A Vetustissimus Arabic Treatise on the Quadrans Vetus

See my article “Rub” [= quadrant] in The encyclopaedia of Islam, new edn (10 vols to date, Leiden, 1960–), via (1995), 574–5 and 4 plates, and David A. King, Studies in astronomical timekeeping in medieval Islam (hereafter SATMI), 12 parts (Leiden, in press), VIII: Astronomical instrumentation in the Islamic world. Section 6, for overviews of the early history of the different kinds of quadrant in the Middle Ages. Peter Schmalzl, Zur Geschichte des Quadranten bei den Arabern (Munich, 1929), is still useful. The Baghdad treatise was announced already in King, “Early Islamic astronomy”, Journal for the history of astronomy, xii (1981), 55–59, p. 59. The findings of this paper are summarised in idem, “The neglected astrolabe”, in Mathematische Probleme in Mittelalter: Der lateinische und arabische Sprachbereich, ed. by M. Folkerts (Wiesbaden, 1996), 45–55, pp. 48 and 51–52, and idem, “Astronomical instruments between East and West”, in Kommunikation zwischen Orient und Okzident: Alltag und Sachkultur, ed. by Harry Kühnel (Vienna, 1994), 143–98, p. 165.

See ref. 10 below. In the modern literature the best overview of the markings and their history is Archinard M. , “The diagram of unequal hours”, Annals of science, xlvii (1990), 173–90.

The seasonal day hours of Antiquity and the Middle Ages are the one-twelfth divisions of the length of daylight; they are hence dependent on terrestrial latitude and solar longitude.

On the astrolabe see North John D. , “The astrolabe”, Scientific American, ccxxx (1974), 96–106, repr. in idem, Stars, minds and fate: Essays in ancient and medieval cosmology (London and Ronceverte, W.V.), 1989, no. 14, and King, “Astrolabe” (ref. 1), the latter dealing, in particular, with the origin of some of the components.

King, SATMI (ref. 1), IXa: On the universal horary quadrant for timekeeping by the Sun. This includes studies of the development of the shadow-square and the universal sundial on astrolabe alidades.

On universal instruments and tables see further King, SATMI (ref. 1), VI: Universal solutions in Islamic astronomy. Alas the universal horary quadrant was omitted from the earlier studies (first published in the 1980s and repr. in King, Astronomy in the service of Islam (Aldershot, 1993), VI and VII) on which the new versions are based. On the concept of universality in relation to the standard astrolabe see idem, “Bringing astronomical instruments back to Earth: The geographical data on medieval astrolabes (to ca. 1100)”, in Between demonstration and imagination: Essays in the history of science and philosophy presented to John D. North, ed. by A. Vanderjagt and L. Nauta (Leiden, 1999), 3–53, pp. 6–11 and 14–17.

Likewise I use the expression “universal horary dial” to refer to the kind of markings found, i.a., on the medieval instrument known as the navicula or the Uhrtäfelchen of Regiomontanus: They too are universal, they serve the determination of the (equinoctial) hours using the exact formula, and they consist of a set of fixed horary markings to be used in conjunction with an ingenious movable device with thread and movable bead attached with which one can enter the local latitude and the solar declination. This seems preferable to various other expressions in the modern literature, some with good historical backgrounds, such as “Uhrtäfelchen” or “cadran solaire rectiligne” or “rectilinear dial” or “universal sundial”. The correct use of the universal horary dial on the navicula is nowhere described in the modern literature. See further the parallel study listed as King, “14th-century England or 9th-century Baghdad? New insights on the origins of the elusive astronomical instrument called navicula de Venetiis”, to appear in a special issue of Centaurus in honour of Bernard R. Goldstein, and a more detailed investigation in idem, SATMI (ref. 1), IXb: On universal horary dials for determining time by the sun and stars. There is a sense in which the quadrant and the dial serve the same purpose: Both serve essentially only timekeeping by the sun; both are universal, but one is approximate and the other is accurate; one is very easy to use and the other requires some dexterity, especially to arrive at the exact solution. A more complex mathematical device is required for timekeeping for any latitude by the stars, since this also involves the sun and various kinds of stellar data must be accessible. Such a device was invented in Baghdad in the ninth century: See King, World-maps for finding the direction and distance to Mecca: Innovation and tradition in Islamic science (Leiden and London, 1999), 354–9, for some first reflections; Charette F. and Schmidl P. G. , “A universal plate for timekeeping with the stars by &Hdotabash al-&Hdotāsib: Text, translation and preliminary commentary”, Suhayl—Journal for the history of the exact and natural sciences in Islamic civilisation (Barcelona), ii (2001), 107–59, for the text and an annotated translation; and King, “Navicula”, and a more detailed investigation in idem, SATMI, IXb-13, on the modus operandi. The universal horary quadrant can of course be used to find the time of night from the instantaneous and culminating altitudes of any star, but the time is then given in “stellar hours” related to the arc of visibility of the star in question. Only one medieval astronomer who used such “hours” is known to me (SATMI, I-4.6.1).

