Logical Necessities in Mixed Equations: Chapter IV
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4. The Algebra of ‘Abd Al-Hamîd Ibn Turk and Al-Khwârazmî
The explanation of the double root of x2+ c = bx as given by Gandz is undoubtedly very satisfactory as far as the question of the ultimate sources of Al-Khwârazmî's algebra is concerned. But it cannot be claimed to stand for the answer Al-Khwârazmî himself would have given if the same question were asked from him, and Gandz too, perhaps, does not mean to offer this solution of the problem as necessarily valid in the latter sense.
According to the conclusions reached by Gandz, in a first stage i.e., in the "old Babylonian school," the first six types of equations in two unknowns seen in the list given above were the types in use . Later on the remaining three types of equation with one unknown also came into use, but the type x2 + c = bx was avoided . Gandz considers a new school to have developed directly out of this second stage found in Babylonian algebra. The place and time of its appearance is not known, and its earliest representative known was Al-Khwârazmî. The outstanding characteristic of this new school of algebra is its practice of excluding the six old Babylonian types and of using the three "mixed" equations in one unknown. The old Babylonian attitude is thus seen to have been completely reversed .
The reasons for the disappearance of the avoidance of, or the hesitation felt toward, the type x2 + c = bx are not accounted for in these views advanced by Gandz. Moreover, the fact that the acceptance and free usage of the type x2 + c = bx was accompanied by an aloofness toward the old types and methods suggests that the interpretation of the double root exclusively with the help of the pair of equations x+ y = b and xy = c should not constitute an explanation that could be prevalent and current in the time of Al-Khwârazmî but only an elucidation valid in terms of what may be called a long term history and distant origins.
This is not to say, of course, that Al-Khwârazmî was not familiar with the old types of equations and that he did not know that the two roots of the equation x2 + c = bx correspond to the two unknowns in a pair of equations such as x + y = b and xy = c which leads to the equation x2 + c = bx. The question here concerns the prevalent methods of algebraic reasoning, solution, and explanation, i.e., if the course of development in algebra as traced by Gandz is true, it could be said that Al-Khwârazmî's explanation of the double root of the equation x2 + c = bx and the acceptance of only one solution for each of the other two "mixed" equations, would, most likely, not be in terms of an abandoned type of algebra.
It is of course possible to think that Al-Khwârazmî himself would have given such an explanation for the occurrence of two solutions for the equation in question and only one for the other equations. But the adoption of such a view would imply that Al-Khwârazmî was either the founder of the new school or that in his time the school was relatively new and that its ideas were not as yet entirely formed or sufficiently wide-spread. As we have seen, Gandz is inclined toward such a view .
‘Abd al-Hamîd ibn Turk's text can be of help in elucidating this point. For it seems quite reasonable to think that the answer Al-Khwârazmî himself would have given in connection with the two solutions of the equation x2+. c = bx would have been quite similar to that given by ‘Abd al-Hamîd, in its general lines, if not identical to it.
Gandz says: "Al-Khwârazmî tries hard to break away from algebraic analysis and to give to his geometric demonstrations the appearance of a geometric independence and self-sufficiency. They are presented in such a way as to make the impression that they are arrived at independently without the help of algebraic analysis. It seems as if geometric demonstrations are the only form of reasoning and explanation which is admitted. The algebraic explanation is, as a rule, never given ." It may be added here that Al-Khwârazmî closely associates the "cause" of an equation and its geometrical figure .
These observations of Gandz find full confirmation in ‘Abd al-Hamîd's text. It may also be said that this tendency probably constitutes an even more important characteristic of the new school than its practice of avoiding the old types and transforming them into the three "mixed" equations of the second degree.
In answering our question therefore Al-Khwârazmî would be expected to present us with certain figures serving to illustrate the different cases which occur for the equation x2 + c = bx. For these cases represent, in the terminology of ‘Abd al-Hamîd, the logical necessities and the fixed relations connected with this type of equation, and the figures serve to give the "causes" for each type, according to Al-Khwârazmî himself. Indeed, for each one of the equations x2 + bx = c and x2= bx + c one figure is all that is needed for an unequivocal representation of the solution, whereas for x2 + c = bx it is impossible to represent the solution with a single figure drawn on the same principle. This point is set forth very clearly in ‘Abd al-Hamîd's text.
