The works of Abu Mahmud Hamid ibn al-Khidr al-Khojandi (ca. 985-995 AD)

Welcome to the webpage www.jphogendijk.nl/khujandi.html
On this page you will find links to information about the extant and lost works and the astrolabe of the medieval Islamic mathematician and astronomer Abu Mahmud Hamid ibn al-Khidr al-Khujandi (ca. 985-995 AD). His name is also written as al-Khojandi or Khojandi, meaning: from the city of Khojand, Tajikistan.

Extant instrument: his astrolabe.

Extant works: 1. On the comprehensive instrument.   2. The The determination of the obliquity of the ecliptic and the latitude of a locality.  
Extant fragments of (or information on) lost works:
3. On the elapsed hours of the night   4. Azimuth of the qibla. 5. Construction of passing points of azimuthal circles on the astrolabe. 6. Number theory. 7. Two more titles mentioned by Hajji Khalifa.

Introduction to the webpage: For al-Khujandi's extant works, the following information is provided: (1) an abbreviated title in CAPITALS, the title in Arabic with translation, and bibliographical references with abbreviations (explained on a special page  ), (2) Arabic editions; (3) Translations into languages other than Arabic; (4) Arabic manuscripts (library, manuscript number, if possible with links to digital versions), (5) all commentaries and studied by later medieval Islamic authors, and some (but by no means all) modern studies. For lost works or contributions from lost works, we have provided the evidence from the medieval Islamic tradition together with modern studies if available.
For additions, questions, suggestions and comments please send an email to jphogendijk_at_gmail.com.

Bibliographic articles: Note that the dates of his birth and death are unknown, "exact"dates such as "940" and "1000" are approximate guesses by modern authors. The so-called portraits on the internet are modern artist's impressions only.
Here are some serious bibliographical sources:
Sevim Tekeli in Dictionary of Scientific Biography, available online.
Glen van Brummelen, article Abu Mahmud Hamid ibn al-Khidr al-Khujandi in p. 630-631 of Thomas Hockey et. al. (eds.), The Biographical Encyclopaedia of Astronomers, New York 2007, available online
See also GAS V, pp. 307-8, GAS VI, pp. 220-222, RI no. 269 pp. 100-101 (the abbreviatons GAS, RI, IMA etc. are explained in the abbreviations page)
See also the following references: Qorbani pp. 231-236; Kh.F. Abdulla-zade and N. N. Neghmatov, Al-Khujandi, Dushanbe 1986 [not seen]; Khurshid F. Abdullah-Zadeh, Nu`man N. Ne'matov, Khujandi-Nameh, translated [into Persian] by Baqer Mozaffar-Zadeh, Mirror of Heritage (Ayene-ye Mirath) New Series, vol. 10, supplement no. 26, 2012 (1391 A.H. Shamsi), Tehran, Miras-e Maktoub, ISSN 1561-9400, 98 pp.

THE EXTANT ASTROLABE.
This astrolabe is now in the Museum of Islamic Art (MIA) in Doha, Qatar.
See the description in the collection highlight
Photos of the astrolabe can be downloaded from the MIA website, which should be consulted for all further information. The MIA website displays images of the front,   mater,   back side,   spider,   plate for latitude 21 degrees,   plate for latitude 27 degrees, and alidade. The back side contains (lower left) the name Hamid ibn al-Khidr al-Khujandi and the date 374 H.
Al-Khujandi's astrolabe was first published in detail in David A. King, Early Islamic Astronomical Instruments in Kuwaiti Collections, pp. 76-96 in Arlene Fullerton and Géza Fehévári, eds., Kuwait: Art and Architecture: Collection of Essays, Kuwait 1995.
A very detailed description of al-Khujandi's astrolabe can be found in David A. King, In Synchrony with the Heavens vol. 2, Leiden 2005, pp. 503-517. In his excellent description, King argues that the astrolabe was made in Baghdad, but his argument seems to be based only on the fact that the plates of the astrolabe are for cities with geographical latitudes 21, 27, 30, 33, 36, 39, 42 and 66;27. I think that this is in itself no sufficient evidence to show where the astrolabe was made.
David A. King's works can be accessed through his academia.edu-account.

