An analysis of Ibn al-Shāṭir’s star table

Abstract

Ibn al-Shāṭir’s (1306–1375/1376 AD) star table in his Jadīd zīj, comprising of the equatorial coordinates and magnitudes of 89 stars, is edited and analyzed in this paper on the basis of the extant manuscripts going back to the late 14th and early 15th centuries. It established a new tradition of arranging the celestial coordinates in the star tables in Egypt and Syria after him. The right ascensions (mean absolute error MAE = 20.0′, mean error μ = –2.8′, standard deviation σ = 29.1′) and the declinations (MAE = 21.1′, μ = –3.2′, σ = 29.3′) are nearly of the same degree of precision. The stars in the region RA ~ 120°–180° generally have the least errors in both declination and right ascension. The declinations of the southern stars were measured more precisely than those of the northern ones. The values for the declinations of the stars in the region δ ~ –30°–0° (the middle of the sky towards the south of the horizon of Damascus) are significantly the most accurate. A systematic northward shift can be seen in the declinations of the southern stars. The declinations of 15 of 18 stars spreading out in RA ~ 67°–121° show a southerly, downward shift. More than 65% of the stars have the errors in both the declination and right ascension less than 32′. No outlier in the tabular coordinates exceeds ±98′. Also, Ibn al-Shāṭir measured the magnitudes of, at least, a few stars: he assigned a correct magnitude of +4 to λ Ori, a component of the star cluster in the Orion that was considered a nebulous object in the Almagest star catalogue, and presented more precise values for the magnitudes of α Sco, α Oph, β Cas, κ Ori, γ Gem, and β CMi than Ptolemy and al-Ṣūfi.

Keywords

Ibn al-Shāṭir medieval islamic astronomy observational astronomy star tables stellar astronomy

Introduction

It will not be far from the truth to say that in medieval Islamic astronomy, the knowledge of the fixed stars was not an astronomical affair of the first rank, and, relatively, little attention was paid to systematically observing and continuously recording them. Most of the star tables scattered in medieval treatises and handbooks were updated from the Almagest star catalogue (henceforth, ASC) either directly or indirectly via media of ‘Abd al-Raḥmān al-Ṣūfī’s (903–986 AD) Ṣuwar al-kawākib al-thābita (Book of the fixed stars) by adding to Ptolemy’s longitudes the increments of precession in intervening time intervals. For instance, in his al-Qānūn al-mas’ūdī, Abū al-Rayḥān al-Bīrūnī (973–1048 AD) remarks that “until now, it has not happened to me to know something about the situations of the fixed stars, except for Spica.”¹ This work was written in the 1030s, when Bīrūnī was in his old days, and had already carried out a good number of the observations of his own (mainly, of the Sun). The observation of Spica was made in 1009 AD, about 3 years after the appearance of the supernova of 1006 AD, and thus, it does not come as a surprise why he could not discover the appearance of this guest star. For the majority of Islamic astronomers, stellar astronomy appears to have been a marginal branch of astronomy: Bīrūnī treats its place in astronomy similar to that of pharmacology in medicine²; and Naṣīr al-Dīn al-Ṭūsī (1201–1274 AD) is somewhat undecided to count it either as “a science/scholarly discipline (‘ilm), or as a technique/art (fann)”, which is “independent of cosmography (hay’a)”.³

The seven important non-Ptolemaic star tables/catalogue we know from medieval Middle Eastern astronomy are as follows:

1. The star table in the Mumtaḥan zīj, which stemmed from the observational programs carried out at Baghdad and Damascus during 829–832 AD. It contains the ecliptical coordinates of 24 stars together with their declinations and degrees of transit (i.e. the points on the ecliptic that transit the local meridian simultaneously with the stars). Epoch: 214 al-Hijra (hereafter, denoted by H), whose beginning is 17 Bahman (the 11th month) 197 Yazdigird (afterwards, indicated in Y) = 10 March 829, JDN 2023919.⁴

2. The star table in Ibn Yūnus’s Ḥākimī zīj (Cairo, before 1009 AD) includes ecliptical coordinates of 62 stars. Epoch: the end of 400 Yazdigird = 7 March 1032, JDN 2098062.⁵

3. The star table in the Īlkhānī zīj (Maragha, the 1260s–1270s AD) contains the ecliptical coordinates of 16 stars. Epoch: the beginning of 601 Y = 18 January 1232, JDN 2171063.⁶

4. The star list in Muḥyī al-Dīn al-Maghribī’s Talkhīṣ al-majisṭī (Compendium of the Almagest) (d. June 1283) contains the celestial and ecliptical coordinates of eight bright stars observed at the Maragha observatory in 1267–1268 AD.⁷

5. The star catalogue produced from the observational program conducted by a certain Jamāl al-Dīn Muḥammad b. Ṭāhir b. Muḥammad al-Zaydī of Bukharā (now in Uzbekistan) and other Persian astronomers in the service of the Mongolian Yuan dynasty of China (1260–1368 AD) at Beijing in the latter part of the 13th century. It has been preserved partly in a Chinese translation of a Persian zīj written there (the so-called Huihui li, “Islamic astronomical system”) and a Korean revision of it, including a catalogue of the ecliptical coordinates and magnitudes of 277 zodiacal stars with the latitudes bounded within ~ ±9° (Epoch: 1396 AD),⁸ and partly in a table of the ecliptical coordinates and magnitudes of 48 stars (Epoch: 764 H/1362–1363 AD) in the Sanjufīnī zīj written by Abū Muḥammad ‘Aṭā’ b. Aḥmad b. Muḥammad Kh^wāja Ghāzī al-Samarqandī al-Sanjufinī in northwestern Tibet in 1366 AD.⁹ These two sources have 11 stars in common. The original epoch is not exactly known, although it appears to have fallen somewhere in the 1260s AD (probably, 661 H/1262–1263 AD).

6. The star table in Ibn al-Shāṭir’s Jadīd zīj (Damascus, 1306–1375/1376 AD) includes the equatorial coordinates of 89 stars with the magnitudes for 84 ones. Epoch: 765 H, whose beginning is 9 October 1363 AD, JDN 2219175. It is the main focus of the present paper.

7. The star catalogue of Ulugh Beg’s Sulṭānī zīj (Samarqand, before 1449 AD) includes the ecliptical coordinates of the total number of the stars in the ASC with the magnitudes borrowed from al-Ṣūfī’s Ṣuwar. Epoch: 841 H, whose beginning is 4 July 1437, JDN 2246107.¹⁰

Besides these, the observation reports of the bright stars are scattered in the various treatises and astronomical handbooks with tables either in relation to the determination of the rate of precession¹¹ or among the planetary observations.¹²

In this paper, we investigate in detail Ibn al-Shāṭir’s star table, which is important for a variety of reasons. To the best of our knowledge, besides Muḥyī al-Dīn’s small star list, it is the only star table presumably prepared on the basis of new observations that gives the equatorial coordinates. The direct measurement of the celestial coordinates is relatively easier than that of the ecliptical ones: it only needs a meridian instrument (like a mural quadrant) used to measure the altitude of a star when transiting the local meridian, and, knowing the latitude of the place, the declination can be derived from it; the right ascension can be measured from the time elapsed from the meridian passage of a reference celestial body. The ecliptic coordinates can be directly measured with the aid of an armillary spheres. Alternatively, the celestial coordinates can first be measured and then are transformed to the ecliptical ones; this is what al-Battānī tells us to have done for the measurement of the ecliptic positions of the three bright stars in 879–880 AD¹³ or what Muḥyī al-Dīn explains in detail to have carried out at the Maragha observatory in order to determine the ecliptical coordinates of eight bright stars about a century before Ibn al-Shāṭir.¹⁴ Accordingly, in Ibn al-Shāṭir’s star table, we are very likely confronted with the first-hand observational data that are only affected by errors in the latitude of the place and the solar theory. In addition, since Ibn al-Shāṭir’s treatise containing his observational records was seemingly permanently lost, his star table represents the only surviving part of his observational data, a fact that enhances its importance. The analysis of this table will reveal how accurate the experimental and practical skills of one of the great theorists of medieval astronomy might have been. Lastly, it appears to have inaugurated the establishment of a specific tradition of arranging the celestial coordinates in the star tables in Egypt and Syria since the 15th century (see Section “The tradition of the star tables with the equatorial coordinates”).

The paper is organized to contain the following sections. In Section “Ibn al-Shāṭir’s star table”, we outline the main features of Ibn al-Shāṭir’s fruitful astronomical career and his star table in addition to the introduction of the manuscripts consulted for the present study. Sections “Identification of the stars in Ibn al-Shāṭir’s star table” and “Problematic stars in Ibn al-Shāṭir’s star table” deal with, respectively, the identification of the stars and the problematic stars in Ibn al-Shāṭir’s table. A thorough analysis of the degree of precision attained in Ibn al-Shāṭir’s measurements of the stellar declinations, right ascensions, and magnitudes as well as the correlation between the errors in declination and magnitude are presented in Section “Statistical analysis of the accuracy of the table”. Section “Test of authenticity” contains a simple test of authenticity in order to show that Ibn al-Shāṭir’s star table is independent of the ASC. The establishment of the tradition of the arrangement of the celestial coordinates in the star tables in the medieval Egypt and Syria after Ibn al-Shāṭir is highlighted in Section “The tradition of the star tables with the equatorial coordinates”. Final concluding remarks and comparative statements are summarized in Section “Discussion, comparison, and conclusion”.

