Logical Necessities in Mixed Equations: Chapter II
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2. The Author, ‘Abd Al Hamîd Ibn Wâsî' Ibn Turk
This text gives, as far as our present information goes, the only work of the author that has come down to us, and very little is known concerning the author himself. Certain sources refer to him as the grandson of "the Turk from Jîl." Jîl, Jîlan, or Gîlan, is a district to the south of the Caspian Sea. Others have the word Khuttalî, i.e., "from Khuttal," a region around the sources of the Oxus River, to the south of Farghana and west of Chinese Turkistan. Still another formal possibility is from Jabalî . Jabalî could refer to several places, most of them being in Syria. In the Arabic script these three words could be easily mistaken for one another through the omission or addition of dots. As manuscript A that is quite old has the form Jîlî, this may be said to constitute rather strong evidence in favour of this version, but the possibility of the form Khuttalî cannot be ruled out .
The titles Ibn Turk and Ibn Turk al-Jîlî (or Khuttalî) indicate that 'Abd al-Hamîd's grandfather was called "the Turk from Jîl (or Khuttal)" and therefore that 'Abd al-Hamîd was Turkish or of Turkish descent. 'Abd al-Hamîd's grandson or great grandson Abû Barza  too kept the title Ibn Turk, indicating that the family remained to be Turkish. It is of interest in this connection that Al-Khwârazmî too was from the district of Turkistan.
Among the earlier scientists of medieval Islam a large number are seen to have originated from districts to the northeast of Persia. As Turks formed a part of the population of these districts , it is reasonable to think that a considerable number among this group of scientists were Turkish or of Turkish ancestry, although it is generally difficult to speak of the nationality of such scientists individually with any degree of certainty. But a few of them are seen to have been given the title "The Turk" or "Turkish," just as a few scientists bore the title "Al-Farsi," i.e., Persian, or from the region of Fars. For example, the two or three scientists of the Amajur Family (fl. 885-933) , distinguished philosopher Abû Nasr al-Fârâbî (d. 950-951)  and the famous lexicographer Abû Nasr Isma'il al-Jawharî from Fârâb (d. 1002)  had the title Al-Turkî. 'Abd al-Hamîd ibn Wâsi' ibn Turk is apparently one of the earliest among this category of Turkish scientists.
Information concerning 'Abd al-Hamîd ibn Turk is given by Ibn al-Nadîm, Ibn al-Qiftî, and Hajji Khalîfa. As this information is completed and partly given also in connection with 'Abd al-Hamîd's grandson or great grandson Abû Barza, who was also a mathematician, there is some necessity of taking up these two scientists together.
Figure 2: Al-Fārābi depicted on a postage stamp of Kazakhstan.
Ibn al-Nadîm says concerning 'Abd al-Hamîd, "He is Abu 'l-Fadl 'Abd al-Hamîd ibn Wâsi' ibn Turk al-Khuttalî (or, al-Jîlî), the calculator, and it is said that he is surnamed Abû Muhammad, and of his books are The Comprehensive Book in Arithmetic which contains six books (chapters) and The Book of Commercial Transactions."
This item occurs under the general heading "The Calculators and the Arithmeticians." 'Abd al-Hamîd ibn Turk is the first item and the second item concerns Abû Barza, and information on Abû Kâmil Shujâ' ibn Aslam follows it immediately. On Abû Barza, Ibn al-Nadîm writes, "Abu Barza Al-Fadl ibn Muhammad ibn 'Abd al-Hamîd ibn Turk ibn Wâsî' al-Khuttalî (or Jîlî), and of his books are The Book of Commercial Transactions and the Book of Mensuration ."
Ibn al-Qiftî gives the following information concerning 'Abd al-Hamîd ibn Turk:
"'Abd al-Hamîd Ibn Wâsi' Abu 'l-Fadl. He is a calculator learned in the art of calculation (hisâb) having antecedence in the field, and the people of that profession mention him. He is known as Ibn Turk al-Jîlî, and he is surnamed also as Abû Muhammad. In the field of arithmetic he has well known and much used publications. Among them is the Comprehensive Book in Arithmetic which comprises six books, and The Book of Little-Known Things in Arithmetic and the Qualities of Numbers ."
The same author has the following item on Abû Barza:
"Abû Barza, the calculator in Baghdad and busied himself with the science of arithmetic, its subtleties, its fine points, and with the discovery of its peculiarities and rare qualities. He is the author of books in that field and has made original contributions to the subject. He died in Baghdad on the twenty seventh of the month of Safar in the year two hundred and ninety eight (November, 910 A.D.) ."
