Logical Necessities in Mixed Equations: Chapter V
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5. The Origins And Sources Of The Algebra Of Cabd Al Hamîd Ibn Turk And Al Khwârazmî
One of the most important and influential books in the history of algebra is Diophantos' Arithmetica. Apparently, the algebra of Diophantos is directly related to and derived from the old Babylonian school. But Diophantos did not influence the algebra of Al-Khwârazmî and ‘Abd al-Hamîd ibn Turk. His influence was felt in Islam after his book was translated by Qustâ ibn Lûqâ (d. ca. 912), but this was after the times of ‘Abd al-Hamîd and Al-Khwârazmî.
I have outlined above the development of algebra, or the transformations it underwent, from its old Babylonian origins up to the time of Al-Khwârazmî, as conceived and brought to light by Gandz. The first stage was characterized by the usage of certain types of equations. At an intermediary stage the three "mixed" equations also came to be used, but the type x2 + c = bx was avoided. At the stage represented by Al-Khwârazmî the old Babylonian types became excluded and the "mixed" equations began to be used.
A question that naturally comes to the mind is the reason for this reversal of attitude. Let us hear its answer from Gandz.
He says, "In Al-Khwârazmî's algebra we may easily discern the reverse of the Babylonian attitude. Here we find that the three Arabic types are used, regularly and exclusively. The old Babylonian types and methods are entirely rejected and repudiated. They never occur; they are thoroughly discarded, while the three Arabic types, formerly the struggling and tolerated methods, are now the dominating ones to the complete exclusion of the old Babylonian methods. In Al-Khwârazmî's Algebra we, very frequently, find the same problems as in the Babylonian texts. But the Babylonian methods of solution though near at hand and though very convenient, are systematically avoided. Herein lays the great merit of Al-Khwârazmî, his great contribution to the progress of algebra. He does away with all those brilliant ideas, ingenious devices, and clever tricks adopted by the Babylonians for the solution of their individual problems. He entirely spurns this romanticism and individualism in the algebra, and instead, he introduces and originates what we may call the classic period in algebra. The methods of solution are, so to say, standardized. There are only, three types and all the quadratic equations and all the problems may be reduced to these standard types and solved according to their rules. 'And we found,' so he says at the outset of his book, on p. 15, after the six types (bx2 = ax, bx2 = a, bx = a, x2 + ax = b, x2 + b = ax, x2 = b + ax) have been described and explained by him, 'that all the problems handled by algebra will necessarily be reduced to one of these six types just described and commented upon. So bear them in mind.' There is a sharp point of polemics in these words. He means to say: You must not waste your time with the study and practice of all those antiquated Babylonian types and of all the innumerable devices and tricks to be employed in order to reduce the problems to these types, of equations. It is enough to study these three types that I have just expounded, and you will be in a position to solve all the problems. And the rest of his algebra is planned to bear out this statement. The problems are selected from the great storehouse of Babylonian mathematics and are all reduced to the three Arabic types. Thus the uselessness of the Babylonian methods and the usefulness of the Arabic methods are fully demonstrated ."
As to the relation between the algebra of the time of Al-Khwârazmî and Greek geometric algebra as developed by the Pythagoreans and found in Euclid, Gandz is of the belief that although this algebra too is based' upon and derived from the Babylonians no direct relations or connections exist between it and the algebra of Al-Khwârazmî.
Speaking of geometrical demonstrations and comparing Euclid and Al-Khwârazmî, Gandz says, "Euclid demonstrates the antiquated old Babylonian algebra by a highly advanced geometry; Al-Khwârazmî demonstrates types of an advanced algebra by the antiquated geometry of the ancient Babylonians.
"The older historians of mathematics believed to find in the geometric demonstrations of Al-Khwârazmî the evidence of Greek influence. In reality, however, these geometric demonstrations are the strongest evidence against the theory of Greek influence. They clearly show the deep chasm between the two systems of mathematical thought, in algebra as well as in geometry ."
And concerning Diophantos he says, "Both, Al-Khwârazmî and Diophantos, drew from Babylonian sources, but whereas Diophantos still adheres to old Babylonian methods of solution, Al-Khwârazmî rejects those old methods and introduces the more modern methods of solution ."
