Mathematics

Algebra

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

Previous | 1 | 2 | 3 | 4 | 5 | 6

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.

Let us first suppose Q to be on BZ. We draw QT and complete the quadrilateral KQ. This quadrilateral is then equal to twenty five. The line QT is equal to QD, and BD is equal to NQ. The line TN is therefore equal to each of the lines BQ and NA. We mark off from the line KT, which is equal to QT, a section equal to NQ, i.e., the line KL, and we draw LM. The remaining section LT is thus equal to TN and the quadrilateral on KM equal to the quadrilateral on NB, and the quadrilateral LN is equilateral. But the quadrilaterals HQ and QA together are twenty one, and the quadrilateral NB is equal to the quadrilateral KM, while the quadrilateral HQ is common between them. The quadrilaterals KM and HQ equal therefore twenty one. But the quadrilateral KQ was equal to twenty five. The quadrilateral LN which is their difference is thus equal to four. Each one of its sides is its root. The line TN is therefore equal to two. But TN was equal to QB. The line QB is thus equal to two. When we subtract two, which is the value of QB, from five, which is the length of QP, we obtain the value of the line BD, which is the root of the square quantity, as equal to three. And this is the shape of the figure.

Large image

Figure 16

If, on the other hand, the midpoint of ZD falls within the section BD, then the line QT is shorter than the line AB, and it does not cut the quadrilateral AD. For the line TQ is equal to the line QD, and the line AB is equal to the line BD, while the line BD is greater than the line QD. The line AB is thus lengthier than the line TQ. Let then the point Q be on the line BD. We draw the line QT and complete the quadrilateral KQ, and it is equal to twenty five. Now, the line NB is equal to the line QD. The line AN is therefore equal to the line BQ. But the line BQ is equal to NT. The line NT is thus equal to the line AN, and the line KT is equal to the line TQ. We mark off from the line TQ a section equal to the line KN, i.e., the section TL, and we draw the line LM. There thus remains the line LQ, which is equal to the line TN, and the line LM is equal to the line TN. The quadrilateral on the line MT is therefore equal to the quadrilateral on the line HN, and the quadrilateral on the line MQ is equilateral. But the two quadrilaterals HN and NZ are equal to the quadrilaterals NZ and MT. The quadrilaterals NZ and MT together are therefore equal to twenty one. Now, the quadrilateral KQ was equal to twenty five. When we subtract from it the quadrilaterals NZ and MT, which are equal to twenty one, the remaining quadrilateral MQ is seen to equal therefore four. This latter being equilateral, each one of its sides is its root. The line BQ is then equal to two. When we add the line BQ to the line QD, which is equal to five, we obtain seven, and this is the root of the square quantity. The square quantity is therefore forty nine, and when twenty one is added to it becomes seventy.

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

Likewise, let the point Q fall within the section BD. We draw the line QT at right angles to ZD and of the same length as each one of the lines ZQ and QD, and we complete the quadrilateral KQ, which thus has the value twenty five. Conditions such as those satisfied here indicate that the quadrilateral LQ is greater than the quadrilateral HL. We consider the quadrilateral KB as added to both these quadrilaterals. The quadrilaterals KB and BT together are thus greater than the quadrilaterals KB and KA taken together, now, the quadrilaterals KB and KA together had the value thirty and the quadrilaterals KB and BT together twenty five. Twenty five therefore becomes greater than thirty. But this is absurd and impossible. And this is the shape of the figure.

Large image

Figure 20

The case of equality of a numerical quantity and a certain number of roots to one square quantity. Thus, when we say that four roots and five dirhams are equal to one square quantity, we set the square quantity equal to a plane quadrilateral figure of equal sides and right angles. Let this be the surface AD. Each of its sides is the root of the square quantity. Within it we draw the line HZ parallel to the lines AB and CD both, and we set the surface AZ equal to five. The remaining surface HD is thus equal to four roots. As the line CD is the root of the square quantity and the surface HD is equal to four roots, the line HC becomes equal to four. At the point Q we divide the line HC into two equal parts, and we draw the line QT at right angles to it and equal to each one of the two lines HQ and QC. Its length is thus equal to two. We complete the quadrilateral KQ, which is equal to four. We then extend the line QT to the point L, and we set the line TL equal to each one of the lines AH and BZ. We draw the line LM at right angles to the line QL. The line AQ is thus equal to the line MA. The line QC is therefore equal to MB. But the line QC is also equal to the line LN. The line LN is thus equal to MB. And each one of KN and TL is equal to each one of MN and BZ. The quadrilateral MZ is therefore equal to the quadrilateral KL. In our construction the quadrilateral AN is contiguous to both these quadrilaterals. The quadrilaterals AN and BN together therefore equal the quadrilaterals AN and NT. But the quadrilaterals AN and JMB together are equal to five. The quadrilaterals AN and NT together are thus equal to five. But the quadrilateral KQ is equal to four. The quadrilateral AL has therefore the value nine. Each one of its sides is its root. Thus, the line AQ is equal to three. Now, the line QC was equal to two. The whole line AC is therefore equal to five, and this is the root of the square quantity. And this is the shape of the figure.

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.

