A remarkable mathematical and astronomical brass instrument signed by Nastulus, one of the leading astronomer-craftsmen of late-9th and early 10th-century Baghdad

This highly sophisticated mathematical instrument made in Baghdad by the Muslim astronomer known as Nastûlus who was particularly active between 890 and 930 brings our knowledge of the activities in that flourishing scientific centre a substantial step further. This type of instrument was until now not known to exist, although sundials based on the same principle are described in Arabic treatises datable to ca. 950 and ca. 1280. It is essentially a mathematical device providing a graphic solution to a problem that was of interest to Muslim astronomers, namely, the determination of the solar altitude as a function of time throughout the year, here specifically for the latitude of Baghdad. Whilst it is of limited practical use, the instrument reveals a level of mathematical competence and sophistication that is at first sight astounding. However, with a deeper understanding of the scientific milieu from which it came, it is fully within the theoretical competence of the scientists of that environment. Nevertheless, the spectacular accuracy of the engraving of the principal curves on the instrument is completely unexpected.

(In the sequel, an asterisk * indicates that the subject is treated in more detail in the technical commentary that follows.)

the historical background

By the 9^th century, the new Abbasid capital of Baghdad was the world centre of scientific activity, especially in astronomy and mathematics.[1] Scholars in that flourishing cosmopolitan city had adopted the knowledge they found in Indian, Persian and Greek sources. Also, in the mid 8^th century, the Muslims encountered the astrolabe, a two-dimensional representation of the three-dimensional heavens, and by the early 9^th century they were devising new types of instruments (many of which are currently thought to have been European inventions). With remarkable rapidity they created a new science that well merits the appellation “Islamic science” for two reasons.[2] Firstly, the principal universal language of serious innovative science from the 9^th century until the 15^th was Arabic, and secondly, some of the interests of the Muslim scientists were directed towards the complicated problems of regulating the Muslim lunar calendar, praying at precise times that depend on astronomical phenomena and local latitude and vary from one day to the next, and determining a sacred direction (qibla) towards the Kaaba in Mecca. In particular, the times of Muslim prayer are defined in terms of phenomena that depend on the altitude of the sun above or below the local horizon.[3]

From the 8^th century to the early modern period, Muslim astronomers were foremost in the science of astronomical timekeeping (‘ilm al-mîqât in Arabic), of which the determination of the times of prayer constitutes a small but highly significant part.[4] The tools of the astronomer were, in addition to the mathematics of spherical and planetary astronomy, handbooks full of astronomical tables and explanatory text, and astronomical observational instruments and mathematical computational devices. In particular, Muslim astronomers produced all sorts of tables and instruments relating to timekeeping, some serving specific latitudes and others serving all latitudes. This new mathematical instrument was surely made using one of these tables, specifically for Baghdad, which has not survived.* Two such tables are, however, known from 13^th-century Yemen, computed for Sanaa and Taiz.* Sundials based on the same principle, with similar curves, are known from treatises compiled by al- Sûfî (Shiraz, ca. 950) and al-Marrâkushî (Cairo, ca. 1280).*

The Muslim astronomer who called himself Nastûlus has caused scholars in the Middle Ages and in recent times some problems. His signatures on all three of his surviving instruments are always without a diacritical point on the first letter, which could in theory be read as an “n” or a “b”. His name appears to be related to Nastûrus, “Nestorius”, which suggests a Christian (Nestorian) connection.[5] Yet the medieval Arabic textual sources call him Muhammad ibn Muhammad or Muhammad ibn ‘Abdallâh, “known as Nastûlus”.[6] No biographical information on him is available. Until 2005, he was known to us from a small number of citations in later Arabic scientific literature and two standard astrolabes signed by him that are now preserved in Kuwait and Cairo,[7] both of which are engraved in the same distinctive kûfî script and with the standard alphanumerical notation for numbers.[8] Nastûlus is credited with the invention of two of the non-standard types of astrolabe that Muslim scholars developed in the 9^th and 10^th centuries, partly because the standard astrolabe, brilliant in conception as it was, was considered somewhat mundane.[9]

