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Logical Necessities in Mixed Equations: 'Abd Al-Hamîd Ibn Turk and the Algebra of his Time

This article was first published as a part of the book entitled Abdülhamid Ibn Türk'ün Katişik Denklemlerde Mantikî Zaruretler Adli Yazisi ve Zamanin Cebri (Logical Necessities in Mixed Equations By ‘Abd Al-Hamîd Ibn Turk and the Algebra of His Time) (Ankara: Türk Tarih Kurumu Basimevi 1985). We are publishing only an English translation of this book by the permission of Turk Tarih Kurumu. Thanks to the Prof. Yusuf Halacoglu for giving us permission.

The Arabic text presented further below in this volume together with its Turkish and English translations, i.e., Logical Necessities in Mixed Equations by ‘Abd al-Hamîd ibn Wâsi' ibn Turk, is based on two manuscripts, both preserved in Istanbul libraries. One of these, referred to by the letter A in the edition given below, is in the Millet Library: section Carullah, No. 1505; It forms part of a collection of short treatises bound together and occupies pages 2a to 5a in the volume. Brockelmann mentions this manuscript [1]. The second copy, referred to by the letter B in the critical text given below, is in the Suleymaniye Library: section Husrev Pasa, No. 257. This manuscript too forms part of a collection and occupies pages 5b to 8a in the volume.

None of these manuscripts or collections bears any dates of transcription. Manuscript A is quite old and may be guessed to be from the twelfth century. Manuscript B is relatively recent. It must be several centuries later than A. The manuscript texts themselves have no title. I have added the title on the basis of the copyist's note seen at the end of the text given below.

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Figure 1: A line sketch depicting Al-Biruni in a Turkish stamp. The stamp reads: "Ebû Reyhan El-Bîrûnî 973-1059".

A comparison of the two manuscripts shows them to agree on many points of detail. Notes 3, 8, n, 15, 23, 35, 36, 37, 45, 48, 51, 62, and 63 in the Arabic text indicate grammatical errors which are common to both. Moreover, at two common points they are both of doubtful reading, as indicated by notes 4 and 66. And as seen from note 33-34, in one point of the text they contain an almost identical small lacuna that had to be reconstructed and filled in with the help of the context, while note 41 indicates a common numerical error identical or closely similar in both manuscripts.

From these details one gains the impression that B was copied from A, or, at least, that they are very closely related. The alternative that B may not be a mere copy of A is supported especially by their difference from the standpoint of diacritical marks and the slightly different manner in which their figures are lettered.

Manuscript A contains but few dots, while in B the letters are dotted in an almost complete manner. In a few cases the dotting of verbs as found in B are clearly incorrect, as may be seen in footnotes 39 and 42, e. g. At points where I have not followed the manner in which B is dotted, I have indicated the differences in the footnotes though the disagreement may at times be of little significance. The word mas'ala is written in varying orthography in both manuscripts. I have adopted a unified spelling for this word in the text given below.

Some doubt exists concerning two words of the text. One is the word muzâd wherein the z is undotted, i. e., written as r even in B [2]. Most dictionaries do not have the fourth form of zâda, but as Dozy has it [3], I "have decided to read the word in question as the past participle of the fourth form of the verb zâda which fits well into the context.

The word darûrât, the plural of darûra, is of crucial importance to this text. A perusal of the text will show that at the beginning of most paragraphs the word sayrûra occurs. In one paragraph the word darûra is seen, and at the end of the text the statement supplying us with the title of the text contains the word darûrât. The words darûra and sayrûra could easily be confused with one another, especially in a manuscript like A where letters are rarely dotted. The possibility comes therefore to mind that all these may be the same word and its plural, i. e., either sayrûra or darûra. But manuscript B that is clearly dotted gives them as seen in the edited text below, and in manuscript A too they would seem to agree with the forms given in B. Moreover, the adopted forms seem to represent the most reasonable possibility, as will be further explained below.