The following passage in Hajji Khalîfa's Kashf al-Zunûn is of great interest although its reference to 'Abd al-Hamîd is quite incidental.
Hajji Khalîfa writes, "Abu Kâmil Shujâ' ibn Aslam says in his Kitâb al-Wasâyâ bi'l-Jabr we'l-Muqâbala: ‘I have written a book known as Kamâl al-Jabr wa-Tamâmuhu wa'z-Ziyâdatu fî Usûlihi and in its second book I have proved the priority and antecedence of Muhammad ibn Mûsâ. [Al-Khwârazmî] in algebra and have refuted the assertion of the professional (?) [Mathematician] known as Abû Barza in what he makes go back to ‘Abd al-Hamîd, who, he claims, is his ancestor (or grandfather). And when I made clear his shortcoming and his deficiency in what he traces back and attributes to his ancestor, I decided to write a book in the subject of legacies treated by the way of algebra (al-wasâyâ bi'l-jabr wa'l-muqâbala).." .
Most likely, the Kamal al-Jabr wa-Tamâmuhu ... contains further information of interest, but this book is probably lost. Apparently a copy of Abû Kâmil's Kitâb al-Wâsâyâ exists in Musul ; I have not been able to consult it. Of course Abû Barza's book too, in which he presumably set forth his claim, if found, would shed much needed light on this controversial question.
Figure 3: Manuscript of Maqalah fi al-jabr wa-al-muqabalah by 'Umar al-Khayyam, hand written manuscript on paper, 56 leaves, copied in Lahore in the 13th century. Property of Columbia University Library, RBML, Smith Oriental, MS 45. (Source).
It is seen clearly from the passage quoted by Hajji Khalîfa that there was rivalry between ‘Abd al-Hamîd ibn Turk and Al-Khwârazmî in the matter of antecedence and priority in the publication of books on algebra, or, at least, that such an issue was raised by Abû Barza and that the question was taken very seriously by Abû Kâmil Shuja'.
Salih Zeki considers Abû Kâmil Shujâ' to be a contemporary of Al-Khwârazmî and concludes from the passage in the Kashf al-Zunûn that Abû Barza himself claimed priority over Al-Khwârazmî in the writing of a book on algebra. Under these circumstances, our author ‘Abd al-Hamîd would be two or three generations before Al-Khwârazmî, and his priority in the matter would be securely established .
According to Aldo Mieli, however, Abû Kâmil Shujâ' flourished toward the year 900 , while Sarton places him between the years 850 and 955 , These latter dates are based on the fact that in his Algebra Abû Kâmil speaks of Al-Khwârazmî as a mathematician of the past, while a commentary to Abû Kâmil's Algebra was written by ‘Ali ibn Muhammad al-‘Imrânî al-Mawsilî who died in the lunar year 955-956 A.D .
From the phraseology and tone of Abû Kâmil's statement quoted above from the Kashf al-Zunûn it may be concluded that he lived at a somewhat later date than Abû Barza ; and at the most, he could be a contemporary of Abû Barza. We have seen, on the other hand, that Ibn al-Qiftî gives the exact date of Abû Barza's death as 910 A.D. It seems quite clear therefore that Abû Kâmil's lifetime must correspond approximately to the first half of the tenth century, extending, very likely, back to the end of the ninth century.
Now, ‘Abd al-Hamîd was two or three generations before Abû Barza, and Al-Khwârazmî is said to have been among the group of astrologers who had gathered at the death-bed of the caliph Al-Wathiq, who died in 847. Al-Khwârazmî must therefore have lived beyond this date, but there is some doubt on this point . There can be no certainty therefore that ‘Abd al-Hamîd's lifetime was before that of Al-Khwârazmî. It is much more reasonable to assume that they were, roughly speaking, contemporaries. And this should be a safe assumption, as it may be concluded, on the basis of Ibn al-Qiftî's statement quoted above, that ‘Abd al-Hamîd wrote his book on arithmetic and the art of calculation before Al-Khwârazmî's book on the same general subject.
Moreover, the passage quoted by Hajji Khalîfa is quite clear in that Abû Barza made the claim of priority of publication in the field of algebra over Al-Khwârazmî not for himself but in behalf of his grandfather or ancestor ‘Abd al-Hamîd. As we have seen, however, Abû Kâmil Shujâ' rejects this claim rather violently; and he asserts the priority of Al-Khwârazmî once more in his Algebra .