Extant works

1. ON THE COMPREHENSIVE INSTRUMENT GAS VI p. 221 no. 1, RI p. 100 no.A1.
Arabic editions: Not available because the treatise has not been published.
Translations: Not available because the treatise has not been translated.
Richard Lorch (1942-2021) worked on an edition and translation but these are not mentioned in his list of publications (see Historia Mathematica 58 (2022), pp. 7-16 https://doi.org/10.1016/j.hm.2022.01.003)
Arabic manuscripts: GAS and RI mentions the following manuscripts:
Oxford, Bodleian Library, Huntington 566, more information
Birmingham, 560, more information
Bursa, Haracci 1217, for reproduction of a few leaves see F. Sezgin, Science and Technology in Islam: Catalogue of the Collection of Instruments, Frankfurt 2010, vol. 2, pp. 152-153.
Kairo, Dar al-Kutub, miqat 970, for more information see D.A. King, A Survey of the Manuscripts in the Egyptian National Library, Winona Lake 1986, p. 40 (no. B50) and p. 276 for reproduction of one leaf.
Tehran, private library of Fakhraddin Nasiri, copied by Abu al-Qasim Hibatallah ibn al-Husayin al-Asturlabi (see below). I have been unable to locate this manuscript in DENA and in F. Ghassemlou, F. Payervand Sabet, A comprehensive catalogue of astronomical manuscripts in the libraries of Iran[in Persian], A.H. (solar) 1391.
The treatise was summarized and commented upon by later authors.
1-1. A commentary by Abu al-Qasim Hibatallah ibn al-Husayin al-Asturlabi (RI no. 438 p. 174) who apparently modified the instrument and made it suitable for all latitudes, whereas al-Khujandi had designed the instrument for an individual latitude. This commentary is extant in two manuscripts which also contain the original al- by Khujandi: Birmingham 560, see here, and an autograph manuscript in Tehran, private library of Fakhraddin Nasiri (see Sezgin GAS VI, 221).
A manuscript of the same or another text by Hibatallah, in which al-Khujandi is also mentioned, is Oxford, Bodleian Library, Marsh 663, ff. 163-189, For the identification of the author see F. Rosenthal, Al-Asturlabi and al-Samaw'al on Scientific Progress, Osiris 9 (1950), pp. 555-564, availabe via jstor https://www.jstor.org/stable/301858. More information on the manuscript.
1-2. In Section 3 of Subdivision 5 (on the construction of spherical instruments) of part 32 (on constructions) of the work on timekeeping by al-Marrakushi, Comprehensive Collection of Principles and Objectives in the Science of Timekeeping (jAmi` al-mabAdi' wa-l-ghAyAt fI `ilm al-mIqAt,) facsimile-edition, ed. Fuat Sezgin, Frankfurt 1984, vol. 2 pp 14 line 18 - p. 19 line 14 and for the application p. 245 line 22 - p. 246 line 19, see scan of these pages. The instrument is apparently presented in the version of Hibatallah, but no reference is made to either Hibatallah or al-Khujandi. The passage in al-Marrakushi was summarized in an unclear way in pp. 148-149 and figures 21-23 (see scan of these pages and figures) in L.A. Sédillot in Mémoire sur les instruments astronomiques des Arabes, in: Mémoires présentés par divers savants à l'Académie des inscriptions et belles-lettres de l'Institut de France. Première série, vol. 1, 1844. pp. 1-229.
1-3. In Chapter 5 (ff. 67b-71a) of the anonymous Arabic treatise: Summary on the construction of some observational instruments and the work with them (mukhta.sar fI .san`at ba`.d al-AlAt al-ra.sadiya wa-al-`amal bihA), manuscript Berlin, Ahlwardt 5857 (Sprenger 1877), digital version (ISMI website, the reference to Abu Ja`far al-Khazin's Zij Safaih is erroneous), Chapter 5 is found in images nos. 140-148 in the digital version. Acccording to the Ahlwardt catalogue. the work was derived from al-Marrakushi's Comprehensive Collection of Principles and Objectives in the Science of Timekeeping . No reference is made to Hibatallah or al-Khujandi.
Josef Frank, Über zwei astronomische arabische Instrumente, Zeitschrift für Instrumentenkunde 41 (1921), pp. 193-200, reprinted in Sezgin, IMA vol. 88, pp. 63-69, is based on the Berlin manuscript of this anonymous Arabic treatise Summary on the construction... 1-3. Schirmer was able to identify the author of the instrument as al-Khujandi because he had found in the Ta'rikh al-Hukama a note on Hibatallah, where al-Khujandi is mentioned as inventor of the instrument. See Ibn al-Qifti, Ta'rikh al-Hukama, ed. Lippert p. 339-340, scan.
Reconstruction of the instrument in the original version of al-Khujandi in F. Sezgin, Science and Technology in Islam: Catalogue of the Collection of Instruments, Frankfurt 2010, vol. 2, pp. 151-153, scan.

