Biruni’s Telescopic-Shape Instrument for Observing the Lunar Crescent

by S. Mohammad Mozaffari, Georg Zotti Published on: 17th October 2020

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This paper deals with an optical aid named barbakh that Abū al-Rayḥān al-Bīrūnī (973–1048 CE) proposes for facilitating the observation of the lunar crescent in his al-Qānūn al-Mas‘ūdī VIII.14. The device consists of a long tube mounted on a shaft erected at the centre of the Indian circle (a circle on a plane ground with marks indicating the four cardinal points), and can rotate around itself and also move in the vertical plane. The main function of this sighting tube is to provide an observer with a darkened environment in order to strengthen his eyesight and give him more focus for finding the narrow crescent near the western horizon about the beginning of a lunar month. We first briefly review the history of altitude-azimuthal observational instruments, and then present a translation of Bīrūnī’s account, visualize the instrument in question by a 3D virtual reconstruction, and comment upon its structure and applicability.

Figure 1. A possible reconstruction of Bīrūnī’s sighting tube.

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Note of Editor: This paper appeared for the first time in Suhayl: International Journal for the History of the Exact and Natural Sciences in Islamic Civilisation, Vol. 14 (2015), pp. 167–88. The PDF can be retrieved online via this link.

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Altitude-azimuthal instruments either are used to measure the horizontal coordinates of a celestial object or to make use of these coordinates to sight a heavenly body. They belong to the “empirical” type of astronomical instruments.[1] None of the classical instruments mentioned in Ptolemy’s Almagest have the simultaneous measurement of both altitude and azimuth of a heavenly object as their main function.[2] One of the earliest examples of altitude-azimuthal instruments is described by Abū al-Rayḥān al-Bīrūnī for the observation of the lunar crescent near the western horizon (the horizontal coordinates are deployed in it to sight the lunar crescent). In this paper, we attempt to make a virtual reconstruction of this instrument and discuss the aspects of its efficiency and applicability from a technical point of view. Another early example is that proposed by Bīrūnī’s contemporary, Ibn Sīnā (Avicenna), who presumably built a model of it in his observatory at Iṣfahān (central Iran) between 1024 and 1037 CE (this was used in order to measure the altitude and azimuth of an object in any direction in the sky).[3]

Figure 2. The observer has prepared a pole on the expected azimuth of the lunar crescent along the outer rim of the Indian Circle and observes from its center.

Figure 3. A more accurate line of sight can be achieved with another pole in the center of the Indian Circle and observing over both poles.

Nearly three centuries later, four altitude-azimuthal instruments appeared in the first period of activity at the Maragha observatory (1259–1283 CE). These were constructed by Mu’ayyad al-Dīn al-‘Urḍī (d. 1266 CE), the main instrument maker of the observatory at the time. We provide the names and brief descriptions of these instruments for later reference:

(1) Ālat dhāt al-rub‘ayn, “Instrument having two quadrants” or, as al-Kāshī (1380–1429 CE) calls it, dhāt al-samt wa-l-irtifā‘, “(Instrument) having azimuth and altitude”,[4] a double azimuthal quadrant built from copper inside a circular wall, capable of measuring, at the same time, azimuth and altitude of two celestial objects;[5]

(2) Dhāt al-jayb wa-l-samt, “(Instrument) having sine and azimuth”, an instrument  to determine the sine of the zenith distance of a heavenly body and its azimuth using a wooden bar rotating on an iron axis inside a circular wall, on one end of which the alidade can slide, the other end sliding up a vertical central pillar;[6]

(3) Dhāt al-jayb wa-’l-sahm, “(Instrument) having sine and versed sine”, a similar instrument to determine the sine and versed sine of the altitude and/or zenith distance of a celestial object;[7] and

(4) Āla al-kāmila, “Perfect instrument”, consisting of a rotating parallactic rule inside a circular wall.[8] Al-‘Urḍī mentions that he had built a model of his “Perfect instrument” for Malik Manṣūr, the ruler of Ḥimṣ (now Homs, Syria), in 650 H/1252–3 CE, in presence of the latter’s vizier, Najm al-Dīn al-Lubūdhī.[9] (1) and (3) are described by al-Kāshī.

