The Samarkand school in astronomy and mathematics (ca. 1420 CE)

Introduction. This website is about the works of the mathematicians and astronomers who worked together in Samarkand (Uzbekistan) around 1420 CE. At that time the area was ruled by Ulugh Begh, also a mathematician and astronomer himself (no. 3 below). Under his guidance, Samarkand became an extraordinary scientific center which deserves to be better known.
The purpose of this website is to provide easy access to some the works of these mathematicians and astronomers. We provide links to digital versions of manuscripts in Persian and Arabic, to editions in Arabic and Persian, and to translations into Western languages, Russian, Uzbek and Turkish. Note that most of the works of these authors and also most of the extant Arabic and Persian manuscripts are not available in digital versions. Information about such manuscripts can be found in bibliographies such as RI and Suter and in catalogues of the libraries all over the world in which these manuscripts are preserved.
Abbreviated references (such as RI, Suter, BEA) are explained at the end of this page. We have added references to commentators of the most famous Samarkand mathematicans and astronomers, and a few modern studies. The website is not exhaustive in this regard, it is intended to provide access to the source materials available in digital form and to translations of these sources.
The website is under construction and will remain so for some time to come. Additions and corrections are welcome and can be mailed to J.P.Hogendijk_at_gmail.com
We have added scans of sources which are not in copyright to the best of our knowledge (if one of these sources is in copyright we ask the publisher to let us know so we can remove the file immediately).
Arabic or Persian titles of works have been transcribed in self-explanatory ways, for example: A is long a, I is long i, s is sin, .s is sad.

General information on the school of Samarkand:
Ihsan Fazlioglu, The Samarqand Mathematical and Astronomical School A Basis for Ottoman Philosophy and Science, Journal for the History of Arabic Science 14 (2008), pp. 3-68, download (islamsci.mcgill.ca)
see the webpages by Dr Rob van Gent (Utrecht) which can be accessed via his main webpage on Ulugh Beg.

List of mathematicians and astronomers: 1. (al-)Kashi (or Kashani)   2. Qadi-Zadeh al-Rumi   3. Ulugh Beg     4. Al-Qushji     5. Miram Celebi     6. (al-)Birjandi   7. Anonymous  

1. KASHI or AL-KASHI (died ca. 1429). Full name: Ghiyath al-Din Jamshid ibn Mas`ud (al-)Kashi (or kashani), born in Kashan in Iran. See RI no. 802, pp. 269-271, Suter p. 173 no 429. On his biography: see the following:
article by Petra Schmidl in BEA, pp. 613-615;
Article Al-Kashi by B.A. Rosenfeld and A.P. Juschkewitch, originally from C.G.Gillispie, ed., Dictionary of Scientific Biography,, vol. 7, New York 1973, pp. 255-262.
A.Q. Qorbani, Kashani-nameh, second edition, Tehran 1368 A.H. Solar [in Persian]
Mohammad K. Azarian, A summary of the mathematical works of Ghiyath ud-Din Jamshid Kashani, Journal of Recreational Mathematics 29 (1996), 32-42Scan
Works:

Works by al-Kashi:

1. Treatise on the Circumference of the Circle. Arabic title: al-risAla al-mu.hI.tiyya. RI no. 802, p. 270 no. M3. In Arabic.
Arabic edition with German translation in Paul Luckey, Der Lehrbrief über den Kreisumfang (ar-risala al-Muhitiya) von Gamsid b. Mas`ud al-Kasi übersetzt und erläutert. Herausgegeben von Albred Siggel. Berlin 1953 [Arabic text and German translation], reprinted in IMA56, pp. 227-329, scan.
Russian translation {\em Dzhemshid Giyaseddin al-Kashi,} {\em Klyuch Arifmetiki, Traktat ob Okruzhnosti}, Per. B.A.~Rosenfeld, comm. A.P.~Yuschkevitch, B.A.~Rosenfeld, Moskva: Gosudarstvennoe Izdatelstvo, 1956. [Russian translation]
Arabic manuscripts: At least eight manuscripts are preserved to date. The following are available:
Istanbul, AM 756. Facsimile edition of this manuscript in pp. 388-424 of Dzhemshid Giyaseddin al-Kashi Klyuch Arifmetiki, Traktat ob Okruzhnosti, Per. B.A.~Rosenfeld, comm. A.P.~Yuschkevitch, B.A.~Rosenfeld, Moskva: Gosudarstvennoe Izdatelstvo, 1956.
Mashhad, Astan-e Qods (Holy Shrine Library) 5389, facsimile in Jan P. Hogendijk, Al-Kashi's determination of pi to 16 decimals in an old manuscript, Zeitschrift für Geschichte der arabisch-islamischen Wissenschaften 19 (2009), 73-152.   Download. At that time I believed that the manuscript is not an autograph by al-Kashi: arguments for the authenticity were presented in (ADDREF)
Tehran 642/4
Studies: Mohammad K. Azarian, Al-Risala al-muhitiyya: a summary. Missiouri Journal of Mathematical Sciences 22 (2010), pp. 64-85, Scan; Mohammad K. Azarian, The Introduction of Al-Risala Al-Muhitiyya: An English Translation, International Journal of Pure and Applied Mathematics 57 (2009), 903-914. [based on Qorbani's Persian translation] Scan; Mohammad K. Azarian, Al-Kashi's Fundamental Theorem, International Journal of Pure and Applied Mathematics, 14 (2004), 499-509. Scan