2. CORRECT DETERMINATION fI ta.s.h.I.h al-mayl wa-`ar.d al-balad ba`d .hu.sUl irtifA`At ni.sf al-nahAr al-mu.haqqaqa `inda al-inqilAbayn: On the correct determination of the (total) declination (i.e., obliquity of the ecliptic) and the latitude of the locality after obtaining the corrected meridian altitudes at the two (summer, winter) solstices. GAS VI p. 221 no. 2, RI p. 100 no. A2.
Arabic edition in L. Cheikho, Risalat al-Khojandi fI l-mayl wa-`ar.d al-balad, Mashriq 11 (1908), 61-69, scan.. The text by al-Khojandi is followed by a brief description of the Fakhri Sextant by al-Biruni, who had seen it with his own eyes (pp. 68-68).
Translations:
German translation in pp. 63-79 of Oskar Schirmer, Studien zur Astronomie der Araber, Sitzungsberichte der Physikalisch-medizinischen Sozietät zu Erlangen 58-59 (1926-27), pp. 33-88, download (zobodat.at). The article was reprinted in F. Sezgin, IMA vol. 22, pp. 1-56.
Persian translation by Baqir Mozaffarzadeh, published in pp. 84-93 in Khojandi-Nameh, Ayene-ye Miras New Series, Supplement 26 (1391 A.H. solar, 2012). The translation is based directly on the Arabic and it includes the appendix by al-Biruni and the list of names of geometers.
Russian translation by H F Abdulla-Zade, Traktat ob utochenii naibol'shego skloneniya i shiroty goroda, in Izv. Akad. Nauk Tadzhik. SSR Otdel. Fiz.-Mat. Khim. i Geol. Nauk 1(75) (1980), 17-22 (no further information available), and also in pp. 81-92 of Kh.F. Abdulla-zade and N. N. Neghmatov, Al-Khujandi, Dushanbe 1986 [not seen]
Arabic manuscript: Beirut, St. Joseph 223/1.
Scan of the manuscript (very poor quality; including the treatise by al-Biruni and the subsequent list of names of geometers)
Further literature: Al-Biruni visited the observatory of Khujandi and wrote a brief report about it, for the Arabic text see Cheikho's edition near the end, scan. See E. Wiedemann, Kleinere Mitteilungen: Über den Sextant des al-Chogendi, Archiv für die Geschichte der Naturwissenschaften und der Technik 2 (1910) pp. 149-151, reprinted IMA 92 pp. 55-57 (based on the description by al-Biruni)
See col. 133-134 in Joh. A. Repsold, Zur Geschichte der astronomischen Messwerkzeuge, Nachträge zu Band 1 (1908). II. Alte arabische Instrumente, Astronomische Nachrichten 206 (1918), Col. 125-138, reprinted in Sezgin, IMA vol. 88, pp. 16-22.
See the commentary in pp. 134-136 of Paul Luckey, Thabit b. Qurra's Buch über ebene Sonnenuhren, Quellen und Studien zur Geschichte der Mathematik, Astronomie und Physik. Abt. B. Studien 4 (1938), 95-148. Reprinted in IMA vol. 22 pp. 141-194. Scan.
See Al-Biruni's analysis of Khujandi's observations and computations is found in the Tahdid Nihayat al-Amakin, pp. 102-109 (scan) in the Arabic edition by P. Bulgakov and I. Ahmad, Cairo 1962, reprinted Frankfurt: Institute for History of Arabic-Islamic Science, 1992, Islamic Geography vol. 25; in the English translation by Jamal Ali pp. 70-77 (scan), The Determination of the Coordinates of Cities: Al-Biruni's Tahdid al-Amakin, Beirut 1967. See the commentary by E.S. Kennedy on al-Biruni's analysis of al-Khujandi's observations and computations in pp. 43-48 (scan) of A Commentary upon Biruni's Kitab Tahdid al-Amakin, Beirut 1973.
For a model of the reconstructed Fakhri sextant in F. Sezgin, Science and Technology in Islam: Catalogue of the Collection of Instruments, Frankfurt 2010, vol. 2 p. 25, scan.

