Born: 989 in Cordoba, Spain
Died: after 1079 in possibly Jaén, Andalusia (now Spain)

Little is known of al-Jayyani's life. Even the identification of al-Jayyani the mathematician with al-Jayyani the Spanish scholar who was born in Cordoba in 989 is not absolutely certain. Everything points to this identification being correct except one (possible) problem.

The Spanish scholar who was born in Cordoba has exactly the same name as the mathematician, and the Spanish scholar is described as an expert in the Qur'an, also being knowledgeable in Arabic philology, inheritance laws and arithmetic. Al-Jayyani, the mathematician, is described as a judge and a jurist in one of his treatises. The only possible problem to the identification is that al-Jayyani wrote a treatise on the total solar eclipse which occurred in Jaén on 1 July 1079. The identification means that he was over ninety years old when he wrote this treatise which, although certainly not impossible, casts a slight doubt.

The only other facts known about al-Jayyani's life are that he lived in Cairo from 1012 to 1017 and that he must have undertaken most of his work in Jaén, the city at the centre of the Moorish principality of Jayyan. This cannot only be deduced from his name "al-Jayyani" which means "from Jaén", but also from the fact that the astronomical tables that he produced were for the longitude of Jaén. Certainly he observed the solar eclipse in Jaén in 1079.

Al-Jayyani's work On ratio is almost certainly his most interesting mathematical work. An English translation of this remarkable treatise is given in [2]. In this work al-Jayyani sets out to defend Euclid's Elements Book V. In [7] Vahabzadeh writes:-

Euclid's definition, in Book V of his "Elements", of the proportionality of four magnitudes gave rise to numerous commentaries. Of these we have selected two [one being al-Jayyani's] whose goal was not to criticise Euclid's point of view but rather to justify it by trying to make explicit the assumptions underlying Euclid's argument.

Al-Jayyani states that he is writing the treatise On ratio (see for example [1]):-

... to explain what may not be clear in the fifth book of Euclid's writing to such as are not satisfied with it.

There are five magnitudes that, according to al-Jayyani, are used in geometry; number, line, surface, angle, and solid. Neither Euclid nor any other Greek mathematician would have considered "number" as a geometrical magnitude, but al-Jayyani needs the notion for his definition of ratio which follows the Arabic idea of number. After assuming that every intelligent person has a basic understanding of ratio, al-Jayyani deduces further properties based on this "commonly understood definition". To justify his approach he writes:-

There is no method to make clear what is already clear in itself.

He then connects this idea of ratio with that given by Euclid. The authors of [1] write:-

Al-Jayyani shows here an understanding comparable with that of Isaac Barrow, who is customarily regarded as the first to have really understood Euclid's Book V.

Another work of great importance is al-Jayyani's The book of unknown arcs of a sphere, the first treatise on spherical trigonometry. The work, which is published together with a Spanish translation and a commentary in [3], contains formulae for right-handed triangles, the general law of sines, and the solution of a spherical triangle by means of the polar triangle. Proofs are sometimes only given as sketches. Debarnot, in his review of [3], argues however that Villuendas:-

... in his commentary ... fails to take the originality of the Determination of the magnitudes sufficiently into account.

Al-Jayyani was to have a strong influence on European mathematics. In addition to translations of his works from the Arabic, his work influenced certain European mathematicians. The article [4] argues that one of Regiomontanus sources was The book of unknown arcs of a sphere. Among the similarities between al-Jayyani's treatise and that of Regiomontanus are the definition of ratios as numbers, the lack of a tangent function, and a similar method of solving a spherical triangle when all sides are unknown.

However, the author of [4] remarks that there are some marked differences in approach between al-Jayyani and Regiomontanus, such as the proof of the spherical sign law. Although it is certain that Regiomontanus based his treatise on Arabic works on spherical trigonometry it may well be that al-Jayyani's work was only one of many such sources.

The article [6] describes the treatise Kitab al-asrar fi nata'ij al-Afkar (The book of secrets about the results of thoughts), attributed to al-Jayyani on the basis of internal evidence together with its date. The work studies hydraulics and water clocks.

Work by al-Jayyani on astronomy was also important. He wrote on the morning and evening twilight, computing the fairly accurate value of 18° for the angle of the sun below the horizon at the start on morning twilight and at the end of the evening twilight.

1. Y Dold-Samplonius, H Hermelink, Biography in Dictionary of Scientific Biography (New York 1970-1990). 
   http://www.encyclopedia.com/doc/1G2-2830902173.html

Books:

1. E B Plooij, Euclid's conception of ratio (Rotterdam, 1950).
2. M V Villuendas, La trigonometria europea en el siglo XI : Estudio de la obra de Ibn Mu'ad, 'el Kitab mayhulat' (Barcelona, 1979).

Articles:

1. N G Hairetdinova, On spherical trigonometry in the Medieval Near East and in Europe, Historia Math. 13 (2) (1986), 136-146.
2. H Hermelink, Tabulae Jahen, Arch. History Exact Sci. 2 (1964/1965), 108-112.
3. D R Hill, A treatise on machines by Ibn Mu'adh Abu 'Abd Allah al-Jayyani, J. Hist. Arabic Sci. 1 (1) (1977), 33-46.
4. B Vahabzadeh, Two commentaries on Euclid's definition of proportional magnitudes, Arabic Sci. Philos. 4 (1) (1994), 7; 181-198.