2. Key to Arithmetic. Arabic title: MiftA.h al-.hisAb. RI no, 802, p. 269-270 no. M1. In Arabic.
Arabic editions:
Al-Kashi, Miftah al-Hisab, ed. Muhammad Hamdi al-Hifni Shaykh, A.S. Dimirdash and Abd al-Hamid Lutfi, Cairo, Dar al-Katib al-Arabi li-l-Tiba `a wa'al-Nashr 1967 (not seen).
Nader Nabulsi, Miftah al-Hisab d'Al-Kashi, Damascus, University of Damascus, 1977.
Translations:
English translation in: Nuh Aydin, Lakhdar Hammoudi, Al-Kashi's Miftah al-Hisab, Volume 1: Arithmetic, Cham (Switzerland) etc., Springer/Birkha&uum;user, 2019, https://doi.org/10.1007/978-3-030-14950-5 and Nuh Aydin, Lakhdar Hammoudi, Ghada Bakbouk, Al-Kashi's Miftah al-Hisab, Volume 2: Geometry, Cham (Switzerland) etc., Springer/Birkha&uum;user, 2020, https://doi.org/10.1007/978-3-030-61330-3 The translation is based on the manuscript Istanbul, Süleymaniye Library, Nurosmaniye 2967.
Russian translation in Dzhemshid Giyaseddin al-Kashi Klyuch Arifmetiki, Traktat ob Okruzhnosti, Per. B.A.~Rosenfeld, comm. A.P.~Yuschkevitch, B.A.~Rosenfeld, Moskva: Gosudarstvennoe Izdatelstvo, 1956.
Arabic manuscripts: More than 20 manuscripts have been preserved. A few are available in a digital version:
London, British Library, Oriental Manuscripts, Add. Ms. 7470, written in 997 H. Digital version (Qatar Digital Library)
London, India Office, 1039 (= Loth no 756), ff. 15b-122a, Digital version (Qatar Digital Library)
Paris, 5020, Digital version (gallica.fr)
Leiden Or. 185. Facsimile edition of this manuscript in pp. 428-566 of Dzhemshid Giyaseddin al-Kashi Klyuch Arifmetiki, Traktat ob Okruzhnosti, Per. B.A.~Rosenfeld, comm. A.P.~Yuschkevitch, B.A.~Rosenfeld, Moskva: Gosudarstvennoe Izdatelstvo, 1956. The manuscript has also been digitized by Brill Publishers, series Middle Eastern Manuscripts Online 1.
Other manuscripts, not yet digitized: Berlin Ahlwardt 599 = Spr. 1824, another manuscript in Berlin is Spr. 1824bis (see Ahlwardt catalogue vol. 5 p. 344)
Berlin, Institute for History of Science (before World War 2) 1, 2 (what manuscripts were these? could they have been photocopies of other manuscripts)
Princeton Yehuda 1189
Istanbul, Süleymaniye Library, Nurosmaniye 2967: this is not digitally available but it is the basis of the English translation by Aydin-Hammoudi-Bakbouk, who reproduce photos of a few pages.
Other manuscripts are mentioned in RI no, 802, p. 269-270 no. M1, and in Aydin-Hammoudi-Bakbouk vol. 1 pp. 10-11.
An abbreviation of the Key to Arithmetic is mentioned in RI p. 270 no. M2; a manuscript in London, British Library, India Office, Islamic 1210 (Loth 757) is available in a digital edition (Qatar Digital Library)
Studies:
Paul Luckey, Die Ausziehung der n-ten Wurzel und der binomische Lehrsatz in der islamischen Mathematik, Mathematische Annalen 120 (1948), 217-274, reprinted in IMA56, pp. 11-68. Download in pdf (GDZ).
Paul Luckey, Zur islamischen Rechenkunst und Algebra des Mittelalters, Forschungen und Fortschritte 24 (1948), 199-204, reprinted in IMA56, pp. 69-74.   Scan.
Paul Luckey, Die Rechenkunst bei Gamsid b.; Mas`ud al-Kasi mit Rückblicken auf die ältere Kunst des Rechnens, Wiesbaden 1952, reprinted in IMA56, pp. 75-226, scan.
Paul Luckey, Beiträge zur Erforschung der islamischen Mathematik II, Orientalia N.S. 22 (1953), pp. 166-189, reprinted in IMA56, pp. 330-353 (on Kashani see the second half of this article). Scan
Baldassare Boncompagni, Intorno alla somma delle quarze potenze dei numeri naturali. Bullettino di bibliographia e di storia delle scienze matematische e fisische (Rome) 10 (1877), pp. 294-302, reprinted in IMA56, pp. 1-10.   scan.
Yvonne Dold-Samplonius,, The XVth Century Timurid Mathematician Ghiyath al-Din al-Kashi and his Computation of the Qubba. Amphora, Festschrift for Hans Wussing on the Occasion of his 65th Birthday, S. S. Demidov, M. Folkerts, D. E. Rowe, Ch. J. Scriba, Eds., Basel, Birkhäuser, 1992, pp. 171-181.
Yvonne Dold-Samplonius, Practical Arabic mathematics: Measuring the Muqarnas by al-Kashi, Centaurus 35 (1992), pp. 193-242. Scan
The Volume of Domes In Arabic Mathematics. in: Vestigia Mathematica, Festschrift for H. L. L. Busard. M. Folkerts and J.P. Hogendijk, Eds., Amsterdam, Rodopi B.V., 1993, pp. 93-107.
Yvonne Dold-Samplonius, How al-Kashi Measures the Muqarnas: A Second Look, in M. Folkerts (Ed.), Mathematische Probleme im Mittelalter: Der lateinische und arabische Sprachbereich, Wolfenbütteler Mittelalter-Studien Vol. 10, 1996, pp. 56 - 90, Wiesbaden.
Yvonne Dold-Samplonius, Video: Qubba for Al-Kashi, 18 minutes, Providence: Americal Mathematical Society, 1997.
Mohammad K. Azarian, Meftah al-Hesab: a summary. Missiouri Journal of Mathematical Sciences 12 (2000), pp. 75-95. Scan
Jan P. Hogendijk, Structuur en schoonheid in de Sleutel tot de Rekenkunde van al-Kashi (ca. 1425), in Vakantiecursus 2004: Structuur in schoonheid, Amsterdam: Centrum voor Wiskunde en Informatica, 2004, CWI Syllabus 53, pp. 111-123, 77. Ook verschenen als "De sleutel tot de Rekenkunde van al-Kashi" in Nieuw Archief voor Wiskunde, vijfde serie, deel 6, nummer 1 (maart 2005), pp. 32-39.   Download
Yvonne Dold-Samplonius, Al-Kashi's Method to Calculate Arches, in: Mélanges offerts à Hossam Elkhadem par ses amis et élèves, Eds. Frank Daelemans, Jean-Marie Duvosquel, Robert Halleux, David Juste Brussels, Archives et Bibliothèques de Belgique, Numéro Spécial 83, 2007, pp.
(Jan P. Hogendijk) Unpublished Dutch translation of Chapter 9, "On the measurement of buildings" of Treatise 4, "On Measurement" of Kashi's Key to Arithmetics (this is about measurement of arcs and vaults).