Of the substantial literature on the Latin quadrans vetus I cite only Knorr W. R. , “The Latin sources of quadrans vetus, and what they imply for its authorship and date”, in Sylla E. and McVaugh M. (eds). Texts and contexts in ancient and medieval science: Studies on the occasion of John E. Murdoch's seventieth birthday (Leiden, 1997), 23–67, and idem, “Sacrobosco's quadrans: Date and sources”, Journal for the history of astronomy, xxviii (1997), 187–222, where the reader will find references to earlier works. On the Middle Castillian treatise in the Libros del saber (ed. Rico y Sinobas (Madrid, 1873), v, 285–316), see Vallicrosa Millás J.-M. , “La introducción del cuadrante con cursor en Europa”, Isis, xvii (1932), 218–58, repr. in idem, Estudios sobre historia de la ciencia espanola (Barcelona, 1949; repr. Madrid, 1987), 65–110, pp. 66–67, etc.

Likewise on the quadrans novus I cite only Poulie E. , “Le quadrant nouveau médiéval”, Journal des savants, 1964, 148–67 and 182–214. On the surviving instruments see Dekker E. , “An unrecorded medieval astrolabe quadrant from c. 1300”, Annals of science, lii (1995), 1–47.

Thus John North, after finding these markings on about 40 of some 130 astrolabes in the Museum of the History of Science, Oxford, wrote (“Astrolabes and the hour-line ritual”, Journal of history of Arabic science (Aleppo), v (1981), 113–14, repr. in idem, Essays (ref. 4), no. 15): “At best, the lines (on the back of an astrolabe) can give the unequal hour with an accuracy only about half as great [!] as that given by the conventional astrolabe itself. At worst, the lines are carelessly drawn, unnumbered, very small indeed, and — worst of all — not associated with an auxiliary scale of solar positions…. Not a single medieval (astrolabe) has survived in a form which would suggest that the unequal-hour lines were used meaningfully.” The correct use of these markings — with a mark on the alidade to represent the solar meridian altitude — is described in medieval and Renaissance treatises, but it is not found in the standard modern literature on the astrolabe. See further King, SATMI, IXa-9.

An exception is Franco Garcia S. , Catálogo crítico de los astrolabios existentes en España (Madrid, 1945), 218–21, p. 219, where the application of ink to mark the meridian position on the alidade is mentioned.

For example, in the popular Latin astrolabe treatise attributed to Messahalla but actually based on the Arabic of Maslama al-Majrīṭī (c. 1000) ( Gunther R. T. , Early science in Oxford, v: Chaucer and Messahalla on the astrolabe (Oxford, 1929), 221 and 174); the astrolabe treatise of Jean Fusoris, compiled in Paris c. 1400 (E. Poulie, Un constructeur d'instruments astronomiques au 15e siècle — Jean Fusoris (Paris, 1963), 116); the early sixteenth-century astrolabe treatise of Johannes Stöffler (Elucidatio, 1524 edn, Oppenheim, fol. 28r); and an English astrolabe treatise partly based on Stöffler, The travailer's ioy and felicitie, by Robert Tanner (facsimile in Gunther R. T. , The astrolabes of the world (2 vols, Oxford, 1932; repr. in 1 vol., London, 1976), ii, 511).