The first algebra which made its appearance in Islam becomes, therefore, according to this theory, a direct-line descendent of Babylonian algebra without any intervening or interfering side-influences. A crucial test and one of the most weighty arguments Gandz offers for this thesis rests on his ability to account for the occurrence of the double root of the equation x2 + c = bx. Let us examine then this thesis of Gandz a little more closely.
As we have seen, there is ample evidence that Al-Khwârazmî, and therefore also ‘Abd Al-Hamîd ibn Turk, knew that the two solutions x1 and x2 of an equation x2 + c = bx often did not correspond to the solutions x and y of a set of equations F(x, y) = 0 and f(x, y) = 0 which leads to the equation x2 + c = bx. It does not seem very satisfactory to think therefore that their explanation of the double root was made through reference to the system x+y = b; xy = c.
In fact, had such been the case, Al-Khwârazmî's text would very likely have revealed it unambiguously. For, in connection with the double root, clear though implicit reference is made, as we have seen, to the sets of equations F(x, y) = 0 and f(x, y)> =0, when the equation x2 + c = bx is derived from such a set. This reference is made, however, not in order to explain the occurrence of two solutions for a single equation but in order to make a choice, if necessary, between the two solutions. Moreover, references of such a nature are necessary to a set of equations in its more general form which would not serve to explain the occurrence of two solutions, and not to the special case x + y = b; xy = c.
The explanation offered by Gandz could therefore be completely valid only in an indirect manner, when the matter is traced back to its origins in the past. The answer that Al-Khwârazmî would have given for the occurrence of the double root would, furthermore, have to be also and especially in terms of the equation x2+ c = bx itself and not merely in terms of a set of equations F(x, y) = 0 and f(x, y) = 0 leading to it.
That answer is ready at hand in ‘Abd al-Hamîd's text, as we have seen, and it concerns the equation x2+ c = bx itself, as it, at least partly, should, and not a set of simultaneous equations from which x2 + c = bx may be considered derived. This manner of accounting for the double root constitutes therefore the valid answer to our version of the question, i.e., the answer not pertaining to distant origins but the one Al-Khwârazmî himself would have supplied, and it serves, appropriately, to bring the method of geometrical demonstration well into the foreground, as it has been pointed out before.
We are, moreover, here in the presence of an algebra which accepted the double root of x2 + c = bx without any hesitation. Would not its explanation by referring it directly to an algebra which felt uneasy toward the double root somewhat miss the point? The required explanation should also elucidate the passage and transition between the two types of algebra. There is obviously an important missing link in Gandz' explanation, arid it is essential, or highly desirable at least, not to bypass it.
What, then, was the nature of the hesitation felt toward the equation x2 + c = bx and why and how did that hesitation disappear? Speaking of the Babylonian algebra and the equation x2 + c = bx to which he refers by the symbol A II, Gandz writes as follows.
"With regard to type A II, however, the writer's theory is that it was never made use, of. It must have been well known to the Babylonian mathematicians, but all kinds of ingenious devices were used to avoid this type of equations. So far, no Babylonian text came to my knowledge which would plainly, expressly and unequivocally exhibit this type of equation and the instruction for its solution, as was the case with the two Arabic types in BM 13901. Indeed, the most remarkable thing of this old document, seems to me to be that in its first part, dealing with equations of one unknown, it has no example of the type x2 + b = ax, whereas the other two types are distinctly taught in several instances. The lesson it teaches us is plainly that in the mathematical school from which this text comes such a type of equation was not recognized at all.
"This lesson is repeated and further corroborated by several other texts. The problems treated in those texts could be very simply and easily solved, if they were reduced to the equa¬tion x2 + b = ax. Instead of that, however, all kinds of tricks and ingenious devices are employed in order to reduce them to the type of an equation with two unknowns x+ y = a; xy = b. Our modern students of Babylonian mathematics explained these equations of type B I by reducing them to the form x2 + b = ax. The Babylonian mathematicians, however, proceeded quite in the opposite way. They made all efforts to transform the equations of the type x2 + b = ax into the type x + y = a; xy = b. The reason for it is clear and was already mentioned in this paper (§ 3, p. 415). The type was well known to them and it was also known to them that it leads to two solutions and two values. This idea of two values for one and the same quantity seems to have been very embarrassing. It was re-garded as an ambiguity, as an illogical absurdity and as nonsense. Hence all the ingenious devices were invented in order to circumvent, dodge and forestall the use of this embarrassing type ."
