**Al-Uqlidisi** is a mathematician who is only known to us through two manuscripts on arithmetic, *Kitab al-fusul fi al-hisab al-Hindi* and *Kitab al-hajari fi al-hisab.* Despite this he is a figure of some importance and has prompted an interesting scholarly argument among historians of science.

The manuscript of the *Kitab al-fusul fi al-hisab al-Hindi* which has survived is a copy of the original which was made in 1157. An English translation of this work has been published by Saidan [4]. The manuscript gives al-Uqlidisi's full name on the front page as well as the information that he composed the text in Damascus in 952-53. In the introduction al-Uqlidisi writes that he travelled widely and learnt from all the mathematicians he met on his travels. He also claimed to have read all the available texts on arithmetic. Other than being able to deduce a little of al-Uqlidisi's character from his writing, we have no other information on his life.

The *Kitab al-fusul fi al-hisab al-Hindi* of al-Uqlidisi is the earliest surviving book that presents the Hindu system. In it al-Uqlidisi argues that the system is of practical value [4]:-

Most arithmeticians are obliged to use it in their work: since it is easy and immediate, requires little memorisation, provides quick answers, demands little thought ... Therefore, we say that it is a science and practice that requires a tool, such as a writer, an artisan, a knight needs to conduct their affairs; since if the artisan has difficulty in finding what he needs for his trade, he will never succeed; to grasp it there is no difficulty, impossibility or preparation.

This treatise on arithmetic is in four parts. The aim of the first part is to introduce the Hindu numerals, to explain a place value system and to describe addition, multiplication and other arithmetic operations on integers and fractions in both decimal and sexagesimal notation. The part second collects arithmetical methods given by earlier mathematicians and converts them in the Indian system. For example the method of casting out nines is described.

The third part of the treatise tries to answer to the standard type of questions that are asked by students: why do it this way ... ?, how can I ... ?, etc. There is plenty of evidence here that al-Uqlidisi must have been a teacher, for only a teacher would know understand the type of problem that a beginning student would encounter.

The fourth part has considerable interest for it claims that up to this work by al-Uqlidisi the Indian methods had been used with a dust board. A dust board was used because the methods required the moving of numbers around in the calculation and rubbing some out as the calculation proceeded. The dust board allowed this in the same sort of way that one can use a blackboard, chalk and a blackboard eraser. However, al-Uqlidisi showed how to modify the methods for pen and paper use.

Al-Uqlidisi's work is historically important as it is the earliest known text offering a direct treatment of decimal fractions. It is here that the scholarly argument referred to above arises. At one time it was thought that Stevin was the first to propose decimal fractions. Further research showed that decimal fractions appeared in the work of al-Kashi, who was then credited with this extremely important contribution. When Saidan studied al-Uqlidisi's *Kitab al-fusul fi al-hisab al-Hindi* in detail he wrote [6]:-

The most remarkable idea in this work is that of decimal fraction. Al-Uqlidisi uses decimal fractions as such, appreciates the importance of a decimal sign, and suggests a good one. Not al-Kashi(d.1436/7)who treated decimal fractions in his "Miftah al-Hisab", but al-Uqlidisi, who lived five centuries earlier, is the first Muslim mathematician so far known to write about decimal fractions.

Following Saidan's paper, some historians went even further in attributing to al-Uqlidisi the complete credit for giving the first complete description and applications of decimal fractions. Rashed, however, although he does not wish to minimise the importance of al-Uqlidisi's contribution to decimal fractions, sees it as [2]:-

... preliminary to its history, whereas al-Samawal's text already constitutes the first chapter.

The argument depends on how one interprets the following passage in al-Uqlidisi's treatise. He explains how to raise a number by one tenth five times [4]:-

... we want to raise a number by its tenth five times. We write down this number as usual; write it down again below moved one place to the right; we therefore know its tenth, which we add to it. So was have added its tenth to this number. We put the resulting fraction in front of this number and we move it to the unit place after marking it[with the ' sign he uses for the decimal point]thus. We add its tenth and so on five times.

Saidan (writing in [1]) sees in this passage that al-Uqlidisi has fully understood the idea of decimal fractions, saying that earlier authors:-

... rather mechanically transformed the decimal fraction obtained into the sexagesimal system, without showing any sign of comprehension of the decimal idea. ... In all operations where powers of ten are involved in the numerator or the denominator,[al-Uqlidisi]is well at home.

On the other hand Rashed sees this passage rather differently [2]:-

... unlike al-Samawal, al-Uqlidisi never formulates the idea of completing the sequence of powers of ten by that of their inverse after having defined the zero power. That said, in the passage just quoted, three basic ideas emerge whose intuitive resonance may have misled historians; what they thought was a theoretical exposition was merely understood implicitly, and, as a result, they have overestimated the author's contribution to decimal fractions.

The two points of view are almost impossible to decide between since what we are looking at is the development of the idea of decimal fractions by different mathematicians, each contributing to its understanding. To take a particular text as the one where the idea appears for the first time in its entirety must always be a somewhat arbitrary decision. There is no disagreement on the fact that al-Uqlidisi made a major step forward.

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

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