Ibn al-Shāṭir’s star table

Ibn al-Shāṭir (15 Sha‘bān 705/2 March 1306–777 H/1375–1376 AD) is best known for his non-Ptolemaic solar, lunar, and planetary models which are described in his Nihāyat al-su’l fī taṣḥīḥ al-uṣūl (A final inquiry on the rectification of [astronomical] hypotheses) and especially for their significant structural similarities with Copernicus’s heliocentric models in the De revolutionibus.¹⁵ The epoch of this work is 701 Y = 24 December 1331, JDN 2207563.

In several places in his Nihāya, he speaks of his astronomical observations and refers to another treatise of his own, titled Ta‘līq al-arṣād (Accounting for the [astronomical] observations), which presumably contained his observation reports (today lost). It is curious that in Nihāya I.5, he says nothing of his stellar observations; it is even strange that he speaks of the five nebulous objects (as is in the ASC), which apparently indicates his neglecting (not being aware of?) the one al-Ṣūfī discovered in Andromeda some 400 years before him.¹⁶

In his Jadīd zīj, Section 19, seemingly written after the Nihāya, Ibn al-Shāṭir says that he observed many of the fixed stars and laid down their ecliptical coordinates in a table for the year 760 H, whose beginning is 2 December 1358; but the star tables found near the end of the work in the various manuscripts altogether comprises the equatorial coordinates of 89 stars with magnitudes for 84 of them for 765 H,¹⁷ whose beginning is 9 October 1363 (JDN 2219175). The equatorial coordinates are suitable for timekeeping purposes, whereas the ecliptical coordinates are more appropriate for astrological operations.

The manuscripts of the Jadīd zīj consulted for this study go back to the late 14th and early 15th centuries, referred here by the sigla assigned in note 17: O: 796 H (1393–1394 AD), L1: 813 H (1410–1411 AD), and L2: 813 H (1410–1411 AD).¹⁸ In MS. O, the star no. 11 is located after no. 12; nos. 58–62 are located in the end of the table between nos. 80 and 84; and, more importantly, nos. 63 and 81–83 are missing. As a result, the table in MS. O has 80 entries. There is an updated version of Ibn al-Shāṭir’s star table in MS. Paris, Bibliothèque Nationale, ar. 2558, ff. 18v–20r (henceforth indicated by siglum P*). The date of copying is 11 Jumādā I 811 H (= 2 October 1408) on f. 64r and Jumādā II of the same year (22 October–19 November 1408) on f. 105v. This manuscript includes a collection of the calendrical and solar tables by Muḥammad b. Muḥammad b. Abī Bakr al-Tīzīnī (Tīzīn is a village in the vicinity of Aleppo) (Rajab 828/May–June 1425–3 Ṣafar 911/6 July 1505), a muwaqqit in the Umayyad Mosque in Damascus and the spherical astronomical, auxiliary trigonometric, and timekeeping tables by Shams al-Dīn al-Khalīlī (ca. 1375 AD), a colleague of Ibn al-Shāṭir and a muezzin in the Umayyad Mosque in Damascus.¹⁹ In this table, only the equatorial coordinates of 81 stars of Ibn al-Shāṭir’s star table are tabulated: six stars are missing from P* (nos. 22, 63, 65, 69, 74, 81–83), but five ones are added to it (nos. 85–89). The declinations are the same as in the original, but the right ascensions are increased by 0;15°. It seems that someone (perhaps, al-Khalīlī?) crudely update the right ascensions in the table by simply adding the precessional amount to them.

Ibn al-Shāṭir’s star table is edited in Table 1. The chart in Figure 1 displays the distribution of these 89 stars according to Ibn al-Shāṭir’s values for their celestial coordinates (indicated in close circles) among the true positions of the stars included in the Bright Star Catalogue (YBS) for the epoch 1363.0 AD (shown as open circles).²⁰ Ibn al-Shāṭir’s values for the right ascension are counted from the Head of Capricorn with RA = 270° (shown as RA), but the modern rights ascensions are reckoned from the Head of Aries with RA = 0° (indicated in RA if necessary). In this article, we show the errors in the declination by dδ and those in the right ascension by dRA; Δ denotes the angular separation, and m stands for the visual magnitude. The modern values are in boldface. ASC-n-X means the nth star in the constellation X in the ASC, and ASC-n means the nth star in its whole.

Table 1.

Ibn al-Shāṭir’s star table for the year 765 H (1−1−765 H = 9−10−1363 AD, JDN 2219175).