Again, Gandz says, "In the selection and arrangement of these 8 problems there is plan and method and no mere chance and accident. Evidently the formulation and the plan-full arrangement of these examples have the aim of furnishing instances for the two fundamental Babylonian types and the three Arabic types. Most characteristic are especially the two last examples which demonstrate the two possible solutions of type A H, or else, let us say, the confusion arising out of the use of this type. We have here before us a regular and systematically lesson in the five fundamental types of the quadratic equations. The lesson gradually progresses from the plain and simple to the more difficult and more complicated tasks. The probability is that the Babylonian teacher chose these examples and this arrangement in order to demonstrate through them the great practicability and the usefulness of the old, traditional Babylonian methods. He most probably wanted to show that all these new fangled Arabic types may be reduced to the old Babylonian types, thus saving us from the confusion and duplicity involved in the use of types all ." The term "Arabic types" refers here to the "mixed" equations x2+ bx = c, x2 + c = bx, and x2 = bx + c.
On still another occasion Gandz expresses his ideas on this matter in the following manner: "But a dualism of value and of solution for one and the same quantity must have appeared to the Babylonian mathematician as a strange thing. That one and the same quantity should be the length and breadth of a rectangle, should amount to 3 and to 7 at the same time, must have been regarded by him as an illogical nonsense; he must have shunned it with abhorrence as an absurdity and monstrosity, belonging into the art of magic rather than into the science of mathematics ."
We have seen that according to Gandz the reasons leading to the exclusive adoption of the "mixed" equations by the school represented by AI Khwârazmî were akin to a principle of economy; these mathematicians wished to standardize the solutions and to make algebra less dependent upon ingenious devices and clever tricks. As to why such an attitude was not adopted by the Babylonians, we now see him give the reason that they did not feel quite at home with the idea of two solutions for one and the same quantity. Presumably therefore what prevented an earlier adoption of the principle of economy in algebra was the hesitation felt toward the double solution of the equation x2 + c = bx.
At least from the standpoint of the algebra of Al-Khwârazmî it seems proper to distinguish between two cases in connection with the hesitation felt for this double root. The case where the equation was directly given; and the case of its derivation from a pair of simultaneous equations in two unknowns. It could be conjectured that for the Babylonian mathematician there should have been less room for hesitation when x1 and x2 corresponded to the solutions sought for x and y. On the other hand, the derivation of two distinct values for one and the same quantity, both satisfying the equation and not traceable to two unknowns entering the problem to be solved, may have produced some kind of a psychological difficulty.
In the algebra of Al-Khwârazmî the double solution seems to have been looked upon as something quite normal when the equation was directly given. But when the equation was derived from a pair of simultaneous equations, there was the possibility that both solutions of x2 + c = bx would not correspond to the two unknowns of the simultaneous equations, and additional considerations had to come into play.
This being the situation, the turning point and the major characteristic of the algebra of the school to which Al-Khwârazmî belonged must have been closely related with the disappearance of the hesitation felt for the double solution of the equation x8 + c = bx when this equation was directly given. As we have seen, it was the method of geometrical demonstration that made this double solution seems quite natural and understandable. It is to be concluded therefore that it was through the superposition of this method of geometrical demonstration on the Babylonian algebra that the new algebra came into being. It was not a direct-line descendent of the Babylonian algebra as found in the cuneiform tablets; but a side influence responsible for the adoption of the geometrical method of demonstration was indispensable for its coming into being.
As we have seen, Gandz finds in this method of geometrical demonstration "the strongest evidence against the theory of Greek influence ." What is it that makes Gandz think so? He mentions two reasons. He believes the geometrical figures of Al-Khwârazmî's algebra to be essentially different from the corresponding figures found in Euclid; he sees a great difference between the geometries underlying these two types of demonstration.