Nastûlus’ “Kuwait” astrolabe, dated 315 Hijra = 927/28, is a standard astrolabe but very much more precisely made than the earliest surviving astrolabe with inscriptions in Arabic, which dates from late-8^th-century Baghdad,[10] yet it lacks the spectacular decoration and mathematical complexity of the astrolabe made by the astronomer al-Khujandî in Baghdad in 374 Hijra = 984/95.[11] His “Cairo” astrolabe is missing the rete and the plates but includes a geographical gazetteer on the mater and quadrants for trigonometric calculations on the back, such as were invented in Baghdad in the early 9^th century.[12]

From the available textual references, it is clear that he was an extremely innovative astronomer-mathematician, and from his two surviving astrolabes, it is clear that he was a highly competent craftsman. We also know from a textual reference that Nastûlus made a plate that he had invented for computing the times and stages of eclipses in 280 Hijra = 893/94,[13] so we may wonder why he was making a standard astrolabe as late as 927/28. The great Muslim scientific polymath al-Bîrûnî (Ghazna, Afghanistan, ca. 1025) mentioned that Nastûlus had written a treatise on a gear mechanism for reproducing the apparent relative motions of the sun and moon, and al-Bîrûnî presented his own mechanism with a different set of gears.[14] This has been published in some detail, together with the only surviving example of such a mechanism made by a Muslim craftsman in Isfahan in 614 Hijra = 1223/24 that is now preserved in the Museum of the History of Science at Oxford.[15] However, until 2005, Nastûlus’ treatise was thought to be lost. al-Bîrûnî also mentioned that Nastûlus had written on a sundial but that its underlying principle was very incorrect (fâsid). It seems that al-Bîrûnî was looking at a copy of Nastûlus with perhaps one page missing.

In a manuscript in a private collection that was studied for the first time in 2005, a text that we can without any doubt attribute to Nastûlus describes a universal sundial for finding the time of day in seasonal hours for any latitude. Several such sundials have survived from Greek, Roman and Byzantine cultures, but no texts are known.[16] The same manuscript contains Nastûlus’ description of a luni-solar gear mechanism such as is also found inside some of the surviving sundials. Nastûlus’ account appears to be based on his correct understanding of some Byzantine instrument that he actually saw, although he does not mention this. In any case, his treatise is certainly not a translation from any Greek text. His account of this kind of sundial seems to have borne no fruit, except in the form of the simple universal sundials on the alidades of astrolabes, the use of which is based on the same principle.[17]

description of the instrument

This new piece advances considerably the history of scientific instrumentation. It is a kind of device that was previously unknown, even from any texts. It is unrelated to the astrolabe except for its basic shape, which is simply a circular disc with a throne and suspensory device at the top, and for the alidade or sighting device on the back that can rotate over an altitude scale. We do not know what Nastûlus called it, but he may have used the expression safîhat al-sâ’ât, “plate for the hours”.

It is signed by Nastûlus in the same way as his two surviving astrolabes, without diacritical points, simply:

sana'hu nastulus

“constructed by Nastûlus”.

The front plate has a solar / calendrical scale around the rim, on which the equinox is on the right hand side and the scales are arranged counter-clockwise.  The names of the signs of the zodiac are standard,[18] and the months are the standard ones of the Julian calendar (Âdhâr = March, etc.).[19] Each set is divided and labelled for each 5° of each zodiacal sign or for each 5 days (with appropriate adjustments for months with more or less than 30 days), with subdivisions for each 1° and each day.

Within these is a family of six lemon-shaped rings engraved for the seasonal hours from 1 on the innermost one to 6 on the outermost one in words using the classical Arabic forms: sa'a sa'ataan thalath arba' khamis sit.

A radial rule is engraved uniformly for each 5° up to 80°, with subdivisions for each 1°. The space near the circumference of the circular disc at the base of this may have been labelled ufuq, “horizon”, but corrosion has apparently eliminated this.

The back bears no markings beyond an altitude scale marked and labelled for each 5° of solar altitude, subdivided for each 1°. The alidade or sighting device is unduly wide, but accurately aligned.

The function of the instrument is explained in the inscription on the front, which reads:

li-ma'rifa mawdi' al-shams wa li-ma'rifa ma mada wa ma baqa min al-nahar min sa'a zamaniya wa aydan li-ma'rifa irtifa' al-sa'at al-zamaniya wa irtafa' nisf al-nahar li-madinat al-salam wa haytha yakun al-'ard lj

'For finding the longitude of the sun and for finding how many seasonal hours have elapsed (since sunrise) or remain (until sunset) and also for finding the solar altitude at the seasonal hours and its altitude at midday throughout the year for the “city of peace” (i.e., Baghdad), and wherever the latitude is 33°.'