The Arabic text of Al-Khwârazmî's Algebra, as it has come down to us in F. Rosen's edition, contains only one figure which concerns the solution obtained by the subtraction of the square root of the discriminant, and this figure is exactly the same as the figure given by ‘Abd al-Hamîd ibn Turk for this solution. The Latin translation of Al-Khwârazmî's Algebra by Robert of Chester contains the figure reproduced here which is obtained by the superposition of the two figures for each one of the solutions . The sides of the square AD represent the value of x obtained by subtracting the square root of the discriminant, and the sides of the square OD represent the other solution for x.
This figure corresponds to the superposition of the two separate figures given by ‘Abd al-Hamîd for the two cases in question. It probably is a later addition. For Al-Khwârazmî's text does not take up the second case in detail arid in a systematic manner, and none of the figures seen in Al-Khwârazmî and ‘Abd al-Hamîd are of such a composite nature.
In conformity with his treatment of the double solution of the equation x2 + c = bx, Gandz feels the need of explaining Al-Khwârazmî's references to the cases of x = b/2 and (b/2)2 < c in a non-geometrical form by tracing them back to old Babylonian analytical methods . But again, these explanations are not in harmony with his observations just quoted to the effect that geometrical demonstrations are the only form of algebraic reasoning and explanation admitted by Al-Khwârazmî. Their clear explanation based on the principle of geometrical demonstration is found in cAbd al-Hamîd ibn Turk's text, and this should correspond pretty nearly, or exactly, to the way Al-Khwârazmî himself would have accounted for them.
In explaining the solution of equations of the type x2 + c = bx Al-Khwârazmî takes the example x2 + 21 = 10x and first gives the solution x = 5 - Ö(52-21) =3. Then he remarks that the root may also be added if desired, thus giving the solution x = 5 + 2 = 7. This special example is followed by the general instruction to try the solution obtained by addition first and if this is found unsatisfactory to resort to the method of subtraction . There is thus a reversal of order, in which the two alternative solutions are recommended to be found, between the example of solution presented and the general instruction given.
Gandz sees a contradiction in this and decides that the general instruction must be a gloss by a later commentator . For he observes that in nine out of a total of ten examples of the solution of equations of this type given by Al-Khwârazmî the solution obtained by subtracting the square root of the discriminant is preferred . Gandz uses the word preference here in a composite sense. At times he refers by it to the rejection or omission of the other root, and at times he means by this word merely a preference in the order of calculation.
Gandz considers this "preference" for the method of subtraction to find confirmation also by the facts that Al-Khwârazmî's geometrical demonstration for the solution of this type of equation concerns the alternative of the method of subtraction  and that Weinberg's translation, from the Hebrew, of Abû Kâmil Shujâ's Algebra, where the author quotes this part of Al-Khwârazmî's text, does not contain the recommendation of first trying, the method of addition .
Gandz says on this occasion, "It is therefore a well established rule with Al-Khwârazmî to prefer the negative root. He teaches this rule expressly in his exposition of the types, on p. 7, and he adheres to it also in his geometric demonstration of the type. There must have been a special reason for this preference and partiality exercised in favour of the negative root. Now if we were to consider the type in its later Arabic form, x2 + b = ax, there would be, to the writer's knowledge, no plausible explanation for this preference. But if we trace the type back to its historical old-Babylonian origin, there appears to be a very good and plausible reason for this strange procedure. The original form was: x + y = a; xy = b. The first part of it is still preserved in Al-Khwârazmî's examples, which have the condition x + y = 10. But we have seen that in five examples the positive root would lead for x to a value x > 10 ."
These five examples, as they are numbered by Gandz, are:
5) x + y= 10; xy/(y-x)=21/4
6) x + y= 10; 5x/2y+ 5x = 50,
7) x + y = 10; y2 = 81x,
8) x + y = 10; 10x = y2,
9) x2 + 20 = 12x.
The solutions for example No. 6 are x1 = 8, x2 = 50/4, and for No. 7 they are x1 = 1 and x2 = 100. Al-Khwârazmî takes only the smaller values in both cases, obviously because both solutions x2 exceed 10.
Al-Khwârazmî himself gives no solutions for examples No, 5 and 8 above , but Gandz includes them in his list. The conclusion properly to be drawn here is that, at times, Al-Khwârazmî leaves the final calculation of the answers to his reader. The solutions here are x1 = 3 and x2 = 70/4 for No. 5, and x1 = 15 - Ö125 and x2 = 15 + Ö125 for No. 8. Gandz says that Al-Khwârazmî would have been obliged to choose the smaller values in both cases, had he given the solutions , and this seems quite acceptable.