Now, are the figures of Al-Khwârazmî's algebra essentially different from the corresponding figures found in Euclid? Corresponding to x-x' = b; xx' = c Euclid has the figure presented here (Figure 9) . The successive steps of solution may be represented as follows:
EFDHKA = LMBA=xx'=c (2)
EA= Ö((b/2)2+c (4)
x= EA – b/2 = Ö((b/2)2+c) – b/2 (5)
x= EA + b/2 = Ö((b/2)2+c)+b/2 (5)
In Al-Khwârazmî and ‘Abd al-Hamîd ibn Turk the equation x'2 + bx' = c does not contain x. The rectangle LMEF is therefore not needed. In fact such is exactly the figure found both in Al-Khwârazmî and ‘Abd al-Hamîd. In solving the equation x'2 + bx' = c, for similar reasons equality (2) above will naturally not occur. Instead, one will have EFDHKA = x'2 + (b/2)x' + (b/2), x' = x'2 + bx' = c, and the remaining relations (3), (4), and (5), will be identical. This is actually seen to be the case in Al-Khwârazmî and ‘Abd al-Hamîd ibn Turk.
That the solution and figure given here by ‘Abd al-Hamîd ibn Turk and Al-Khwârazmî are in harmony with Euclid's way of thinking, as far as the difference of the algebraic background implied by the distinction between the "mixed" equations and the old Babylonian types is concerned, may be considered further corroborated by the fact that theorem 4, e.g., of book II in the Elements can easily be brought into direct correspondence with equation x'2 + bx' = c in one unknown considered here. Its figure is quite similar to the corresponding figure of ‘Abd al-Hamîd and Al-Khwârazmî, and the proof of this theorem, likewise, is essentially the same as the solution for this equation found in ‘Abd al-Hamîd ibn Turk and Al-Khwârazmî.
For the geometrical figure illustrating the solution of x2 = bx + c, the longer straight line x in fig. 2 has to be preserved.
The rectangle xx' is also preserved. The side x' therefore remains in the new figure automatically although it does not appear in the equation. The modifications seen in the figure of Al-Khwârazmî and ‘Abd al-Hamîd ibn Turk, as compared to Euclid's figure, are, firstly, that the square of x is drawn, and secondly, that the rectangle ACHK appears attached to the left side of the square drawn on b/2 instead of being on its right side.
None of these two modifications introduce anything essentially new as compared to the figure found in Euclid. To compare the figure of Al-Khwârazmî and ‘Abd al-Hamîd (Figure 10) with that of Euclid (Figure 10), we may show the first modification, as well as a missing x' of Euclid's figure, in dotted lines. The main difference between the two figures, then, is that in the new figure the square of the unknown is shown. This is natural, as here x is the only unknown and is derived independently.
These trivial alterations in Euclid's figure do not result, moreover, in any change in the manner of geometrical reasoning. The solution of the unknown is based on exactly the same kind of geometrical demonstration. Here too the main relation utilized is c + (b/2)2=(x-b/2)2.
Euclid's figure for x + x' = b; xx' = c (Figure 12)  is, as will readily be seen, very similar to the figure (Figure 11) given for x'2 + c = bx'; x' < b/2 in the algebra of ‘Abd al-Hamîd and Al-Khwârazmî. And the principle of geometrical proof of the solution is likewise identical. The main relation utilized is
(b/2)2 = xx'+(b/2-x')2, or (b/2)2=c+(b/2-x')2.
The first form of this relation containing the term xx' of course does not appear in the solution of the equation x'2 + c = bx.'
For x2 + c = bx; x > b/2, it may be guessed from the previous example concerning the equation x2= bx + c that a larger square, i.e., the square of x will appear in the figure. Again, here too, the part of Euclid's figure (Figure 11) representing x'2 and (b/2)x' jointly should naturally play a secondary part. This portion of the figure is seen to be transposed in a manner corresponding exactly to the transposition found in the figure for x2 = bx + c. The resulting figure (Figure 13) is that found in ‘Abd al-Hamîd ibn Turk's text. For facilitating comparison the new parts and a missing x' are shown in dotted lines in this figure (Figure 13).
The alterations in the figure are thus trivial. The principle of geometrical demonstration of the solution remains, moreover, exactly the same. For the main relation utilized is (b/2)2=c+(x-b/2)2.