The word darûra is obviously not used here in its usual and well-known meaning of social and economic need and necessity. In the Muhît the following meaning is found for this word: "With the logicians it consists of the impossibility of the separation of the predicate from the subject [4]." Several other dictionaries, which I have consulted, do not give such a meaning. In this somewhat unusual meaning, therefore, the word darûra refers to a fixed and necessary relation expressed by certain propositions, or to apodeictic or assertorical necessity.

I have looked into several Arabic texts of algebra with the hope of finding the word darûra in a clearer context, but I have not seen it elsewhere. Al-Khwârazmî's Algebra, which is of greatest significance to this matter because of its chronological proximity to our text, has the word idtirâr that is from the same root but in the eighth form [5]. Gandz quotes the statement of Al-Khwârazmî wherein this word occurs, and he translates it as "logical or algebraic necessity" and takes it to mean "algebraic analysis" in the sense of algebraic reasoning [6]. These passages seem quite clear. The translation "logical necessity" fits the context, and this meaning goes well with the word idtirâr. Al-Khwârazmî's usage of this word too is helpful therefore in the interpretation of the word darûra in our present text.

The word darûra apparently refers in our text to each one of such equations as x² + bx = c or x²=bx + c and such cases as when the discriminant is equal to zero or x < b/2, x > b/2 in the equation x² + c = bx. But it may possibly refer only to these three types of equation.

‘Umar Khayyâm, in his book on algebra makes such statements as 'unknown quantities related to known ones in a manner such that they can be determined' and 'the relations which connect the data of the problems to the unknown [7].' This could be a kind of fixed and necessary relation to which reference could be made in medieval algebra in connection with its equations. In this alternative meaning the word darûrât could be translated as determinate equations.

The name muqtarinât was given to second-degree equations with more than one term on one side of the equality, the other side never being zero. Thus, in contrast to the mufradât, i.e., "simple equations" such as ax² = b, such an equation as x² + bx - c is an example of the muqtarinât, i.e., "mixed equations." The title would thus refer, with this alternative meaning of the word darîrât to the determinate types of "mixed equations."

The translation of darûrât as determinate equations could imply a knowledge of indeterminate equations in Islam before the translation of Diophantos into Arabic. Such a possibility should not be ruled out altogether especially as evidence has been brought to light that examples of such equations are found in the cuneiform tablets [8].

The adoption of this meaning for the word darûra does not seem to be entirely satisfactory, however, for several reasons. Firstly, the idea of fixed relation is emphasized and the idea of logical necessity is pushed to the background. But this latter idea is prominent both in the dictionary meaning of this word and in the closely related term as used by Al-Khwârazmî. Secondly, when the word darûra is translated as determinate equation in the ordinary sense of the term one is tempted to change the words sayrûa into darûra in the text given below [9]; and this would run contrary to manuscript B and would not be the preferred reading in manuscript A. Finally, the title would have to be translated, as stated above, as "the determinate types of mixed equations," and the preposition fî in the Arabic title would not seem to constitute the best choice for conveying such a meaning.

I have therefore preferred the first alternative and taken the word darûra to mean logical necessity without however excluding the meaning 'fixed or uniquely determined relation.' It seems to me that this word is used by ‘Abd al-Hamîd ibn Turk in a slightly different sense as compared with Al-Khwârazmî's usage of the word idtirâr, a difference which is consonant with their dictionary meanings. Al-Khwârazmî's usage of this latter word refers to obvious steps of analytical procedure such as those involved in the operations of "completion" and "reduction." It would therefore seem to approach the idea of analytical treatment, though perhaps not of proof, whereas ‘Abd al-Hamîd appears to use the word darûra in reference to special cases, i.e., all the distinct types of relationships between the unknowns and the coefficients. Thus the equation x² + c = bx is split up into several special cases, whereas no special cases are given for the other two types of "mixed" equations; this is obviously because no subclasses of the other two types of equation present themselves by logical or algebraic necessity. It would seem that darûra does not refer here to geometrical solutions of equations just as Al-Khwârazmî does not use idtirâr in such a sense.