These four examples hardly reveal anything beyond the simple fact that of the solutions x1 and x2 those which fit the initial conditions such as x + y = b were naturally preferred. In the examples considered here the acceptable solutions have to satisfy the condition x < 10. That is why in two out of the four examples the smaller values were chosen by Al-Khwârazmî, and in the remaining two also the smaller roots should have been chosen as asserted by Gandz. I shall take up example No. 9 later on.
We may now consider example No. 10 which appears on Gandz' list. It is (x – (x-x/3-x/4-4)2 = x + 12. This leads to the equation x2 + 576/25=624x/25. Its two solutions are x1 = 312/25-288/25=24/25 and x2 = 312/25-288/25=24. Al-Khwârazmî gives only the value 24, and Grandz remarks that he is compelled to do so because if the value 24/25 is substituted in the equation (x-x/3-x4-4)2 = x + 12, the value obtained inside the parenthesis would be 10/25 - 4, i.e., a negative quantity .
It is thus seen that the solution of smaller value was rejected and the larger one accepted without any hesitation when the nature of the problem required-such a choice. There should therefore be no bias in Al-Khwârazmî against the solution obtained by the method of addition.
We have so far looked into five out of the ten examples mentioned by Gandz. I shall now take up examples 1 and 2. In these two examples Al-Khwârazmî is seen to mention both solutions.
Example No. 1 is x + y = 10; x2 + y2 = 58. It gives the equation x2 + 21 = 10x. Its two solutions are x1 = 3 and x2 = 7, and Al-Khwârazmî gives both solutions , apparently because x1 and x2 both are quite admissible, i.e., they are not only the roots of the equation x2 + 21 =10x but they also give satisfactory values for x and y in the original system x+y = 10, x2 + y2 = 58.
Example No. 2 is x + y = 10; xy = 21. This too leads to the equation x2 + 21 = 10x. Again Al-Khwârazmî gives both solutions. He first mentions 3, and then he says, "and this is one of the parts, and if you wish you may add the root of four to half the coefficient of the unknown (i.e., to 5) and you will obtain seven, and this also is one of the parts." By the word "parts" he refers to x and y, or to x and 10 - x, i.e., to the two parts into which 10 is divided. Then he adds, "And this is a problem, in which one operates both by addition and by subtraction" .
In these two examples there are no conditions which would necessitate the rejection of one of the two solutions x1 = 3 and x2 = 7. And, indeed, Al-Khwârazmî is seen not to make any such choice. The conclusion that the solution which fits the conditions presented by, the problem is selected is therefore seen to be applicable to all cases and to both solutions. The rule, then, is that the choice of x1 or x2, or both, is governed by whether xl, x2, or both x1 and x2, happen to satisfy the problem.
It may be noted that the statements of Al-Khwârazmî quoted above in connection with example No. 2 are quite similar to those wherein Gandz has detected a contradiction. Here too, at the beginning, Al-Khwârazmî mentions and finds the smaller value first and then he gives the second solution, but then, at the end, he mentions addition first and subtraction afterward.
Al-Khwârazmî says, "And this is a problem, in which one operates both by addition and by subtraction," and, "and if you wish you may add the root of four . . . and you will obtain seven." The same peculiarity of expression is partly found also in the passage where Gandz has found a contradiction. This phraseology suggests, firstly, that it could be known before the actual derivation of the values of x1 and x2 whether both solutions would be acceptable or not, and secondly, that when both solutions were acceptable it was considered just as natural to derive both x1 and x2 from the equation x2 + c = bx or to derive only one of them from that equation; the second root could then be derived from one of the equations leading to x2 + c = bx. This second point may be -said to be a direct consequence of the symbolism implicit in Al-Khwârazmî's formulation of equations. For it may be said that in Al-Khwârazmî equations in two unknowns are a bit in the background. From an initial and ephemeral x and y he passes immediately to x and 10 - x or b - x. I shall dwell on the first point in greater detail a little further below .
The expression "if you wish you may. . ." which we have just met and certain other peculiarities encountered here and there in. this algebra make it seem probable that there was in Al-Khwârazmî's algebra a tendency of being satisfied with a single solution of the equation x2 + c = bx even when both solutions were acceptable. That such was not the case is indicated, however, by certain examples of solutions given, as well as by Al-Khwârazmî's clear reference to the type of problems in which one operates both by addition and by subtraction as seen in the quotation made from him above to which footnote 78 has been attached. Hence, our interpretation in the preceding paragraph.