Nos.	Name			Declination, δ		dδ	Right Ascension, RA		dRA	Magnitude
				ISh	Modern		ISh	Modern		ISh	al-Ṣūfī	Ptolemy	Modern
1	Sahm al-rāmī min al-na‘ā’im al-ṣādira	φ	Sgr	−27;20°	−27;23°	+ 3′	1;10°	271;27°	− 17′	3	4 ↑	4	+3.2	+3.4
2	al-Nasr al-wāqi‘	α	Lyr	+38;30	+38;20	+ 10	4; 0	273;52	+ 8	1	1	1	+0.0	+0.0
3	Dhanab al-nasr al-ṭā’ir	ζ	Aql	+13; 5	+13; 6	− 1	8;20	279; 3	− 43	3	3	3	+3.0	+3.0
4	Minqār al-dajāja	β¹	Cyg	+27; 0	+26;47	+ 13	16;50	286;16	+ 34	3	3 ↓	3	+3.1	+3.1
5	al-Nasr al-ṭā’ir	α	Aql	+ 7; 5	+ 7;23	− 18	20; 0	289;55	+ 5	2	2 ↑	2 ↑	+0.8	+0.8
6	Janāḥ al-dajāja al-ayman	δ	Cyg	+43;40	+43;42	− 2	21;10	291;17	− 7	3	3	3	+2.9	+2.9
7	Janūbī thalātha qarn al-jady	β	Cap	−17; 0	−16;35	− 25	26;30	296;15	+ 15	3	3 ↓	3	+3.1	+3.2
8	Ṣadr al-dajāja	γ	Cyg	+38;10	+38;21	− 11	30;10	299;52	+ 18	3	3 ↑	3	+2.2	+2.2
9	Dhanab al-dulfīn	ε	Del	+ 9;15	+ 9;18	− 3	31; 5	300;42	+ 23	3	4 ↑	3 ↓	+4.0	+4.1
10	Munīr sa‘d bula‘	ε	Aqr	−11;15	−11;39	+ 24	33; 0	303;15	− 15	3	4 ↑	3	+3.8	+3.9
11	al-Ridf wa huwa dhanab al-dajāja	α	Cyg	+42;30	+43; 7	− 37	35;10	304;57	+ 13	2	2	2	+1.3	+1.3
12	Janāḥ al-dajāja al-aysar	ζ	Cyg	+27;50	+27;45	+ 5	41;40	311;29	+ 11	3	3	3	+3.2	+3.2
13	Shamālī thalātha sa‘d al-su‘ūd	β	Aqr	− 8; 0	− 8;14	+ 14	45; 0	314;28	+ 32	3	3 ↓	3	+2.9	+3.0
14	Aṣl dhanab al-jady	γ	Cap	−19;10	−19;24	+ 14	46; 0	316; 7	− 7	3	3 ↓	3	+3.7	+3.8
15	Ṭaraf dhanab al-jady	δ	Cap	−18;35	−18;53	+ 18	47;45	317;54	− 9	3	3	3	+2.9	+3.0
*16	Janūbī thalātha sa‘d al-su‘ūd	46	Cap	−11;50	−11;52	+ 2	47;50	317;42	+ 8	3	5	5	+5.1	+5.2
17	al-Jaḥfalat wa huwa fam al-faras	ε	Peg	+ 7;10	+ 7; 5	+ 5	48;10	318;14	− 4	2	3	3 ↑	+2.4	+2.4
18	Wasaṭ thalātha sa‘d al-akhbiya	ζ²	Aqr	− 3;10	− 3;11	+ 1	58;40	329; 0	− 20	3	3 ↓	3	+4.4	+4.5
19	Fam al-ḥūt al-janūbī	α	PsA	−34;30	−32;55	− 95	65;30	335;29	+ 1	1	1	1	+1.2	+1.5
20	Shamālī al-far‘ al-muqaddam wa huwa mankib al-faras	β	Peg	+24; 0	+24;41	− 41	68;20	338;19	+ 1	2	2 ↓	2 ↓	+2.4	+2.4
21	Janūbī al-far‘ al-muqaddam wa huwa matn al-faras	α	Peg	+10;50	+11;50	− 60	68;25	338;18	+ 7	2	2 ↓	2 ↓	+2.5	+2.5
22	al-kaff al-khaḍīb	β	Cas	+55;50	+55;39	+ 11	83;50	354;15	− 25	2	3	3	+2.3	+2.3
23	Shamālī al-far‘ al-mu’akhkhar wa huwa surrat al-faras	α	And	+26; 0	+25;35	+ 25	84;10	354; 0	+ 10	2	2 ↓	2 ↓	+2.1	+2.1
24	Janūbī al-far‘ al-mu’akhkhar wa huwa janāḥ al-faras	γ	Peg	+11;30	+11;39	− 9	85; 5	355;11	− 6	2	2 ↓	2 ↓	+2.8	+2.8
25	Shu‘ba qayṭus al-shamāliyya	ι	Cet	−12;25	−12;22	− 3	87; 0	356;44	+ 16	3	3 ↓	3 ↓	+3.6	+3.7
26	Shu‘ba qayṭus al-janūbiyya	β	Cet	−21;25	−21;31	+ 6	92;50	2;52	− 2	3	3 ↑	3	+2.0	+2.2
27	Nayyir baṭn al-ḥūt	β	And	+32;30	+32;11	+ 19	99; 0	8;44	+ 16	2	2 ↓	3	+2.1	+2.1
28	Jasad qayṭus	τ	Cet	−19;15	−19;22	+ 7	110; 0	18;37	+ 83	3	3 ↓	3	+3.5	+3.7
29	Janūbī al-sharaṭayn	γ	Ari	+16;20	+16; 5	+ 15	110; 5	19;47	+ 18	3	3 ↓	3 ↓	+4.8	+4.8
30	Shamālī al-sharaṭayn	β	Ari	+17;50	+17;36	+ 14	110;45	20; 0	+ 45	3	3	3	+2.6	+2.6
31	al-Nāṭiḥ	α	Ari	+20;20	+20;21	− 1	112;50	22;58	− 8	3	3 ↑	3 ↑	+2.0	+2.0
32	Fam qayṭus	γ	Cet	− 0;55	+ 0;25	− 80	122;30	32;39	− 9	3	3	3	+3.5	+3.5
33	Ra’s al-ghūl	β	Per	+38;50	+38;19	+ 31	127;15	36;57	+ 18	3	2 ↓	2	+2.1	+2.1
34	al-Kaff al-jadhmā	α	Cet	+ 1;40	+ 1;27	+ 13	127;20	37;20	0	3	3	3	+2.5	+2.6
*35	Ākhir al-nahr wa huwa al-ẓalīm	θ¹	Eri	−47; 0	−42;58	−242	129;30	38;32	+ 58	1	1	1	+3.2	+4.1
*35	Ākhir al-nahr wa huwa al-ẓalīm	θ¹	Eri	[−42; 0] ?	−42;58	+ 58	129;30	38;32	+ 58	1	1	1	+3.2	+4.1
36	Anwar al-thurayyā	HR 1188		+23; 0	+23;27	− 27	137;50	48; 9	− 19	4	4	4	+5.3	+5.3
37	al-Dabarān wa huwa ‘ayn al-thawr	α	Tau	+15;10	+15; 1	+ 9	149;50	59;56	− 6	1	1	1	+0.9	+0.9
38	al-‘Ayyūq	α	Aur	+44;40	+45; 3	− 23	157;40	67;34	+ 6	1	1	1	+0.1	+0.1
*39	Rijl al-jawzā’ al-yusrā	β	Ori	− 5;10	− 9; 8	+238	160;50	71; 1	− 11	1	1	1	+0.1	+0.2
*39	Rijl al-jawzā’ al-yusrā	β	Ori	− 9;30✓	− 9; 8	− 22	160;50	71; 1	− 11	1	1	1	+0.1	+0.2
40	Qarn al-thawr al-shamālī	β	Tau	+27;25	+27;49	− 24	161;40	71;35	+ 5	3	2	3 ↑	+1.7	+1.7
41	Mankib al-jawzā’ al-aysar	γ	Ori	+ 4;55	+ 5;33	− 38	162;40	72;47	− 7	2	2	2 ↑	+1.6	+1.7
42	Haq‘a	λ	Ori	+ 8;55	+ 9;17	− 22	165;30	75; 4	+ 26	4	Nebula	Nebula	+3.5	+3.6
43	Wasaṭ al-minṭaqa	ε	Ori	− 2; 5	− 1;49	− 16	166;15	76; 1	+ 14	2	2	2	+1.7	+1.8
44	Rijl al-jawzā’ al-yumnā	κ	Ori	−10;25	−10; 5	− 20	169;50	79;25	+ 25	2	3 ↑	3 ↑	+2.1	+2.2
45	Mankib al-jawzā’ al-ayman	α	Ori	+ 6;10	+ 7; 4	− 54	170;10	80;12	− 2	1	1 ↓	1 ↓	+0.5	+0.5
46	Mirzam al-yamāniya	β	CMa	−17;45	−17;49	+ 4	178;55	88;41	+ 14	3	3	3	+2.0	+2.1
47	Munīr al-han‘a	γ	Gem	+16; 5	+16;42	− 37	179;40	90;14	− 34	2	3	3	+1.9	+1.9
48	Suhayl al-yaman	α	Car	−51;50	−52;26	+ 36	181;15	92;28	− 73	1	1	1	−0.7	+2.1
49	al-Shi‘rā al-yamāniya	α	CMa	−15;45	−16; 1	+ 16	184;20	94;16	+ 4	1	1	1	−1.5	−1.3
50	Mirzam al-ghumayṣā	β	CMi	+ 9; 5	+ 9;22	− 17	191;30	103; 8	− 98	3	4	4	+2.9	+2.9
51	Ra’s al-taw’am al-muqaddam	α	Gem	+32;45	+33; 2	− 17	193;10	103;24	− 14	2	2	2	+2.0	+2.0
52	al-Shi‘rā al-ghumayṣā	α	CMi	+ 6;30	+ 6;39	− 9	196;20	106;28	− 8	1	1	1	+0.4	+0.4
53	Ra’s al-taw’am al-mu’akhkhar	β	Gem	+29; 0	+29;20	− 20	196;40	106;30	+ 10	2	2	2	+1.1	+1.1
54	Ma‘laf al-saraṭān wa huwa al-nathra	M44		+21;25	+21;36	− 11	210;40	120;54	− 14	—	Nebula	Nebula
*55	Ṭaraf min Jabha al-asad	λ	Leo	+25;25	+25;37	− 12	223;40	133;44	− 4	2	4	4	+4.3	+4.3
*56	‘Unuq al-shujā‘ wa huwa al-fard	α	Hya	− 3;50	− 6; 2	+132	224; 5	134; 4	+ 1	2	2	2	+2.0	+2.1
*56	‘Unuq al-shujā‘ wa huwa al-fard	α	Hya	[− 6;50] ?	− 6; 2	− 48	224; 5	134; 4	+ 1	2	2	2	+2.0	+2.1
57	Qalb al-asad wa huwa al-malikī	α	Leo	+15; 5	+14;58	+ 7	233;30	143;33	− 3	1	1	1	+1.4	+1.4
58	Mankib al-asad	γ	Leo	+23;10	+22;57	+ 13	236; 0	146; 7	− 7	2	2	2	+2.6	+2.6
59	Nayyir al-kharatān	δ	Leo	+23;50	+23;57	− 7	250; 0	159;58	+ 2	2	2	2 ↓	+2.6	+2.6
60	al-Ṣarfa	β	Leo	+18;10	+18; 7	+ 3	259; 5	169; 5	0	1	1	1 ↓	+2.1	+2.1
61	Minqār al-ghurāb	α	Crv	−21; 5	−21;11	+ 6	264;30	174; 0	+ 30	3	3 ↓	3	+4.0	+4.2
62	Zāwiya al-‘awwā	γ	Vir	+ 2;15	+ 2; 4	+ 11	272;30	182;22	+ 8	3	3	3	+3.7	+3.7
*63	al-‘Anāq min dhanab al-dubb	ε	UMa	+58;25	+59;27	− 62	276; 9	186;18	− 9	2	2	2	+1.8	+1.8
64	al-Samāk al-a‘zal	α	Vir	− 7;30	− 7;46	+ 16	283;20	192;59	+ 21	1	1 ↓	1	+1.0	+1.1
65	al-‘Anāq	ζ	UMa	+57; 0	+58;18	− 78	283;20	194;27	− 67	2	2	2	+2.3	+2.3
66	al-Qā’id	η	UMa	+52;35	+52;34	+ 1	290; 5	200;31	− 26	2	2	2	+1.9	+1.9
67	al-Samāk al-rāmiḥ	α	Boo	+23;30	+22;36	+ 54	297; 0	206;39	+ 21	1	1	1	+0.0	+0.0
68	Kaffat al-mīzān al-janūbiya	α²	Lib	−13;20	−13;15	− 5	304; 5	214; 1	+ 4	2	3 ↑	2	+2.8	+2.9
*69	Kaffat al-mīzān al-shamāliya	β	Lib	−18;30	− 6;52	−698	310;35	220;46	− 11	2	3 ↑	2	+2.6	+2.7
*69	Kaffat al-mīzān al-shamāliya	β	Lib	[− 6;30] ?	− 6;52	+ 22	310;35	220;46	− 11	2	3 ↑	2	+2.6	+2.7
70	Anwar al-farqadayn	β	UMi	+76;40	+76;45	− 5	313;40	223;53	− 13	2	2	2	+2.1	+2.2
71	Munīr al-fakka	α	CrB	+29;30	+29; 0	+ 30	317; 0	226;57	+ 3	2	2	2 ↑	+2.2	+2.2
72	‘Unuq al-ḥayya	α	Ser	+ 8;40	+ 8;36	+ 4	317;30	228;16	− 46	2	3	3	+2.7	+2.7
73	Wasaṭ thalātha jabha al-‘aqrab	δ	Sco	−20;20	−20;36	+ 16	321;30	230;47	+ 43	3	3	3	+2.3	+2.5
74	Aqrab kawkab fī al-‘aqrab ’ilā al-minṭaqa	ω¹	Sco	−18;20	−18;44	+ 24	321;50	232;30	− 40	3	4	4	+4.0	+4.1
75	Munīr al-jabha al-‘Aqrab	β	Sco	−17;35	−17;51	+ 16	322;15	232;12	+ 3	3	3	3	+2.6	+2.8
76	Qalb al-‘aqrab	α	Sco	−24;15	−24;47	+ 32	328; 0	237;42	+ 18	1	2	2	+1.0	+1.2
*77	Rukbat al-ḥawwā al-yumnā	η	Oph	+14;30	−14;42	+ 12	328;45	248;31	+ 14	2	3	3	+2.4	+2.6
*77	Rukbat al-ḥawwā al-yumnā	η	Oph	−14;30✓	−14;42	338;45✓	328;45	248;31	+ 14	2	3	3	+2.4	+2.6
78	Ra’s al-jāthī	α	Her	+15;45	+15;18	+ 27	341; 0	251;26	− 26	2	3 ↓	3	+3.5	+3.5
79	Tālī al-shawla	λ	Sco	−36; 0	−36;22	+ 22	341;50	252;40	− 50	3	3	3	+1.6	+2.1
80	Ra’s al-ḥawwā	α	Oph	+13;25	+13;13	+ 12	347; 0	256;22	+ 38	2	3	3 ↑	+2.1	+2.1
81	Mankib al-ḥāwī al-ayman	β	Oph	+ 4;25	+ 5; 2	− 37	347;10	258; 2	− 52	2	3 ↓	4 ↑	+2.8	+2.8
*82	Kaff al-ḥawwā	τ	Oph	− 8;10	− 7;57	− 13	352;10	262; 7	+ 3	2	5	4	+5.2	+5.3
*83	al-Iklīl al-janūbī	ε	Sgr	−34;15	−34;25	+ 10	355;10	265;29	− 19	2	3 ↑	3	+1.9	+2.3
*84	Rā‘ī al-na’ā’im	λ	Sgr	−22; 0	−25;31	+211	357;15	267;10	+ 5	3	3	3	+2.8	+3.0
*84	Rā‘ī al-na’ā’im	λ	Sgr	[−25; 0] ?	−25;31	+ 31	357;15	267;10	+ 5	3	3	3	+2.8	+3.0
85	Munīr al-buṭayn	41	Ari	+23;11	+24;29	− 78	123;15	33;18	− 3	—	—	—	—	—
86	Munīr al-safīna	ζ	Pup	−38;36	−38;20	− 16	205; 0	115;19	− 19	—	—	—	—	—
87	Rukbat al-ḥawwā al-yusrā	ζ	Oph	− 8;30	− 9;04	+ 34	330;45	240;35	+ 10	—	—	—	—	—
88	Awwal al-shawla	υ	Sco	−35; 5	−36;31	+ 86	340;45	251;57	− 72	—	—	—	—	—
*89	Awwal [al-na‘ā’im] al-wārida	γ²	Sgr	−20; 0	−30;10	+610	350;25	261;15	− 50	—	—	—	—	—
*89	Awwal [al-na‘ā’im] al-wārida	γ²	Sgr	[−30; 0]✓	−30;10	+ 10	350;25	261;15	− 50	—	—	—	—	—