Ibn Khaldun says: "The first to write on this discipline (algebra) was Abû ‘Abdullah al-Khwârazmî. After him, there was Abû Kâmil Shujâ' ibn Aslam ." According to Salih Zeki, Shihâb al-Din ibn Bahâim too, who was a contemporary of Ibn Khaldun, says, in the commentary he wrote to the book called Yâsamînîya, the same thing about Al-Khwârazmî . Finally, Hajji Khalîfa also states, just prior to the words of Abû Kâmil quoted from him above, that Al-Khwârazmî was the first to write a book in the subject of algebra .
There should be little doubt that Hajji Khalîfa's authority for this statement is Abû Kâmil Shujâ', and the phraseology of Ibn Khaldun too suggests such a possibility. It thus seems that it was through Abû Kâmil that this assertion gained its circulation in certain sources. But Abû Kâmil was certainly far from being impartial toward Abû Barza and his family. He speaks in somewhat derogatory terms about Abû Barza and his knowledge of mathematics, and he is clearly contradicted in this by Ibn al-Qiftî; he expresses doubt concerning Abû Barza's relation to ‘Abd al-Hamîd, and such a relationship is confirmed by the texts of both Ibn al-Nadîm and Ibn al-Qiftî.
It appears quite possible therefore that Abû Barza's claim on behalf of ‘Abd al-Hamîd is not without foundation. The following items of circumstantial evidence may in fact be brought to support it.
It seems likely that the lifetime of ‘Abd al-Hamîd was somewhat earlier than that of Al-Khwârazmî, for, as mentioned before, it appears from the statement of Ibn al-Qiftî quoted above that ‘Abd al-Hamîd preceded Al-Khwârazmî in writing a book on arithmetic . It is even possible to interpret Ibn al-Qiftî's words to mean that ‘Abd al-Hamîd wrote his Algebra before that of Al-Khwârazmî. For the "art" in which, according to Ibn al-Qiftî, ‘Abd al-Hamîd was a pioneer, is the field of "calculation" or "arithmetic" in a general sense, i.e., hisâb. Algebra too is a type of hisâb, and, in fact, the expression "the hisâb of jabr and muqâbala" occurs in the title of Al-Khwârazmî's Algebra . It should perhaps be considered permissible, therefore, to see in Ibn al-Qiftî's statement a partial evidence in favour of ‘Abd al-Hamîd's priority over Al-Khwârazmî in the field of algebra.
‘Umar Khayyâm speaks merely of the existence of special cases of the equation x2 + c = bx, but he does not feel the need of dwelling upon them or even of enumerating them, saying that they are evident . Apparently he considers them to be sufficiently well-known. Al-Khwârazmî too perhaps did not feel the need of going into details for explaining these special cases because they were available in an earlier text. For he is seen, to touch most of these cases; but he does not do so in a systematic manner, and his reference to some of the cases is only implicit .
It may be added also that Abû Kâmil Shujâ', in his words quoted above, seems to treat our author somewhat lightly and that our present text does not support his attitude but shows Ibn al-Qiftî's statement concerning ‘Abd al-Hamîd to be much more accurate. The fact that apparently Abû Kâmil Shujâ' did not attempt to refute Abû Barza's claim on the basis of any considerations pertaining to chronological impossibility or anachronism should likewise constitute a point in favour of an earlier date for ‘Abd al-Hamîd ibn Turk's lifetime as compared to that of Al-Khwârazmî.
It may be conjectured, moreover, that were Al-Khwârazmî's Algebra the first written in Arabic this would have been somehow revealed in the text. Al-Khwârazmî seems to have written his book without any claims of originality of any kind.
In the introductory-section of his Algebra Al-Khwârazmî writes as follows:
"The learned in times which have passed away, and among nations which have ceased to exist, were constantly employed in writing books on the several departments of science and on the various branches of knowledge, bearing in mind those that were to come after them, and hoping for a reward proportionate to their ability, and trusting that their endeavours would meet with acknowledgment, attention, and remembrance- content as they were even with a small degree of praise; small, if compared with the pains which they had undergone, and the difficulties which they had encountered in revealing the secrets and obscurities of science.
