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Little is known of alJayyani's life. Even the identification of alJayyani the mathematician with alJayyani 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. AlJayyani, the mathematician, is described as a judge and a jurist in one of his treatises. The only possible problem to the identification is that alJayyani 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 alJayyani'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 "alJayyani" 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.
AlJayyani'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 alJayyani 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 alJayyani'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.
AlJayyani 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 alJayyani, 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 alJayyani 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, alJayyani 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:
AlJayyani 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 alJayyani'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 righthanded 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.
AlJayyani 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 alJayyani'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 alJayyani 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 alJayyani's work was only one of many such sources.
The article [6] describes the treatise Kitab alasrar fi nata'ij alAfkar (The book of secrets about the results of thoughts), attributed to alJayyani on the basis of internal evidence together with its date. The work studies hydraulics and water clocks.
Work by alJayyani 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.
In the Tabulae Jahen alJayyani gave data to enable the calculation of the time of day, the calendar, the new moon, eclipses and information required for the timing and direction for prayers. As was common at this time, not only was there astronomical information in the work but also astrological information on horoscopes. AlJayyani seems to have considerable respect for alKhwarizmi's astronomical data, which he freely used, but he rejects the ideas of alKhwarizmi on astrology. Much of alJayyani's astrology is based on Hindu sources.
Article by: J J O'Connor and E F Robertson
List of References (7 books/articles)
 
Mathematicians born in the same country

JOC/EFR © November 1999 Copyright information 
School of Mathematics and Statistics University of St Andrews, Scotland  
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