In the second period of activity at the Maragha observatory (1283–ca. 1320 CE), 12 instruments were invented by Ghāzān Khān, the seventh ruler of the Īlkhān dynasty of Iran (r. 1295–1304 CE) on the basis of a totally novel approach. 11 of these instruments are altitude-azimuthal (the remaining is a pinhole image device used to observe solar eclipses).[10]

The altitude-azimuthal instruments appear to have reached their most evolved stage at the Maragha observatory, since no significant innovation in the construction and design of these instruments can be recognized at the Samarqand and Istanbul observatories. In Book III of his Sidrat al-muntahā, Taqī al-Dīn Muḥammad b. Ma‘rūf, the director of the short-lived observatory at Istanbul (1526–1585 CE), introduces some unprecedented observational instruments, of which his improved models of the Ptolemaic parallactic instrument (one is called dhāt al-jayb, “(Instrument) having the sine” and the other, dhāt al-shu‘batayn, “(Instrument) having two branches/legs”,  – which is the standard name for the antique parallactic instrument in Islamic sources, too –  can be classified as altitude-azimuthal instruments. The eight instruments built at the Istanbul observatory are described in detail in an anonymous treatise in Turkish named the Ālāt-i raṣadiyya li Zīj-i Shāhanshāhiyya.[11]

Figure 4. Bīrūnī solely refers to the horizontal and vertical alignments of the horns of the lunar crescent with respect to the line joining the two luminaries; what he intends to say can be simply illustrated with this figure.

Concerning the history of the altitude-azimuthal type of observational instruments prior to the beginning of the past millennium, namely, before Bīrūnī’s time, we should first consider the fact that, although not a single example of such instruments can be found in Greek and Roman sources,[12] various examples can be traced in Indian astronomical sources from the time before the rise of astronomy in medieval Islam. For instance, in the Súrya Siddhánta and in Brahmagupta’s Brāhmasphuṭasiddhānta (628 CE), some prototypes of altitude-azimuthal instruments can be recognized that enabled observers to measure altitude, azimuth, time, the solar declination and amplitude, and the longitudes of the sun, moon, and planets. For example, cakra was a wooden wheel suspended by a string and graduated to 360 degrees that could be used for measuring the altitude and zenith distance of a heavenly object at any azimuth, the angular distance between the sun and the moon, and the time elapsed from sunrise or remaining till sunset. Also, the longitude of a planet can be measured by this instrument from the longitude of a reference star already known (two other variants with 180º and 90º graduations are also described, which are called, respectively, dhanuṣ, “semi-circle”, and turyagola, “quadrant”). Horizontal variants of the same instruments were also used: pīṭha was a horizontally placed cakra on whose circumference the cardinal points are marked off, and in whose centre a vertical axis, equal in length to the radius of the circle, is erected. Kapāla and kartarī were two variants of dhanuṣ; the first consists of a semicircle installed to the north of the east-west line with a vertical gnomon in its centre, and the latter consists of two semicircles located in the planes of the equator  and the meridian. These instruments were used in timekeeping. A similar instrument was yaṣṭi, a staff which can be utilized with two circles engraved on the ground, of which one represents the azimuthal circle and the other stands for the sun’s diurnal circle at a given day.[13] In later Hindu sources, based on Brahmagupta, instructions are given how to convert the ortive amplitude of the sun, as observed at sunrise/sunset or as derived from three observations of the solar altitude at a given day by means of this instrument, to its declination, and then to its longitude.[14] It is worth mentioning that Brahmagupta’s Brāhmasphuṭasiddhānta and Khaņḍakhādyaka (665 CE) were translated into Arabic or adapted in the early ‘Abbāsid period.[15]

In the second place, it seems necessary to discuss some anachronistic misunderstandings attached to the history of altitude-azimuthal instruments in the early Islamic period, especially in the two systematic astronomical observational programs associated with the rule of al-Ma’mūn (‘Abbāsid caliph from 812 to 833 CE) in Baghdad and Damascus. In the following, we will evaluate whether such instruments were available at the time and/or were employed in the observations.

