ar. Comm., p. 244-271 (Manitius).
Ar. Comm., p. 124/5.
This is again a term of Indian astronomy (cf., e.g., Pc.-Sk. IV, 27 and 28).
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 ).
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).
Cf. above Table 2, p. 44 (from Alm. II, 6).
This also holds for the intermediate values at 1/4h and 1/2”, not listed in Table 31. 3° Cf. above p. 304.
Cf. below IV D 3.
Eratosthenes, however, can probably be excluded since his distancesdeviate from the Hipparchian (cf. Fig. 291, p. 1313 ).
Cf. also below p. 1014.
Cf., e.g., the treatise by Hypsicles (below IV D 1, 2 A) who is about contemporary with Hipparchus.
E.g. column J of the lunar theory of System B. Cf. below II B 3, 5 B.
Vettius Valens, Anthol. IX, 11 (ed. Kroll, p. 354, 4-6); also above p. 263.
Cf. above p. 58.
Rome [1950], p. 214f.
Cf. above p.55.
Cf. below p.971, n. 21.
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.
Cf. Fig. 286 as compared with Fig. 53 (p. 1221 ).
Nallino, Batt. I, p. 43 f.
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).
Alm. III, 4 (Manitius I, p. 166); cf. Fig. 53, p. 1221.
Alm. III, 4 (Manitius I, p. 170); also Theon of Smyrna (Dupuis, p. 218/219), etc.
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.
Cf. above p. 297 (2) or below p. 310 (5).
Cf. also above p. 55. note 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.
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.
From Suidas, ed. Adler II, p. 657, 27 f.
Cf. below I E 5, 2 A.
Kugler, Mondrechnung (1900), p. 111; cf. p.348 ff.
Cf. below p. 310.
Cf. above p. 271.
Alm. V, 2 and V, 3; cf. above p. 84 and p. 89.
Manitus I, p. 197f.; cf above p. 69 (1) to (3).
Cf. below p. 483 (3); p.478(2 c); p. 523 (2c) or ACT, p. 75 (20).
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].
Alm. IV, 2 (Manitius I, p. 195).
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).
Copernicus, De Revol. IV, 4, silently correcting Ptolemy’s value (Gesamtausg., p. 215, 31 f).
ACT I, p. 272.
This was pointed out by A. Aaboe [1955].
Cf. below p. 378 (15h). p. 396 (5b), and p. 496 (20)
Cf., e.g., above p. 125 f.
Cf. above p. 306.
The proper understanding of this passage in relation to Ptolemy’s method is due to Olaf Schmidt [1937].
We shall discuss these parameters later on (below p.325).
Ptolem. Opera II, p. 153, 18-20. The ratio (6) is also used in Tamil eclipse computations; cf. Neugebauer [1952], p. 272 (3).
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).
Actually the magnitudes are 1.5 and 2.8, respectively, (P. V. Neugebauer, Kanon d. Mondf.). Alm. VI, 9 (Manitius I, p. 394 - 396 ).
Alm. IV, 9 (Manitius I, p. 238-241).
Cf. above p. 81.
Cf. above p. 84.
Manitius I, p. 245 f; cf. also Man. I, p. 212, 25.
Below p. 318.
A denominator 4;45 would have been slightly better because 1,0/4;45
Cf. above p.84.
In Alm. IV, 11; cf. below p.316.
Rome, CA I, p. 68; translation Toomer [1967], p. 147.
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.
Cf. below I E 5, 4 B.
These observations antedate Hipparchus’ lifetime.
For a Babylonian record of this eclipse cf. Schaumberger, Erg., p. 368, note 1.
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.
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).
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.
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°.
Manitius I, p. 247, 5.
Taking the equation of time into consideration does not help matters.
Toomer [1973], p.9-16. Cf. also above p. 299f.
Above p.74f. and Figs. 65 to 67 there.
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).
Cf., e.g., Rome [1950].
Below II B 6 and II B 7.
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.
Aim. III, (Man. I, p. 183, 5 ).
Cf. above p. 73 ff.
Cf. e.g., above p.72; p. 77.
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).
Below p. 322.
Above p. 310 (3).
Published in CCAG 8, 2, p. 126 to 134.
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).
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.
CCAG 8, 2, p. 126, 21 to 28.
NH II, 57 (Budé II, p. 25 f.).
Above IB6,4.
Below p. 523 (2 c).
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.
Maass, Comm. Ar. rel., p. 47, 13; also below p. 666.
Cf. below I E 5, 4 B.
Cf. for this method, e.g., above p. 295.