It is therefore quite clear that the geometrical figures of Al-Khwârazmî's algebra, far from being totally different from the corresponding figures found in Euclid's Elements, are essentially the same as the latter, and the nature of the geometrical demonstrations in the two cases arc, for all intents and purposes, and as geometrical demonstrations, identical. Why, then, does Gandz believe the two geometries underlying these two types of demonstration to be very different from one another?
Gandz says, "His (Euclid's) figure has nothing in common with the two, respectively three, figures of Al-Khwârazmî. The latter one proves two Arabic types independent of each other.-Euclid prove the ancient Babylonian type B II. Algebraically, AI Khwârazmî is ahead of Euclid with 1000 years, geometrically, he is behind of Euclid with 1000 years. His demonstrations are based entirely upon intuition. He has nothing else to base them upon. In his geometry the Euclidean axioms, definitions, theorems, and propositions are entirely ignored ."
On another occasion Gandz reaches the conclusion that Al-Khwârazmî knows nothing of the special importance of the problem of cutting a given straight line in extreme and mean ratio .
Speaking of the equation x2 + c = bx, in a passage partly quoted before, Gandz writes, "In. spite of their apparent similarity, the two figures of Euclid and Al-Khwârazmî are basically and intrinsically different. They are proving different cases by different methods. Euclid proves and demonstrates a case of ancient Babylonian algebra; Al-Khwârazmî demonstrates a case of modern Babylonian school whose algebra came to be regarded as Arabic algebra. Euclid demonstrates the antiquated old Babylonian algebra by a highly advanced geometry; Al-Khwârazmî demonstrates types of an advanced algebra by the antiquated geometry of the ancient Babylonians.
"The older historians of mathematics believed to find in the geometric demonstrations of Al-Khwârazmî the evidence for Greek influence. In reality, however, these geometric demonstrations are the strongest evidence against the theory of Greek influence. They clearly show the chasm between the two systems of mathematical thought, in algebra as well as in geometry ."
Speaking of the part of Al-Khwârazmî's Algebra dealing with menstruation, Gandz says, "Now, if Al-Khwârazmî had really studied Greek mathematics, we were certainly justified in the expectation to find some traces of the content or terminology of Euclid's Elements in his geometry. The fact, however, is that there are no such traces of Euclid in Al-Khwârazmî's geometry. Euclid's Elements in their spirit and letter are entirely unknown to him. Al-Khwârazmî has neither definitions, nor axioms, nor postulates, nor any demonstration of the Euclidean kind. He has just a plain treatise on menstruation, a compilation of popular rules for the practical purpose of land surveyors ."
Each of Euclid's figures concern two unknowns and are thus equivalent to two figures in Al-Khwârazmî's or ‘Abd al-Hamîd ibn Turk's algebra. Thus Euclid's figures correspond to the Babylonian equations with two unknowns while those of Al-Khwârazmî and ‘Abd al-Hamîd correspond to the "mixed" equations and their special cases. But these do not constitute basic and essential differences as far as the type of geometry going into the demonstrations is concerned.
It should not be safe, furthermore, to conclude that Al-Khwârazmî was not familiar with the Euclidean geometry just because the part of his Algebra dealing with menstruation is not based on Euclidean methods of treatment. Menstruation was apparently a subject conceived as serving the need of surveyors in a practical manner by supplying them with ready formulas and instructions. Al-Khwârazmî must have written this part of his book in conformity to a set prototype. Abû Barza ibn Turk wrote an independent book on this subject .
Euclid's Elements is said to have been made available in Islam already during the reign of the Abbasid caliph Al-Mansur (754-775). This item of information which is accepted by Kapp  is derived apparently from Ibn Khaldun. According to Ibn Khaldun, upon Al-Mansur's request the Byzantine emperor sent him a number of books among which was that of Euclid . This seems reasonable. For there is information that a translation of Euclid into Arabic was made by Hajjaj ibn Yusuf ibn Matar around the year 790 for Harun al-Rashid (786-809) . This was probably its first translation made in Islam. Hajjaj ibn Yusuf made a second translation of this book for Al-Mamun (813-833). This translation has come down to us partly, and, moreover, Sanad ibn ‘Alî, one of Al-Mamun's astronomers, is said to have written a commentary on it .