The above-mentioned misunderstanding is related to the Ma’mūnic observational program made in Damascus, and is rooted in some later Islamic sources belonging to the period from the turn of the fifteenth century onwards, the writings of the already mentioned al-Kāshī and Taqī al-Dīn Muḥammad b. Ma‘rūf. From these writings, the misunderstanding was taken over into the modern scholarly literature. When describing two of al-‘Urḍī’s altitude-azimuthal instruments, al-Kāshī says that these did not exist and were not employed in ancient (i.e., pre-Islamic) observations,[16] and adds that dhāt al-samt wa-l-irtifā‘ (i.e., al-‘Urḍī’s azimuthal double quadrants) had been built first in Damascus (without specifying at what time) and then at Maragha. Also, in his Sidrat III, Taqī al-Dīn says that this instrument is an invention of Islamic observational astronomy.[17] In the Ālāt-i raṣadiyya li Zīj-i Shāhanshāhiyya, it is added that it was designed by Islamic astronomers in Damascus, next brought over to Maragha, and then used by Ibn al-Shāṭir in Damascus.[18] Sayılı argues that the first statement refers to the Ma’mūnic observations in Damascus.[19] But, as Kennedy remarks,[20] al-Kāshī’s reference to Damascus may merely be a case of confusion with al-‘Urḍī’s “Perfect Instrument” which, as mentioned above, he says to have built at Homs before joining the Maragha team. It is quite probable that al-Kāshī was a source for Taqī al-Dīn, as his other works were read and annotated by the latter,[21] and that al-Kāshī’s mention of Damascus, just after his mention that the instrument was not available in antiquity, might have led Taqī al-Dīn to ascribe the instrument to the early Islamic observers in Damascus. After all, it does not seem reasonable to assume that these ambiguous and very late sources can be taken as reliable testimony for the existence of an azimuthal quadrant in Damascus in the ninth century.

Nevertheless, a lost episode of altitude-azimuthal instruments in medieval Middle Eastern astronomy appears to pertain to the Ma’mūnic observational program carried out by Yaḥyā b. Abī Manṣūr in Baghdad. It merits noting that both observational programs carried out in Baghdad and Damascus in the first half of the ninth century concentrated on the determination of the solar parameters,[22] and that the most important result achieved was the discovery of the motion of the solar apogee.[23] Bīrūnī reports four values measured for the obliquity of the ecliptic[24] and also the times estimated for five autumnal equinoxes (829–832, and 843 CE) from these observations.[25] In general, three sets of values for the solar orbital elements were measured in the observational programs in Baghdad and Damascus during 829–832 CE and, some 11 years later, again in Baghdad in 843–844 CE.[26] Bīrūnī associates an armillary sphere and a “circle” (about whose nature he says nothing further) with the Baghdad observations and a 10-cubit (~ 5 m) gnomon made of iron and a quadrant made of marble with a radius of the same length with the Damascus observations. Bīrūnī mentions the existence of a defect in an instrument Yaḥya made use of, but does not specify which instrument this was. The only deficiency Bīrūnī mentions about the instruments employed in the Ma’mūnic observations is related to the gnomon erected in the monastery Murrān near Damascus;[27] as Bīrūnī says, its length changed by one sha‘īra (1/144 cubit ~ 3.5 mm) between early morning and evening, because of the decrease in temperature during the night, and consequently, it did not allow an accurate measurement of the true length of the year.[28]