3. Two letters to his father on scientific life in Samarkand. RI p. 269, no HS1, HS2. The letters were written in Persian, al-Kashi's native language.
First letter, written around 1423, two years after al-Kashi's arrival in Samarkand. This letter was discovered by Dr Mohammad Bagheri in 1994, in a manuscript in Tehran.
Persian edition with commentary in Mohammad Bagheri, Az Samarqand be Kashan: NamehA-ye Ghiyath al-Din Jamshid Kashani be pedaresh, Tehran: Scientific and Cultural Publications, 1375 A.H.(Shamsi), 1996, see pp. 35-50.
English translation: Mohammad Bagheri, A newly found letter by al-Kashi on scientific life in Samarkand, Historia Mathematica 24 (1997), pp. 241-256, download via the page of articles in English by Mohammad Bagheri.
Persian manuscript: Tehran, Majlis Library, 5138/142
Second letter. This letter was known to modern researchers before the first letter.
Persian edition first published in lithograph volume A.H. 1318 (= A.H. Shamsi 1279, corresponding to 1900-1901 A.D.), on this volume and the reprints (not seen by me) see Bagheri, Az Samarqand beh Kashan p. 28.
Persian edition by M. Tabataba'i in Amuzesh wa-Parvaresh 10 (1319 A.H Shamsi, ca. 1940 A.D.) pp. 9-16, 59-61 (not seen)
Persian edition, with English and Turkish translations, in Aydin Sayili, Ulug Bey ve Semerkanddeki ilim faaliyeti hakkinda Giyasüddin-i Kasi'nin mektubu (Ghiyath al-Din al-Kashi's letter on Ulugh Bey and the scientific activity in Samarqand), Ankara 1960, 115 pp, reprinted in IMA56, pp. 361-473.
Critical Persian edition with commentary in Mohammad Bagheri, Az Samarqand be Kashan: NamehA-ye Ghiyath al-Din Jamshid Kashani be pedaresh, Tehran: Scientific and Cultural Publications, 1375 A.H.(Shamsi), 1996, see pp. 61-74.
English translations: E.S. Kennedy, A Letter of Jamshid al-Kashi to his father: Scientific Research at a Fifteenth Century Court, Orientalia 29, 1960, pp. 191-213, reprinted in SIES pp. 722-744 (translated from Tabataba'i's edition)
Arabic translation by A.S. Dimirdash in Journal for the History of Science 1963 (not seen), reprinted in Al-Kashi, Miftah al-Hisab, ed. Muhammad Hamdi al-Hifni Shaykh, A.S. Dimirdash and Abd al-Hamid Lutfi, Cairo, Dar al-Katib al-Arabi li-l-Tiba `a wa'al-Nashr 1967 (not seen).
Russian translation by G. Sabirov and N. Babaev, Dushanbe 1973 (not seen).
Russian translation by Dilaram Yusupova, Iz Istorii Nauki epokhi Ulugbeka Tashkent 1979. (not seen)
Uzbek translation by Dilaram Yusupova (no further information known to me)
Persian manuscripts: Tehran, Sepahsalar 2016, f. 510a-517b. Sayili says that the manuscript also contains an extract from another letter on f. 517b-518b but does not give further information. On this and the other manuscripts of the second letter see Bagheri, Az Samarqand beh Kashan pp. 33-36.