Aim. V, 19, Heiberg I, p. 450, 1 and 4 (rzapaaLlcocrncâ).
Rome CA I, p. 150, 20-155, 27.
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.
Above p. 90.
Cf. above I A 5, 5.
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).
Cf. above p. 304 ff.
Alm. V, 19, Manitius I, p. 329, 24 - 29.
Cf. below p. 324.
Actually Ptolemy’s own methods are just as crude (cf. above I B 5, 6).
Cf. the apparatus to Alm. II, 13 in Heiberg, p. 181, 28.
Rome, CA I, p. 152, note (2); similarly p. 168, note (1).
Alm. V, 8 gives for AL =30 the latitude 2;30°. Thus the latitude at L should be greater. Hultsch [ 1900 ].
Swerdlow [1969].
Swerdlow [1969], p. 297/298.
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).
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).
Dupuis, p. 318/319; ed. Hiller, p. 197, 9.
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.
Cf below p. 962.
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.
Cf. for this eclipse (of —189 March 14) above p. 316, n. 9.
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.
Cf above p.296.
Alm. IX, 3 (Man. II, p. 99). Cf. above I C 1, 4; also Neugebauer [ 1956 ], p. 295.
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).
Ptolemy, Alm. IX, 2, Man. II, p. 96.
Cf. below p. 823 f.
Below V A 1.
Goldstein [1967]; cf. below V B 7.
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.
Cf. above p. 327.
Cf. above p. 291.
Wessely [ 1900 ]; Neugebauer [1962, 3], p. 40, col. II, 7-10. Cf. for this text also below p. 737 (n).
Cf. below p.693.
Cf. above p. 306 and below p. 823.
NH II, 95, Loeb I, p. 239, Budé II, p. 41.
Continued in the passage discussed above p. 285.
AG, p. 543/4; also Pfeiffer, Sterngl., p. 115.
Cumont [1909], p. 268; similar Eg. astrol., p. 156, etc.
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.
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).”
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.
Mentioned by Ptolemy in his “Analemma”; cf. below p.1380 and Figs. 26 and 27.
Engelbrecht, p. 47, 20.
Engelbrecht, p. 60, 30.
These passages from Hephaistio are conveniently tabulated in a monograph by Karl Trüdinger, Studien zur Geschichte der griechisch-römischen Ethnographie (Basel 1918), p. 84.
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.
Cf. above p.283; p. 287.
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”.
Even a hemispherical cupola is by no means an a priori concept; cf. below p. 577.
Cf. above I E 2, 1 A.
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).
Cf. below p.934; also above p.280.
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.
Honigmann, SK (widely accepted); Dicks [1955] opposing Honigmann.
Reinhardt (cf., e.g., Honigmann, SK, p. 8f.).
Cf., e.g. below IV D 1, 3.
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 ).
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.
Cf. above p. 305, n. 27.
It is, of course, absurd to give latitudes to seconds (rounded!), as, e.g., in the Loeb translation.
Cf. above p. 305.
Cf. below p. 653.
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.
Cf. below p. 746, n. 3.
Alm. VI, 11 (Heiberg I, p. 538/539). Introduction to the Handy Tables (Opera II, ed. Heiberg, p. 174, 17 ).
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).
Above p. 43 f. and Table 2.
Ptolemy, Opera II ed. Heiberg, p. 4, 3-20; below p. 928.
Above p. 304.
Cf. p. 1313, Fig. 291.
Strabo II 5, 35 (Loeb I, p. 507; Budé I, 2, p. 119).
Above p. 290.
AIm.II, 6 gives M =12;45” for 1p =12;30°. The 8800” from the equator would correspond to cp =12;34°.
Aratus Comm., p. 82, 24 f. Actually S x 31;17° in —125.
E.g. Manitius in Ar. Comm., p. 301, n. 28. Cf., however, Schjellerup [1881], p. 30 about q Cas.
Manitius, p. 120, 18.
Cf. above p. 335.
Strabo II 5, 38 (Loeb I, p. 511, Budé I, 2, p. 120).
Mich. Pap. III, p. 316.
Strabo II 5, 41 (Loeb I, p. 514, Budé I, 2, p. 122).
Strabo II 5, 34 (Loeb I, p. 503, Budé I, 2, p. 117).
Ptolemy, Geogr. I 4, 2 (Nobbe, p. 11; Milk, p.21).
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.”
Strabo, Geogr. I 1, 12 (Loeb I, p. 25; Budé I, 1, p. 74); cf also below p. 667.