We will consider Yaḥyā’s instrument simply named “circle” in more detail. Since the observers in Baghdad measured the obliquity of the ecliptic (see note 24), this instrument may have been a solstitial armilla, maybe similar to the “Two Circles” in Almagest I.12 or to later alterations that substituted an alidade for its inner circle.[29] In the primary sources earlier than Bīrūnī, e.g., in the prologue of the two extant versions of the Mumtaḥan zīj and in Ḥabash’s Kitāb al-ajrām wa ’l-ab‘ād (“Book on the volumes and distances [of celestial bodies]”), the “circle” Yaḥyā made use of is unspecifically named dā’irat al-Shammāsiyya, the “Shammāsiyya circle”.[30] These sources also mention that in his observation of the vernal equinox of 829 CE by means of this instrument, Yaḥyā measured, in Baghdad, a longitude of 179;43º for the sun at sunrise on Sunday, 19 September 829 (JDN 2024112), from which the time of the autumnal equinox was deduced as 4/5 hours afternoon on this date.[31] Bīrūnī only mentions the time but not the solar longitude. In the Talkhīṣ al-Majisṭī (Compendium of the Almagest), Muḥyī al-Dīn al-Maghribī, the most prominent astronomer of the Maragha observatory (d. 1283 CE),[32] correctly derives from Yaḥyā’s value for the solar longitude the time of the autumnal equinox of 829 CE as 6;54 hours after the moment of the observation or 0;2,15 days afternoon.[33] He also provides us with a clue for understanding the nature of Yaḥyā’s “circle” by calling it al-dā’irat al-samtiyya, the “azimuthal circle”.

Muḥyī al-Dīn is presumably the earliest source referring to Yaḥyā’s instrument as an “azimuthal circle”. Of course, this can neither be verified, nor is it sufficient in itself to decide on the type and form of the instrument. Moreover, it is nearly impossible to decide which reading of the name of the instrument is correct, because the names may be scribal confusions of each other (سمتیّة ↔ شمّاسیّة).(34) However, the principal question is how Yaḥyā could have been able to measure both the obliquity of the ecliptic and the longitude of the sun at sunrise on an equinoctial day by a single “circle” or “ring”. There are only two possibilities: either the instrument was a circle suspended by a string or rotating about a vertical axis and could thus be used for observing altitude, zenith distance, the obliquity of the ecliptic, etc., or it consisted of a circle installed parallel to or on the ground and thus allowed measuring, for example, the azimuth of a celestial object when it was rising or setting.

We present a hypothesis on the nature of Yaḥyā’s circular instrument based upon the assumption of Indian influence. As mentioned earlier, Brahmagupta’s Brāhmasphuṭasiddhānta had already been translated into Arabic or adapted in Islamic astronomy in the early ‘Abbāsid period. It contains detailed descriptions of some circular altitude-azimuthal instruments. The procedure for measuring the ortive amplitude of the sun by means of a horizontally installed circle, and then converting the result to the solar longitude is described in it as well as in other Indian sources in connection with circular altitude-azimuthal instruments. If this hypothesis were true, it can be said that Yaḥyā’s “circle” is probably the first example of a horizontal instrument in Islamic astronomy.[35] However, further concrete data is needed to establish the structure and application of Yaḥyā’s circular instrument with certainty.

The PDF can be retrieved online via this link.

Figure 5. A possible reconstruction of Bīrūnī’s sighting tube.