1. Ibn al-Shāṭir’s values for the right ascension are counted from the Head of Capricorn with RA = 270;0°, but the modern rights ascensions are reckoned from the Head of Aries with RA = 0°.

2. The additional stars in MS. P*: For nos. 85−89: the tabular RA value minus 0;15°. Arrangement: No. 85: between nos. 32 and 33. No. 86: between nos. 53 and 54. No. 87: between nos. 76 and 77. No. 88: between nos. 77 and 78. No. 89: between nos. 81 and 82.

Magnitudes: (4) L1: +2. (5) L1: +3. (11) L1: +3. (12) L1: +2. (25) L1&L2: +2. (84) L1&L2: +2.

Figure 1.

The distribution of 89 stars according to Ibn al-Shāṭir’s values for their celestial coordinates (denoted by the filled circles) among the true positions of the stars included in the Bright Star Catalogue (YBS) for the epoch 1363.0 AD (shown as the open black circles). The circles are comfortably sized on the basis of the modern magnitudes from −1 (the biggest) to +6 (the smallest).

Ibn al-Shāṭir occupied the position of muwaqqit of the Umayyad Mosque in Damascus at least from 733 H (1332–1333AD) until his death and carried out his astronomical observations from a minaret in the Umayyad Mosque in Damascus. Three are three minarets in the mosque: the Minaret of the Bride (Mi’dhanat al-‘arūs), on which there is a 19th-century replica of the sundial constructed by Ibn al-Shāṭir,²¹ the Minaret of Jesus (Mi’dhanat al-‘Īsā), and the third one was built on the southwest tower of the mosque.²² The southernmost star in the table is α Car (no. 48); with a declination of ~ –52;26° at the time, it transited the meridian of Damascus in a low altitude of ~ 4;14°.²³ The geographical barriers around Damascus in the southern direction do not cause any limitations more than a single degree when an observer at the Umayyad Mosque is elevated 50 m from the ground (the greatest altitude limitation in the south is due to Mt. al-Māni‘, about 21 Km distant from the mosque: 0.85° in an azimuth ~ 5°). So, α Car could be observed from a minaret of the mosque.

Identification of the stars in Ibn al-Shāṭir’s star table

A four-condition criterion for the identification of the stars in a stellar table in medieval Islamic astronomy comprises of (1) tradition, (2) position, (3) magnitude, and (4) spatial pattern. By “tradition”, we mean the names, specific descriptions, and relative position of the stars in the canonical and authoritative texts, that is, the ASC and al-Ṣūfī’s Ṣuwar al-kawākib. Most of the stars in Ibn al-Shāṭir’s table can be conveniently identified with relative ease on the basis of their given names and tabular coordinates. An example is given below:

No. 36: Ptolemy gives the coordinates and magnitudes of the four faint stars marking the boundaries of the Pleiades, which can be identified as 19 Tau (m = 5; +4.3), 23 Tau (m = 5; +4.2) (or 17 Tau, m = 5; +3.7),²⁴ 27 Tau (m = 5; +3.6), and HR 1188 (m = 4; +5.3). So, according to the tradition, the latest is the brightest of these four stars, although there is some ambiguity with respect to the description given in the ASC and the magnitude value registered for it.²⁵ However, in reality, η Tau (Alcyone), locating approximately in the middle of the Pleiades, is the brightest star of the cluster (m = +2.9), and the first three boundary stars, in addition to 20 Tau, are placed after it in order of magnitude. It is curious that, seemingly, Alcyone even escaped from al-Ṣūfī’s attention, since he comments that “the stars of the Pleiades are more than those four, but they are limited to those four, because the stars in it are densely close to each other [so that they cannot be distinguished from each other] and those four stars are greater in magnitude than others”.²⁶ In Ibn al-Shāṭir’s brief description “the brightest of the Pleiades”, it is not known whether he refers to the brightest of the four boundary stars or the brightest of the whole star cluster. The first seems quite probably to be the case: the tabular magnitude is traditionally in agreement with the fourth boundary star, and also, the registered coordinates mark a spot very close to HR 1188 with Δ ≈ 33′. In contrast, not only the tabulated magnitude is not compatible with that of Alcyone, but also the spot the tabular coordinates point towards is relatively far distant from it (~ 66′).

Nevertheless, we are confronted with a few problematic stars in Ibn al-Shāṭir star table, marked by the asterisks in Table 1. They can be divided into three distinct groups with respect to the sources of confusion:

(A) In seven cases, the problem arises from the values registered for their declinations, nos. 35, 39, 56, 69, 77, 84, and 89. The problems with nos. 39 and 77 can be readily solved with the aid of another piece of evidence, that is, MS. P*, which helps us to solve the dilemma caused by the scribal errors committed in writing down the Abjad numerals. So, we take the two into account in the evaluation of the general accuracy of the table. Of the five remaining stars, the textual change under the assumption of occurring possible scribal errors is plausible with a higher degree of confidence only in the case of no. 89. So, we take its modified declination value into account in the final analysis of the accuracy of the table, but completely exclude the four remaining stars, that is, nos. 35, 56, 69, and 84, in our statistical analysis.

(B) In two cases, nos. 63 and 83, the confusion arises from the names, so that the coordinates given in each case closely fall in the vicinity of a star, but it is not identical to that defined in the tradition. We can safely take into account the tabular coordinates of these stars in our evaluation of the accuracy of the table.

(C) There are 19 stars in Ibn al-Shāṭir’s table that the tabular magnitudes for them are in agreement neither with Ptolemy, nor with al-Ṣūfī. Regardless of the one-third bins used by Ptolemy and al-Ṣūfī in their magnitude records, the differences between the tabular values and Ptolemy’s, al-Ṣūfī’s, or modern ones amount to a single unit in the cases of 16 stars, on which we will discuss at some length in the sequel (see below, Subsection “Magnitudes” in Section “Statistical analysis of the accuracy of the table”), but are larger than it for three remaining stars: nos. 16, 55, and 82. These three are problematic, because the registered magnitudes can by no means be attributed to the observational errors. Like Group (B), we consider the tabular coordinates of these stars in our evaluation of the accuracy of the table.

Fortunately, none of the values for the right ascension show the egregious errors of the same size as those in the declination. The extreme errors in both the declination and right ascension in the case of the identifiable stars do not exceed ±98′ (= dRA for β CMi, no. 50).

