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
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 (
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 article by Petra Schmidl in BEA, pp. 613-615
Works: see separate webpage.

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.

  • 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
    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
    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.

  • 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) ( 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 ( (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 ( This is Sédillot's manuscript E.
    Another (?) Arabic translation Paris, B.N., Fonds Arabe 2535, Download ( 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 (
    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

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.

  • 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. 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.

  • 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 ( 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 (
    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 ( 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.

  • 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)
  • Commentary on the Tadhkira (by Nasir al-Din al-Tusi), in Arabic. This commentary was translated into Sanskrit in the 17th century. Chapter 7 on the so-called Tusi couple was editedin Arabic and translated into English in Takanori Kusuba and David Pingree, Arabic astronomy in Sanskrit: AL-Birjandi on Tadhkira II, Chapter 11 and its Sanskrit translation, Leiden: Brill, 2002.

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
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).

BEA = Thomas Hockey et al. (eds.). The Biographical Encyclopedia of Astronomers, Springer Reference. New York: Springer, 2007.
ISMI = Islamic Scientific Manuscripts Initiative, website
Pertsch = Die Handschriften-Verzeichnisse der königlichen Bibliothek zu Berlin. Vierter Band: Verzeichnis der persischen Handschriften von W. Pertsch. Berlin 1888. Download (
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>

feb 2021