Logical Necessities in Mixed Equations: Chapter III
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3. ‘Abd Al Hamîd Ibn Turk's Logical Necessities in Mixed Equations
The knowledge of the history of algebra underwent a great transformation when it was shown, by O. Neugebauer, about thirty years ago, that the Mesopotamian cuneiform tablets gave proof of the existence of a rich knowledge of algebra as far back as two thousand years before our era , This was bound not only to change the perspective of Greek mathematics in general and of Greek algebra in particular but also to show Al-Khwârazmî's place in the history of algebra in a different new light.
The importance of Al-Khwârazmî's place in the history of algebra may be said to rest upon two considerations, although his algebra is more primitive in some respects than certain earlier phases of "algebra and even though no originality can be claimed for him in this branch of mathematics. He is, or was, thought to have written the first treatise or systematic manual on algebra, or, at least, to have been the first to do so in Islam; his book on algebra played a great part in the transmission of the knowledge of algebra to Europe, being also instrumental in giving this discipline its European name .
‘Abd al-Hamîd ibn Turk's Algebra constitutes a serious challenge to the first of these two items in Al-Khwârazmî's title to fame, still leaving him as an influential figure in the history of algebra.
It may be surmised from the statements of Abû Kâmil Shujâ' that there possibly was some kind of rivalry between Al-Khwârazmî and ‘Abd al-Hamîd himself in the field of algebra. Indeed, Abû Barza may not have been the originator of the controversy but someone who renewed it in an intensified form. It is likewise possible that this rivalry did not consist merely of a question of priority but that also certain differences of approach existed between them. 'For Abû Kâmil Shujâ', in the beginning of his Algebra too, praises Al-Khwârazmî for the solidity and superiority of his knowledge of the subject .
Gandz sees in a certain statement of Al-Khwârazmî "a sharp point of polemics." He refers here first to the old Babylonian tendency to reduce problems to certain types of equations and to avoid the types found in Al-Khwârazmî. He then explains that Al-Khwârazmî represents the complete reversal of this tendency. The "point of polemics" in question refers to Al-Khwârazmî's instruction and recommendation to reduce all problems to one of the "mixed" equations. Gandz sometimes speaks of Al-Khwârazmî merely as a representative of this new school in algebra, but apparently he considers him as one who at least played an important part in making the ideas or practices of this school predominant .
The question comes to mind therefore whether any divergences having to do with such matters may have existed between Al-Khwârazmî and ‘Abd al-Hamîd ibn Turk. A brief perusal of ‘Abd al-Hamîd's text shows clearly that there were no major divergences of such nature between the two authors. It is possible that the slight difference seen in their terminologies already referred to may be of interest in this respect. Differences of pedagogical presentation too should be likely. At any rate, it seems quite reasonable to think that any differences which may have existed between them were of such a nature as not to seem of any considerable magnitude from our distance.
As the text of ‘Abd al-Hamîd that is available at present is only a part, and probably a relatively small part, of his book, the information available to us is insufficient to answer such questions. In summarizing and analyzing this text, therefore, I shall keep this problem in the background and shall prefer to utilize this text, as much as possible, to the end of increasing our knowledge of the algebra of the time of Al-Khwârazmî and ‘Abd al-Hamîd.
The present text of ‘Abd al-Hamîd ibn Turk which is of about fourteen hundred words is probably a chapter of ‘Abd al-Hamîd's Algebra. It may possibly consist of only a part of one chapter, but in that case it forms a well-rounded section, complete in itself with its beginning and end.
Our text begins with the equation x2 = bx. This is apparently placed at the beginning of the section as an introductory passage. For this equation is not a "mixed" equation, i.e., of the type called muqtarinât unless the definition of this type of equation may be extended to include equations in x2 and x. His explanation of the solution suggests that, just like Al-Khwârazmî , he does not think of dividing such an equation as x2 = bx through by x. This is a clear sign of the predominance of the geometrical way of thinking, as contrasted to the analytical, in this algebra.
In Nesselmann's classification, the present text represents the rhetorical stage of algebra. The equations dealt with, besides the above-mentioned x2 = bx, are x2 + bx = c, x2 + c = bx, and x2 = bx + c. Their solutions are based on geometrical reasoning. The idea of negative quantity does not exist, and in the representative types of equation none of the terms is subtracted. These three types of second degree equation, therefore, taken together, do not add up quite to the general case ax2 + bx +c = 0. As would be expected, in all these general characteristics ‘Abd al-Hamîd ibn Wâsî's text shows no differences with that of Al-Khwârazmî.