4. Treatise on the sine of one degree. Arabic title: risAla fI al-watar wa'l-jayb, "Letter on the chord and the sine." RI p. 270 no. M4. In Arabic. We know that al-Kashi wrote this treatise because he refers to it in his Key to Arithmetic.
No Arabic manuscripts of al-Kashi's treatise on the sine of one degree are known to exist. The contents survive to some extent in later commentaries and summaries by Qadi Zadeh Rumi, Miram Celebi, an anonymous author, and Ali Qushchi, see below.
According to RI, the Arabic text of the treatise is found in a lithograph edition printed in Tehran in 1881-2 CE. This lithograph edition does not contain the original text by al-Kashi but a commentary by Qadi Zadeh Rumi. See the facsimile of the lithograph edition in
B.A. Rosenfeld, Jan P. Hogendijk, A mathematical treatise written in the Samarqand observatory of Ulugh Beg, Zeitschrift für Geschichte der arabisch-islamischen Wissenschaften 15 (2003), 25-65.   Download.
Studies:
The Russian and French translations mentioned in RI are made not from the lost text by al-Kashi but from the commentary by Miram Celebi to the Zij of Ulug Beg.
The article: Asger Aaboe, Al-Kashi's iteration method for the determination of sin 1. Scripta Mathematica 20 (1954), 24-29, reprinted in IMA56, pp. 354-360   (scan) is also based on Miram Celebi.

5. Khaqani zij. This astronomical handbook in Persian with tables is still unpublished. See the summary in p. 164-166 of E.S. Kennedy, A survey of Islamic astronomical tables, second edition, Philadelphia, American Philosophical Society, no date [first edition 1956], available through Jstor https://www.jstor.org/stable/1005726.
More than ten Arabic manuscripts are listed in RI no. 802, A1, p. 270-271; more than 20 manuscripts in ISMIISMI
The Arabic manuscript London, India Office, Ethe 2232 (Loth 430) on which Kennedy's summary is based, is not yet digitally available (although one expects it to appear some time in the Qatar Digital Library)
Studies
E.S. Kennedy, Horoscope determination in Kashi's Zij-i Khaqani, Second International Symposium for History of Arabic Science, Proceedings, Aleppo 1979, p. 61.

  • E.S. Kennedy, Spherical astronomy in Kashi's Khaqani Zij, Zeitschrift für Geschichte der arabisch-islamischen Wissenschaften 2 (1985), pp. 1-46.
    E.S. Kennedy, Treatise V of Kashi's Khaqani Zij: Determination of the Ascendent, Zeitschrift für Geschichte der arabisch-islamischen Wissenschaften 10 (1995-6), pp. 123-145.
    E.S. Kennedy, On the Contents and Significance of the Khaqani Zij by Jamshid Ghiyath al-Din al-Kashi. Frankfurt: Institute for the History of Arabic-Islamic Science, 1998. 66 pp. Series Islamic Mathematics and Astronomy vol. 84.  
    Download.
    Javad Hamadanizadeh, The trigonometric tables of al-Kashi in his Zij-i Khaqani Historia Mathematica 7 (1980), 35-45, with erratum in Historia Mathematica 7 (1980), 468. Compare also: Javad Hamadanizadeh, A second-order interpolation scheme described in the Zij-i Ilkhanī. Historia Mathematica12 (1985), 56–59.
    Glen van Brummelen, Taking latitude with Ptolemy: Jamshid al-Kashi's novel geometric model of the motions of the inferior planets, Archive for History of Exact Sciences 60 (2006), 353-377.

    6. Nuzhat al-Hada'iq (Delight of Gardens) or Zubdat al-Hada'iq (Essence of Gardens).
    This treatise exists in an original Arabic version by Kashani, which he wrote in Kashan in 1416, and to which he made additions in Samarkand in 1426, and in a revised Persian version made by an anonymous astronomer in Istanbul around 1490.
    Kashani's original and his additions seem to be extant in various manuscripts and they are printed at the end of a lithograph edition of Kashani's Miftah al-Hisab, Tehran 1888.
    Here are scans from the lithograph edition:
    scan of Kashani's Nuzhat al-Hada'iq from 1416 (lithograph edition pp. 250-288)
    scan of the additions (Ilhaqat) from 1426 (lithograph edition)

    The revised Persian version of Kashi's Nuzhat al-Hada'iq was published and translated in E.S. Kennedy's book The Planetary Equatorium of Jamshid Ghiyath al-Din al-Kashi, Princeton University Press 1960. This book is in the public domain now. Scans: Introduction by E.S. Kennedy   Persian manuscript   English translation by E.S. Kennedy   Commentary by E.S. Kennedy (including indices and bibliography).

