Ghiyath al-Din Kashani, or al-Kashi (ca. 1440)

This page is now part of the website www.samarkand.eu on the Samarkand school in mathematics and astronomy in the fifteenth century CE. This page contains information on the following works by al-Kashi: 1. Treatise on the circumference of the circle. 2. Key fo Arithmetic. 3. Two letters to his father on scientific life in Samarkand. 4. Treatise on the sine of 1 degree (lost). 5. Khaqani Zij (astronomical handbook with tables) 6. Delight of Gardens (astronomical). 7. Stairway to Heaven. 8. Depression of the visible horizon. 9. Another brief astronomical treatise. We have only listed treatises by al-Kashi on which we have information to offer. Al-Kashi authored other works which are listed in RI pp. 269-271.
Note on names: Iranian reseachers call him Kashi or Kashani (without the Arabic adjective al-), meaning from Kashan, a city in Iran. The full name was ghiyAth al-dIn jamshId ibn mas`Ud kAshi/al-kAshI/kAshAnI.
Abbreviations such as RI, BEA, IMA56, SIES are explained at the bottom of this page.

General information:
Biographical article on al-Kashi's astronomical work 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 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.
  Scans: Part 1: pp. v-vi and 1-32, introduction and German translation; Part 2: pp. 33- 71, German commentary.   Part 3: pp. 72- 95. Arabic text.
Translations: German translation see above.
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:
Part 1, pp. 75-109, introduction and Ch. 1-3;
Part 2, pp. 109-135, Ch. 4-5;
Part 3, pp. 136-172, Ch. 6-7;
Part 4, pp. 172-209, Ch. 8-11;
Part 5, pp. 209-225, Ch. 12: Arabic texts.
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 for these authors the website www.samarkand.eu.
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.
    Scan of the Introduction by E.S. Kennedy
    Scans of the Persian manuscript which Kennedy translated:
    Part 1   Part 2   Part 3   Part 4   Part 5
    Scans of Kennedy's English translations of this Persian manuscript:
    Part 1   Part 2   Part 3   Part 4   Part 5
    Kennedy's commentary:
    Part 1   Part 2   Part 3 (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

    (identified by Dr M. Bagheri). 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.

    Abbreviations:
    IMA56 = F. Sezgin, ed., Al-Kashi: Texts and Studies, Frankfurt: Institut für Geschichte der arabisch-islamischen Wissenschaften, 1998, series Islamic Mathematics and Astronomy, vol. 56.
    SIES = E.S. Kennedy, Colleagues and Former Students, Studies in the Islamic Exact Sciences, Beirut: American University of Beirut, 1983.
    BEA = Thomas Hockey et al. (eds.). The Biographical Encyclopedia of Astronomers, Springer Reference. New York: Springer, 2007.
    RI = B. Rosenfeld, E. Ihsanoglu, Mathematicians, Astronomers and Other Scholars of Islamic Civilization and their works (7th-19th century), Istanbul 2003.

    Note: This page was intitially prepared for a conference on Kashani in Kashan in the spring of 2012. It was incorporated into the website www.samarkand.eu in 2021.