The numerical example ‘Abd al-Hamîd ibn Turk gives for the equation x2 + bx = c is x2 + 10x = 24. He is thus seen not to use here the famous equation x2 + 10x = 39 found in Al-Khwârazmî, Al-Karkhî, or Al-Karajî, ‘Umar Khayyâm, Fibonacci, and others. The geometrical figure used for this equation is formed by adding two rectangles of sides x and b/2 to the adjacent sides of a square representing x2 and then completing the larger square of side x + b/2 by the addition of the square (b/2) 2. He thus has x + bx + (b/2) 2 = c + (b/2) 2, and this gives the solution x = Ö((b/2)2+c-b/2). This familiar figure is found, e.g., in Al-Khwârazmî but not in ‘Umar Khayyâm. This is the only figure used by ‘Abd al-Hamîd. Al-Khwârazmî has a second figure which is also found in ‘Umar Khayyâm's Algebra .
For equation x = bx + c ‘Abd al-Hamîd ibn Turk gives the example x2 = 4x + 5. On the same occasion Al-Khwârazmî uses the equation x2 = 3x + 4 and ‘Umar Khayyâm x2 = 5x + 6 . In his geometrical figure, ‘Abd al-Hamîd starts with the square x2 and subtracts from it a rectangle equal to c. The remaining rectangle equals bx. One side of this rectangle being equal to b, the square figure (b/2) 2 is drawn on its side adjacent to the rectangle c. Two sides of this square are then lengthened by the amount x - b, thus forming the square [b/2+(x-b)]2=x2-bx+(b/2)2. This square is equal to c+(b/2)2. Each of its sides therefore is equal to Ö((b/2)2+c, and to obtain x we have to add b/2 to this quantity.
The treatment of the type x2 + c = bx constitutes the most interesting part of ‘Abd al-Hamîd's text. The examples he gives for this equation are x2 + 21 = 10x with its double solution, x2 + 25 = 10x and x2 + 9 = 6x for the case x = b/2, and x2 + 30 = 10x for the case with no solution. The equation x2 + 21 = 10x is also found in Al-Khwârazmî  and Al-Karkhî, or Al-Karajî . ‘Umar Khayyâm mentions the terms x2 and 10x but leaves the constant term undetermined, saying that it can be varied for the different cases that occur . There is thus a parallelism also between ‘Umar Khayyâm and ‘Abd al-Hamîd in this respect.
Figure 6: A stamp issued by Iran in 1980, depicting al-Farabi, al-Biruni, and Avicenna. (Source).
The geometrical scheme followed in the solution of this type of equation is to represent c by a rectangle one side of which is equal to x and to juxtapose x2 and c in such a manner that their equal sides are superposed. They thus form one single rectangle together. Then the square (b/2) 2 is drawn in such a manner that one of its angles coincides with one of the angles of the rectangle c which are not adjacent to x2.
In the general case where b/2≠x and (b/2)2 > c, there are two possibilities. The case x < b/2 in which one has to find the value of the geometrical square (b/2-x)2, and the alternative x > b/2 in which case one has to find the value of the geometrical square (x-b/2)2. The procedure followed is to find geometrically the value of the square Ö((b/2)2+c which is equal to (b/2)2-bx+x2, i.e., equal to (b/2-x)2 for x < b/2 and to (x-b/2)2 for x > b/2. Thus for x < b/2, b/2-x=Ö((b/2)2-c), and x = b/2 - Ö((b/2)2-c); and for x > b/2, x - b/2 = Ö((b/2)2-c) and x = b/2 + Ö((b/2)2-c). Hence, two values are found for x, and this demonstration of two solutions for x2 + c = bx is based on a purely geometrical reasoning.
The case x = b/2 is also considered, and it is shown geometrically that in this case c must be equal to (b/2)2 in order to obtain a solution for the unknown x.
Thinking in terms of our general equation and formula ax2 + bx + c = o and x = [-b±Ö(b2-4ac)]/2a, the equation of the type x2 + c = bx represents the case of positive c and positive a. This is therefore the type in which imaginary, roots may occur. ‘Abd al-Hamîd discusses this also, and shows with the help of two geometrical figures that in case c > (b/2)2 the equation has no solution regardless of whether we imagine x < b/2 or x > b/2.
The geometrical scheme of demonstration adopted by ‘Umar Khayyâm for the type x2 + c = bx is seen to be different from that of ‘Abd al-Hamîd . Al-Khwârazmî uses a figure which is identical with that of ‘Abd al-Hamîd for the case c < (b/2)2 and x < b/2. For the case c < (b/2)2 and x > (b/2)2 the extant Arabic text of Al-Khwârazmî, as it has come down to us in Rosen's edition, contains no figure and no special treatment, but its Latin translation by Robert of Chester has a figure that is different from that given by ‘Abd al-Hamîd, in that it is of a composite nature .