Euclid's Arabic translation was later improved especially by Hunayn ibn Ishaq, Ishaq ibn Hunayn, and Thabit ibn Qurra. But the details presented above indicate that Euclid's Geometry became, known to the mathematicians of Islam during the reign of Harun al-Rashid. Al-Khwârazmî, on the other hand, was attached especially to the Bayt al-Hikma whose most prominent function was its being the centre of translation activity . It seems unlikely therefore that Al-Khwârazmî should not have been familiar with the Euclidean geometry.
But it is not really relevant to our subject whether Al-Khwârazmî knew Euclidean geometry or not, and this is apparently the point on which we must dwell for a moment. For the difference between the conclusions we have reached and that of Gandz has its roots in this point.
From the quotations just given, it is seen that Gandz looks upon the Euclidean geometrical demonstrations of the algebraic equations in question as forming an inseparable part of the Euclidean geometry with its definitions, axioms, postulates, and proofs. It is true of course that the material in question appears in Euclid in the form of theorems. But these Euclidean theorems could, together with their proofs, be very well taken out of their Euclidean context and placed within a less advanced type of geometry. And they could likewise be presented in the form of problems, as they were in their Pythagorean origin, without altering their contents or details of procedure in any essential manner.
These geometrical demonstrations of Al-Khwârazmî and ‘Abd al-Hamîd ibn Turk, as well as the corresponding Euclidean propositions considered in their historical background, resemble the solutions of the second degree equation by the analytical method of completing squares as seen in modern textbooks of algebra. They are proofs in the sense that they prove the correctness of the solutions of these types of problems treated with a geometric scheme of approach. These proofs are based upon the knowledge of certain geometrical relations, but they do not necessarily presuppose a logical system of geometrical reasoning based on definitions, axioms, postulates, and theorems. The nature of these geometrical demonstrations, even in their Euclidean form, is such that, in a sense, they need not be considered as necessarily partaking of the logical and systematic perfection of the Euclidean geometry. For they are based on very simple geometrical knowledge.
As they occur in Euclid's text, the underlying geometrical relations are of course based on axioms, postulates, and theorems. But, on the other hand, there is nothing forbidding us from assuming that the situation is the same in the case of Al-Khwârazmî. The point, however, is that this particular question is not very relevant here.
The geometrical demonstrations of Euclid as well as those of Al-Khwârazmî and ‘Abd al-Hamîd ibn Turk stand indispensably in need only of the type of geometry which was developed, or, at any rate, represented, by the Pythagoreans. Whether, as Gandz says, this also corresponds to Babylonian geometry, or whether it was derived from it or was similar to it, is still another question.
It would seem quite safe to say that the geometrical knowledge necessary for the geometrical demonstrations found in, Al-Khwârazmî and ‘Abd al-Hamîd ibn Turk was not beyond the reach of the Babylonian mathematicians. But there apparently is no documentary evidence indicating that such geometrical demonstrations were used by the Babylonians in their algebra.
As we have seen, Al-Khwârazmî speaks, in his Algebra, of the learned men "in times which have passed away and among nations which have ceased to exist" who "were constantly employed in writing books on several departments of science and various branches of knowledge ," Looking at the first part of this quotation, it seems probable that Al-Khwârazmî himself associated his algebra with the Babylonians. But it is also possible to see here an allusion to the Greeks. Moreover, the second part of the quotation refers to books written on various scientific subjects, and this applies more readily to the Greeks. This seems especially likely when we take into consideration the fact that the Arabic text contains reference also to books written on philosophy, or wisdom, (Hikma) , a word which does not appear very clearly in Rosen's translation quoted above.
The interpretation of these statements of Al-Khwârazmî cannot be made with certainty, and, moreover, he may not have been well-informed on the history of his subject. His statement seems, nevertheless, to support the theory of Greek influence.
In short, it seems quite reasonable to see in the geometrical demonstrations of the algebra of Al-Khwârazmî and ‘Abd al-Hamîd ibn Turk a clear sign of Greek influence. Thus, the Greek influence in question could have acted on the Babylonian algebra of the later school to produce the algebra of ‘Abd al-Hamîd and Al-Khwârazmî, if we think  in terms of the course of development sketched by Gandz. But it is also probable that the algebra of Al-Khwârazmî and ‘Abd al-Hamîd was a direct outgrowth of the Greek geometric algebra, possibly with the superposition of a supplementary influence from the later Babylonian school.