Problematic stars in Ibn al-Shāṭir’s star table

Explanations for the problematic stars Group (A)

*No. 35 (see Figure 2): The declination of −4 7 ;0° for θ¹ Eri, as given in all four MSS consulted, suffers from a striking error of ~ −4°, which is not in accord with the precision of the coordinates of the nearby stars. It seems a scribal error in the form 42/ → 47/ to have occurred in the integer part of the declination value.

Figure 2.

The constellations of Cet and Eri (nos. 25, 26, 28, 32, 34, and 35).

*No. 39 (see Figure 3): The declination of – 5 ; 10 ° for the bright star β Ori, as given in MSS. L1, L2, and O, has a large error of ~ +4°, which is not compatible with the precision attained in stellar coordinates in the rest of the table (cf. Figure 3). Moreover, all stars spreading out in the range of the right ascension from 67° to 81°, including the five stars in the constellation Orion in addition to α Aur and β Tau (corresponding to nos. 38 and 40–45) show a southerly, downward, shift, reaching to a maximum error of –54′ in the case of the first magnitude star α Ori. The table in MS. P* tabulates δ = –9;30°, which is in all likelihood the correct value, having an error of ~ –22′. It is apparent that a scribal error ه ی → ط ل occurred in a prototype from which the first three manuscripts were copied.

Figure 3.

The constellation of Ori (nos. 39, 41, 42, 43, 44, and 45).

*No. 56 (see Figure 4): The star is identical to α Hya, but the tabular value for its declination suffers from a large error ~ +132′, which is not compatible with the general accuracy of the table, while its right ascension is only ~ +1′ off. Particularly, none of the nearby stars shows such a striking error in the declination. The tabular coordinates point towards the faint star HR 3750 (m = +5.4) = ASC-11-Hya, with δ ≈ –3;25° and RA ≈ 134;3° at the time. If this star is taken as no. 56 in Ibn al-Shāṭir’s star table, the error in the declination reduces to –25′, while the error in the right ascension very slightly increases to +2′. However, this star does not seem in all probability to be identical to no. 56, because neither the traditional name nor the magnitude is in agreement with the table. We suggest a minor change in the integer part of the declination value from 3 to 6, assuming a scribal error in the form ﺠ → و to have occurred in an early prototype from which the extant MSS were descended. If this is in reality the case, the error in the declination reduces to −48′.

Figure 4.

The constellations of Leo, Cnc, and Hya (nos. 54, 55, 56, 57, 58, 59, and 60).

*No. 69 (see Figure 5): It is a matter of surprise and confusion that the star on the northern claw of Libra (β Lib = no. 69) is given a more southerly declination than that imagined on the southern claw (α² Lib = no. 68), which puts the star in excess of 11.5° far from its correct place. It is almost certain that a scribal error should have been committed here, although it is rather difficult to reconcile the registered value of –18;30° with the true value of –6;52° at the time. Maybe for this reason, the complier of the table in MS. P* has excluded it. The tabular coordinates for α² Lib show the trivial errors dδ = –5′ and dRA = +4′, and also for β Lib: dRA = –11′. Accordingly, the error in the declination of β Lib should not be too far from such a range of errors. We assume that the integer digit was mistakenly copied in the form /6 → /18, which reduces the error in the declination value to +22′.

Figure 5.

The constellation of Libra (nos. 68 and 69).

*No. 77 (see Figure 6): The star on the right knee of Ophiuchus is η Oph, a star in the south of the equator and in the north of the ecliptic, but MSS L1, L2, and O give a northern declination of + 14;30° together with a right ascension RA = 3 2 8;45° for it. This position is far distant from η Oph, but falls in the south of the ω Her (m = +4.6) = ASC-30-Her (see Figure 6). It seems in all likelihood that we are confronted here with two errors: the first in registering the declinational direction, which should be southern, and the other in the right ascension, which is modified as RA = 338;45°, under the assumption that a scribal error in writing down the Abjad numerals with similar forms (شکح مه → شلح مه) to have been committed in a very early prototype of which the preserved MSS were descended. The modified coordinates, indeed, mark a point in the vicinity of η Oph, with the small positive errors in the coordinates not more than a quarter of a degree. The support comes from P*, in which both the declination of the star is southern and an updated RA = 339;0° is given for it. MS. P* also gives the coordinates of the star on the “left” knee of Ophiuchus, ζ Oph (no. 87).

Figure 6.

The constellations of Oph and Her (nos. 77, 78, 80, 81, and 82).

*No. 84 (see Figure 7): The four stars γ², δ, ε, and η Sgr are located in the vicinity of the Milky Way. As al-Ṣūfī explains, to the pagan Arabs the Milky Way resembled a river and these four stars four she-ostriches recently arrived at the riverbank (al-na‘ā’im al-wārid; the first word is the plural of na‘āma); by analogy, φ, σ, τ, and ζ Sgr, located to the east of the first four stars, were imagined as the four she-ostriches exiting the river (al-na‘ā’im al-ṣādir); and the two stars λ and μ Sgr, located to the north of these eight stars, were visualized as the two he-ostrich (ḍalīm). The starless region between the two four-star groups was called the “junction” (al-waṣl),²⁷ which formed the 20th lunar mansion. The star no. 84 in Ibn al-Shāṭir’s table in MSS. L1, L2, and O is called rā‘ī al-na‘ā’im, “the herdsman of the ostriches”,²⁸ and in MS. P*, qubbat al-na‘ā’im, “the dome of the ostriches”,²⁹ both of which were the alternative names for λ Sgr; but the coordinates given for it falls in a starless region in the east of the line joining μ and λ Sgr and in the south of 21 Sgr, a faint star (m = +4.8) not recorded in the ASC; its distances from these three stars are, respectively, about 195′, 211′, and 85′. Despite its closeness to 21 Sgr, the latter does not seem to actually be the star Ibn al-Shāṭir recorded in the last entry of the table, not only because of violating the tradition, but also because it is extremely unlikely that Ibn al-Shāṭir recorded a relatively fainter star, but paid no attention to the ten brighter stars located around it. If traditionally identified as λ Sgr, the value registered for the right ascension of this star is nicely accurate (error ~ +5′), but its declination suffers from an egregious error ~ +211′, which is not compatible at all with the general accuracy of the table. In order to resolve the problem, we suggest a change in the declination value from –2 2 ;0° to –25;0°, assuming a probable scribal error in the form of کب → که to have occurred in an early archetype from which the extant MSS were descended. By this alteration, the error in the declination of λ Sgr considerably reduces to +31′.

Figure 7.

The constellation of Sgr (nos. 1, 83, 84, and 89).

*No. 89 (see Figure 7): The star awwal al-wārida, that is, the first star of al-na‘ā’im al-wārid, should be γ² Sgr (see above, *No. 84), but the tabular coordinates points towards a spot more than 10° to the north of γ² Sgr within a starless region to the unaided eye. It seems likely a scribal error /30 → /20 to have occurred in the integer part of the declination in all preserved MSS. By this change, the error in the declination significantly decreases to ~ +10′ (the close circle in the vicinity of γ² Sgr in Figure 7 indicates the spot marked on the basis of this modification).

Explanations for the problematic stars Group (B)

*No. 63: The three circumpolar stars on the tail of Ursa Major, ε, ζ, and η UMa (corresponding to ASC-25–27-UMa), are called, respectively, al-jawn,³⁰ al-‘anāq, and al-qā’id in Arabic. In Ibn al-Shāṭir’s table, nos. 63 and 65 are called by the same name, ‘anāq; the first is supplied with an additional description: min dhanab al-dubb, “[the star] on the tail of the Bear”. No. 63 is missing in MS. O. In MS. L2 the scribe or a later commentator crossed out the whole entry; this was seemingly because it was thought that no. 63 is a duplication of the star no. 65, despite the fact that their coordinates are evidently different from each other. The additional description and magnitude given for the star no. 63 together with its spatial pattern with respect to nos. 65 and 66 on the basis of the coordinates given in the table obviously show that no. 63 should be ε UMa. However, it is not know why it is given the name of ζ UMa. It is curious that the declination values registered for ε and ζ UMa suffer from the relatively large negative errors of the same order (respectively, ~ –62′ and –78′), while the error amounts to only +1′ in the case of η UMa.

*No. 83: 13 stars forming Corona Australis in the ASC were spread out, at Ibn al-Shāṭir’s time, from a right ascension RA ≈ 264.9° (α Tel)³¹ to RA ≈ 276.5° (α CrA), and from a declination δ ≈ –37.7° (ε and γ CrA) to δ ≈ −46.0° (α Tel); so a point marked by δ = –34;15° and RA = 355;10°, as given in the table, obviously falls distantly to the north of this zone, but, surprisingly, very close to ε Sgr, defined in the ASC as the star in the southern portion of the bow in the Sagittarius. In this case the tabular coordinates by no means agree with the traditional name, and there is no way to manipulate the numerical data to conform to the tradition. Perhaps, Ibn al-Shāṭir wished to mark the brightest star in the region, so that the relatively faint stars of Corona Australis can be found in the southeast of it (?).

