Abū’l-‘Abbās al-Faḍl ibn Ḥātim al-Nairīzī (Arabic: أبو العباس الفضل بن حاتم النيريزي, Latin: Anaritius, Nazirius, 865–922) was a Persian mathematician and astronomer from Nayriz, Fars Province, Iran.
He flourished under al-Mu’tadid, Caliph from 892 to 902, and compiled astronomical tables, writing a book for al-Mu’tadid on atmospheric phenomena.
Nayrizi wrote commentaries on Ptolemy and Euclid. The latter were translated by Gerard of Cremona. Nairizi used the so-called umbra (versa), the equivalent to the tangent, as a genuine trigonometric line (but he was anticipated in this by al-Marwazi).
He wrote a treatise on the spherical astrolabe, which is very elaborate and seems to be the best Persian work on the subject. It is divided into four books:
- Historical and critical introduction.
- Description of the spherical astrolabe; its superiority over plane astrolabes and all other astronomical instruments.
He gave a proof of the Pythagorean theorem using the Pythagorean tiling.
Ibn al-Nadim mentions Nayrizi as a distinguished astronomer with eight works by him listed in his book al-Fihrist.
Al-Nayrizi was probably born in Nayriz which was a small town southeast of Shiraz now in central Iran. Certainly he must have been associated with this town in his youth to have been called al-Nayrizi. Little is known of his life but we do know that he dedicated some of his works to al-Mu’tadid so he almost certainly moved to Baghdad and worked there for the caliph.
The period during which al-Nayrizi was growing up was a turbulent one in the region in which he lived. Following the assassination of the caliph al-Mutawwakil in 861 there was a period of anarchy and civil war. The Caliph al-Mu’tamid and his brother al-Muwaffaq who was a military leader, reunited the empire from 870 but a rebellion was eventually put down in 883 only after many years of military campaigns by al-Muwaffaq and his brother al-Mu’tadid. Al-Mu’tamid died in 892 and, since al-Mu’tadid had forced him to disinherit his own son, al-Mu’tadid became caliph in that year.
Al-Mu’tadid reorganised the administration and reformed finances, and he demonstrated great skill and ruthlessness in dealing with the many factions that had arisen. There followed a period of great cultural activity, with Baghdad home to many intellectuals. Al-Nayrizi must have worked for al-Mu’tadid during his ten year of rule, for he wrote works for the caliph on meteorological phenomena and on instruments to measure the distance to objects. If al-Mu’tadid’s reign had begun with political intrigue then it seemed to end in the same way, the general opinion being that, in 902, al-Mu’tadid was poisoned by his political enemies. Al-Mu’tadid’s son al-Muktafi became caliph in 902 and ruled until 908. It seems likely that al-Nayrizi would continue to work in Baghdad for the new caliph since the same support for intellectuals in Baghdad continued.
The Fihrist (Index) was a work compiled by the bookseller Ibn an-Nadim in 988. It gives a full account of the Arabic literature which was available in the 10th century and in particular mentions al-Nayrizi as a distinguished astronomer. Eight works by al-Nayrizi are listed in the Fihrist. A later work, written in the 13th century, described al-Nayrizi as both a distinguished astronomer and as a leading expert in geometry.
Al-Nayrizi’s works on astronomy include a commentary of Ptolemy’s Almagest and Tetrabiblos. Neither have survived. He is most famous for his commentary on Euclid’s Elementswhich has survived. The Leiden manuscript referred to in the title of  contains the revision by al-Nayrizi of the second Arabic translation of Euclid’s Elements by al-Hajjaj. The translation by al-Hajjaj has not survived and the article  examines to what extent al-Nayrizi changed the translation, arguing that indeed he made considerable changes. The paper  looks at different manuscripts containing versions of al-Nayrizi’s commentary, some in Arabic, one a Latin version.
In dealing with ratio and proportion in his commentary on the Elements, al-Nayrizi adopts concepts proposed by al-Mahani who had worked in Baghdad, probably before al-Nayrizi arrived there. Al-Nayrizi wrote a work on how to calculate the direction of the sacred shrine of the Ka’bah in Mecca (to was important for Muslims to be able to do this since they had to face that direction five times each day when performing the daily prayer). In this work he effectively uses the tan function, but he was not the first to use these trigonometrical ideas.
The article  is a translation into Russian of the short treatise by al-Nayrizi on Euclid’s fifth postulate. In his work on proofs of the parallel postulate, al-Nayrizi quotes work by a mathematician named Aghanis. In  Sabra argues convincingly that Aghanis is the Athenian philosopher Agapius who was a pupil of Proclus and Marinus and taught around 511 AD.