Example No. 4 in Gandz is x + y = 10; x/y+y/x=13/6. It gives the equation x2 + 24 = 10x, and its two solutions are x1 = 4, x2 = 6. Al-Khwârazmî mentions only x1 = 4 . It is obvious that he accepts x2 = 6 also. We must conclude therefore that, at times, he mentions one of the solutions only, when both are acceptable. As we have just seen, his phraseology too shows this to be quite permissible. He leaves it to the reader to find the other answer either directly from the equation x2 + 24 = 10x or, by subtraction, from x + y = 10, i.e., from the relation 10 - x = y.
We now come to example No. 3 in Gandz' list. It is x + y = 10; x2 + y2 + (y - x) = 54. Or we may write it directly as (10 - x) 2 + x2 + [(10 - x) -x] = 54. It is seen that in this equation the relation y > x or 10 – x > x is assumed to hold by the very formulation of the problem. The equation leads to x2 + 28 =11x, and the solutions are x1 = 4, x2 = 7. Al-Khwârazmî gives only x1 = 4 . The reason for his not accepting both values is obviously that 4 + 7 ≠ 10. But 7 too is less than 10. Why then did he not choose x2 = 7? The reason for this must simply have been that only x1 satisfies the relation 10 - x > x which was assumed in the formulation of the problem.
Gandz asks the following additional question: Why is it that Al-Khwârazmî did not formulate the equation in the form x2 + (10 - x) 2 + [x - (10 – x)] = 54, assuming that x- (10 - x) > 0? His answer is that in that case the equation x2 + 18 = 9x would have been obtained, and as the two solutions of this equation are x1 = 3 and x2 = 6, Al-Khwârazmî would have been compelled to choose x2 = 6 because x1 = 3 would not satisfy the relation x- (10 - x) > 0. But Al-Khwârazmî according to Gandz feels inertia for accepting the larger answer; therefore he chose to formulate his problem as he did .
But how can we be sure that Al-Khwârazmî chose to formulate his problem, as he did because he had a feeling against the acceptable solution resulting from the alternative formulation? He has the problem (10 - x) 2 - x2 = 40, e.g., in his book, just before our present example No. 3. It implies the relation 10 – x > x, and leads directly to the simple equation x = 3. He could have formulated it as x2 - (10 - x) 2 = 40 leading to the equation x2 + 10x = 70. Could we conclude from here that Al-Khwârazmî avoids equations of the type x2 + bx = c? It is much more reasonable to answer Gandz' additional question by saying that, had Al-Khwârazmî chosen the second formulation, he would have accepted x2 = 6, because this is the answer which satisfies the condition x > (10 - x).
We now come to example No. 9, the only one remaining from Gandz' list. This is the equation x2 + 20 = 12x. It is directly given, i.e., it is not derived from other relations imposing such conditions as x + y = 10. Here the solutions are x1 = 2, x2 = 10, and Al-Khwârazmî gives only x1 = 2 .
Why does he not mention x2 = 10? To answer this question Gandz assumes that Al-Khwârazmî associated this example in his mind with a system in the form of x + y = 10; x2 = 4xy occurring in another part of Al-Khwârazmî's text . He changes the second member of this pair into y2 = 4xy and says that the equation x2 + 20 = 10x must in reality be the second degree equation derived from x + y = 10; y2 = 4xy. For one thus obtains (10 - x) 2 = 4x (10 - x), or x2 + 20 = 12x. The condition x + y = 10 being thus introduced into the equation x2 + 20 = 12x, the solution x2 = 10 can be said to be undesirable because it leads to the value zero for y. This, according to Gandz, is the reason why Al-Khwârazmî passes over x2 = 10 with silence .
The equations x2 + 20 = 12x and x + y = 10; x2 = 4xy are separated in Al-Khwârazmî's book by twenty five intervening isolated examples, and Al-Khwârazmî gives no inkling of the connection suggested by Gandz. Moreover, the equations x + y = 10; x2 = 4xy, as treated by Al-Khwârazmî, become transformed into x2 = 8x. This is an example of "simple" equations (mufradât), and not-of the "mixed" equations where the example x2 + 20 = 12x occurs. Al-Khwârazmî is thus seen to deal with these two examples in two different parts of the section on problems, in his book, treating two different categories of equations. What conceivable reason could there possibly be under these circumstances for seeing the system x + y = 10x; x2 = 4xy, and in its version proposed by Gandz, loom suddenly behind the equation x2 + 20 = 12x?
As we have seen it was Al-Khwârazmî's intention to write a book that could easily be understood . And Gandz himself claims that Al-Khwârazmî represented the tendency of the simplification and standardization of algebraically procedures and that his school did away with all solutions and transformations depending upon subtly concocted relations and ingenious devices . How then could Al-Khwârazmî expect his readers to jump back twenty five examples in his book to make such a circuitous interpretation of his mere silence?