    The following articles by E.S. Kennedy are mostly based on the revised Persian version of Kashi's Nuzhat al-Hada'iq:
    E.S. Kennedy, al-Kashi's Plate of Conjunctions, Isis 38 (1947), pp. 56-59, reprinted in SIES pp. 448-451. Download at http://www.jstor.org/stable/225450
    E.S. Kennedy, A Fifteenth-Century Planetary Computer: al-Kashi's Tabaq al-Manateq, I. Motion of the Sun and Moon in Longitude. Isis 41, 1950, pp. 180-183, reprinted in SIES pp. 452-455. Download at http://www.jstor.org/stable/227189
    E.S. Kennedy, A Fifteenth-Century Lunar Eclipse Computer. Scripta Mathematica 17, 1951, reprinted in SIES pp. 456-462.
    E.S. Kennedy, An Islamic Computer for Planetary Latitudes, Journal of the American Oriental Society 71, 1951, pp. 13-21, reprinted in SIES pp. 463-471. Download at http://www.jstor.org/stable/595221
    E.S. Kennedy, A Fifteenth-Century Planetary Computer: al-Kashi's Tabaq al-Manateq, II. Longitudes, Distances and Equations of the Planets. Isis 43, 1952, pp. 42-50, reprinted in SIES pp. 472-480. Download at http://www.jstor.org/stable/227133.
    E.S. Kennedy, The Planetary Equatorium of Jamshid Ghiyath al-Din al-Kashi, Princeton University Press 1960.

    7. Stairway towards Heaven (Ladder towards Heaven) In Arabic, title: Sullam al-samA', on sizes and distances of the planets, RI no. 802 p. 271 A4.
    Arabic manuscript London, British Library, Delhi Arabic 1925, ff. 1b-14a, is available in a digital version (Qatar Digital Library)

    8. Treatise on depression of the visible horizon
    See Mohammad Bagheri, A new treatise by al-Kashi on depression of the visible horizon, in: From China to Paris: 2000 years transmission of the mathematical ideas, ed. Y. Dold et al., Franz Steiner Verlag , Stuttgart, 2002, pp. 357-368, download via the page of articles in English by Mohammad Bagheri.

    9. Another treatise on astronomical instruments
    See E.S. Kennedy, Al-Kashi's Treatise on Astronomical Instruments, Journal of Near Eastern Studies 20 (1961), pp. 98-108, reprinted in SIES pp. 394- 404. [With facsimile of the Arabic ms. Leiden, Or. 945, 12r-13r]. Download at http://www.jstor.org/stable/543721.

    2. QADI-ZADEH al-RUMI (died after 1440). Full name: Salah al-Din Musa ibn Muhammad ibn Mahmud Qadi-Zadeh al-Rumi. See RI no.808, pp. 272-274, Suter p. 174 no. 430.
    On his biography, see the article by Jamil Ragep in BEA, p. 972.
    Works:

    • Commentary by Qadi Zadeh on the Summary on cosmology (mulakhkha.s fI al-hay'ah) by Mahmud al-Chaghmini (died 1221), in Persian. RI no. 808 p. 273-274 no. A1. The commmentary exists in numerous manuscripts. The following are available in digital versions:
      Manuscript Washington DC, USA, Library of Congress, no further details known to me, Download
      Three manuscripts in Philadelphia USA, University of Pennsylvania, Laurens J. Schoenberg Collection: no. 408, digital version.   no. 410, digital version.   no. 474, digital version.
      Pages of a manuscript (Ms. 232) in Winchester college, Winchester, UK.
      Apparently the text was printed in al-Rumi,Sharh al-Mulakhkhas al-Chaghmini, Kalkata 1290 H. (1873) (not seen)
      Note: A manuscript of the original Summary on cosmology by Mahmud al-Jaghmini (See Ri p. 198 no. A1) is also in the Laurens J. Schoenberg Collection, no. 388, digital version. For a German translation of the work see Rudloff and Hochheim, Die Astronomie des Mahmud ibn Omar al-Gagmini, Zeitschrift der deutschen morgenländischen Gesellschaft 47, 1893, pp. 213-275, download (University library, Halle, Germany)
    • Treatise in Arabic on the sine of 1 degree. Full title: " A treatise on the determination of the sine of one degree with true precision, determined by the most perfect of the geometers, Jamshid al-Kashani, edited and revised in this letter by Qadi-Zadeh al-Rumi, the author of the commentary on al-Chaghmini" (risAla fI istikhrAj jayb daraja wA.hida `alA ta.hqIq al-h.aqIq istakhrajahu fa.dl al-muhandisIn ghiyAth al-dIn jamshId al-qAsAnI .harrarahu wa-naqqa.hahu fI hadhihi al-risAla qA.dI zAdah al-rUmi mu`allif shar.h al-chaghmInI).
      Critical Arabic edition with [modern] Persian translation in Fateme Savadi, Risala fI istikhraj jayb daraja wahida, Musa ibn Muhammad Qazizadah Rumi (ca. 1365-1437 AD) Tehran, Miras-e Maktub 2009, ISBN: 978-964-9700-67-1, available via https://mcgill.academia.edu/FatemeSavadi
      Facsimile edition of the Arabic 1881-2 lithograph edition with English translation and commentary in: B.A. Rosenfeld, Jan P. Hogendijk, A mathematical treatise written in the Samarqand observatory of Ulugh Beg, Zeitschrift für Geschichte der arabisch-islamischen Wissenschaften 15 (2003), 25-65.   Download.
      This treatise has come down to us in Arabic manuscripts in Berlin (in the Humboldt Universität, a handwritten copy of the Cairo manuscript made in the 1920s for Carl Schoy), Cairo (Dar al-Kutub, Mustafa Fadil Riyada 37), Istanbul (Kandilli 76, Hüseyin Celebi 751/3), umerous manuscripts in Iran,, and a lithograph edition printed in 1881-2 in Tehran, Iran.
      B.A. Rosenfeld and others have misattributed the Arabic text in these manuscripts to Ulugh Beg. The confusion was cleared up by means of new evidence in the following article by Fateme Savadi: naqdI bar estedlal Rozenfeld dar bab-e entesab yek resala-ye riyazi beh Ulugbeg (d. 853 Q) [A criticism of Rozenfeld's argumentation on the attribution of a mathematical treatise to Ulugh Beg], Ta'rikh-e Elm 4 (1384 A.H. solar = 2005-2006), pp. 85-103 [in Persian], download at https://mcgill.academia.edu/FatemeSavadi
      Studies: Selim Zaki Athar-e Baqiyeh, Istanbul 1911, vol. 1, pp. 133-139, see scan [Turkish in Arabic script]; B.A. Rosenfeld and A.P. Yuschkevich, O traktate Kazi-Zade ar-Rumi ob opredelenii sinusa odnogo gradusa, Istoriko-Matematicheskie Issledovaniya13 (1960), 533-538 [not seen]; A. Ahmedov and B.A. Rosenfeld, Kto byl avtorom "Traktata ob opredelenii sinusa odnigi gradusa?", ONU 10 (1975), 51-53 [not seen, wrongly attributes the text to Ulugh Beg]; A. Ahmedov and B.A. Rosenfeld, The mathematical treatise of Ulugh Beg, Science in Islamic Civilization,Istanbul, 2000, pp. 143-150 [not seen, again the wrong attribution to Ulugh Beg].
      Note: RI p. 273 no. M3 mentions a treatise in Persian on the determination of the sine of 1 degree in the Berlin manuscript Pertsch 339, and wrongly attributes it to Qadi-Zadeh. This treatise was not written by Qadi-Zadeh but by his grandson Miram Celebi (see below no. 5); it is a commentary to the astronomical tables of Ulugh Beg, which includes much more than the determination of the sine of one degree.