7. References

• Adivar, Abdülhak Adnan, Hârizmî, Islam. Ansiklopedisi, vol. 5, No. 42, 1949, pp. 258-259.
• Anbouba, Adel, Al-Karaji, Etudes Littéraires, University of Lebanon, été et automne 1959, pp. 69-70, 76-77.
• Anbouba, Adel, Ihyâ al Jabr, Manshûrât al Jâmi'a al Lubnânîya, Qism al Dirâsât al Riyâdîya, Beyrut 1955.
• Brockelmann, Carl, Geschichte der Arabischen Literatur, supplement vol. I, p. 383.
• Dozy, R., Supplément aux Dictionnaires Arabes, Leiden 1881, vol. 1, p. 617.
• Dunlop, D. M., a Source of Al-Mas'udi: The Madînat al-Fâdilah of Al-Fârâbî, Al-Mas'udi Millenary Commemoration Volume, ed. S. Maqbul Ahmad and A. Rahman, Aligarh Muslim University 1960.
• Frye, Richard N. and Aydin Sayili, Turks in the Middle East Before the Saljuqs, Journal of the American Oriental Society, vol. 63, No. 3, 194.3, pp. 194-207.
• Gandz, S., Studies in Babylonian Mathematics, I, Indeterminate Analysis in Babylonian Mathematics, Osiris, vol. 8, pp. 12-40.
• Gandz, S., The Origin and Development of the Quadratic Equations in Babylonian, Greek, and Early Arabic Algebra, Osiris, vol. 3, pp. 515-516.
• Gandz, S., The Sources of Al-Khwârazmî's Algebra, Osiris, vol. i; 1936, p. 273; Gandz, The Origin and Development of the Quadratic Equations, Osiris, vol. 3, p. 535.
• Hajji Khalîfa, Kashf al Zunûn, art. Kitab al Jabr wa'l Muqabala and art. Kitab al Wâsâyâ, ed. Yaltkaya, Istanbul 1943.
• Heath, T. L., Diophantus of Alexandria, A Study of the History of Greek Algebra, Cambridge 1910.
• Høyrup, Jens, "Algebraic Traditions Behind Ibn Turk and Al-Khwârizmî," pp. 247–268 in Acts of the International Symposium on Ibn Turk, Khwârezmî, Fârâbî, and Ibn Sînâ (Ankara, 9–12 September, 1985). (Atatürk Culture Center Publications, No: 41. Series of Acts of Congresses and Symposiums, No: 1). Ankara: Atatürk Supreme Council for Culture, Language and History, 1990.
• Høyrup, Jens, "Al-Khwârizmî, Ibn Turk, and the Liber Mensurationum: on the Origins of Islamic Algebra." Erdem 2 (Ankara 1986), 445–484.
• Ibn al Nadîm, Kitâb Fihrist al ‘Ulûm, ed. Flugel, vol. 1, 1871.
• Ibn al Qiftî, Tarîkh al Hukamâ, ed. Lippert, Berlin 1903, p. 230.
• Ibn Khaldun, Muqaddima, tr. F. Rosenthal, vol. 3, London 1958.
• Kapp, A. G., Arabische Vbersetzer und Kommentatorm Euklids I, Isis, vol. 22, 1934-35- p- 170.
• L'Algèbre d'Omar Alkhayyâmî, ed. and tr. F. Woepke, Paris 1851.
• Mieli, Aldo, La Science Arabe, Leiden 1939.
• Muhammad ibn Mûsâ al Khwârazmî, Kitâb al Mukhtasar fî Hisâb al Jabr wa'l Muqabala, ed. and tr. F. Rosen, London 1830, 1831.
• Neugebauer, Otto, Studien zur Geschichte der Antiken Algebra, Quellen und Studien zur Geschichte der Mathematik, Aslmnomie, und Physik, series B, vol. 2, 1932, pp. 1-27.
• Rodet, Leon, L'Algèbre d'Al-Khârizmî, Journal Asiatique, series 7, vol. 11, 1878, pp. 90-92.
• Rosenfeld, B. A.,-E. Ihsanoglu, Mathematicians, Astronomers and other Scholars of Islamic Civilisation and their works (7th – 9th c.). Istanbul: Research Center for Islamic History, Art and Culture, 2003.
• Salih Zeki, Athâr-i Bâqiye, vol. 2, Istanbul 1913, p. 246.
• Sarton. George. Introduction to the History of Science, vol. 1, Baltimore 1927, p. 630.
• Sayili, A., The Observatory in Islam, Ankara 1960, p. 101.
• Stern, S. M., Al-Mas'udî and the Philosopher Al-Fârâbî, Al-Mas'udi Commemoration Volume, p. 40.
• Suter, Heinrich, Die Mathematiker und Astronomen der Araber und ihre Werke, Abhandlungen zur Geschichte der Mathematischen Wissenschaften, Leipzig 1900, 1902.
• Umar Khayyâm, Algebra, (L'Algebre d'Omar Al-Khayyâm), ed. and tr. F. Woepke, Paris 1851.
• Vogel, Kurt, Bemerkungen zu den Quadratischen Gleichungen der Babylonischen Mathematik, Osiris, vol. 1, 1936, p. 703.
• Wiedemann, E., Khwârizmî, Encyclopaedia of Islam, vol. 2.
• Woepke, F., Extrait du Fakhrî, Traité d'Algèbre par Abou Bekr Mohammed ben Alhaçan Alkarkhî, Paris 1853, pp. 8, 67-71.

*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.

Previous | 1 | 2 | 3 | 4 | 5 | 6

by: FSTC Limited, Thu 15 January, 2009