Moreover, this kind of explanation by Gandz is, if not contradictory to his general theory concerning this type of equation, at least not in harmony with it. For Gandz' own general theory directs us, in the present example, e. g., to see the system x + y = 12 and xy = 20 behind the equation x2 + 20 = 12x, and not the system x + y = 10; y2 = 4xy, in order to render the occurrence of two solutions intelligible. But the reasoning that x = 10 gives y = 0 and is therefore undesirable as a solution would not work in that case.
As we have seen, Al-Khwârazmî occasionally leaves the calculation of the result, whether it is the larger or the smaller root, to his readers  and his phraseology suggests that he should not necessarily be expected to mention two solutions when they are both acceptable . It should therefore be quite possible that Al-Khwârazmî left it to the reader to figure out the value of x2 in the present example also. It may be supposed too, although such a supposition would represent an extreme attitude and is really unnecessary, that the solution x2 = 10 is missing due to some kind of an oversight on the part of some copyist or of Al-Khwârazmî himself. At any rate, a single exceptional case, even if it were in existence here, should not justify such an attempt to establish a complicated rule.
It does also seem a bit exaggerated to see in the occurrence of the equation x + y = 10 in any specific and isolated problem a meaningful survival of one half of the standard Babylonian equation type I even if the second member of the pair bore no resemblance to that type. It is hardly necessary to invoke the old Babylonian practices for the clarification of the individual solutions of the equation x2 + c = bx found in Al-Khwârazmî. It can be said with little hesitation that these examples contain no puzzles and that they require no complex explanations.
Three cases occur with regard to the acceptance or rejection of the two solutions of the equation x2 + c = bx. The solution obtained by the method of subtraction may be accepted; the solution obtained by the method of addition may be accepted; both solutions may be accepted. The special conditions contained in the problem solved dictate the choice between these three cases.
These general conclusions may be said not to be in any essential disagreement with those of Gandz. Gandz claims, however, that the procedures followed in the choice and preference of these roots cannot be made intelligible unless we consider them in the light of their distant Babylonian origins. This certainly does not seem to be true. The treatment of these examples rather indicates that the algebra of Al-Khwârazmî was quite self-sufficient in explaining its methods and the procedures it employed. Moreover, strict dependence upon geometrical reasoning was a prominent feature of this algebra, and it will have to be brought well into the foreground.
We come now to the question of preference, if any, in the order in which the two solutions were derived even if both solutions were to be accepted, as in the case when the equation x2 + c = fax was directly given or when both solutions were expected or foreseen to be acceptable.
It will be observed from the foregoing details that Al-Khwârazmî's text contains five examples which may serve the elucidation of this question in an unequivocal manner. Once he proceeds to teach the solution of x2 + c = bx and uses the method of subtraction first, then he adds the second solution. Immediately after, he gives a general instruction in which he speaks of the method of addition first . There are, in addition to these, two problems which could be expressed in a pair of equations in two unknowns and leading both to the equation x2 + 21 = 10x. These are examples No. 1 and 2 in the list mentioned above. In one of these he derives first the smaller answer obtained by subtraction and then finds the other solution . In the other example too first the method of subtraction is used and then the method of addition, but immediately after he adds that the equation "should be solved both by addition and by subtraction ," thus once more mentioning the method of addition first.
In these five cases therefore Al-Khwârazmî is seen to mention three times the method of subtraction first and twice the method of addition first. These examples are too few to constitute a basis for a general conclusion. It should be safe to decide, however, that Al-Khwârazmî exercises no preference or partiality in the matter of the order of derivation of the two solutions of the equation x2 + c = bx.
It is true nevertheless that the smaller solution is seen to occur more frequently in the list made by Gandz and that the smaller root of the equation is observed to constitute the satisfactory solution in the majority of the examples where only one solution is acceptable.
If we add to Gandz' list the two general statements by Al-Khwârazmî concerning the double roots and the example connected with his teaching how to solve x2 + c = bx we will have thirteen cases. Six or seven of these concern or exemplify cases where both roots are acceptable. In five of them Al-Khwârazmî speaks of both roots. Twice he speaks of the larger root first and three times of the smaller root first. In the remaining two examples he mentions only the smaller root of the equation. There are in addition six examples wherein only one solution is acceptable. In one of these the acceptable solution is the larger one and in the remaining five the smaller one. Al-Khwârazmî mentions the former one and three out of the latter five.