3. BOOK ON THE ELAPSED HOURS OF NIGHT GAS VI p. 222 no.4, RI p. 101 no. A6. This work is now lost. The title is mentioned in the anonymous Collection of Rules of the Science of Astronomy (jAmi` qawAnIn `ilm al-hay'a), RI p. 141, (no. 341) no. M5. In the introduction to that work, the anonymous (12th-century?) author mentions as one of the important writers on the preliminary theorems (muqaddamAt) of astronomy "Abu Muhammad Hamid ibn al-Khidr al-Khujandi in his book on the elapsed hours of night (Wa-abU mu.hammad .hAmid ibn al-khi.dr al-khujandI fI kitAbihi fi al-sA`At al-mA.diya min al-layl),without further information. See manuscript Topkapi Serayi Ahmet III 3342 f. 1b line 13.
This work by al-Khujandi must have contained the spherical sine theorem, which is as follows: if in a triangle of great circle arcs the sides are a, b, c and the angles are A, B, C (where a is opposite A, b is opposite B, c is opposite C), then sin(a)/sin(A)=sin(b)/sin(B)=sin(c)/sin(C). The proof by al-Khujandi is cited by the following authors, with references to al-Khujandi:
3-1. The Collection of Rules of the Science of Astronomy cites the "method of al-Khujandi" followed by the "method of Kushyar and perhaps it is by al-Khujandi", see manuscript Topkapi, Ahmet III 3342 ff. 40a-41a. The information may be based on al-Biruni, see 3-2. On this work see N. G. Khairetdinova, trigonometricheskiy traktat isfahanskogo anonyma, Istoriko-matematicheskie issledovaniya17 (1966), pp. 399-464.
3-2. Al-Biruni (973-ca. 1048), Keys to Astronomy (maqAlId `ilm al-hay'a), see pp. 138-142 in the Arabic edition of Marie-Thérèse Debarnot, Al-Biruni, Kitab maqalid `ilm al-hay'a, La trigonométrie sphérique chez les Arabes de l'Est à la fin du Xe siècle. Damascus: Institut Français de Damas, 1985, PIFD 114.
Al-Biruni presents al-Khujandi's proof and adds that al-Khujandi called the theorem " the rule of astronomy " (qAnUn al-hay'a), and that al-Khujandi proved it in a work on " the procedures at night with the fixed stars" (a`mAl al-layl bi'l-kawAkib al-thAbita, see Debarnot p. 100-101). I take this as a description of the contents of the Book on the Elapsed Hours of Night. Al-Biruni says that al-Khujandi proved the theorem only for the special case of spherical triangles with one angular point the vernal or autumnal equinox and whose sides are an arcs of the equator, an arc of the ecliptic and a declination arc perpendicular to the equator. According to Biruni (see Debarnot pp. 144-145), al-Khujandi did not realize that the theorem is general for all spherical triangles. In the same work, al-Biruni says that he met al-Khujandi in Rayy, and that al-Khujandi also met Kushyar ibn Labban al-Jili (Debarnot pp. 100-101, 102-103). See scan of the relevant pages (100-103, 138-145).
3-3. Al-Khujandi's proof of the sine theorem is also cited (this time in general form) in the anonymous treatise Various geometrical problems (masA'il handasiyya mutafarriqa), which has been preserved in f. 168b-169a of the manuscript Cairo, Dar al-Kutub, Mustafa Fa.dil riya.da 41m, ff. 165b-170b (See RI p. 100 no. 1 (where the entire work is incorrectly attributed to al-Khujandi). and GAS V, p. 308 no. 1 and p. 391 (where the manuscript number is incorrectly given as 40m/1). The various geometrical problems is an anonymous collection of geometrical problems and solutions by other mathematicians and astronomers as well, such as Ibn al-Haytham). Scan of f. 168b-169a containing the proof.
German translation in pp. 261-263 of Carl Schoy, Behandlung einiger geometrischen Fragepunkte durch muslimische Mathematiker, Isis 8 (1926), pp. 254-263, available through jstor https://www.jstor.org/stable/223641, reprinted in IMA vol 58 pp. 84-93.
3-4. Al-Khojandi's proof is also cited in the treatise by Nasir al-Din al-Tusi (1201-1274 AD) on the transversal theorem (kitAb shakl al-qa.t.tA`), RI p. 214 no. M14, see Arabic edition (p. 117) and French translation (pp. 151-152) in Traité du Quadrilatère attribué à Nassiruddin-El-Toussi traduit par Alexandre Pacha Caratheodory, Constantinople 1891, reprinted in IMA vol. 47. AL-Khujandi is mentioned two more times (Arabic pp. 108 line 11 and p. 125 line 13, French translation pp. 140, 162). The information in al-Tusi's work was probably based on the Keys of Astronomy of al-Biruni, which al-Tusi mentions by name.