It will be seen from footnote 40 of the Arabic text that although I have adopted the form qawl, manuscript B has the form qawluhu, and this is perhaps true for manuscript A also. One reason for my preference of qawl is that otherwise the third word following it, i.e., ‘Isrîn, would have to be corrected in both manuscripts and made ‘Isrîn. Moreover, the pronoun at the end of qawluhu stands, so to say, in the air, the text containing nothing to which this pronoun can refer. It could be conjectured, however, that the missing earlier parts of the text might possibly be of such a nature as to make this usage meaningful. In that case, ‘Abd al-Hamîd may have referred previously to a certain author or authority and done so several times, but such a speculation is of course meaningless in the absence of the needed text.

The word murabba' seems to be used in slightly varying senses in the text. At times it is used in the meaning of quadrilateral. For when speaking of the geometrical square, murabba' often occurs together with the adjectives equilateral and rectangular. On the other hand, however, murabba' is also used without further specification when referring to rectangles, and at times to squares [10]. I have translated this word as quadrilateral and have generally given a literal rendering of the text although this has made the translation somewhat clumsy at points. I have thereby aimed to give a translation that, in technical points, resembles the original text. The need is felt, e.g., to distinguish between the geometrical square and the algebraic term standing for the square of the unknown quantity.

The term mâl, standing for x², used in Arabic books of algebra, does not, as a word, contain the idea of square. In its ordinary dictionary meaning it is merely a quantity, money, capital, possession, and the like. In the present text the word mâl is used throughout for expressing x², and jadhr, to express x; jadhr means root, or "a quantity which is multiplied by itself," as it is sometimes defined. There are other Arabic terms also for these quantities, but the consistent use of this pair of terms should by no means be unusual.

In our present text x² is seen to come to the foreground, as an unknown, almost as prominently as x, and this observation may be said to be applicable to Al-Khwârazmî as well. It almost seems as if ‘Abd al-Hamîd thinks in terms of an equation of the form X +b ÖX= c, rather than x² + b x = c, X being the real unknown and ÖX the square root of the unknown.

It is of interest in this connection that Al-Khwârazmî occasionally uses the term mâl also for the unknown in the first power [11], and that in his geometrical demonstrations, or solutions of equations, Al-Karkhî (Al-Karajî? [12]) lets line segments represent x² as well as x [13].

Apparently, the question of the Arabic terminology of algebra has interested not only recent historians of mathematics [14] but also the mathematicians of Islam. Thus, there is a marginal note in an Istanbul manuscript of Al-Karkhî's, or Al-Karajî's, algebra called Al-Fakhrî [15]. The writer of this note refers to the three terms shay, i.e., thing, meaning also the unknown x, jadhr, and mâl. He points to slightly varying usages of these terms and to different shades of meaning between them, quoting certain authors as authorities for his statements.

One distinction between shay and jadhr mentioned in this marginal note is that shay refers to the unknown, while the same thing is called jadhr when its value is determined. And another distinction made between them is contained in a statement concerning the terms shay and mâl. It is said that shay and mâl are not used together; implying that if a term in x² exists then the name, jadhr and not shay is given to x.

It is also asserted in this marginal note that shay stands for "the unknown" whether the unknown be in the form of x or x². This is reminiscent of 'Abd al-Hamîd ibn Turk's text, and the following statement in the same marginal note is likewise of interest in this connection. It is said, namely, that the word shay stands both for the "absolute unknown, i.e., the thing whose solution is required, and the unknown which is multiplied by itself; thus, in the second case it is a name given to the root and in the first case a name given to the unknown.

It will be noted that, in the text of 'Abd al-Hamîd ibn Turk, the letters of one of their diagonals generally indicates rectangles. This is quite usual, but the expression "the murabba'" of AB, e.g., is also frequently found used in the same sense, AB referring to the diagonal of a rectangle. I have translated these as "the quadrilateral drawn on" AB, or an equivalent expression. Such examples too make the translation of murabba' as quadrilateral preferable to its translation as square.