Because of the hole at the end of the first line, only the horizontal line of the jîm (for three) of the lâm-jîm (for 33) is visible, and this means that the latitude was taken as 33°, and that the other common value 33°25', which is actually more accurate, was not used.* The appellation “city of peace” was common for Baghdad, even though life there in the 9^th and 10^th centuries was not always free from political and religious strife.[20]

[Caption:] _{Computer-generated curves based on the accurate formula (red) and on the standard medieval approximate formula (green). When these are superposed on an image of Nastûlus’ curves the agreement is astounding.}

These markings are of considerable historical interest, in that they are the earliest known graphical representation of curves on metal or on stone or marble that are neither circles nor conics. Their execution is spectacularly accurate. What is particularly remarkable is that the curves are engraved smoothly. The “curves” on one horary quadrant for Baghdad (on the back of al-Khujandî’s astrolabe dated 984/85) and one sundial for Cordova (datable ca. 1000) that were made by leading astronomers are constructed by joining the base points with line segments. It is by no means clear how Nastûlus achieved this feat, and it is anticipated that this will prompt some scholarly debate. A microscopic analysis would surely yield some clues to the way in which Nastûlus engraved them.

Similar curiously-shaped markings are found on sundials described by the astronomers ‘Abd al-Rahmân al-Sûfî (Shiraz, ca. 1000) and Najm al-Dîn al-Misrî (Cairo, ca. 1325), but these are graphical representations in the same format of the shadows at the hours, and the outer lemon-shaped ring is much larger than all of the others, so that the instruments look rather strange. There are no surviving examples of these sundials, and the diagrams in the manuscripts in which they are illustrated are not drawn with any great precision.*

Tables of time as a function of solar altitude and solar meridian altitude for each degree of both arguments are known from 10^th-century Baghdad, some specifically for the latitude of Baghdad and based on the accurate formula, others for all latitudes and based on the approximate formula, and here we have a graphical representation of the inverse problem: the determination of the solar altitude as a function of time throughout the year, using one or other of these formulae.

In brief, this extraordinary instrument raises all sorts of technical questions, but also some other issues. None of the surviving astrolabes from the 9^th and 10^th century, mostly from Baghdad, bear any markings relating to the times of prayer. Nastûlus’ new instrument conforms to this tradition. It would have involved rather little additional work to have included a curve for the time of the ‘asr prayer, though this would have had a shape that was different from that of the hour curves. It has been incorrectly argued that Islamic astronomy developed simply because of the importance of astronomy for religious purposes. Here we see a Muslim astronomer devising yet another instrument purely for scientific purposes, which include mathematical experimentation.

The piece is not an instrument suited for timekeeping as such, in spite of what Nastûlus wrote in his inscription. To find the time from the solar altitude one would have to interpolate between the hour curves and lose all control of accuracy. For that purpose various varieties of astrolabes and quadrants and sundials were available, in addition to all sorts of tables, as well as a mysterious one known only by the name lawh al-sâ’ât, “board for the hours”.[21] It is rather an excès de délicatesse, in the sense that it provides information that one does not really need to know. It is a mathematician’s delight, and proves that its maker had the wits and the time to conceive it and to make it well. And one may well wonder what else he made.

The instrument within the context of an advanced astronomical tradition – a technical commentary

In the early 9^th century Muslim scholars, as part of their geodetic measurements, had determined the latitude of Baghdad (accurately 33°20'). The first group derived 33°, which is one value that remained popular, especially in the instrument tradition of al-Khwârizmî. Other astronomers preferred 33°25'.[22] (Readers should bear in mind that scientific knowledge was not cumulative in pre-modern science; one professional group or regional school might favour a certain tradition and sustain it regardless of the opinions of others).