In Al-Khwârazmî's examples therefore the smaller root of this type of equation is encountered, in one way or another, more frequently than the larger solution. Is there a reason, for this? It is possible that Al-Khwârazmî has a tendency or an inclination to set the unknown he is going to eliminate as greater than the unknown he lets remain in his equations, i.e., to set y > x, or, for x + y = 10, e.g., to suppose x < 10 - x. For the examples found in his text suggest such likelihood. This may partly explain why the smaller solution is more prominently represented in his examples. But the relation x < 10-x cannot be said to constitute an established rule with Al-Khwârazmî. The first example in the group of problems found in his book is x2 = 4x (10 - x), and here x > 10 – x .
Our foregoing conclusions may be summarized or formulated as follows. When a system of two equations in two unknowns F(x, y) = 0 and f(x, y) = 0 leads to an equation of the type x2 + c = bx, the two roots x1 and x2 of the latter equation will in general result in two new values y1 and y2 for y. It is only in the special case wherein both F(x, y) = 0 and f(x, y) = 0 are symmetrical with respect to x and y that x1 = y2 and y1 = x2. The solutions x1 and x2 will therefore stand in such a case for the solutions of x and y in the system F(x, y) = 0 and f(x, y) = 0.
Now, as in Al-Khwârazmî's algebra the solutions of x2 + c = bx were in principle conceived to stand not only for the solutions of this equation itself but also for the solutions of x and y in F(x, y) = 0 and f(x, y) = 0 when the equation x2 + c = bx was derived from such a system of simultaneous equations, and as the examples of simultaneous equations used were not always both symmetrical with respect to x and y, it was natural that, occasionally at least, the two solutions x2 and x2 should be found not to be both satisfactory. One of them had to be accepted and the other rejected. Special conditions contained in F(x, y) = 0 and f(x, y) =0, depending also upon certain algebraic conceptions of the time, governed the choice between x1 and x2 in such cases.
In short, Al-Khwârazmî is seen not to contradict himself, and his general instruction need not be discarded. There is no conclusive evidence indicating that he exhibited any partiality in the order of derivation of the two roots. He may be said, however, not to treat this question in a well ordered fashion. ‘Abd al-Hamîd ibn Turk's text, on the other hand presents the matter in a more orderly and complete fashion, and as it was, likely, written before Al-Khwârazmî's Algebra, the latter may not have felt the need of clearer explanation because he did not consider the subject, in its details, as one that was unknown or obscure.
The additional light shed on the subject by the text of ‘Abd al-Hamîd ibn Turk was of some help in arriving at the conclusions presented above concerning Al-Khwârazmî's algebra. There should not be much need therefore to insist on their applicability to ‘Abd al-Hamîd's text also.
‘Abd al-Hamîd ibn Turk applies the rules of subtraction and addition both to the same example, namely to the familiar equation x2 + 21 = 10x, and this implies the permissibility of accepting both roots. He, moreover, clearly distinguishes between the cases b/2 > x and b/2 < x, with positive discriminant. In the first case the square root of the discriminant has to be subtracted from half of b and in the second case it has to be added to the same quantity.
This mode of expression seems a bit awkward from the standpoint of the solution of the equation though not from the viewpoint of geometrical representation. For, from the standpoint of the solution of the equation, this phraseology amounts merely to saying that if x is smaller than half of b then subtract the square root of the discriminant from half of b and in the opposite case add that amount to half of b. One is tempted to interpret this to mean that first both values of x are found, and if it is seen that the problem requires a solution which satisfies the condition x < b/2 the solution which fulfils this condition is chosen, and in the opposite case the alternative solution is accepted.
‘Abd al-Hamîd ibn Turk leaves the impression of having been a person who attached some importance to formal and logical order of presentation and exposition, however, and he would be expected to avoid such a circular manner of expression. There is evidence that Al-Khwârazmî too knew and used some, at least, of these rules and ways of classification . It may be said therefore that these rules and modes of expression had been developed apparently by the school of algebra of which ‘Abd al-Hamîd ibn Turk and Al-Khwârazmî are the earliest representatives known to us, and that these rules and expressions had probably received the approval of several other mathematicians.
Moreover, ‘Abd al-Hamîd is seen to use an identical expression when speaking of the case of negative discriminant, and it could not be, said that in this case too he means to refer to the actual comparison of x and b/2. He could only be interpreted in this case to mean that c > (b/2)2 leads to the impossibility of solution whether we imagine the relation b/2 > x or b/2 < x to hold. It is thus seen that the same mode of expression has to be interpreted in a certain manner for the case of positive discriminant and in another manner for that of negative discriminant.