"Some applied themselves to obtain information which was not known before them, and left it to posterity; others commented upon the difficulties in the works left by their predecessors, and defined the best method (of study), or rendered the access (to science) easier or placed it more within reach; others again discovered mistakes in preceding works, and arranged that which was confused, or adjusted what was irregular, and corrected the faults of their fellow-labourers, without arrogance towards them, or taking pride in what they did themselves.
"That fondness for science, by which God has distinguished the Imam Al-Mamun, the Commander of the Faithful (besides the caliphate which He has vouchsafed unto him by lawful succession, in the robe of which He has invested him, and with the honours of which He has adorned him), that affability and condescension which he shows to the learned, that promptitude with which he protects and supports them in the elucidation of obscurities and in the removal of difficulties,- has encouraged me to compose a short work on calculating by (the rules of) Completion and Reduction, confining it to what is easiest and most useful in arithmetic, such as men constantly require in cases of inheritance, legacies, 1 partition, law-suits, and trade, and in all their dealings with one another, or where the measuring of lands, the digging of canals, geometrical computation, and other objects of various sorts and kinds are concerned - relying on the goodness of my intention therein, and hoping that the learned will reward it, by obtaining (for me) through their prayer the excellence of the Divine mercy: in requital of which may the choicest blessings and the abundant bounty of God be theirs! My confidence rests with God, in this and in every thing, and in Him I put my trust. He is the Lord of the Sublime Throne. May His blessing descend upon all the prophets and heavenly messengers!" .
These details given by Al-Khwârazmî indicate that his Algebra was written during the reign of Al-Mamun, i.e., between the years 813 and 833. From the words of its author the main purpose for its composition was to make available a book which could serve practical needs and one that would be easy to follow. Such reasons for writing a book are frequently encountered in Islam. There is no suggestion that this was the first book on algebra in Arabic.
Al-Khwârazmî states his book to be a short work, and the exact title of Al-Khwârazmî's Algebra too, as it has come down to us in its printed edition, contains the word mukhtasar, i.e., abridged . It may be contended that, considering the conditions prevalent at the time, such a word would not be very likely to be applied to the first book written in Arabic on algebra.
Figure 5: The planetarium of al-Khawarizmi Astronomy Complex in Melaka, Malaysia (Source), and in the left top corner the depiction of al-Khawarizmi on a Soviet stamp.
From the considerations dwelled upon above it seems quite likely that Abû Barza's claim was not without foundation and that ‘Abd al-Hamîd ibn Wasi' ibn Turk, and not Muhammad ibn Mûsâ al-Khwârazmî, was the first to write an Arabic book on algebra in Islam.
Al-Khwârazmî's Algebra contains a very short section on commercial transactions . It may be noted that according to Ibn al-Nadîm, ‘Abd al-Hamîd ibn Turk wrote an independent book devoted to this subject . It seems quite certain that in the field of algebra itself too, just as in the field of commercial transactions, it was ‘Abd al-Hamîd ibn Turk who wrote the longer and more detailed treatise.
 See Heinrich Suter, Die Mathematiker und Astronomen der Araber und ihre Werke, Abhandlungen zur Geschichte der Mathematischen Wissenschaften, Leipzig 1900, 1902, p. 17, note.
 Brockelmann accepts the form Khuttalî (Gesch. d. Arab. Lit., S. vol. 1, p. 383), and both Suter and Flügel accept it as the preferred form (See Suter, op. cit., p. 17).
 For more information about Abû Barza see: B. A. Rosenfeld-E. Ihsanoglu, Mathematicians, Astronomers and other Scholars of Islamic Civilisation and their works (7th – 9th c.). Istanbul: Research Center for Islamic History, Art and Culture, 2003, No. 115. Also see for Ibn Turk: Jens Høyrup, "Al-Khwârizmî, Ibn Turk, and the Liber Mensurationum: on the Origins of Islamic Algebra." Erdem 2 (Ankara 1986), 445–484; Jens Høyrup, "Algebraic Traditions Behind Ibn Turk and Al-Khwârizmî," pp. 247–268 in Acts of the International Symposium on Ibn Turk, Khwârezmî, Fârâbî, and Ibn Sînâ (Ankara, 9–12 September, 1985). (Atatürk Culture Center Publications, No: 41. Series of Acts of Congresses and Symposiums, No: 1). Ankara: Atatürk Supreme Council for Culture, Language and History, 1990.
 Richard N. Frye and Aydin Sayili, Turks in the Middle East Before the Saljuqs, Journal of the American Oriental Society, vol. 63, No. 3, 194.3, pp. 194-207.