Another evidence in favour of Greek influence may be found in the fact that, as Gandz points out, Al-Khwârazmî does not reject irrational numbers as solutions. This may be said to apply to ‘Abd al-Hamîd also. For, had he rejected irrational solutions, the extant part of his book would have been the very place to speak of it. Alongside of the special case and logical necessity represented by the imaginaries, he would have to have another special case of impossibility of solution corresponding to results involving irrational quantities. Diophantos did not admit irrational solutions .
The acceptance of irrational quantities in the algebra of Al-Khwârazmî and ‘Abd al-Hamîd ibn Turk is very likely a result of Greek influence. For, as we have seen, complete reliance on reasoning in terms of geometrical demonstrations was apparently a very prominent and characteristic feature of this algebra, and this points to an awareness of the difficulties presented by analytical methods of treatment based purely on number and discrete quantity.
It would seem therefore that the algebra of Al-Khwârazmî was the outcome of a tendency of simplification and economy in procedure and method coupled with the disappearance of the hesitation felt toward the double root of the equation x2 + c = bx. The ambiguity surrounding this question of the double root was apparently dissipated through the adoption of the method of geometrical demonstration, this method being a modification of the method of geometrical demonstration of the Pythagoreans and Euclid. The nature of the modification was determined by the need for its adaptation to the tendency for the exclusive use of the "mixed" equations. The geometrical method of demonstration may also have commended itself because of the need of avoiding difficulties arising from the irrationals. The tendency for the exclusive use of the "mixed" equations at the stage of solution was probably a Babylonian development, and the adoption of the new method of geometrical demonstration was apparently a result of influence deriving from the Greek geometric algebra.
This seems to be the most reasonable picture of the course of the developments leading to the algebra of ‘Abd al-Hamîd ibn Turk and Al-Khwârazmî. Chronological and geographical details of a substantial nature are entirely lacking. Moreover, this picture is only partly based on direct and conclusive evidence. For the relevant documents available at present leave serious lacunas, and these have to be filled by carefully thought out guesses. Greater certainty on these points of detail will have to await future research based on fresh documentary evidence.
 Gandz, the Origin and Development . . ., Osiris, vol. 3, pp. 509-510.
 Gandz, the Origin and Development ..., Osiris, vol. 3, pp. 523-524.
 Gandz, the Origin and Development ..., Osiris, vol. 3, p. 527.
 Gandz, The Origin and Development . . -, Osiris, vol. 3, p. 480.
 Gandz, The Origin and Development ..., Osiris, vol. 3, p. 496.
 Gandz, The Origin and Development . . ., Osiris, vol. 3, p. 415.
 See above, note 98.
 Euclid, Elements, book II. Proposition G.
 Euclid, II, 5.
 Gandz, The Origin and Development . . ., Osiris, vol. 3, p. 519.
 Gandz, The Origin and Development .... Osiris, vol. 3, p. 531.
 Gandz, The Origin and Development ..., Osiris, vol. 3, pp. 523-524.
 Gandz, The Sources of Al-Khwârazmî's Algebra, Osiris, vol. 1, p. 265.
 Ibn al Nadîm, Fihrist, vol. 1, p. 281.
 A. G. Kapp, Arabische Vbersetzer und Kommentatorm Euklids I, Isis, vol. 22, 1934-35- p- 170.
 Ibn Khaldun, Muqaddima, tr. F. Rosenthal, vol. 3. London 1958, pp. 115-116.
 Kapp, pp. 133, 164.
 Kapp, pp. 166, 170.
 Sayili, The Observatory in Islam, pp. 53-56.
 See above, p. 94 and footnote 42.
 Al-Khwârazmî, Algebra, Rosen, text, p. 1.
 Gandz, The Origin and Development .. ., Osiris, vol . 3, pp. 534-536.
 Gandz, The Origin and Development . .., Osiris, vol. 3. pp. 534-536.
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by: FSTC Limited, Thu 15 January, 2009