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Logical Necessities in Mixed Equations: Original Text (English Translation) and References

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6. Original text: English translation

Logical Necessities in Mixed Equations: From The Kitâb Al Jabr Wa'l Muqâbala of Abû 'l-Fadl ‘Abd Al-Hamîd Ibn Wâsi' Ibn Turk Al-Jîlî

With the name of God the merciful and compassionate. Blessing and peace be upon Muhammad, the master of the prophets, and on all his descendents.

The case of the equality of [a certain number of] square quantities to a number of roots (i.e., root of the square quantity). When we say, e.g., that one square quantity equals three roots; we represent the square quantity by the area of a plane quadrilateral figure with equal sides and right angles. Let ABCD be this figure. Each one of its sides is the root of the square quantity. The line AB is therefore the root of the square quantity. But the quadrilateral ABCD equals three roots, and AB is the root of the square quantity. The line BD is therefore numerically equal to three. For when we multiply it by AB, which is the root of the square quantity, it gives us the quadrilateral ABCD which is equal to three roots. But BD is the root of the square-quantity. The root of the square quantity is therefore three. And the square quantity is nine. And this is the shape of the figure.

 Large image Figure 14

The case of equality of one square quantity and a number of roots to a certain number. Thus, when we say that one square quantity and ten roots equal twenty four, we represent the square quantity by a plane quadrilateral figure with equal sides and right angles. Let this be the surface AD. Each one of its sides is the root of the square quantity. We add to this figure the rectangular surfaces QD and DH, and we set the length of each numerically equal to five and their breadth equal to that of the surface AD, i.e., equal to the root of the square quantity. Each one of these two rectangular surfaces is therefore equal to five roots. The three surfaces AD, QD, and DH are thus equal to ten roots and one square quantity, i.e., twenty four. In order to complete the larger figure AK the product of ZD with DT is needed. Each one of these lines is equal numerically to five, and the quadrilateral formed by them, i.e., the surface DK, is equal to twenty five. This twenty five becomes juxtaposed upon the twenty four consisting of the surfaces DH, DQ, and DA, and the whole thing thus adds up to forty nine. This is the greater surface AK. We take its root, which is seven, and this is the value of each one of its sides. When we subtract from this seven, which is the line AH, the extended line CH which is equal to five, the line AC, which is the root of the square quantity, remains, and it, is found to be equal to two. The root, of the square quantity is therefore two and the square quantity itself four. And when ten roots are added to it the quantity twenty four is obtained. And this is the shape (of the figure).

 Large image Figure 15

The case of equality of a square quantity and a given number to a number of roots. Thus, when we say that one square quantity and twenty one equal ten roots, we represent the square quantity by a plane quadrilateral figure of equal sides and right angles, and this is the surface AD. Each one of its sides is the root of the square quantity. We add to it the rectangular area HB and set it equal to twenty one. Each one of the lines HC and DZ is therefore equal to ten. For the line CD is the root of the square quantity, and the areas ZA and AD are equal to ten roots. At the point Q, we divide the line ZD into two equal parts, and we draw at right angles to it the line QT equal in length to both ZQ, and QD. The point Q, which is the midpoint, will either fall on the line ZB or on the line BD. This point of equal division cannot in this example be the point B. For if B were located at the middle of the line ZD, BD would be equal to the line BZ. And as the line BD is of the same length as AB, the line AB would equal BZ, and the quadrilateral built on HB would thus equal twenty five. But we know this not to be so. For its value was supposed to be twenty one. And in case the point Q, which is the midpoint of ZD, is located on the line ZB, then the line QT must surely cut the quadrilateral HB. For QT is of the same length as QD and QP is longer than BD, while BD is equal to AB. The line QT is therefore longer than the line AB.

 Large image Figure 16

 Large image Figure 17

As to the intermediate case of equality, this obtains when the root of the square quantity is equal to half the number of the roots. This will not occur except in an example wherein half of the number of the roots is multiplied by its equal, this product being the numerical quantity which is with (on the same side of the equality as) the square quantity. Such is the case when it is said that one square quantity and twenty five equal ten roots, or one square quantity and nine equal six roots, and similar examples. These conditions being satisfied, we represent the square quantity by a plane quadrilateral figure of equal sides and right angles. Let this be the surface AD. We add to it the surface HB which we set equal to twenty five. The surface HD thus becomes one square quantity and twenty five, and this is equal to ten roots. Each one of the lines HC and DZ is therefore equal to ten. When we divide the line ZD into two equal parts at B and draw from this midpoint a perpendicular line the length of which is five, i.e., equal to each one of the halves, and square it, we obtain twenty five. AB is this line and it is equal to the line HZ. Its square is the surface HB. For it is equal to twenty five. Half the number of the roots is therefore the root of the square quantity. And this is the shape of the figure.

 Large image Figure 18

There is the logical necessity of impossibility in this type of equation when the numerical quantity which is with (on the same side of the equality as) the square quantity is greater than half the number of the roots multiplied by its equal. Thus, when we say that one square quantity and thirty dirhams equal ten roots, we represent the square quantity by an equilateral plane quadrilateral figure. Let this be the surface AD. We add to it the rectangular figure HB, and we set it equal to thirty. The surface HD is thus equal to ten roots and each one of the lines HC and ZD have the value ten. We divide the line ZD into two equal parts at the point Q.

Let us first consider the case wherein the point Q is located on the line ZB, as was done before. We draw the line QT at right angles to ZD and of the same length as each one of the lines ZQ and QD, i.e., equal to five, and we complete the quadrilateral KQ, which is thus equal to twenty five. The line TQ is equal to the line DQ. Therefore the line TL is equal to the line LA. As the line LQ is equal to the line AC and the line KT is equal to the line TQ, the quadrilateral KL is greater than the quadrilateral LB; and we add the quadrilateral HQ to both these quadrilaterals. The quadrilaterals KL and HQ together are therefore greater than the quadrilaterals HQ and QA taken together. Now, the quadrilaterals HQ, and QA together were equal to thirty, and the quadrilaterals KL and HQ together were equal to twenty five. Twenty five becomes therefore greater than thirty. But this is absurd and impossible. The logical necessity of impossibility in this type of equation has thus come into appearance. And this is the shape of the figure.

 Large image Figure 19

 Large image Figure 20

 Large image Figure 21

Here ends "Logical Necessities in Mixed Equations" taken from the book of Al-Jabr wa'l Muqâbala by ‘Abd al Hamîd ibn Wâsî' al Jîlî, may God's blessing be upon him.

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*Professor Aydin Sayili (1913-1992) hold the Chair for History of Science in Ankara University since 1952. He was the first Turkish historian of science trained in "History of Science." He completed his Ph.D in 1942 at Harvard University with George Sarton.

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by: FSTC Limited, Thu 15 January, 2009