 See A. Sayili, The Observatory in Islam, Ankara 1960, p. 101.
 See D. M. Dunlop, a Source of Al-Mas'udi: The Madînat al-Fâdilah of Al-Fârâbî, Al-Mas'udi Millenary Commemoration Volume, ed. S. Maqbul Ahmad and A. Rahman, Aligarh Muslim University 1960, pp. 69, 70. See also, S. M. Stern, Al-Mas'udî and the Philosopher Al-Fârâbî, Al-Mas'udi Commemoration Volume, p. 40.
 See the partial edition of Al-Jawharî's dictionary by Everardus Scheidius, 1774.
 Ibn al Nadîm, Kitâb Fihrist al ‘Ulûm, ed. Flugel, vol. 1, 1871, p. 273.
 Ibn al Qiftî, Tarîkh al Hukamâ, ed. Lippert, Berlin 1903, p. 230, See also, below, pp. 92-93 and note 39.
 Ibn al Qiftî, p. 406.
 Hajji Khalîfa, Kashf al Zunûn, art. Kitab al Jabr wa'l Muqabala and art. Kitab al Wâsâyâ, ed. Yaltkaya, Istanbul 1943, vol. 2, columns 1407-1408, 1469-1470. See also, Salih Zeki, Athâr-i Bâqiye, vol. 2, Istanbul 1913, p. 246.
 See Brockelmann, Gesch. Arab. Lit. S. vol. 1, p. 390.
 Salih Zeki, op. cit, vol. 2, p. 246.
 Aldo Mieli, La Science Arabe, Leiden 1939, p. 108.
 George Sarton, Introduction to the History of Science, vol. 1, Baltimore 1927, p. 630.
 See Suter, op. cit., pp. 43, 56-57; Salih Zeki, op. cit., vol. 2, p. 255.
 That Abû Ka.mil was of a somewhat later date than Abû Barza may be said to be indirectly confirmed by the fact that Abû Kâmil's name comes after that of Abû Barza's in the Fihrist of Ibn al Nadîm (vol. 1, p. 281). Adel Ambouba associates Abû Kâmil approximately with the year 900 (op. cit., 1959, p. 73).
 E. Wiedemann, Khwârizmî, Encyclopaedia of Islam, vol. 2; Abdülhak Adnan Adivar, Hârizmî, Islam. Ansiklopedisi, vol. 5, No. 42, 1949, pp. 258-259.
 See manuscript in the Bayezit General Library in Istanbul, No. 19046 (or, Kara Mustafa Pasa, No. 379), p. 2a.
 Ibn Khaldun, Muqaddima, tr. F. Rosenthal, vol. 3, London 1958, p. 125.
 Salih Zeki, op. cit., vol 2, p. 248 and footnote.
 Kashf al Zunûn, ed. Yaltkaya, vol. 2, column 1407.
 See above, p. 89 and note 24.
 See e.g., Gandz, The Sources of Al-Khwârazmî's Algebra, Osiris, vol. 1, p. 274. See also, Adel Ambouba, Ihyâ al Jabr, Manshûrât al Jâmi'a al Lubnânîya, Qism al Dirâsât al Riyâdîya, Beyrut 1955, pp. 8-9.
 ‘Umar Khayyâm, Algebra, ed. and tr. Woepke, text, p. 14, tr., p.23.
 Al-Khwârazmî, Algebra, Rosen, text, pp. 7-8, tr., pp. 11-12; Gandz, The Origin and Development ..., Osiris, vol. 3, pp. 519-523, 533-534.
 Al-Khwârazmî, Algebra, tr. Rosen, pp. 2-4.; also, text, p. 2. See also, Adel Ambouba, op. cit., 1955, p. 23.
 See Rosen's edition referred to above. This edition is based on an Oxford manuscript. Mr. Adel Ambouba has kindly informed me that according to the Revue of the Institute of Arabic Manuscripts of Cairo of November 1956 there is a second copy of Al-Khwârazmî's Algebra in Cairo. Mr. Ambouba himself has discovered a third copy in Germany.
 Al-Khwârazmî, Algebra, Rosen, text, pp. 48-50, tr., pp. 68-70. See also, Adel Ambouba, op. cit., 1955, p. 11.
 Fihrist, vol. 1, p. 273. See also above, p. 89 and note 23.
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by: FSTC Limited, Mon 19 January, 2009