Explanations for the problematic stars Group (C)

*No. 16: The brightest of the three stars β and ξ Aqr and 46 Cap, extending from the north to the south, which form the asterism Sa‘d al-su‘ūd in the Arabic folk star nomenclature,³² is the first, with a magnitude of ~ +3 (no. 13). The second is of magnitude +4.7 (Ptolemy and al-Ṣūfī: 5), and the third, of +5.1 (Ptolemy and al-Ṣūfī: 5). There is no brighter star in the vicinity of the southernmost star of this asterism, 46 Cap (no. 16), to be confused with it. So it seems in all likelihood that the tabular magnitude for this star is in error.

*No. 55: Ṭaraf is the name of the ninth lunar mansion, consisting of κ Cnc and λ Leo (the latter received its modern name “Alterf” from this tradition).³³ The coordinates given for no. 55 finely points towards a point in the vicinity of λ Leo with Δ ≈ 12.5′, which is in agreement with the precision attained in the stellar location in the case of the other four stars of Leo registered in Ibn al-Shāṭir’s table (the maximum separation Δ ≈ 14.5′ is the case with γ Leo); but λ Leo is a star of magnitude +4.3, and both Ptolemy and al-Ṣūfī have a magnitude of 4 for it, not of 2 as written down in Ibn al-Shāṭir’s table. The nearest brighter star to λ Leo is ε Leo (with m = +3.0), which is sizably distant, with Δ ≈ 3.5°, from the spot marked by the tabular coordinates of no. 55. Consequently, neither the coordinates, nor the magnitude, nor the traditional name suggests ε Leo as an alternative for no. 55, and therefore, the tabular magnitude value 2 for no. 55 should be taken as a textual error.

*No. 82: The two stars τ and ν Oph were imagined to be located in the right hand of Ophiuchus. The first has a magnitude of +5.3 and the second: +3.3. Ptolemy has, respectively, 4 and 4↓, which al-Ṣūfī substantially improves, respectively, as 5 and 4↑.³⁴ The magnitude value of +2 in the table agrees with ν Oph, especially considering the fact that Ibn al-Shāṭir decreased the magnitudes of the stars in Ophiuchus and Hercules by one (see below; Subsection “Magnitudes” in Section “Statistical analysis of the accuracy of the table”), but the point marked by the tabular coordinates fall in a distance Δ ≈ 1.7° from ν Oph with outliers dδ = +78′ and dRA = +69′. It is very close to τ Oph with Δ ≈ 13.3′, dδ = –13′, and dRA = +3′, which are also in agreement with the errors found in the registered coordinates of the other stars of Ophiuchus. Therefore, it is evident that in this case, the coordinates should be preferred to the magnitude.

Statistical analysis of the accuracy of the table

Coordinates

Before delving in depth into the analysis of Ibn al-Shāṭir’s table, the two notes are worth to mention: first, of the suggested modifications in the Abjad numerals for the declination values in the previous section, we only take into account that put forward in the case of no. 89. As noted earlier, there is no difficulty with the values for the right ascension. Thus, our statistical results of the errors in the declinations, dδ, are for the 85 stars and those in the right ascensions, dRA, for all 89 stars. Second, we have modified the values of dδ given in Table 1 by: (a) the atmospheric refraction R (Figure 8), which should be added to the tabular values of dδ in the case of the 74 stars passing the meridian in the south direction and subtracted from them in the case of the other 11 stars transiting the meridian in the north direction. It amounts to 11′ in the case of α Car and 4′ for θ¹ Eri, but for the rest of the stars, it does not exceed 3′, and thus is negligible; and (2) the trivial error of –1′ in Ibn al-Shāṭir’s value for the geographical latitude of Damascus, φ = 33;30° N (33;31° N),³⁵ which is added to the tabular values of dδ.

Figure 8.

The atmospheric refraction R with respect to the declination (x-axis bottom) and the true (“airless”) altitude h (x-axis top) in Ibn al-Shāṭir’s stellar observations.

The values of dδ and dRA, are plotted in Figures 9–12 against the modern values of δ and RA in two ways: in each couple of the figures, the graph labeled by (a) shows the errors for each star. That labeled by (b) depicts the mean errors, the mean absolute errors, and the standard deviations (indicated in the error bars) for the specific sky regions, as demarcated below, which are useful for the evaluation of the variations in the sizes of the errors in each region. In the case of the declination, these regions are arranged in the four zones from the south to the north, Zone 1: –60° ⩽ δ < –30° (8 stars), Zone 2: –30° ⩽ δ < 0° (29 stars), Zone 3: 0° ⩽ δ < +30° (39 stars), and Zone 4: +30° ⩽ δ < +90° (13 stars). In the case of the right ascension, they are classified into the six zones with intervals of 60° each (roughly, equivalent to two zodiacal signs) with the number of the stars in each zone n = 13, 17, 8, 15, 19, and 17. In Figures 9(a) and 10(a), the close circles indicate the errors in the case of the northern stars, and the open ones stand for those of the southern stars.

Figures 9–12.

(a) The errors in Ibn al-Shāṭir’s values for the declinations (dδ)/right ascensions (dRA) of the stars as individually plotted against the modern declinations/right ascensions (in Figures 10 and 11, the close circles, •, show the southern stars, and the open circles, ○, stand for the northern stars); (b) The mean error (•), mean absolute error (○; their values are given inside the boxes), and standard deviation (indicated in the error bars) of the declinations/right ascensions of the stars as classified into the zones of the declination (1: –60° ⩽ δ < –30°, 2: –30° ⩽ δ < 0°, 3: 0° ⩽ δ < +30°, and 4: +30° ⩽ δ) and of the right ascension (in intervals of 60°).

Figure 13 shows the distribution of the values of dδ with respect to dRA.

Figure 13.

The errors in Ibn al-Shāṭir’s values for the stellar declinations with respect to the corresponding errors in the right ascension.

Figures 14(a) and 14(b) display the boxplots for dδ and dRA (the whiskers in both graphs represent the lowest datum still within 1.5 IQR of the lower quartile and the highest datum still within 1.5 IQR of the upper quartile; NB. IQR: interquartile range), in which the outliers can be recognized: we encounter five outliers in dδ (nos. 19, 32, 65, 85, and 88) and seven outliers in dRA (nos. 28, 35, 48, 50, 65, 79, and 88).

Figure 14.

The boxplots for Ibn al-Shāṭir’s errors in (a) the declination and (b) the right ascension.

The histograms in Figures 15(a) and 15(b) represent the distribution of dδ and dRA with the bin widths of 10′.

Figure 15.

The histograms representing the distribution of Ibn al-Shāṭir’s errors in (a) the declination and (b) the right ascension.

The general statistical results for dδ and dRA are as follows. The average of dδ is μ = –3.2′ with the standard deviation σ = 29.3′ and the mean absolute error of dδ (i.e. the average of the values of |dδ|) MAE = 21.1′. The stars nos. 19 and 88 have the greatest errors (respectively, –96′ and +84′), whereas nos. 3 and 31 are error-free. For dRA: μ = –2.8′ with σ = 29.1′ and MAE = 20.0′. Nos. 28 and 50 have the extreme errors (respectively, +83′ and –98′), while nos. 34 and 60 have no error. So, the right ascensions and the declinations are nearly of the same degree of precision. The same can also be seen in the histograms: in the case of the distribution of the errors within ±20′, the declination is 54 (63.5%), and the right ascension is 62 (69.7%). In the case of the declination, the errors of +10′–+20′ are the most concentrated, but, for the right ascensions, the errors of 0′–+10′ are the most concentrated.

Specifically for dδ (Figures 9(a)–9(b)):

For the 33 southern stars (except for the four problematic stars): μ = +5.9′ with σ = 27.4′ and MAE = 19.1′. Only nine of these stars have the negative errors in the declination. In a sense, a systematic northward shift appears to have occurred in Ibn al-Shāṭir’s measurement of the declinations of the southern stars. In the case of the 52 northern stars, μ = –8.9′ with σ = 29.3′ and MAE = 22.4′. Therefore, it can be seen that the declinations of the southern stars were measured somewhat more precisely than those of the northern ones. The errors dδ for the northern stars are nearly evenly distributed with respect to their directions (22 stars with the positive errors, 28 with the negative ones, and the two error-free). The negative mean error in the declinations of these stars arises from the facts that the three outliers have the quite large negative errors (–77′ for no. 85 and –80′ for no. 32, both in Zone 3, and –77′ for no. 65 in Zone 4), which are totally responsible for about a half of the negative average error, and the negative values of dδ for these stars are appreciably greater in size than the positive ones (excluding the outliers, in Zone 3 : Σdδ = −432′ for 18 stars and +274′ for 17 stars, and in Zone 4: –149′ for seven stars and +77′ for five stars).

Zone 2 is significantly the most accurate. It falls in the middle of the sky towards the south of the local horizon of Damascus. Zone 1 has the least precise values, which is principally due to the fact that we have fewer data in this zone than the other zones together with the two major outliers (–96′ for no. 19 and +84′ for no. 88), which inevitably make the variation of the errors in it quite large. Zones 3 and 4 are approximately of the same degree of accuracy.

Considering the values of dδ with respect to the RA zones (Figure 10(a)–10(b)):

Zone 3, having the fewest number of the stars and mainly including those of Leo, is the most precise. The negative mean error in Zone 1 is mainly because of the two large outliers (–80′ for no. 32 and –77′ for no. 85). Of the remaining ten stars in this zone, only no. 36 has a negative moderate error of –26′. Ibn al-Shāṭir’s values for the declinations of 15 of the 18 stars spreading out in RA ~ 67°–121°, including Zone 2 (the constellations Aur, Ori, Gem, CMi, Pup, and the star β Tau), except for the two stars of CMa and α Car, in addition to one object from Zone 3 (M44) show a southerly, downward shift, which is responsible for the negative mean error of dδ in Zone 2.