[Caption:] _{The celestial sphere as understood by medieval astronomers, and the same reduced to two dimensions by means of an analemma. (From King, In Synchrony with the Heavens.)}

The astronomy and mathematics necessary to understand Nastûlus’ instrument is not trivial. The first diagram shows the three-dimensional celestial sphere as the Muslim astronomers thought of it in mathematical terms. The observer is at O with his horizon NESW. The celestial axis is OP and the celestial equator is EQW. The local latitude is equal to the arc NP. The sun is shown here at an arbitrary position in the morning, X, in the summer. Its declination is XT, and the limits of this are +23½° in summer and -23½° in winter (this quantity is known as the obliquity of the ecliptic). The sun appears to rise at A, culminate at B and set at C. The length of daylight is twice the arc AB. The positions of the sun at the six seasonal hours of the morning, equal divisions of the time between sunrise and midday, are shown as red dots. The altitude of the sun at X above the horizon is the arc XK. The problem is to find the varying altitude of the red dots throughout the year.

The second diagram shows the same scene reduced to the meridian plane by the procedure known as the analemma, adopted by the Muslims from Greek sources.[23] It was such a procedure that was adopted by the earliest Muslim astronomers, before they applied spherical trigonometry to solve the same problems in completely different ways. The various quantities that we shall encounter below are highlighted in the diagram.

In this discussion, we shall use medieval versions of the relevant formulae, which are actually simpler than the modern versions. In all medieval Arabic texts the formulae are written out in words, although abbreviations and special terms were used. The tables for timekeeping that are discussed below are mainly extant in unique manuscripts that have survived the vicissitudes of time and are now scattered in libraries around the world.

We use the following notation, and the corresponding Arabic terms:

λ       solar ecliptic longitude (tûl or mawdi’ al-shams)

δ      solar declination (al-mayl)

e      obliquity of the ecliptic (al-mayl al-a’zam)

ф       local latitude (‘ard al-balad)

h        instantaneous solar altitude (irtifâ’ al-shams)

H       solar meridian altitude (irtifâ’ nisf al-nahâr)

D       time in equatorial degrees from sunrise to midday or from midday to sunset (nisf al-nahâr)

T        time in equatorial degrees since sunrise or time remaining until sunset (al-dâ’ir min al-falak)

°        equatorial degrees, where 360° is equivalent to 24 hours, that is, 1° is equivalent to 4 minutes (darajât)

sdh    seasonal day hours, that is, the length of daylight 2D divided by 12 (sâ’ât zamâniyya)

vers   the versed sine function,[24] where vers q = 1 - cos θ (al-sahm)

To start with, we need to calculate for each few degrees of λ, the functions D (λ) and H(λ,ф) and D (λ,ф), using:

D(λ,ф) = sin λ sin ε ; H(λ,ф) = 90° - ф + δ(λ) ; and

D(λ,ф) = 90° + arc sin { tan δ(λ) tan ф }.

The formulae that were generally used for calculating the time from the solar altitude were equivalent to:

T = D - arc vers { vers D ( 1 - [ sin h / sin H ] ) } .

Not only did the Muslims know of this formula already by the 9^th century, but in the early 14^th century an Egyptian astronomer Najm al-Dîn al-Misrî even tabulated T(H,D,h) for each 1° of each argument – a total of about 440,000 entries.[25]

An approximate formula that worked well for all latitudes from the Yemen to N. Iraq was:

T^sdh = 1 /15 arc sin { sin h / Sin H }

This very simple formula is accurate at the equinoxes for all latitudes, but expressing the result in seasonal hours stabilizes the result for all latitudes, and the error is minimal for the latitudes of Muslim centres of astronomical activity.[26] It is important to realize that this simple approximation was, in medieval terms, “universal” (medieval Arabic ufuqî, “serving all latitudes”).[27] The formula was adopted by the Muslims from Indian sources, but the Greeks knew it also (and used it to devise a universal sundial that was known to Nastûlus).

The inverse problem of finding the solar altitude at the seasonal hours is most simply solved using the approximate formula. First we find the length of any seasonal hour T_n(λ,ф) (n = 1, 2, ... ,6) for latitude ф, in equatorial degrees, using:

T_n(λ) = n x D(λ,ф) / 6         (T₆ = D) ,

and then the altitude of the sun at the n^th hour, h_n, is:

h_n(λ) = arc sin { sin H x sin T_n } .

The use of the accurate formula is a little more cumbersome, but that would not have stopped any of the best Muslim astronomers from using it. The formula would be:

sin h_n(λ) = sin H(λ) x { 1  -  vers [ D(λ) - T_n(λ) ] / vers D(λ) }.