    3. ULUGH BEG (1394-1449). Full name: Mirza Muhammad ibn Shahrukh ibn Timur Ulugh Beg Guragan, see Ri no. 816, pp. 277-279. On his biography: see the article by Benno van Dalen in BEA, pp. 1157-1159.
    Works:

    • 1. Astronomical handbook of Ulugh Beg: Persian title: Zij-i Ulugh BEg, zIj-i sul.tAn-i or Zij-i gUrkAnI, that is Zij of Ulugh Beg, Sultani Zij or Gurkani Zij. RI no. 816 p. 278 no. A1. Consists of an introduction and tables.
      Persian edition and French translation of the introduction (without the tables): L. P. E. A. Sédillot, Prolégomènes des Tables Astronomiques d'Oloug-Beg, Paris 1847. 1853. Download the French translation (Paris 1853) (archive.org). The Persian edition is not yet available in a digital version.
      The two volumes were reprinted as vol. 52 and 53 in the series Islamic Mathematics and Astronomy, ed. F. Sezgin, Frankfurt: Institute for the History of Arabic-Islamic Science, 1998.
      Facsimile edition of a Persian manuscript (See below) and Turkish translations of the introduction plus transcriptions of the tables, and Turkish commentary, in Mustafa Kacar, Attila Bir, eds., Ulug Bey'in Astronomi CetvelleriL Zic-i Ulug Bey, Ankara 2012, T.C. Kültür ve Turizm Bakanligi, ISBN 978-975-17-3593-5
      According to RI there is a Russian and Uzbek translation by A. Akhmedov: no further information available to me. Perhaps the Uzbek translation is identical to:
      Ilmi nujum: ziji jadidi Kuragoniy tórtinchi kitobi, ed. Ashraf Ahmad. Toskent: Abdolla Qodiriy Nomidagi Xalq Merosi Nahriyoti, 1994 (not seen)
      Manuscripts of the Zij of Ulugh Beg:
      Persian original version:
      Paris, Bibliothèque Nationale (B.N.), Ancien Fonds Persan 172, download (gallica.fr) (see Catalogue des manuscrits Persans, vol. 1, Ancien Fonds, par Francis Richard, 1989, p. 189). This is Sédillot's manuscript B.
      Paris, B.N., Ancien Fonds Persan 164 (Sédillot's manuscript A), and Persan, Suppl. Pers, 366, 1341, 1881, have apparently not yet been digitised
      London, British Library, India Office Islamic 1148, 68b-135b (apparently incomplete)Digital version (Qatar Digital Library)
      Istanbul, Topkapi Seray, Revan 1714: excellent color facsimile edition in vol. 1 of Mustafa Kacar, Attila Bir, Ulug Bey'in Astronomi CetvelleriL Zic-i Ulug Bey (see above)
      Leiden, Or. 165: this (incomplete Persian) manuscript has been digitized by Brill, accessible via the beginning screen of Middle Eastern Manuscripts Online: Pioneer Orientalists
      Oxford Bodleian Library, Greaves 5, only one leaf available in a digital version.
      Yale University Library, Arabic Mss. Supp. 952, only one leaf available in a digital version.
      First page of a splendid manuscript of the Zij Ulugh Beg (location unknown)
      Berlin, Staatsbibliothek, Ms. Orient Fol. 280, Pertsch no. 337 p. 358. Digital version not yet available.
      Berlin, Staatsbibliothek, Ms. Orient Fol. 228, Pertsch no. 338 p. 360. Digital version not yet availale.
      Arabic translations:
      Arabic translation by al-Rifa`i on behalf of Ibn Abi'l-Fath al-Sufi:
      Manuscript Cairo, Egyptian National Library, no. 5205? Download (world digital library). The second part of the manuscript seems to be part of a more recent Arabic translation.
      Paris, B.M., Fonds Arabe 2534, digital version (gallica.fr). This is Sédillot's manuscript E.
      Another (?) Arabic translation Paris, B.N., Fonds Arabe 2535, Download (gallica.fr). This is Sédillot's manuscript F.