Footnotes

[1] Here, we follow the classification set forth in Charette 2006, p. 123. Research on medieval Islamic observational instruments is insufficient from many aspects (King 2004/5, Vol. 2, pp. 11–27 gives a general perspective of it), while other types of operational and demonstrational instruments such as various models of astrolabes, sundials, portable quadrants, and so on, have been investigated to the finest details (e.g., see King [1987] 1995; Charette 2003).
[2] Of the seven antique observational devices, only Two Circles, Mural Quadrant (Almagest I.12), and Parallactic Instrument (Almagest V.12) are installed in the plane of the meridian and are used for measuring the altitude/zenith distance of a celestial object when it passes the meridian (see Toomer 1998, pp. 61–63, 133, 217–219, 244–247, 252 and 404–407).
[3] Ibn Sīnā describes this instrument in his Maqāla fi ’l-ālāt al-raṣadiyya (“Essay on observational instruments”); concerning it, see Wiedemann and Juynboll 1926; for a figure of the instrument, see Sezgin and Neubauer 2010, Vol. 2, pp. 26–27. In spite of what Sayılı ([1960] 1988, pp. 156–158) and the other sources blindly following him (e.g., Sezgin and Neubauer 2010, loc cit.) claim, Ibn Sīnā’s observatory was located at Iṣfahān, not in Hamadhān, as his pupil, Abū ‘Ubayd al-Jūzjānī, says definitely that Ibn Sīnā “wrote the Maqāla fi ’l-ālāt in Iṣfahān during his observations for ‘Alā’ al-Dawla [d. 1041 CE, Daylamī military leader and founder of the short-lived but significant Kakuyid dynasty]” (Gohlman 1974, pp. 104–105).
[4] Kāshī, Sharḥ, S: ff. 13r–14r, P: pp. 23–26, M: pp. 36–38, T: f. 116r; Kennedy 1961, pp. 101–103, 106.
[5] Seemann 1929, pp. 72–81.
[6] Seemann 1929, pp. 87–92.
[7] Seemann 1929, pp. 92–96.
[8] Seemann 1929, pp. 81–87, 96–104. For an illustration of the instruments of the first period of the Maragha observatory, see Sezgin and Neubauer 2010, Vol. 2, pp. 38–52.
[9] Seemann 1929, p. 97.
[10] About them, see our extensive study in Mozaffari and Zotti 2012; 2013.
[11] Taqī al-Dīn invented and built some other new observational instruments, which were set up at the Istanbul observatory in addition to the classical instruments (e.g., a mural quadrant with a radius of 13 cubits ~ 6.5 m that he installed at the observatory on 22 April 1574, and an armillary sphere with a radius of its meridian ring equal to 91/6 cubits ~ 4.6 m) and those already invented by his Muslim predecessors, like al-‘Urḍī’s “Two Quadrants”. Taqī al-Dīn gives a detailed account of his instruments in his Sidrat (K: f. 14v–15r). An illustrated version of the treatise Ālāt-i raṣadiyya li Zīj-i Shāhanshāhiyya is also appended to MS. K of Taqī al-Dīn’s Sidrat (ff. 48v–50r); it is clear that it is not Taqī al-Dīn’s since he is referred to as mawlānā, “our master”, therein. About his instruments, see Tekeli 1963; Sezgin and Neubauer 2010, Vol. 2, pp. 53–61. About his observations carried out in Istanbul in the 1570s, see Mozaffari and Steele 2015.
[12] See, e.g., Evans 1999; 2015.
[13] Súrya Siddhánta XIII.20: [1860] 1997, pp. 306–307; Brāhmasphuṭasiddhānta XXII.8–45: see Sarma 1986/7, pp. 68–69.
[14] E.g., Siddhánta Śiromaņi XI.10–15, 28–39 ([1861] 1974, pp. 212–213, 218–221).
[15] Pingree 1973, p. 37.
[16] Kāshī, Sharḥ, S: ff. 13v–14r, P: p. pp. 25–26, M: pp. 37–38, T: f. 116r; Kennedy 1961, pp. 101, 103.
[17] Taqī al-Dīn, Sidrat, K: f. 15v.
[18] Taqī al-Dīn, Sidrat, K: f. 49r.
[19] Sayılı [1960] 1988, pp. 73–74.
[20] Kennedy 1961, p. 106.