See ref. 16 on the earliest text, and Viladrich M. , “Medieval Islamic horary quadrants for specific latitudes and their influence on the European tradition”, Suhayl — Journal for the history of the exact and natural sciences in Islamic civilisation, i (2000), 273–355, for an overview. New insights are in F. Charette, “Mathematical instrumentation in 14th-century Egypt and Syria”, doctoral thesis, Johann Wolfgang Goethe University, Institute for History of Science, 2002, Section 3.1.

MS Cairo DM 969, 4, fols. 8v-9v, copied in an orderly but inelegant Persian hand. The manuscript, and also MS DM 970 by the same copyist c. 1800, contains a series of important treatises on instruments from the ninth and tenth century, some, like this treatise on the universal horary quadrant, unique, and all unpublished. For details on the quadrant text see further King D. A. , A catalogue of the scientific manuscripts in the Egyptian National Library (in Arabic) (2 vols, Cairo, 1981 and 1986), i, 168–9, and ii, 540–1 (ad 4.6.12), and idem, A survey of the scientific manuscripts in the Egyptian National Library (in English) (American Research Center in Egypt, Catalogs, v; Winona Lake, Ind., 1987), p. 53 (no. B105) and pi. LIVa (caption on p. 207).

On al-Khwārizmī see article by Gerald Toomer in Dictionary of scientific biography. In King D. A. , “Al-Khwārizmī and new trends in mathematical astronomy in the ninth century”. Occasional papers on the Near East (New York University, Hagop Kevorkian Center for Near Eastern Studies), ii (1983), 43 pp., a study of various treatises attributed to him discovered in the 1970s, mainly dealing with instruments, this anonymous treatise is not mentioned.

On his writings on the horary quadrant for a fixed latitude see King, “al-Khwārizmī” (ref. 15), 30–31. On his writings on the sine quadrant and the purpose for which he invented it, namely, to solve the formula underlying the universal horary quadrant, see ibid., 27–29, and 7 and 10–11. For the text and a new translation, see F. Charette (with P. Schmidl), “Scientific initiative in 9th-century Baghdad: Al-Khwārizmī on astronomical instruments”, to appear, text J. Of course, die universal horary quadrant of type 2 — see Fig. 2(b) — is nothing other than one variety of the sine quadrant. Lorch R. (“A note on the horary quadrant”, Journal of history of Arabic science, v (1981), 115–20 (repr. in idem, Arabic mathematical sciences: Instruments, texts, transmission (Aldershot, 1995), XVII), and “Some early applications of the sine quadrant”, Suhayl—Journal for the history of the exact and natural sciences in Islamic civilisation, i (2000), 251–72) has stressed the origin of the sine quadrant as a device for solving the approximate formula for timekeeping. See also Hogendijk J. P. , “Al-Khwārizmī's tables of the ‘sine of the hours’ and the underlying sine table”, Historia scientiarum, xlii (1991), 1–12.

On the times of prayer in Islam see the article “Mī&kdot āt, ii” [= astronomical timekeeping and the regulation of the times of prayer] in Encyclopaedia of Islam (ref. 1). See further refs 21 and 28 and the commentary to Ch. 4 of the Baghdad treatise.

This parameter, which is not particularly accurate, is used in various other treatises associated with him: See King, “al-Khwārizmī” (ref. 15), 2, and also idem, Astronomy in the service of Islam (ref. 6), XIV, 129. See further the commentary to Chap. 9 of the Baghdad treatise.

On al-Marrākushī see the article in Encyclopaedia of Islam (ref. 1), and Sédillot J.-J. , Traité des instruments astronomiques des Arabes composé au treizième siècle par About Hhassan Ali de Maroc… (2 vols, Paris, 1834–35, repr. in 1 vol., Frankfurt am Main, 1985), and Sédillot L. A. , “Mémoire sur les instruments astronomiques des Arabes”, Mémoires de l'Académie Royale des Inscriptions et Belles-lettres de l'Institut de France, i (1844), 1–229, (repr. Frankfurt am Main, 1989). On some of these references see Charette, “Mathematical instrumentation” (ref. 13), Section 5.1.