4. AZIMUTH OF THE QIBLA. GAS VI p. 222 no. 3, RI p. 101 no. A5. In the Book on the projection of the constellations and the flattening of the sphere (kitab fI ta.s.tI.h al-.suwar wa-tab.tI.h al-kuwar) al-Biruni mentions the approximative construction of the azimuth of the qibla (direction of Mecca) by al-Battani. Al-Biruni says (in Berggren's translation) "and that way of constructing the azimuth of the qibla is a grosss error which all the scholars accused him of in their books on the azimuth of the qibla, e.g., Abu Sa`id Ahmad b. Muhammad b Abd al-Jalil (al-Sijzi), Abu Mansur `Ali b. `Iraq, and Abu Mahmud Hamid b. al-Khidr al-Khujandi". See p. 51 of J. Len Berggren, Al-Biruni on plane maps of the sphere, Journal for the History of Arabic Science 6 (1982), pp. 47-112.

5. CONSTRUCTION OF PASSING POINTS OF AZIMUTHAL CIRCLES IN A PRACTICAL WAY (istikhrAj majAz dawA'ir al-sumUt bi-al-.sinA`aT. GAS V p. 308 no. 2. Two methods from this lost work are cited in a letter by Abu Nasr Ibn `Iraq to Abu Rayhan al-Biruni, Letter on the intersections of the azimuthal circles on the astrolabe, risAla fI majAzAt dawA'ir al-sumUt fi'l-as.turlAb. GAS VI, 243 no. 6. RI p. 114 no. M5. Abu Nasr ibn `Iraq provides his own proofs for the two methods.
See pp. 3-9 of the Arabic edition: Tract no. 14 (27 pp.) in: Abu Nasr Mansur b. Ali b. Iraq, Rasa'il ila'l-Biruni, Hyderabad, Osmaniya Oriental Publications Bureau, 1376 H/1948 AD, scan; Spanish translation in Julio Samsó Estudios sobre Abu Nasr Mansur b. Ali b. Iraq, Barcelona 1969 pp. 89-104; commentary pp. 49-53. Arabic manuscripts: Patna (India), Khuda Bakhsh library, Bankipore 2468 (now 2519), ff. 79b-83b. scan, and Oxford, Bodleian Library, Thurston 3, ff. 124a-126b and Marsh 713, 247a-251a (a later copy of the Thurston 3 manuscript).
Studies: See pp. 320-324 and 335-337 in J. Lennart Berggren, Medieval Islamic methods for drawing azimuth circles on the astrolabe, Centaurus 34 (1991), pp. 309-344, https://doi.org/10.1111/j.1600-0498.1991.tb00864.x There it is explained that one of Al-Khujandi's methods is also mentioned by al-Biruni in his work Full discussion of all possible ways to construct the astrolabe, see p. 56 in the edition by Sayyid Muhammad Akbar Jawadi al-Hoseini.

