Ibn Mu'adh al-Jayyani

Background

Ibn Mu'adh al-Jayyani was born c. 989 in Cordova, al-Andalus (now in Spain). Little is known about his life.

Career

Ibn Bash-kuwal mentions a Koranic scholar of the same name who had some knowledge of Arabic philology, inheritance laws, and arithmetic. Since in his treatise Maqala ft sharh al nisba (On Ratio) al-Jayyani is called qad't (judge) as well as faqih (jurist), he is thought to be identical with this scholar, who was born in Cordoba in 989-990 and lived in Cairo from the beginning of 1012 until the end of 1017. The date of al-Jayyani’s death must be later than 1079, for he wrote a treatise (“On the Total Solar Eclipse”) on an eclipse which occurred in Jaen on July 1, 1079. This means that he took the real astronomical and not the average date according to the ordinary Islamic lunar calendar. In Tabulae Jahen he explains that the difference between these two dates may amount to as much as two days.

“On the Total Solar Eclipse” was translated into Hebrew by Samuel ben Jehuda, as was a treatise entitled “On the Dawn.” A Latin translation of the latter work, Liber de crepusculis, was made by Gerard of Cremona. The Arabic texts of these two works are not known to be extant.

The Liber de crepusculis, a work dealing with the phenomena of morning and evening twilights, was for a long time attributed to Ibn al-Haytham, probably because in some manuscripts it comes immediately after his Perspectiva or De aspect ibus, sometimes without any mention of the name of the author of the second work. In it al-Jayyani gives an estimation of the angle of depression of the sun at the beginning of the morning twilight and at the end of the evening twilight, obtaining the reasonably accurate value of 18°. On the basis of this and other data he attempts to calculate the height of the atmospheric moisture responsible for the phenomena of twilights. The work found a wide interest in the Latin Middle Ages and in the Renaissance.

In the Libros del saber, al-Jayyani is quoted as considering the twelve astrological houses to be of equal length. Other astronomical works by al-Jayyani are the Tabula residuum ascensionum ad revolutiones annorum solarium secundum Multad Arcadius, preserved in Latin translation (possibly a fragment of the Tabulae Jahen), and Matrah shifa'at al-kawakib (“Projection of the Rays of the Stars”).

Several mathematical works by al-Jayyani are extant in Arabic. His treatise Kitdb majhulat qisiyy al-kura (“Determination of the Magnitudes of the Arcs on the Surface of a Sphere”), which is also cited in Saraceni cuiusdam de Eris, is a work on spherical trigonometry.

The treatise On Ratio is a defense of Euclid. Al-Jayyani says in his preface that it is intended “to explain what may not be clear in the fifth book of Euclid’s writing to such as are not satisfied with it.” The criticism of Euclid, to which al-Jayyani objected, was a general dissatisfaction among Arabic mathematicians with Euclid V, definition 5. The cool, abstract form in which the Euclidean doctrine of proportions was presented did not appeal to the Arabic mind, since little or nothing could be deduced regarding the way in which it had come into being. So from the ninth century on, the Arabs tried either to obtain equivalent results more in accord with their own views, or to find a relation between their views and the unsatisfying theory. Those who chose the second way, such as Ibn al-Haytham, al-Khayyami, and al-Tusi, tried to explain the Greek technique of equimultiples in terms of more basic, better-known concepts and methods.

The most successful among them was al-Jayyani. To establish a common base he assumes that a rightthinking person has a primitive conception of ratio and proportionality. From this he derives a number of truths characteristic of proportional magnitudes, without proofs, since “There is no method to make clear what is already clear in itself.” He then makes the connection by converting Euclid’s multiples into parts, so that magnitudes truly proportional according to his own view also satisfy Euclid’s criterion. The converse is proved by an indirect proof much resembling the one of Ibn al-Haytham, being based on the existence of a fourth proportional and the unlimited divisibility of magnitudes. In the third part al-Jayyani deals with unequal ratios.

Achievements

• Ibn Mu'adh al-Jayyani is remembered as an important Arab mathematician and astronomer. He is probably best remembered as a supporter of Euclid, and also for his important commentaries on Euclid's Elements. He is also known for his treatise On the Total Solar Eclipse, and for the treatise on spherical trigonometry. This treatise later had a "strong influence on European mathematics," and his "definition of ratios as numbers" and "method of solving a spherical triangle when all sides are unknown" are likely to have influenced Regiomontanus.

Views

Al-Jayyani was a fervent admirer of Euclid.

References

• Dictionary of Scientific Biography, Volume 7
  1973