All these considerations suggest that we possibly lack the knowledge of certain details needed here and that if the true elements of the particular reasoning involved were known to us the strangeness of this mode of expression would disappear. It may be wondered therefore whether this algebra may have possessed a method for deciding beforehand on the following questions: will both solutions of x2 + c = bx be acceptable; and if not, should the acceptable solution be greater or smaller than b/2, in case x ≠ b/2, the case x = b/2 being detectable through a comparison made between c and b.
As we have seen before, certain statements of Al-Khwârazmî too suggest that he considered it predictable before the solutions were actually derived, whether both roots of the equation would be acceptable or only one of them . Al-Khwârazmî's general instruction, however, is to the effect that one may first try the method of addition and if the result is not satisfactory then subtraction will be sure to give the satisfactory result , and this renders predictions unnecessary. But Al-Khwârazmî may be supposed to give here a short-cut and simplified rule which he considered more practical and preferable from a pedagogical standpoint.
The missing parts of ‘Abd al-Hamîd ibn Turk's Algebra did probably contain information serving to shed light on this question. In the absence of such information, only tentative guesses could be advanced.
It may be conjectured that very frequently one of the equations F(x, y) = 0 and f(x, y) = 0 used, leading to an equation of the type x2 + c = bx, say, F(x, y) = 0, was, e. g., in the form x + y = B. If B was seen to be equal to b, it was decided that both roots of x2 + c = bx would be acceptable. In case b ≠ B, they may have looked at f(x, y) = 0 to see whether it bore more strongly upon x or upon y, and in some such a way it may have been decided whether the acceptable solution of x2 + c = bx should be greater or less than b/2.
This assumption or guess has seemed sensible to me because such a procedure should be traceable to the much used Babylonian and Diophantine methods of transformation of equations by the introduction of new unknown quantities. Thus if F(x, y) = 0 is in the form x + y = B and f(x, y) = 0 is seen to be transformable into the form zy = c or z2 + y2 = c, then the way z is related to x could supply the needed relation between, x and b/2.
It may seem reasonable to think that in his treatment of the solution of x2 + c = bx when (b/2)2 > c, ‘Abd al-Hamîd ibn Turk takes into consideration, although he does not state it explicitly, the possibility of the operations needed for the solution, i.e., in our terminology, both the constant term and the coefficient of x2 being positive, in this case, in the equation of the type x2 + c = bx the square root of the discriminant (b/2)2 - c is smaller than b/2, and the possibility of a negative solution, which was not acceptable, is thus excluded. For this may be said to be implied by his detailed treatment, on this occasion, of the case of negative discriminant.
Such a consideration would constitute and supply an additional arithmetical or operational elucidation of the reason why x2 + c = bx has two solutions while each of the other "mixed" equations has only one. It would serve to supplement the geometrical demonstrations by an additional comparison between the three "mixed" equations. Indeed, the solutions x= Ö((b/2)2+c) ± b/2 of the other two "mixed" equations would both become transformed into negative quantities if one thought of the alternative of subtracting the square root of the discriminant. In other words, both cases would lead to operations impossible to perform. For these two "mixed" equations, therefore, such alternatives had no meaning in the geometrically conceived algebra of the time.
The meaning of the word darûra becomes clearer in the light of ‘Abd al-Hamîd's treatment of these special cases and after the details considered above. The meaning of fixed or uniquely determined relation seems superimposed on the meaning of logical necessity and a meaning close to that of "determinate equation" ensues perhaps because the cases are envisaged, when necessary, as a pair of simultaneous equations. Thus the equation x3 + c = bx may be said not to represent a uniquely determined relation because it admits, in general, two distinct solutions. When one of the additional conditions x < b/2, x > b/2 is also imposed, however, one equation and one inequality together constitute a uniquely determined relation.
This explains why ‘Abd al-Hamîd's reference to the case wherein both solutions may be admitted is indirect and implicit. For in this part of his book his main purpose is to make an exposition of the darûrât. The case x = b/2 ay be looked upon in a like manner when this relation is translated into the condition c= (b/2)2 "Logical necessity" seems essential as a component in the meaning of darûra because it accounts better for the case of negative discriminant for which case too both' alternatives x < b/2, x > b/2 are taken into consideration. Moreover, this meaning of the word possibly may, by extension, refer also to the method of geometrical demonstration or "proof."
We cannot be entirely certain that ‘Abd al-Hamîd ibn Turk wrote his book before Al-Khwârazmî's Algebra, but there seems to be no reason for doubt that the part of his text that has come down to us constitutes the first extant systematic and well-rounded treatment and exposition of the topic it deals with.