The associations of the values of dδ for the adjacent stars can be conveniently recognized; for example, γ Cet and 41 Ari (RA ~ 32°–34°) having the two outliers, δ and β Sco (RA ~ 230°–233°) with dδ = +16′, β Aqr and γ Cap (RA ~ 314°–317°) with dδ = +14′, as well as α and β Peg and α PsA (RA ~ 335°–340°) with the egregious negative errors.

In the case of dRA (Figures 11(a)–11(b)):

Like the declination, Zone 3 is the most accurate, and Zone 6 is ranked second. The large mean errors and standard deviations in the other zones stem from the striking outliers distributed among them (Zone 1: +83′ for no. 28 and +58′ for no. 35, Zone 2: –73′ for no. 48 and –98′ for no. 50, Zone 4: –67′ for no. 65, and Zone 5: –72′ for no. 88 and –90′ for no. 79). No systematic pattern does exist in them.

Considering the values of dRA with respect to the δ zones (Figure 12(a)–12(b)):

Like the declination, Zone 1 has the biggest errors (four outliers). Zone 4 is the most precise. The other two zones are nearly of the same degree of precision.

In Figure 13, the two horizontal and the two vertical lines separate the outliers from the core of the data. If the stellar positions are derived from the precise observations, the distribution of the errors should be concentrated at the centre (0,0). Factually, 58 stars (65.2%) have both |dδ| and |dRA| ⩽ 32′, as encompassed by the circle in the figure.

Magnitudes

The magnitude values from Ibn al-Shāṭir, al-Ṣūfī and Ptolemy are presented in Table 1. Ptolemy and al-Ṣūfī³⁶ give their magnitude values according to the basic traditional system, arranging the magnitudes from 1 to 6 with notations for stars which appear brighter or fainter than an integral magnitude (we indicate them, respectively, with ↑ and ↓ in the table), so that every integral magnitude is divided into the one-third bins. Ibn al-Shāṭir apparently neglects to give the magnitude values in this manner. The last two columns in Table 1 indicates the two sets of the apparent magnitudes, the first from the YBS and the last the same values as adjusted by the atmospheric extinction for an observer at Damascus (φ = 33;31°), both given to a precision of 0.1. The latter set has been computed on the basis of the formula given in Schaefer 2013.³⁷ Needless to say, the differences between the two sets of the magnitudes are greater for the southern stars, amounting to 0.3 for α PsA, 0.4 for ε Sgr, 0.5 for λ Sco, 0.9 for θ1 Eri, and 2.8 for α car; for the rest, they remain ⩽ 0.2, and so are negligible. Because al-Ṣūfī observed from Shiraz with a lower latitude than Damascus, the differences turn out to be smaller, reaching to 0.3 for α PsA and ε Sgr, 0.4 for λ Sco, 0.7 for θ¹ Eri, and 1.5 for α Car.³⁸ For Ptolemy, the maximum differences are 0.3 for β Cet and ε Sgr, 0.4 for α PsA and λ Sco, 1.2 for θ¹ Eri, and 1.9 for α Car. It should be noted that in the case of the 19 stars that Ibn al-Shāṭir’s values for their magnitudes are different from both Ptolemy’s and al-Ṣūfī’s (see below), there is no difference between the modern magnitude values as modified by the atmospheric extinction for Ibn al-Shāṭir’s time at Damascus and for either al-Ṣūfī’s time at Shiraz or Ptolemy’s time at Alexandria to be greater than 0.1; for this reason, in our comparative examination, it is not necessary to tabulate the modified magnitudes in the cases of al-Ṣūfī and Ptolemy.

Regardless of the one-third magnitude bins, Ibn al-Shāṭir’s magnitude values are in agreement with both Ptolemy’s and al-Ṣūfī’s for most cases (actually, 59 stars). Among the stars that Ptolemy and al-Ṣūfī have different values for their magnitudes, Ibn al-Shāṭir’s values agree with Ptolemy’s in five cases (nos. 9, 10, 40, 68, and 69) and with al-Ṣūfī’s, only in a single case (no. 27). In 19 cases (nos. 1, 16, 17, 22, 33, 42, 44, 47, 50, 55, 72, 74, 76–78, 80–83), they agree neither with Ptolemy’s, nor with al-Ṣūfī’s. For 16 of these 19 stars, Ptolemy and al-Ṣūfī have the same values, while only in two cases (nos. 81 and 82) their values are different from each other, and al-Ṣūfī has a more precise value for each. Note that neither Ptolemy nor al-Ṣūfī does assign a magnitude to no. 42. Putting aside the three stars having problematic magnitude values (nos. 16, 55, and 82; see above), we should look for some clues in Ibn al-Shāṭir’s 16 unprecedented magnitude values as to whether he in reality measured them independently of Ptolemy and al-Ṣūfī: unlike Ptolemy and al-Ṣūfī, Ibn al-Shāṭir gives a value for the magnitude of no. 42, and for the remaining 15 stars, the differences between the values recorded in Ibn al-Shāṭir’s table and those of Ptolemy and al-Ṣūfī amount to 1 (except for no. 81, for which Ibn al-Shāṭir’s value is less than Ptolemy’s by 2).

1. No. 42: Of the seven nebulous objects in the ASC, two are in reality asterisms (ω Cyg and ν Sgr). The remaining five ones are deep-sky objects (the open clusters λ Ori, NGC 869 and NGC 884, NGC 2632,a nebulous object in Coma Berenices, which can be identified as Melotte 111, and ASC-22-Sco, which can be identified as either G Sco or NGC 6475.³⁹ Ptolemy assigned a magnitude value to neither the latter five nebulous objects, nor the stars in their vicinity. In fact, there is no star in these sky regions whose magnitude can be measured individually by the naked eye (i.e. m > +6), except for the open cluster λ Ori, which has three stars with m < +5 (λ, φ¹, and φ² Ori), and Melotte 111. What makes the apparent magnitudes of the clusters fall within the range of the visual power of the human eye is the combination of the apparent brightness of the faint stars densely gathered in those small regions. The concept or quantity of the “combined magnitude” was not known to any medieval astronomer, and it was apparently not known how the brightness of a nebulous object to be compared to a reference star in order to estimate its magnitude. Al-Ṣūfī also does not give any magnitude for the nebulous object he discovered in Andromeda (NGC 224, M31).⁴⁰

Al-Ṣūfī identifies the nebulous object in the head of Orion, identical to the Arabic fifth lunar mansion, Haq’a (no. 42), as the three faints conjunct stars in the form of a triangle, like the three didactical points in the Arabic letter thā (ث),⁴¹ without giving any value for their magnitudes, which was followed by Ulugh Beg (1394–1449 AD).⁴² Analogously, following Ptolemy, Nicolaus Copernicus (1473–1543 AD) does not give any magnitude for this nebulous object.⁴³ Ibn al-Shāṭir appears to take the fifth lunar mansion as a single star (λ Ori) and assigns to it a correct magnitude of 4. Thus, he appears to have given for the first time a value for the magnitude of the head of Orion. Nevertheless, he gives no value for the magnitude of the other nebulous object in the table (no. 54, M44), apparently for the same reason mentioned above. More than two centuries later, Tycho Brahe (1546–1601 AD) tabulated these three stars discretely and assigns to the superior of the three conjunct stars in the head of Orion a magnitude of 4↓ and to both the westerly star and the third star in order of rising a magnitude of 5 (respectively, φ¹: m = +4.4 and φ²: m = +4.1).⁴⁴

2. Of the remaining 15 stars, Ibn al-Shāṭir’s unprecedented magnitude values show definite improvements over Ptolemy’s and al-Sūfī’s for at least six stars (especially, in the cases of nos. 47 = γ Gem, 50 = β CMi, 76 = α Sco, and 80 = α Oph, for which al-Ṣūfī’s or Ptolemy’s errors reach ~ +1), while al-Ṣūfī’s or Ptolemy’s values are more precise in 3 cases (especially, for nos. 74 and 78, ω¹ Sco and α Her, for which Ibn al-Shāṭir’s errors are ~ –1. NB. We have excluded β Per for its large variability with a short period,⁴⁵ for which Tycho also gives a magnitude of 3),⁴⁶ and for the five stars, the errors are by-and-large of the same size, but with the opposite signs.

For 14 of the 15 stars in question, Ibn al-Shāṭir’s values are less than both Ptolemy’s and al-Ṣūfī’s by 1 (except for no. 81, for which the difference between Ibn al-Shāṭir’s and Ptolemy’s amounts to 2), and only in one case, Ibn al-Shāṭir’s value is greater than Ptolemy’s and al-Ṣūfī’s (no. 33). Ibn al-Shāṭir’s most critical change in the stellar magnitudes is the case with the large area covered by the adjacent constellations Hercules (no. 78), Serpens (no. 72), Ophiuchus (nos. 77, 80, and 81), and Sagittarius (nos. 1 and 83). He appears to have seen the stars of this region brighter than Ptolemy and al-Ṣūfī did. It seems to have been an inevitable consequence of correctly taking Antares (α Sco, no. 76) as a first magnitude star, while both Ptolemy and al-Ṣūfī have an inaccurate magnitude of 2 for it, and thereafter using it as a standard in order to determine the magnitudes of the nearby stars. However, only the magnitude of one of the stars in Scorpius (ω¹ Sco, no. 74) is decreased, whereas those of the three others (β, δ, and λ Sco, nos. 73, 75, and 79) have remained unchanged. Besides, he made the significant improvements in the measurement of the magnitudes of the individual stars β Cas, κ Ori, γ Gem, and β CMi.