It would have been desirable to translate the word mâl without using the word square. But as such a term which would correspond roughly to mâl is not in use at present in algebra; I have chosen the expression "square quantity," which may be said to be in harmony with the terminology employed by 'Umar Khayyâm who is preoccupied with making similar distinctions [16].

Notes 6, 24, 38, 52, 60, and 75 of the Arabic text indicate differences in letters used in the corresponding figures found in manuscript B. There is disagreement in one letter only in the figures indicated by the notes 6, 52, and 60, while the other figures referred to show a difference of two letters. Note 61 also indicate a certain correction that had to be made in the corresponding figure of both A and B, namely the interchange of the positions of the letters H and K. The sign x in the footnotes of the Arabic text is used to show damaged spots, which exist only in manuscript A.

[1] Carl Brockelmann, Geschichte der Arabischen Literatur, supplement vol. I, p. 383.

[3] R. Dozy, Supplément aux Dictionnaires Arabes, Leiden 1881, vol. 1, p. 617.

[5] Muhammad ibn Mûsâ al Khwârazmî, Kitâb al Mukhtasar fî Hisâb al Jabr wa'l Muqabala, ed. and tr. F. Rosen, London 1830, 1831, text, p. 8.

[6] S. Gandz, The Origin and Development of the Quadratic Equations in Babylonian, Greek, and Early Arabic Algebra, Osiris, vol. 3, pp. 515-516.

[7] 'Umar Khayyâm, Algebra, (L'Algebre d'Omar Al-Khayyâm), ed. and tr. F. Woepke, Paris 1851, text, p. 4, tr., p. 5.

[8] S. Gandz, Studies in Babylonian Mathematics, I, Indeterminate Analysis in Babylonian Mathematics, Osiris, vol. 8, pp. 12-40.

[9] The word darûra should refer in this case to the fixed and "necessary" elation implied by each determinate equation. It is of interest in this connection that Al-Khwârazmî speaks of the "cause" of each type of "mixed" equation and considers its answer to be contained in the geometrical figure corresponding to its solution.

[10] Al-Khwârazmî is seen to use the word murabbac to mean square, although in a few examples he adds to it the adjectives equilateral and equiangular as is done by 'Abd al Hamîd ibn Turk. See Al-Khwârazmî, Rosen, text, pp. 1I, 12, 13, 14, tr., pp. 14, 19. See also, Leon Rodet, L'Algèbre d'Al-Khârizmî, Journal Asiatique, series 7, vol. 11, 1878, pp. 90-92.

[11] See S. Gandz, The Sources of Al-Khwârazmî's Algebra, Osiris, vol. i; 1936, p. 273; Gandz, The Origin and Development of the Quadratic Equations, Osiris, vol. 3, p. 535 and note.

[12] Considerable evidence has been brought to light showing that the correct form of this mathematician's name was probably Karajî and not Karkhi. See Adel Ambouba, Al-Karaji, Etudes Littéraires, University of Lebanon, été et automne 1959, pp. 69-70, 76-77.

[13] F. Woepke, Extrait du Fakhrî, Traité d'Algèbre par Abou Bekr Mohammed ben Alhaçan Alkarkhî, Paris 1853, pp. 8, 67-71.

[14] See e. g., Woepke, Extrait du Fakhrî, p. 48; T. L. Heath, Diophantus of Alexandria, A Study of the History of Greek Algebra, Cambridge 1910, pp. 40-41; Gandz, The Sources of Al-Khwârazmî's Algebra, Osiris, vol. 1, pp. 272-274; Gandz, The Originand Development ..., Osiris, vol. 3, pp. 535-536, note 94.

[16] L'Algèbre d'Omar Alkhayyâmî, ed. and tr. F. Woepke, Paris 1851, see, e.g., text, p. 5, tr., pp. 7-8.