We have seen that Gandz is inclined to believe that Al-Khwârazmî was, if not the founder of the school of algebra which he represents, at least one of its earlier representatives who played a part in the dissemination of its views and its manners of approach. The text of ‘Abd al-Hamîd, however, may be said to corroborate the opposite thesis. The algebra of ‘Abd al-Hamîd and Al, Khwârazmî does not at all seem close to its stage of genesis. It does not have the earmarks of an algebra which had not still finished going through its initial processes of development but of one with well-established rules, traditions, and points of view.
 Gandz, The Origin and Development ..., Osiris, vol. 3, pp. 417-456.
 Gandz, The Origin and Development ..., Osiris, vol. 3, pp. 470-508.
 Gandz, The Origin and Development ..., Osiris, vol. 3, pp. 509-510.
 See above, p. 98 and note 49.
 Gandz, The Origin and Development, . , Osiris, vol. 3, pp. 514-515.
 Gandz, The Origin and Development, . , Osiris, vol. 3, pp. 514-515.
 Gandz, The Origin and Development ..., Osiris, vol. 3, p. 515.
 This figure is taken from Gandz. See Gandz, The Origin and Development ...., Osiris, vol. 3, p. 522.
 Gandz, The Origin and Development ..., Osiris, vol. 3, pp. 533-534.
 Gandzf The Origin and Development . . ., Osiris, vol. 3, pp. 519-520.
 Gandz, The Origin and Development . . ., Osiris, vol. 3, p. 532.
 Gandz, The Origin and Development . . ., Osiris, vol. 3, p. 533.
 Gandz, The Origin and Development ..., Osiris, vol. 3, p. 520, note 84. As we shall see below, there is another statement by Al-Khwârazmî wherein addition is mentioned first, so that the effort of Gandz to discard the former statement does not seem to be justified (see below, pp. 112, 113, 118 and notes 78, 79, and 92).
 Gandz, The Origin and Development ..., Osiris,-'vol. 3, p. 533. See also, pp. 525-532.
 Al-Khwârazmî, Algebra, Rosen, text, pp. 37, 36-37, tr., p. 51.
 Gandz, The Origin and Development .. , Osiris, vol. 3, pp. 528, 530.
 Al-Khwârazmî, Algebra, Rosen, text, pp. 42-44, tr., pp. 60-62; Gandz, The Origin and Development ..., Osiris, vol. 3, pp. 531-532.
 Al-Khwârazmî, Algebra, Rosen text, pp. 28-29, tr., pp. 39-40; Gandz, The Origin and Development ..., Osiris, vol. 3, p. 525.
 Al-Khwârazmî, Algebra, Rosen, text, p, 30, tr., pp. 41-42; Gandz, The Origin and Development . . ., Osiris, vol. 3, pp. 524-525 (the translation given here is mine).
 See below, pp. 122-123 and notes 95, 96.
 Al-Khwârazmî, Algebra, Rosen text, pp. 32-33, tr., pp. 44-45; Gandz, The Origin and Development ..., Osiris, vol. 3, pp. 527-528.
 Al-Khwârazmî, Algebra, Rosen, text, pp. 31-32, tr.3 pp. 43-44.
 Gandz, The Origin and Development ..., Osiris, vol. 3, pp. 526-527.
 Al-Khwârazmî, Algebra, Rosen, text, p. 40, tr., p. 56.
 Al-Khwârazmî, Algebra, Rosen, text, p. 25, tr., pp. 35-36.
 Gandz, The Origin and Development ..., Osiris, vol. 3, p. 531.
 See above; pp. 94-95 and note 42.
 See below; pp. 126-127 and note 97.
 See above, pp. 110-111, 113-114, 114 and note 80.
 See above, p. 113-114 and note 79.
 Al-Khwârazmî, Algebra, Rosen, text, p. 7, tr., p. n. Sec also, above, p. 109 and note 68.
 Al-Khwârazmî, Algebra, Rosen, text, p. 29, tr., p. 40. See also, above, p. 112 and note 77.
 Al-Khwârazmî, Algebra, Rosen, text, p. 30, tr., pp. 41-42. See also, above, p. 112 and note 78.
 Al-Khwârazmî, Algebra, Rosen, text, p. 25, tr., p. 35.
 See above, p. 108 and note 67.
 See above, p. 113-114 and note 79.
 Al-Khwârazmî, Algebra, Rosen, text. p. 7, tr., p. 11. See also, above, p. 109 and note 68.
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by: FSTC Limited, Thu 15 January, 2009