See King, “al-Khwārizmī” (ref. 15), 10–11, on some mysterious tables by him entitled jayb al-sā at, “sine of the hours”, and the rather surprising explanation in Hogendijk, “Al-Khwārizmī's sine of the hours” (ref. 16).

A German translation of the treatise on the use of the instrument was published by J. Frank in 1922. Brief remarks on the treatise on the construction of the instrument are in King, “al-Khwārizmī” (ref. 15), 22–27. See now the edition of both, with translation and commentary, in Charette, “al-Khwārizmī on instruments” (ref. 16). Two reasons why this would have to be a late work of his are: (1) In the treatise on the use of the astrolabe he is still experimenting with new definitions of the times of the żuhr and a⋅r prayers: See the commentary to Chap. 4. (2) In his treatise on the construction of the astrolabe al-Khwārizmī proposes a simple linear horizontal shadow-scale, 12 units below the horizontal diameter, and in this treatise on the universal horary quadrant we are dealing with a more developed shadow-square.

Compare, for example, the Arabic in the texts edited in Charette, “al-Khwārizmī on instruments” (ref. 16).

See ref. 14 above.

King D. A. and Samsó J. , with a contribution by Goldstein B. R. , “Astronomical handbooks and tables from the Islamic world (750–1900): An interim report”, Suhayl — Journal for the history of the exact and natural sciences in Islamic civilisation, ii (2001), 9–105, pp. 56–59. (A summary of this paper is in the article “Zīdj” [= astronomical handbooks with tables] in Encyclopaedia of Islam (ref. 1)).

Most of &Hdot abash's treatises on instruments use 34° for the latitude of the Abbasid capital, Samarra: See King, Mecca-centred world-maps (ref. 7), 40–41 and 345–59.

Drecker J. (Die Theorie der Sonnenuhren, vol. IE of Die Geschichte der Zeitmessung und der Uhren, ed. by von Bassermann-Jordan E. (Berlin and Leipzig, 1925), 86) mentions copies of Sacrobosco's treatise in the Bayerische Staatsbibliothek in Munich in which the text is associated with “Thabit” (Clm 10661) and “Albatenius” (Clm 14583). No such treatise was known to Francis Carmody, who in 1960 surveyed various Arabic and Latin treatises associated with Thābit ibn Qurra (Dictionary of scientific biography, s.v). Likewise, no such treatise was known to Régis Morelon, who in 1987 published various original Arabic texts by Thābit. “Albatenius”, the early tenth-century Syrian astronomer al-Battānī (ibid., s.v.), is not known to have written on instruments other than those in the Almagest (the section on sundials in the unique copy of his astronomical handbook is spurious).

On the calendars used by Muslim astronomers see King and Samsó, “Islamic astronomical handbooks and tables” (ref. 24), 19–20, and the article “Ta'rīkh, iv” [= calendars] by B. van Dalen in Encyclopaedia of Islam (ref. 1).

In Christian Europe, all of the specifically Islamic materials in Islamic astronomical works, such as the treatment of the times of the żuhr and the a⋅r prayers, were invariably suppressed in new Latin versions. This was perhaps unfortunate in the case of the Baghdad treatise, for the procedures advocated here (see the commentary to Chap. 4) would have been very useful for determining the sexts, as well as the terces and nones, but these were surely of less interest at the university scene in Paris than in monasteries.

See already, for example, Delambre J.-B. , Histoire de l'astronomie du moyen âge (Paris, 1819; repr. New York and London, 1965), 243–7; Drecker , Theorie der Sonnenuhren (ref. 26), 86, for a derivation of the formula; and Lorch, “Horary quadrant” (ref. 16), where the hour-angle is used instead of the time of day.

See King, SATMI, VII: An approximate formula for timekeeping from 750 to 1900, and also Charette, “Mathematical instrumentation” (ref. 13), Section 3.2–3.

Compare North, “Hour-line ritual”, 113, quoted in ref. 10 above. The errors using the formula are investigated more fully in King, SATMI, VII-2.3 (ref. 30).

See the discussions in Lorch, “Horary quadrant” (ref. 16), 116–17; King, SATMI, VII-2.2; and Charette, “Mathematical instrumentation” (ref. 13), Section 5.2.