[21] E.g., a note on f. 90v of MS. K of the Sidrat concerning a computational error committed by al-Kāshī.
[22] On the astronomical activities in the Ma’mūnic period, see Sayılı [1960] 1988, chapter 2 (note that this source contains numerous statements that should be considered with caution; e.g., see note 27, below), a brief summary of which is given in Charette 2006, p. 125; about the solar observations carried out in them, see Mozaffari 2013, esp. Part 1, pp. 327–329.
[23] See Mozaffari 2013, Part 1, pp. 322, 326, the discussion of the discovery of the solar apogee’s motion in ibid, Part 2, pp. 403–408.
[24] Yaḥyā, working in Baghdad, measured ε = 23;33º. Concerning Khālid’s observations in Damascus, Bīrūnī, in his Taḥdīd, first mentions that the maximum solar noon-altitude of 80;3,55º was obtained in 217 H/832 CE and the two values 32;56º and 32;55º for the minimum solar noon-altitude were observed, respectively, in 216 H/ 831 CE and 218 H/833 CE, from which he derives ε = 23;33,57,30º and 23;34,27,30º. He also mentions that Sanad b. ‘Alī, who supervised the observations, reported 23;33,52º which is closer to the first value. Bīrūnī follows that in a table in which the solar noon altitudes observed by Khālid in Damascus were written down, he found the following values for the extremal meridian altitudes of the sun: on Sunday–Tuesday, 13–15 Jumādā I 217 H/21–23 Urdībihisht 201 Yazdigird (16–18 June 832, JDN 2025113–5): 80;4,10°, 80;4,30º, and 80;4,28º, and on Monday–Wednesday, 19–21  Dhý al-Qa‘da 217 H/24–26 Ābān 201 Y (16–18 December 832, JDN 2025296–8): 32;55,0º, 32;54,58º, and 32;55,28º. Bīrūnī notices that the max and min noon-altitudes in this table are, respectively, 80;4,30º and 32;54,48º which result in ε = 23;34,51º, but doing some extrapolations between the abovementioned values, he computes 23;34,57,30º (Bīrūnī 1967, pp. 60–64; Bīrūnī 1954–6, Vol. 1, p. 363–4; also, see Kennedy 1973, pp. 32–39. The true modern value at the time ~ 23;35,33º). This seems to have been the authoritative source for the later astronomers to associate the value 23;35º with the Damascus observations. It is noteworthy that Khālid’s extremal altitudes are in error by less than –1′ (see Said and Stephenson 1995, esp. pp. 122, 125).
[25] Bīrūnī 1954–6, Vol. 2, p. 640, nos. 9–13. As he mentions (ibid, Vol. 2, p. 653), Khālid b. ‘Abd al-Malik al-Marwarūdhī, Sanad b. ‘Alī, and ‘Alī b. ‘Īsā al-Ḥarranī were engaged in the observations carried out in Baghdad in 843 for determining the solar orbital elements. See also the next note. Note that the table of vernal equinoxes in the Hyderabad edition of Bīrūnī’s work is in disorder. The correct order can be easily obtained by means of the times of observations as reckoned from the beginning of the Nabonassar era: in nos. 11–13, the years should be 1579, 1580, and 1591, as in the manuscript “B” listed in the preface of the Hyderabad edition (Vol. 1, before p. 1), noted in the apparatus of the table.
[26] Ibn Yūnus (L: p. 104) remarks unspecifically that the time of the autumnal equinox of 844 was observed by “a group of scientists” (see Said and Stephenson 1995, p. 128).
