**Ahmed ibn Yusuf**'s father Yusuf ibn Ibrahim was also a mathematician. Yusuf ibn Ibrahim lived in Baghdad but moved to Damascus in about 839. After a little while he moved again, taking his son Ahmed with him, and went to live in Cairo. Although we are far from certain about the date of Ahmed's birth it is believed to have been before the family moved to Damascus. Again it is unclear exactly when the family moved again to Cairo but as Ahmed became known as "al-Misri " meaning "the Egyptian" it is likely that he lived in Cairo from a fairly young age.

It is worth saying a word or two about Yusuf ibn Ibrahim, Ahmed's father, since scholars have had some difficulty in deciding which texts are due to the father, which to the son, or perhaps to joint work of the two. Yusuf ibn Ibrahim is known to have been a member of a group of scholars and this must have provided a strong intellectual environment for Ahmed. As well as a text on medicine, Yusuf is known to have written a work on astronomy and produced a collection of astronomical tables.

Ahmed was to achieve an important role in Egypt and to understand this we must examine how Egypt achieved relative independence from the Abbasid Caliph. The Caliphs had strengthened their armies in the 9^{th} century with Turkish slaves and began to put their Turkish commanders into positions as governors of certain territories in the Empire. In 868 the Turkish general Babak was put in charge of Egypt and he chose to send his stepson Ahmad ibn Tulun there to take control. Ahmad ibn Tulun soon built up an army under his own control and managed to take control of the finances of the country. Although he never declared complete independence from the Caliph he governed Egypt, and after 878 also Syria which his armies conquered, as an autonomous region.

Ahmad ibn Tulun had a large family who formed the administration of Egypt. Ahmed ibn Yusuf was appointed as a private secretary to the family, in particular he was employed by one of Ahmad ibn Tulun's sons. In 884 Ahmad ibn Tulun died but his family continued to rule Egypt until the 905 when the Caliph sent an army to retake Egypt for the Empire. The period had been a fruitful one for Egypt during which agriculture, commerce and industry flourished. More importantly for Ahmed ibn Yusuf, the learning and scholarship of Baghdad was encouraged in Egypt, and he was able to pursue his mathematical researches while working for the Tulunid dynasty.

We know of a work by Ahmed on ratio and proportion, a book *On similar arcs*, a commentary on Ptolemy's *Centiloquium* and a book about the astrolabe. All these works have survived and historians are confident that they are indeed the work of Ahmed, but several other works which some claim to be due to him are probably by other authors.

Ahmed's work on ratio and proportion was translated into Latin by Gherard of Cremona. The book is largely a commentary on, and expansion of, Book 5 of Euclid's *Elements*. It was a carefully constructed work which influenced early European mathematicians such as Fibonacci. However it was not without its defects and Campanus of Novara pointed out a circular argument which occurs in Ahmed's reasoning.

The book *On similar arcs* was also translated into Latin and influenced European mathematicians. In the treatise Ahmed proves that similar arcs of circles can be equal and not equal. The proof, like that on ratio and proportion, is based on Euclid. This time it is Propositions 20 and 21 of Book III of Euclid's *Elements* which are the main tools used by Ahmed. The complete Arabic text of this treatise is given in [2].

Ahmed ibn Yusuf also gave methods to solve tax problems which appear in Fibonacci's *Liber Abaci*. He was also quoted by Bradwardine, Jordanus and Pacioli.

**Article by:** *J J O'Connor* and *E F Robertson*

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