      Commentaries to or adaptations of the Zij of Ulugh Beg:
      For the commentaries by al-Qushchi, Miram Celebi an (al-)Birjandi, see below, mathematicians/astronomers no. 4, 5, 6.
      Ms. Paris, B.N., Persan 172, Persian version of the Zij of Ulugh Beg with tables adapted to a longitude of 41 degrees 15 minutes East of the Fortunate Islands (i.e., the Canary Islands), pages in disorder (introduction at the end). Download (gallica.fr)
      Studies on the Zij of Ulugh Beg: see the special webpage by Dr Rob van Gent, Utrecht.

    • 2. Treatise on the determination of the sine of 1 degree RI p. 278 no. M1.
      We know that Ulugh Beg wrote this treatise because he refers to it in the chapter on sines in his astronomical tables (Zij). No manuscripts of the treatise by Ulugh Beg on the sine of 1 degree have been found.
      B.A Rosenfeld and other scholars believed that the treatise by Ulugh Beg is extant in several Arabic manuscripts. Fateme Savadi has shown, by means of new evidence, that all these manuscripts contain the treatise by Qadi Zadeh Rumi (mathematician no. 2 above) on the sine of 1 degree, not the treatise by Ulugh Beg. See her article: naqdI bar estedlal Rozenfeld dar bab-e entesab yek resala-ye riyazi beh Ulugbeg (d. 853 Q) [A criticism of Rozenfeld's argumentation on the attribution of a mathematical treatise to Ulugh Beg], Ta'rikh-e Elm 4 (1384 A.H. solar), pp. 85-103 [in Persian], download at https://mcgill.academia.edu/FatemeSavadi

    4. ALI AL-QUSHJI (ca. 1402-1474). Full name: `Ala al-Din `Ali ibn Muhammad al-Qushji, Turkish form: Ali Kusçu. See RI no. 845, pp. 285-287, Suter p. 178 no. 438.
    On his biography, see the article by Ihsan Fazlioglu in BEA, pp. 946-948.
    Works:

    • Commentary on the Zij of Ulug Beg, Sharh-i Zij-i Ulugh Beg, in Persian. RI no. 845 p. 287 A3.
      Manuscript in Philadelphia USA, University of Pennsylvania, Laurens J. Schoenberg Collection, no. 400, digital version.
    • A commentary by Hibatallah al-Hasani al-Husayni Shah Mir to a work by Ali al-Qushji, in in Berlin, Staatsbibliothek, ms. Sprenger 1846, see Pertsch 332 p. 352.digital version
    • According to Pertsch, the Staatsbibliothek in Berlin contains an anonymous "Abhandlung ueber Astronomie und mathematische Geographie" [Treatise on astronomy and mathematical geography] in ms. Sprenger 1845. Pertsch attributes the treatise to al-Qushji (see Pertsch no. 331 p. 351). According to the digital version (ISMI), this manuscript contains a work not by al-Qushji but by Nasir al-Din al-Tusi.

    5. MIRAM CELEBI (1475-1525). Full name: Mahmud ibn Qutb al-Din Muhammad ibn Muhammad ibn Musa Qadi-Zadeh. He was the grandson of (Muhammad ibn Musa) Qadi-Zadeh al-Rumi (no. 2 above) and also via his mother the grandson of Ali Quschi (no. 4 above). RI no. 940 pp. 316-317. On his biography, see the article by Ihsan Fazlioglu in BEA, pp. 788-789.
    Works:

    • Rules for the operation[s] (i.e., computations), and correction of the table, in Persian: DastUr al-`amal va-ta.s.hI.h al-jadwal. This is a commentary on the astronomical tables of Ulugh Beg.
      Persian manuscript: Paris, Biblothèque Nationale, Persan 171, digital version (gallica.fr). This manuscript was used by Sédillot, who called it D.
      Persian manuscript Berlin, Staatsbibliothek, Wetzstein II, 1140, see Pertsch no. 339 p. 360. Digital version not yet available.
      According to Savadi [p. 92 in her article Naqdi, cited under Qadi-Zadeh above], an autograph manuscript is in Istanbul, Hamidiyye library, no. 848-9.
      Persian edition of the section on zones (in lunar theory) on pp. 281-285 of L. P. E. A. Sédillot, Prolégomènes des Tables Astronomiques d'Oloug-Beg, Paris 1853. Download (archive.org).
      French translation of several parts, in the footnotes of pp. 65-83, 88-90, 106-107, 113, 131-133, 13-153, of: L. P. E. A. Sédillot, Prolégomènes des Tables Astronomiques d'Oloug-Beg, Paris 1853. Download (archive.org). The footnotes on pp. 77-83 concern al-Kashi's method for determining the sine of 1 degree.
      Russian translation of the part on sines and the sine of 1 degree: pp. 311-319 of: Dzhemshid Giyaseddin al-Kashi Klyuch Arifmetiki, Traktat ob Okruzhnosti, Per. B.A.~Rosenfeld, comm. A.P.~Yuschkevitch, B.A.~Rosenfeld, Moskva: Gosudarstvennoe Izdatelstvo, 1956.
      Study, based on Sédillot's translation: Asger Aaboe, Al-Kashi's iteration method for the determination of sin 1. Scripta Mathematica 20 (1954), 24-29, reprinted in F. Sezgin, ed., Islamic Mathematics and Astronomy, vol. 56 (Frankfurt 1998), pp. 354-360, scan.

    6. (Al-)BIRJANDI (died 1525). Full name: Abd al-Ali Nizam al-Din Muhammad ibn Husayn (al-)Birjandi, from Birjand in Iran. RI no. 938 p. 315 no. A8. On (al-)Birjandi, see the article by Takanori Kusuba in BEA, p. 127.
    Works:

    • Commentary to the astronomical tables of Ulugh Beg, in Persian: Sharh Zij-i Ulugh Beg. RI no. 938 p. 315 no. A8.
      Manuscript in Philadelphia USA, University of Pennsylvania, Laurens J. Schoenberg Collection, no. 411, digital version. A remark at the end indicates that the manuscript is an autograph, i.e. that it was written in Birjandi's own hand. However, Iranian colleagues have compared the handwriting in the Pennsylvania manuscript with manuscripts in the Marashi library (Qom) about which we are sure that they were written by Birjandi in his own hand. It turns out that the Pennsylvania manuscript cannot be an autograph, so the scribe must have copied the notes by Birjandi in the end.

      Remarks (.hAshiya) by (al-)Birjandi to the commentary by Qazi-Zadeh on the Mulakhkhas by al-Chaghmini, RI no. 938 p. 315 no. A11, manuscript in Qatar National Library, no. 14670, digital version (Qatar Digital Library)

    7. RELEVANT ANONYMOUS WORKS.
    Anonymous Persian treatise on the determination of the sine of one degree.
    Edition with commentary: Fateme Savadi, Resaleh-ye Farsi darbareh muhasebeh-ye jaib-e yek darajah [A Persian treatise on the determination of the sine of one degree], Ta'rikh-e Elm vol. 6, no. 1 (1387 A.H. Shamsi = 2008-9) pp. 69-104. Download the article here or here, or via https://mcgill.academia.edu/FatemeSavadi.
    According to Savadi, this text is based on texts by Ali Quschi and Qadi-Zadeh Rumi.
    Persian manuscript: Berlin, Staatsbibliothek, Landsberg 144 (Ahlwardt 5683), ff. 74b-82a, not mentioned in Pertsch since most of the codex is in Arabic. Digital version This is the only known manuscript of the treatise.
    Anonymous 14th-century treatise on arithmetics (extant in two manuscripts: Istanbul, Süleymaniye Library, AyaSofya 2723 and Alexandria, Egypt, Baladiyya library, 3063/151b-173a), summarized in Ihsan Fazlioglu, Pervaya Matematicheskaya Kniga v Zolotoordynskom Gosudarstve - Shedevr v vychisleternoi matematike (At-Tukhfe Fi `Ilm al-Khisab) (1) (2) (3) [The first mathematical book in the Golden Horde State: A Masterpiece of Computational Mathematics; in Russian] Zolotoordynskoe Obozrenie no. 4 (2014), pp. 57-68, no. 1 (2015), pp. 106-127, no. 2 (2015) pp. 88-102, download part (1)   download part (2)   download part (3).

    ABBREVIATIONS
    BEA = Thomas Hockey et al. (eds.). The Biographical Encyclopedia of Astronomers, Springer Reference. New York: Springer, 2007.
    ISMI = Islamic Scientific Manuscripts Initiative, website https://ismi.mpiwg-berlin.mpg.de
    Pertsch = Die Handschriften-Verzeichnisse der königlichen Bibliothek zu Berlin. Vierter Band: Verzeichnis der persischen Handschriften von W. Pertsch. Berlin 1888. Download (archive.org)
    RI = B. Rosenfeld, E. Ihsanoglu, Mathematicians, Astronomers and Other Scholars of Islamic Civilization and their works (7th-19th century), Istanbul 2003. Apparently not available in a digital version.
    Suter = Heinrich Suter, Die Mathematiker und Astronomen der Araber und ihre Werke, Leipzig 1900. Digital version (University library, Halle, Germany)

    To the website of Jan P. Hogendijk


    Jan Hogendijk <J.P.Hogendijk at uu.nl>

    feb 2022