Even in the 9^th century leading astronomers such as Habash al-Hâsib would have prepared a table of such a function as this by first compiling auxiliary tables of the functions sin H(λ) and vers D(λ) and then their quotient. Their trigonometric tables were, of course, adequate to the task. However, interpolating in tables of the sine and versed sine that only have values for each degree of argument was fraught with risk of error.

Some tables of the relevant functions are known from 9^th- and 10^th-century Baghdad and should be mentioned.[28]

The 10^th-century astronomer ‘Alî ibn Amâjûr compiled a table of T(H,h) with values in equatorial degrees and minutes thereof (1° = 4 minutes of time) for arguments:

H = 21°, 22°, ... , 84° and h = 1°, 2°, ... , H

The values, which number ca. 3,300, are computed with remarkable accuracy for latitude 33°25', and the limits for H indicate that the table was intended for timekeeping by the stars as well as by the sun. The same astronomer also compiled a table of T(H,h) with values in seasonal hours and minutes for arguments:

H = 1°, 2°, ... 90° and h = 1°, 2°, ... , H ,

now based on the approximate formula and hence serving all latitudes.

Values of the solar altitude as a function of time (seasonal hours 1-6) at the solstices form part of a set of tables compiled in Baghdad in the early 9^th century for the construction of horizontal sundials. These tables, for which each subset serves a specific latitude between 21° and 40°, are attributed to al-Khwârizmî, though Habash al-Hâsib is also a candidate. They display not only the solar altitude at the hours, but also the polar coordinates (shadow and azimuth) needed for the construction of the hyperbolae on a sundial. The points given by these coordinates for each hour would then be connected with straight lines to produce the sundial. For the latitude of Baghdad (33°) and Samarra (34°) more values are given, respectively, for each ten minutes and each thirty minutes. Surely Nastûlus had seen these tables, but for his purposes, they would only have given him a set of points along the vertical axis of his new instrument.

A table compiled by al-Khwârizmî displays the function h(T,H) to the nearest degree for each seasonal hour of T and each 1° of H, and this is based on the approximate formula and hence serves all latitudes. If Nastûlus had used this table, it would not have given him a set of smooth curves. The method would have been to calculate H for each few degrees of λ and then derive the value h for each hour from the table.

It is simpler to hypothesize that somebody compiled a table of the function h(T,λ) for each seasonal hour and each degree of λ for latitude 33°, that is, with 2,160 entries. The underlying value of the obliquity of the ecliptic would have been 23°35', used by Nastûlus on his astrolabes. We do know of two such tables from the Yemen, which particularly in the 13^th and 14^th centuries was an important centre of astronomy. These, computed with either the accurate or the approximate formula, probably the latter, serve respectively the latitude of Sanaa by the Rasulid Sultan al-Ashraf himself, and the latitude of Taiz by the court astronomer Abu l-‘Uqûl.[29] Nastûlus probably computed one for Baghdad himself. One thing is for sure: the table he used for his new instrument was accurately computed.

With such a table Nastûlus could have constructed six sets of 360 dots. Any mistakes in the table would have been immediately obvious as each new mark was about to be made and the position of the mark could have been corrected. The problem with this assertion is that it assumes that Nastûlus could have made the marks using a radial rule that was (a) accurately centred, and (b) accurately divided. Any defects in the radial rule would be reflected in the resulting curves, and these show no obvious traces of error.

[Caption:] _{The diagram of a sundial in a 13^th-century copy of al-Sûfî’s astrolabe treatise from the facsimile, with the hour curves completely distorted by the copyist, and the reconstructions of the later sundials of al-Marrâkushî and Najm al-Dîn al-Misrî (from Charette, Mathematical Instrumentation).}

Horizontal sundials based on the same principle were described by ‘Abd al-Rahmân Sûfî (Shiraz, ca. 950), without a table, and al-Marrâkushî (Cairo, ca. 1280), with a table,[30] and although these represent functional instruments giving the time from the gnomon shadow, there were so many more standard varieties available that, until the rediscovery of the present piece, one might have doubted that such complex sundials were ever actually constructed. The available manuscript copies show defective illustrations, and possibly only the author’s autograph copy had reasonably accurate diagrams. The descriptions of the markings in the accompanying Arabic texts have been used to reconstruct them.