[27] None of the later Islamic astronomers, including Bīrūnī (1954–6, Vol. 2, p. 637), say that this monastery was on Mt. Qāsiyūn, as stated commonly in the secondary literature (e.g., Sayılı [1960] 1988, p. 71; Charette 2006, p. 125). According to the available historical sources, the monastery Murrān that was near Damascus (there was another monastery with the same name on a mount overlooking Kafarṭāb near al-Mu‘arra) had nothing to do with Mt. Qāsiyūn; e.g., in his Mu‘jam al-buldān (Vol. 2, pp. 533–534, Vol. 4, pp. 295–296), Yāqūt states that the monastery is on a hill overlooking saffron farms, which is named after the monastery itself by some authors such as Bīrūnī and Ḥabash in his Kitāb al-ajrām wa ’l-ab‘ād (cf. Langermann 1985, p. 120), while Mt. Qāsiyūn overlooks the city itself.
[28] Bīrūnī 1954–6, Vol. 1, p. 363, Vol. 2, pp. 637, 778.
[29] E.g., the ḥalqat al-‘aḍudiyya al-¼ūfī made use of about two centuries later; see Mozaffari and Zotti 2012, p. 402.
[30] See Vernet 1956, p. 508; Langermann 1985, p. 121.
[31] This value is in error by ~ +7 hours (cf., also, Said and Stephenson 1995, p. 128).
[32] About Muḥyī al-Dīn, see Saliba 1983, 1985, 1986 and Mozaffari 2014. A monograph about his unique contribution to observational and practical astronomy at the Maragha observatory on the basis of a thorough analysis of his documented observations in the Talkhīṣ al-Majisṭī is in preparation by one of us (SMM).
[33] Al-Maghribī, Talkhīṣ, f. 58r. On the equinoctial days the true daily motion of the sun is approximately equal to its daily mean motion (about 0;59,8º); the time for the sun to travel 0;17º to reach the point of the autumnal equinox is calculated as 0;17/0;59,8 ≈ 6.9 hours. Note that half the length of an equinoctial day is equal to 6 hours, and so the time of the equinox counted from sunrise minus 6 hours yields the time elapsed from noon to the equinox: 6;54 – 6 = 0;54 hours or 0;2,15 days.
[34] The reasons why the two primary sources, i.e., the available MSS of the Mumtaḥan zīj and Ḥabash’s Kitāb al-ajrām wa ’l-ab‘ād, might be, in our estimate, not much more reliable than al-Maghribī’s 13th-century Talkhīṣ lies in the following facts: the two known MSS of the Mumtaḥan zīj (see Yaḥya b. Abī Manṣūr in the bibliography at the end of the paper; also, e.g., van Dalen 2004; Kunitzsch 2003; Viladrich 1988; Vernet 1956). were compiled after Ibn al-A‘lam (d. 985 CE) and ultimately go back to the same recension of the Mumtaḥan zīj, presumably compiled in the tenth century (see van Dalen 2004, esp. p. 11); consequently, they belong to a time some two centuries after the original. Also, Ḥabash’s treatise is extant in a Judeo-Arabic codex written not earlier than 1144 H/1731 CE (see Langermann 1985, pp. 108, 113). Thus, the possibility of scribal errors and other mistakes in these manuscripts is equally likely as that in the sources available to al-Maghribī.
[35] Under this assumption, Yaḥyā would have read off the ortive amplitude as ~0;10º towards the north from the azimuthal circle. The explanation is that the ortive amplitude η is computed as follows: tan η = tan δ/cos φ, where sin δ = sin λ ´ sin ε, δ is the solar declination, λ the solar longitude, ε the obliquity of the ecliptic (23;33º or 23;35º as measured by Yaḥyā and his colleagues; see above, note 24), and φ the latitude of Baghdad (the historical value used at the time: 33;20º in agreement with the modern measurements). For λ = 179;43º, this yields η = 0;8º.

by Mohammad Mozaffari1 and Georg Zotti2

1 Research Institute for Astronomy and Astrophysics of Maragha, Maragheh University, Iran (ORCID ID: 0000-0001-8677-8718)
2 Ludwig Boltzmann Institute for Archaeological Prospection and Virtual Archaeology, Austria (ORCID ID: 0000-0002-8152-0408)

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