In the light of the abovementioned pieces of evidence, it can be almost certainly said that Ibn al-Shāṭir renewed the measurement of the stellar magnitudes about four centuries after al-Ṣūfī and, for at least 16 stars, estimated the values different from Ptolemy’s and al-Ṣūfī. However, for the large bulk of his star table, either he should have been influenced by Ptolemy’s and al-Ṣūfī’s values and/or his observations should have confirmed that they are reasonably accurate, as in effect they are so to a great extent.

Test of authenticity

Excluding the problematic stars as well as those having the outliers in their positional errors (totally 15 stars), the differences in longitude between Ptolemy’s and Ibn al-Shāṭir’s values range from 17;3° to 19;26°, with μ = 18;8° and σ = 0;26°, and the differences in latitude from –0;73° to +0;76°, with μ = 0° and σ = 0;21°. This finding together with Ibn al-Shāṭir’s unprecedented values for the stellar magnitudes serve as the conclusive evidence that his star table is independent of the ASC.

The tradition of the star tables with the equatorial coordinates

Ibn al-Shāṭir established a new tradition in composing the star tables arranged in the celestial coordinates, which are user-friendly for the timekeeping purposes. The following examples only reflect a small portion of such stellar tables in the works written in Syria and Egypt after Ibn al-Shāṭir: a catalogue of 120 stars in Shihāb al-Dīn al-Ḥalabī’s (d. 1455 AD) al-‘Iqd al-yamānī . . . (The fortunate necklace . . .)⁴⁷ and another table of 84 stars in al-Durr al-fākhir. . . (The excellent pearl . . .) by the same author,⁴⁸ both for the beginning of 840 H/15 July 1436 (JDN 2245753); a catalogue of 302 stars by al-Tīzīnī for the end of 940 H/11 July 1534 (JDN 2281543), which has been published by T. Hyde in 1665 in an appendix to Ulugh Beg’s star catalogue⁴⁹; a catalogue of 112 stars by Muḥammad b. Haykal al-Akhṣāṣī, a 16th-century astronomer at the Azhar mosque at Cairo (Akhṣaṣ, a village in the vicinity of the city of Fayyūm in the Middle Egypt in the southwest of Cairo),⁵⁰ which has been published in Knobel 1895.

Discussion, comparison, and conclusion

In this study, we have edited and analyzed the star table included in Ibn al-Shāṭir’s Jadīd zīj. The main findings and further comparative notes are summarized below.

1. A thorough detailed analysis of the table in Section “Statistical analysis of the accuracy of the table” showed that in Ibn al-Shāṭir’s table, we are mainly confronted with the random errors although some systematic patterns can be distinguished in it. The principal results are summarized in the following.

( i ) The right ascensions (MAE = 20.0′) and the declinations (MAE = 21.1′) are nearly of the same degree of precision.

( ii ) The stars in the region RA ~ 120°–180° generally have the least errors in both declination and right ascension.

( iii ) The declinations of the southern stars were measured a bit more precisely than those of the northern ones.

( iv ) The values for the declinations of the stars in the region δ ~ –30°–0° are significantly the most accurate.

( v ) A systematic northward shift can be seen in the declinations of the southern stars.

( vi ) The declinations of 15 of 18 stars spreading out in RA ~ 67°–121° show a southerly, downward shift.

( vii ) More than 65% of the stars have the errors in both the declination and right ascension less than 32′. No outlier in the tabular coordinates exceeds ±98′.

2. The different patterns of the errors in Ibn al-Shāṭir’s observational data, notably the obvious contrast between the systematic positional errors in some celestial regions and the errors randomly distributed in other sky zones reinforces the idea that the observations were made in different times and by different instruments. However, in order to keep the effect of precession as small as possible, it was reasonable for Ibn al-Shāṭir to observe the stars in a narrow span of time, at best during a single year.

3. In the historiography of astronomy until the turn of the present century, it was accepted that after Ptolemy, al-Ṣūfī made a systematic observational program for the measurement of the stellar magnitudes, which remained unrivalled until Tycho Brahe. It was used without any improvement in Ulugh Beg’s star catalogue, as the column of the magnitudes in it is headed by “the magnitudes on the basis of al-Ṣūfī’s opinion”.⁵¹ Meanwhile, it was known that in the star catalogue prepared by Jamāl al-Dīn al-Zaydī and other Persian astronomers in the service of the Mongolian Yuan dynasty of China, some non-Ptolemaic magnitude values were used, which, in all likelihood, stemmed from a new series of the observations carried out at Beijing in the latter part of the 13th century. And now we know of Ibn al-Shāṭir’s new measurements of the stellar magnitudes, at least, in the case of a few stars about a century later: for the first time, he assigned a correct magnitude of 4 to λ Ori, a component of the star cluster in the Orion that was considered a nebulous object in the Ptolemaic tradition, and also measured more precise values for the magnitudes of α Sco, α Oph, β Cas, κ Ori, γ Gem, and β CMi than Ptolemy and al-Ṣūfi.

A comparison between these two sets of non-Ptolemaic values for the stellar magnitudes shows some interesting common characteristics. The two sources partially containing Jamāl al-Dīn catalogue (cf. “Introduction”, no. 5) have altogether 56 stars in common with Ibn al-Shāṭir’s star table. In 36 cases, Jamāl al-Dīn has the same magnitude values as in Ptolemy’s and Ṣūfī’s catalogues, and in most of these cases the magnitudes are precise. In three cases his values agree only with al-Ṣūfī’s, which are more precise than Ptoelmy’s (ε Del, α² and β Lib), and in one case only with Ptolemy’s, which is less accurate than al-Ṣūfī’s (β Tau). For the rest, he has different values from both Ptolemy and al-Ṣūfī, while both Ptolemy and al-Ṣūfi have the same value; Jamal al-Dīn’s values are imprecise for a half (actually seven stars, putting aside β Per for its variability with a short period), but more accurate than Ptolemy’s/al-Ṣūfī’s for other eight stars: he correctly assigns m = 1 to α Aql and α Cyg, and m = 2 to β Cet, α Ari, and δ Sco (for these five stars, Ibn al-Shāṭir has Ptolemy’s/al-Ṣūfī’s values), and like Ibn al-Shāṭir, m = 1 to α Sco and m = 2 to β Cas and α Oph. This is not the only common feature of the astronomies developed by Jamāl al-Dīn in China and Ibn al-Shāṭir in Syria a century separated from each other; a millennium after Ptolemy, they correctly put a substantial difference between the motion of the solar and planetary apogees and the precession; however, no influence or transmission from the first to the latter seems to have existed in reality.⁵²

4. Although nothing is known on Ibn al-Shāṭir’s observational instruments, he should, at least, have measured the stellar declinations by a meridian instrument equipped with a kind of alidade. His large sundial (length ~ 2 m and width ~ 1 m), “the most sophisticated sundial known from before the late European Renaissance”,⁵³ reveals his skills in devising and constructing astronomical instruments. For measuring the meridian altitude/declination with a precision of 1 minute of arc (the length of each arc-minute subdivisions taken as equal to 1 mm, near the limit of resolution of a typical human eye), a quadrant with a radius of ~ 3.5 m was needed, like the model constructed in the Maragha observatory, of which Muḥyī al-Dīn al-Maghribī used to accomplish his observations.

Both Ibn al-Shāṭir and Muḥyī al-Dīn give the same kind of observational data. For the equatorial coordinates of the eight stars they have in common (nos. 5, 37, 49, 52, 57, 64, 67, and 76), the mean absolute errors in Ibn al-Shāṭir’s values for the declinations and right ascensions are, respectively, 20.3′ (μ = +13.5′ with σ = 22.6′) and 10.5′ (μ = +6.4′ with σ = 11.9′). The corresponding errors found in Muḥyī al-Dīn’s observations are, respectively, 6.2′ (μ = +1.8′ with σ = 7.5′) and 26.4′ (μ = +21.4′ with σ = 21.8′).⁵⁴ Therefore, Muḥyī al-Dīn obtained more precise values for the stellar meridian altitudes/declinations than Ibn al-Shāṭir, while the latter achieved more accurate values for the culmination times/right ascensions. In other words, it may be not far from the truth to state that Ibn al-Shāṭir appears to have had a less accurate meridian instrument, but more precise timekeeping devices, at his disposal than those available to Muḥyī al-Dīn at Maragha.

Footnotes

Acknowledgements

I would like to express my very great appreciation to Benno van Dalen (Munich) and Julio Samsó Moya (Barcelona) for their constructive encouragements, Dennis Duke (USA) for supplying me with the data needed to prepare the star charts, and an anonymous referee for his/her useful suggestions. I would also like to offer my special thanks to Shi Yunli (Hefei, China).

Note on contributor

S. Mohammad Mozaffari is an Iranian historian of medieval astronomy. He has published several papers on observational and mathematical astronomy in medieval Islamic astronomy since 2012. He is currently working on the development of observational astronomy in the medieval Middle East.