The general solution for the second degree equation x = [-b± Ö(b2-4ac)]/2a becomes x = -b/2± Ö((b/2)2-c) for the case a = 1. Since in the times of ‘Abd al-Hamîd ibn Turk and Al-Khwârazmî the negative root was excluded, only a positive root could be conceived as added or subtracted. Now, in Al-Khwârazmî, there is only one solution for each one of the equations x2 + bx = c and x2 — bx + c, namely, x = Ö((b/2)2+c) - b/2 and x = Ö((b/2)2+c)+b/2 respectively, but two solutions for the equation x2 + c = bx, viz., x = b/2±Ö((b/2)2-c).
In the equation x2 + c = bx therefore the positive square root is added to b/2 to obtain one solution and subtracted from the same quantity to obtain the other solution. Gandz has explained this peculiarity by tracing these three equations back to their Babylonian origin.
The types of equations found in the cuneiform tablets are, according to the list given by Gandz , the following:
I. x + y = b; xy = c
II. x — y = b; xy = c
III. x + y = b; x2 + y2 = c
IV. x — y = b; x2 + y2 = c
V. x + y = b; x2 — y2 = c
VI. x — y = b; x2 — y2 = c
VII. x2 + bx = c
VIII. x2 — bx = c
IX. x2 + c = bx
Types I and II lead directly, III and IV with change in the constant term, to the types VII, VIII, and IX; types V and VI become transformed into first degree equations when reduced to one unknown. Types VII, VIII, and IX are of course those found in ‘Abd al-Hamîd and Al-Khwârazmî.
When the pair of equations of type I is reduced to one unknown one obtains two equations of the type x2 + c - bx, one in x and the other in y. Hence, the two solutions for the unknown, adding up to b. That is, the two solutions for x in this equation stand for the two solutions, one for x and the other for y, in the original pair x + y = b; xy = c. When, on the other hand, the second Babylonian type is similarly transformed, one of the two equations obtained is in the form x2 + bx = c and the other in the form x2 = bx + c. Hence, the two single solutions for these two equations correspond to the two solutions, one for x and the other for y, in the original pair x - y = b; xy = c. Thus, the four solutions for the three "mixed" equations are accounted for without having to think in terms of a negative square root .
Such therefore is the historical background of the algebra of ‘Abd al-Hamîd ibn Turk also. In fact, the two cases of the equation x2 + c = bx which are clearly distinguished in ‘Abd al-Hamîd's text may be said to represent a distinction between the two original solutions x and y. This clear twofold classification may therefore be looked upon as a vestige of past his¬tory and may be said to corroborate the explanation given by Gandz.
It may be said, on the other hand, that to distinguish between the cases x < b/2 and x > b/2 is merely tantamount to saying that x has two solutions, and it may be added that the text of ‘Abd al-Hamîd ibn Turk seems to be self-sufficient with respect to the explanation of the occurrence of the double solution so as not to leave much need for referring back to the original forms of the "mixed" equations. ‘Abd al-Hamîd ibn Turk's Logical Necessities in the Mixed Equations is thus seen to have the strange quality of suggesting a need for giving further thought to this matter.
 Otto Neugebauer, Studien zur Geschichte der Antiken Algebra, Quellen und Studien zur Geschichte der Mathematik, Aslmnomie, und Physik, series B, vol. 2, 1932, pp. 1-27; Kurt Vogel, Bemerkungen zu den Quadratischen Gleichungen der Babylonischen Mathematik, Osiris, vol. 1, 1936, p. 703.
 See e. g., Gandz, The Sources of Al-Khuwdrizmi's Algebra, Osiris, vol. 1, p. 270; Gandz, The Origin and Development ..., Osiris, vol. 3, p. 409.
 See manuscript, Istanbul, Bayezjt Library, No. 19046 (or, Kara Mustafa Pasa, No. 379), p. 2a.
 Gandz, The Origin and Development ..., Osiris, vol. 3, pp. 509-511, 542-543.
 See Gandz, The Origin and Development ...., Osiris, vol. 3, p. 538.
 See Woepke, L'Algèbre d'Omar Alkhayyâmî, p. 19.
 See Woepke, ibid., pp. 23-25.
 See Woepke, L'Algèbre d'Omar Alkhayyâmî, p. 22, note.
 Woepke, Extrait du Fakhrî, p. 67.
 Woepke, L'Algèbre d'Omar Alkhayyâmî, p. 21.
 Woepke, L'Algèbre d'Omar Alkhayyâmî, pp. 21-33.
 Gandz, The Origin and Development ..., Osiris, vol. 3, pp. 521-523. See below, p. 108, figure I.
 Gandz, The Origin and Development ..., Osiris, vol. 3, p. 405.
 Gandz, The Origin and Development ..., Osiris, vol. 3, pp, 412-416.
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by: FSTC Limited, Thu 15 January, 2009