The existence of a single instrument bearing such curves strengthens the claim that has been made concerning a different kind of instrument that we know only from three 17^th-century Iranian examples.[31] These are circular brass plates centred on Mecca with a highly sophisticated coordinate grid on which about 150 localities are marked as points indicating their longitude and latitude. The grid is so devised that the radial scale and the circumferential scale show the direction and distance to Mecca. On the three surviving examples, the arcs of ellipses representing the arcs of latitude are sensibly approximated as small arcs of large circles. The mathematics underlying the grids – relating to the determination of the qibla by conic sections – has been located in two treatises from 10^th-century Baghdad and 11^th-century Isfahan,[32] and we can be confident that the three surviving examples are derived from a very much earlier prototype.

Finally, this instrument is important for the history of instrumentation for another reason: it partly resolves the question of the origin of the solar / calendrical scales on Islamic instruments.[33] Some scholars have favoured an Andalusî origin. Others have argued that al-Bîrûnî had mentioned them and also that they occur on various Eastern Islamic instruments that are independent of any Andalusî influence. The Andalusî Abu l-Salt wrote about them in his astrolabe treatise, compiled whilst he was in prison in Cairo ca. 1000, and this treatise was influentual in both the Eastern and Western Islamic worlds. But now we have a clear example of them from early-10^th-century Baghdad. The theory behind such scales is that of Ptolemy’s solar model, updated with the new parameters – longitude of the apogee and eccentricity – that had been derived from the observations of the astronomers of the Caliph al-Ma’mûn in Baghdad in the early 9^th century.[34]

Nastûlus’ instrument is compelling evidence of the sophistication of scientific and technical achievements in Baghdad about 1,100 years ago.