Similar scales for converting λ to δ or sin δ or tan δ are found on other early medieval instruments. An example for δ is found on a universal instrument for timekeeping by the stars by Habash al-Hāsib, a contemporary of al-Khwārizmī (see Charette and Schmidl, “& Hdot abash's universal plate” (ref. 7), 154). An example with which one could find sin δ and cos δ is on the universal horary dial on the sole surviving albion in the Osservatorio Astronomico, Rome (illustrated in North J. D. , Richard of Wallingford: An edition of his writings with introductions, English translation and commentary (3 vols, Oxford, 1976), iii, pi. XXIII). The declination scale on the bottom of the universal horary dial on the medieval English navicular serves to determine δ and that at the side tan δ. See further King, “Navicula” (ref. 7), and idem, SATMI, IXb-7-8.

As, for example, in the fourteenth-century tables for the Yemen and Oxford — see the article “Ta&kdot wīm” [= ephemeris] in Encyclopaedia of Islam (ref. 1), and Eisner S. and MacEoin G. , The Kalendarium of Nicholas of Lynn (Athens, Ga., 1980), respectively. In the tables of solar altitude for the equinoctial hours at the latitude of Baez compiled c. 1435 by pseudo-Enrique de Villena (Tratado de astrologta atribuido a Enrique de Villena, ed. by P. M. Cátedra and Julio Samsó (Madrid, 1980), 171–6), one argument is the dates for which the solar meridian altitude can be rounded to a given degree.

See King, SATMI, I: A survey of Islamic tables for reckoning time by the sun and stars, Section 2.

On some of the terms see Kunitzsch P. , “Glossar der arabischen Fachausdrücke in der mittelalterlichen europäischen Astrolabliteratur”, Nachrichten der Akademie der Wissenschaften in Göttingen, Philologisch-historische Klasse, 1982, no. 11, 455–571 (separatum paginated 1–117).

& Hdot abash uses shażiyya and majrā in his treatise on the universal horary dial for timekeeping by the stars to describe first the movable cursor and then the hollowed groove in which it fits on a diametrical rule: See Charette and Schmidl, “& Hdot abash's universal plate” (ref. 7), 152–3.

See the article “Madjarra” [= Milky Way] by Kunitzsch P. in Encyclopaedia of Islam (ref. 1).

Sédillot-fils, “Mémoire” (ref. 19), 104–5: mujerrih; also Schmalzl, Geschichte des Quadranten (ref. 1), 124–6.

Using the new definitions in terms of shadow increases, the shadows would be given, and one would need to calculate the altitudes.

See further King, “al-Khwārizmī” (ref. 15), 7–9; idem, SATMI, II: A survey of Islamic tables for regulating the times of prayer, Section 3.1; and IV: On the times of prayer in Islam, Section 4.5. The apparent divergence between these definitions — nowhere more apparent than on medieval sundials or astrolabe plates showing both the seasonal hours and the times of the żuhr and a⋅r prayers — is the result of the approximate nature of the Indian formula that was used to relate the time of day to shadow increases.

On shadow scales see ref. 5 above.

See Kennedy E. S. , The exhaustive treatise on shadows by Abū al-Ray&hdotān Mu&hdotammad b. A&hdotmad al-Bīrunī (2 vols, Aleppo, 1976), i, 68–79, and ii, 25–31; idem et al. Studies in the Islamic exact sciences (Beirut, 1983), 23–25; and King, SATMI, III: A survey of Islamic arithmetical shadow-schemes for simple time-reckoning, Section 1.2.

King, “al-Khwārizmī” (ref. 15), 2. See also Kennedy E. S. and Kennedy M. H. , Geographical coordinates of localities from Islamic sources (Frankfurt am Main, 1987), 55.

Illustrated in King, Mecca-centred world-maps (ref. 7), 18–19 and 356.

See further Kennedy and Kennedy , Islamic geographical coordinates (ref. 44), 55–56; King, “Too many cooks … — A newly-rediscovered account of the first Muslim geodetic measurements”, Suhayl—Journal for the history of the exact and natural sciences in Islamic civilisation, i (2000), 207–41, pp. 226–7; and idem, “Geography of astrolabes” (ref. 6), 13–14.