Strabo, Geogr. II, 5, 7 (Loeb I, p. 439; Budé I, 2, p. 86); cf. also below p. 590.
Cf. below V B 4, 1.
Strabo, Geogr. 14, 1 (Loeb I, p. 233; Budé I, 1, p. 167); cf. also below p.652.
Loeb I, p. 202, note b.
Budé II, p. 24. Honigmann, SK, p. 72/73, note 3 obscures the situation with learned irrelevancies. Rehm in RE 8, 2, col. 1666, 46 and 1671, 25, following Heiberg in Ptol. Opera II, Index, p. 276 s.v. Avay paupq.
Alm. III, 1 Heiberg, p. 207, 18.
Halma I, p. 164, Manitius I, p. 145, 23.
Cf. belowVB8,1B.
Cf. Maass, Comm. Ar. rel., p. 330 and Anal. Erat., p. 45-49; p. 139. Cf. also Gudeman in RE 3A, 2, col. 1879f
Comm. in Arist., Vol. VII, p. 264, 25-266, 29. Discussed, e.g., by Duhem, SM I, p. 386, p. 394.
Diels, Dox., p. 404 (also Diels VS15) I, p. 226, 25 or Galen, Opera XIX, p. 307 ed. Kühn).
Plutarch, Moralia 732 F (Loeb IX, p. 196/7) and 1047 C, D. Cf. Biermann-Mau, J. of Symbolic Logic 23 (1958), p. 129-132; also Rome, Annales de la Soc. Sci. de Bruxelles, Sér. A 50 (1930), Mém., p. 101.
Cf. also above p. 293.
In one of his commentaries to Hippocrates (edited in the Corp. Med. Gr. V 9, 2, p. 333, 12-334, 14 =ed. Kühn XVII, 2, p. 240), commenting on a remark by Hippocrates that neither the year nor the lunar month amounts to an integer number of days, Galen says that in particular the question of the length of the interval between consecutive conjunctions of sun and moon requires long discussions and that “Hipparchus wrote a whole book” about it, “such as our work On the length of the year” (I1epì rob éviavaíou Xpóvou aúyypapps gpétapov). The same title, however, is quoted by Hipparchus himself for one of his writings (Alm. III, 1 Heiberg, p. 207, 20; cf above p. 292, n. 3) and Galen in De crisibus III (ed. Kühn IX, p.907, 14-16) again mentions only Hipparchus as having written “one whole book” on the length of the lunar month. All this casts doubts on Galen’s authorship of a book on the length of the year and suggests a corruption of the text in the commentary on Hippocrates. Unfortunately the problem has again been obscured by a conjecture of Bergsträsser; cf. the subsequent note.
Arab. Übers. aus dem Griech., p.226 (from ZDMG 50, 1896, p. 350).
Chronogr. IV, trsl. Budge, p. 29.
Cf., e.g., Diels VS(5) I, p.421, No.5.
Cf. his preface to De revolutionibus (Opera II, p. 3 and p. 30 note, for Copernicus’ Latin translation which he deleted from the printed edition).
Above I E 2, 1 B.
Above p. 283.
VB3,7B.
I E 3, 1, p. 299; VI B 1, 6.
IE2,IC, p.287; IE2,1C1.
E.g. the use of “zodiacal signs” for arcs in any direction (cf. I E 2, 1 A, p. 278).
I E 2, 1 B, p. 283, n.13.
IE2,ID; IE6,1, p.330.
I E 2, 1 C, p. 284f.
IE2,2A.
IE 2, 2C, p.298.
I E 2,2 A, p. 294; I E 2, 2C, p. 298.
IE4.
IE5,1C, p.317.
Cf. below IV C 3, 8.
IE5,1C, p.315.
I E 5,1 C, p. 315 f.
I E 5,2 B and I B 6,4, p. 129.
IE5,1A.
I E 6, 1, p. 329. Also the use of Babylonian units (“cubit” of 2°) points in the same direction (cf.
E 2, 1 A, p. 279; I E 3, 2, p. 304).
I E 3, 2, p. 306.
I E 6, 1, p. 330; also I C 8, 5, p. 261.
TB 5, 4 A, p. 109.
Copernicus found 1179 re ( De revol. IV, 19 ).
IE 5,4 A.
Swerdlow [1969], against Hultsch [1900].
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).
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.
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Neugebauer, O. (1975). The Almagest and its Direct Predecessors. In: A History of Ancient Mathematical Astronomy. Studies in the History of Mathematics and Physical Sciences, vol 1. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-61910-6_2
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