endnotes

[1]      For a bio-bibliographical survey of authors writing in Arabic on mathematics, astronomy and astrology until ca. 1050 see Sezgin, Geschichte des arabischen Schrifttums, vols. V-VII (1974-1979). For a list of publications specifically on astronomical activity in Baghdad see the website “Astronomy in Baghdad” (2003) at the end of the bibliography, which contains many references to research postdating Sezgin’s magnum opus. Reprints of several hundreds of studies from the 19^th and early 20^th centuries are in Sezgin et al., eds., Islamic Mathematics and Astronomy, and Prof. Sezgin’s Institute for the History of Arabic-Islamic Science in Frankfurt has published reprints of dozens of important scientific manuscripts and facsimiles of many more scientific instruments, the latter catalogued in Sezgin & Neubauer, Wissenschaft un Technik im Islam.
[2]      On the scope of the Arabic and Persian literature on mathematical astronomy see the article “Zîdj” (astronomical handbooks with tables) in the Encyclopedia of Islam.
[3]      For brief introductions to the Islamic aspects of Islamic astronomy see the articles “Ru’yat al-hilâl” (sighting the lunar crescent), “Mîkât” (astronomical timekeeping), and “Kibla” (sacred direction).
[4]      On astronomical timekeeping in the Islamic world up to the 19^th century see King, In Synchrony with the Heavens. The history of astronomical timekeeping in Europe from the Middle Ages to the early modern period is poorly documented. For the nature and scope of some of the available materials see ibid., I-10.
[5]      On the Nestorians see the article “Nastûriyyûn” in the Encyclopaedia of Islam.
[6]      On Nastûlus see Sezgin, Geschichte des arabischen Schrifttums, VI, pp. 178-179 and 288, and three articles by Brieux & Maddison, King & Kunitzsch, all summarized in King, In Synchrony with the Heavens, XIIIc-9.
[7]      Both instruments are described in detail ibid.
[8]      On this see the article Irani, “Arabic Numerical Notation”, and the article “Abdjad” in the Encyclopaedia of Islam.
[9]      On these see most recently King, op. cit., X-5.1, and Charette, Mathematical Instrumentation, pp. 63-83.
[10]     This is analyzed ibid., XIIIb.
[11]     See ibid., XIIIc-9.
[12]     On these see ibid., X-6, and now also Charette & Schmidl, “Al-Khwârizmî and Practical Astronomy in Ninth-Century Baghdad”.
[13]     See Sezgin, op. cit., VI, p. 288.
[14]     al-Bîrûnî’s treatise is published in Hill, “Al-Bîrûnî’s Mechanical Calendar”.
[15]     First described in Gunther, Astrolabes of the World, I, pp. 118-120. On the gear mechanism see more recently Field & Wright, “Gears from the Byzantines”; and King, op. cit., II, pp. 65-68.
[16]     See Field & Wright, “Gears from the Byzantines”.
[17]     See King, op. cit., XI, Appendix A.
[18]     See the article “Mintakat al-burûdj” (zodiac) in the Encyclopaedia of Islam.
[19]     See the article “Ta’rîkh, I” (dates and eras) ibid.
[20]     See the article “Baghdâd” ibid.
[21]     It may be that this refers to a highly sophisticated prototype of an instrument for timekeeping by the sun for any latitude that was known in 14^th-century England as the navicula de Venetiis, “little ship of the Venetians”, and from 15^th-century Vienna as Regiomontanus’ Uhrtäfelchen”, “little board for finding the hours”. The latter was extremely popular in Europe for several centuries thereafter. It has recently been shown that Habash invented a more complicated device than this for timekeeping by the stars: see Charette & Schmidl, “Habash’s Universal Plate”. Also, the medieval English tradition of the navicula demonstrates a suspicious lack of familiarity with the correct way to make the instrument: see King, op. cit., XIIb.
[22]     On early values of the latitude of Baghdad (and Mecca) see King, “Too many cooks ... ”, pp. 225-228. The basic reference work on such values is Kennedy & Kennedy, Geographical Coordinates of Localities from Islamic Sources.
[23]     On the analemma in Islamic mathematical astronomy see the studies listed in King, In Synchrony with the Heavens, I, p. 27, n. 40.
[24]     See the article “Sahm” (versed sine) in the Encyclopedia of Islam.
[25]     A detailed analysis is in Charette, “A Monumental Medieval Table for Solving the Problems of Spherical Astronomy for all Latitudes”.
[26]     The history of this formula in the Islamic world and in Europe from 750 to 1900 is documented in King, In Synchrony with the Heavens, XI, also XIIa.
[27]     On the concept of “universality” in Islamic astronomy see ibid., VIa-b.
[28]     These are discussed ibid., I-2.3.1, 2.5.1, 4.1.1 and 4.3.1.
[29]     Ibid., I-4.2.5 and 4.2.6.
[30]     See Sezgin’s facsimile editions of al-Sûfî’s astrolabe treatise, p. 469, and of al-Marrâkushî’s encyclopaedic work on timekeeping, pp. 228-229, and on Najm al-Dîn’s instrument see Charette, Mathematical Instrumentation, pp. 153-155.
[31]     On two of these see King, World-Maps for Finding the Direction and Distance to Mecca.
[32]     On the third example and the newly-discovered textual sources see idem, In Synchrony with the Heavens, VIIIc.
[33]     This is discussed in King, In Synchrony with the Heavens, X-4.7, etc. (see p. 1026 of the index).
[34]     See the article “Shams” (sun) in the Encyclopedia of Islam.

bibliography and bibliographical abbreviations

AIHS: Archives internationales d’Histoire des sciences.

François Charette, Mathematical Instrumentation in Fourteenth-Century Egypt and Syria – The Illustrated Treatise of Najm al-Dîn al-Misrî, Leiden: Brill, 2003.

— , “A Monumental Medieval Table for Solving the Problems of Spherical Astronomy for all Latitudes”, AIHS 48 (1998), pp. 11-64.

François Charette and Petra Schmidl, “Al-Khwârizmî and Practical Astronomy in Ninth-Century Baghdad – The Earliest Extant Corpus of Texts in Arabic on the Astrolabe and Other Portable Instruments”, SCIAMVS 5 (2004), pp. 101-198.

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Kunitzsch: see King & Kunitzsch.

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Website:
“The renaissance of astronomy in Baghdad in the ninth and tenth centuries: A list of publications, mainly from the last 50 years” (2003):
http://web.uni-frankfurt.de/fb13/ign/astronomy_in_baghdad/bibliography.html

Sotheby’s is grateful to Prof. David King of the Institute for the History of Science at the Johann Wolfgang Goethe University in Frankfurt am Main for his comments on this piece. The provisional computer graphics of Nastûlus’ lemon-shaped curves were prepared by Prof. Glen Van Brummelen, also of the Frankfurt Institute during the summer of 2006.