6. NUMBER THEORY see GAS V, 307 and RI p. 100 no. M2.
In the beginning of a treatise on rational right-angled triangles, Abu Ja`far Muhammad ibn al-Husayn al-Khazin refers to the contributions of one "al-Khujandi" see ms. Paris, Bibliothèque Nationale, Fonds Arabe 2457, f. 88v, see scan of the passage. See RI p. 100 no. M2.
Al-Khazin says (in my translation) " I explained that what Abu Muhammad al-Khujandi, may God have mercy on him, proposed in his proof of the (theorem) that the sum of two cube numbers cannot be a cube number, is false and incorrect, and that the rule which he presented for the knowledge of right-angled triangles with rational sides is (a) special (case) and not general".
French translation of the passage see pp. 301-302 in F. Woepcke, Récherches sur plusieurs ouvrages de Léonard de Pise découverts et publiés Par M. le Prince Balthasar Boncompagni et sur les rapports qui existent entre ces ouvrages et les travaux mathématiques des Arabes, Atti dell' Accademia Pontificia de Nuovi Lincei 14 (1860-61), pp. 211-227, 241-269, 301-324, 343-356. See scan of vols. 13-14 of the Atti (google books), pp. 301 and 302 are pp. 886-887 in the pdf.
Woepcke points out that the Paris manuscript 2457 was written in 358 H/ 969 AD - 361 H / 972 AD and that the expression "may God have mercy on him" indicates that this Abu Muhammad al-Khujandi would have died at that time. In that case he cannot have been the same person as Abu Mahmud al-Khujandi to whom this webpage is dedicated. However, in work no. 6 (see above) there is a reference to Abu Muhammad Hamid ibn al-Khidr al-Khujandi. I am unable to solve the mystery. RI p. 100 no. M2 think that Abu Muhammad and Abu Mahmud are the same. ,00
The whole manuscript Paris Bibliothèque Nationale Fonds Arabe 2457 is available online (gallica.fr)
N.B. Abu Ja`far Muhammad al-Husayn and Abu Ja`far al-Khazin are the same person, see GAS VI p. 189, VII, p. 406.
For the Arabic text and a French translation of an extremely defective "proof" by al-Khazin himself (ms. Oxford, Bodleian Library, Thurston 3, f. 140a) of the theorem the sum of two cubes cannot be a cube number, see R. Rashed, L'analyse Diophantienne au Xe siècle: l'exemple d'al-Khazin, Revue d'Histoire des Sciences 32, 1979, pp. 193-220,scan (persee.fr), see the end of the article. Thus it is not clear whether the criticisms of al-Khazin are to be taken seriously. It is unlikely that Abu Muhammad al-Khujandi did have the means to prove the theorem (even the proof which Leonard Euler published in 1770, by means of infinite descent and complex numbers, was incomplete, see Harold Edwards, Fermat's last theorem, New York 1977, pp. 39-54.) See also a modern proof which uses quadratic reciprocity.

The titles of TWO MORE WORKS are mentioned by Katib Celebi (Hajji Khalifa):
7.Treatise of the horizontal plate, called the comprehensive, among the astrolabes, and on its use, by Hamid ibn al-Khidr, known as Ibn Mahmud, al-Khujandi, and it is in sixty chapters, and by someone else in a preliminary and fifteen chapters. (risAla al-.sa.hIfa al-AfAqiyya al-musammAh bi-al-jAmi`a min al-as.tur.AbAt wa-`amluhA li-.hAmid ibn al-khi.dr al-ma`rUf bi-ibn ma.hmUd al-khujandI wa-hiya `alA sittIn bAbaN wa-li-ghayrihi `alA muqaddama wa-khamsa `ashara bAbaN). RI p. 100 no. A4. See Katib Jelebi Lexicon Bibliographicum et Encyclopaedicum, ed. Gustav Flügel, vol. 3, London 1842, p. 416 lines 6-7, scan of the page.
8.Treatise on working with the Zarqala by Hamid ibn al-Khidr, known as Ibn Mahmud, al-Khujandi (kitAb al-`amal bi-al-zarqAla li-.hAmid ibn al-khi.dr al-ma`rUf bi-ibn ma.hmUd al-khujandI). See Katib Jelebi Lexicon Bibliographicum et Encyclopaedicum, ed. Gustav Flügel, vol. 5, London 1850, p. 120 lines 5-6, scan of the page. The title presents a chronological problem since the Zarqala is the universal astrolabe plate invented by al-Zarqali, the Andalusian astronomer who lived in the middle of the eleventh century AD. RI p. 100 no. A3. Rosenfeld and Ihsanoglu in RI p. 100 suggest that one could (hypothetically) assume that the title is correct, and that al-Zarqala might have derived his name from this type of astrolabe (which would have been invented by al-Khujandi). Further evidence for this hypothetical possibility is lacking.
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