952 Abu'l-Hasan al-Uqlidisi

Abu'l-Hasan al-Uqlidisi 952

Abu'l Hasan Ahmad ibn Ibrahim Al-Uqlidisi was an Arab mathematician, who was active in Damascus^[1] and Baghdad.^[2] As his surname indicates, he was a copyist of Euclid's works. He wrote the earliest surviving book on the positional use of the Arabic numerals, Kitab al-Fusul fi al-Hisab al-Hindi (The Arithemetics of Al-Uqlidisi) around 952.^[3]It is especially notable for its treatment of decimal fractions, and that it showed how to carry out calculations without deletions.

While the Persian mathematician Jamshīd al-Kāshī claimed to have discovered decimal fractions himself in the 15th century, J. Lennart Berggrenn notes that he was mistaken, as decimal fractions were first used five centuries before him by al-Uqlidisi as early as the 10th century.^[2]

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.

A second common system was the base-60 numeration inherited from the Babylonians via the Greeks and known as the arithmetic of the astronomers. Although astronomers used this system for their tables, they usually converted numbers to the decimal system for complicated calculations and then converted the answer back to sexagesimals.

The third system was Indian arithmetic, whose basic numeral forms, complete with the zero, eastern Islam took over from the Hindus. (Different forms of the numerals, whose origins are not entirely clear, were used in western Islam.) The basic algorithms also came from India, but these were adapted by al-Uqlīdisī (c. 950) to pen and paper instead of the traditional dust board, a move that helped to popularize this system. Also, the arithmetic algorithms were completed in two ways: by the extension of root-extraction procedures, known to Hindus and Greeks only for square and cube roots, to roots of higher degree and by the extension of the Hindu decimal system for whole numbers to include decimal fractions. These fractions appear simply as computational devices in the work of both al-Uqlīdisī and al-Baghdādī (c. 1000), but in subsequent centuries they received systematic treatment as a general method. As for extraction of roots, Abūʾl-Wafāʾ wrote a treatise (now lost) on the topic, and Omar Khayyam (1048–1131) solved the general problem of extracting roots of any desired degree. Omar’s treatise too is lost, but the method is known from other writers, and it appears that a major step in its development was al-Karajī’s 10th-century derivation by means of mathematical induction of the binomial theorem for whole-number exponents—i.e., his discovery that

During the 10th century Islamic algebraists progressed from al-Khwārizmī’s quadratic polynomials to the mastery of the algebra of expressions involving arbitrary positive or negative integral powers of the unknown. Several algebraists explicitly stressed the analogy between the rules for working with powers of the unknown in algebra and those for working with powers of 10 in arithmetic, and there was interaction between the development of arithmetic and algebra from the 10th to the 12th century. A 12th-century student of al-Karajī’s works, al-Samawʿal, was able to approximate the quotient (20x² + 30x)/(6x² + 12) as

and also gave a rule for finding the coefficients of the successive powers of 1/x. Although none of this employed symbolic algebra, algebraic symbolism was in use by the 14th century in the western part of the Islamic world. The context for this well-developed symbolism was, it seems, commentaries that were destined for teaching purposes, such as that of Ibn Qunfūdh (1330–1407) of Algeria on the algebra of Ibn al-Bannāʿ (1256–1321) of Morocco.

Other parts of algebra developed as well. Both Greeks and Hindus had studied indeterminate equations, and the translation of this material and the application of the newly developed algebra led to the investigation of Diophantine equations by writers like Abū Kāmil, al-Karajī, and Abū Jaʿfar al-Khāzin (first half of 10th century), as well as to attempts to prove a special case of what is now known as Fermat’s last theorem—namely, that there are no rational solutions to x³ + y³ = z³. The great scientist Ibn al-Haytham (965–1040) solved problems involving congruences by what is now called Wilson’s theorem, which states that, if p is a prime, then p divides (p − 1) × (p − 2)⋯× 2 × 1 + 1, and al-Baghdādī gave a variant of the idea of amicable numbers by defining two numbers to “balance” if the sums of their divisors are equal.

However, not only arithmetic and algebra but geometry too underwent extensive development. Thābit ibn Qurrah, his grandson Ibrāhīm ibn Sinān (909–946), Abū Sahl al-Kūhī (died c. 995), and Ibn al-Haytham solved problems involving the pure geometry of conic sections, including the areas and volumes of plane and solid figures formed from them, and also investigated the optical properties of mirrors made from conic sections. Ibrāhīm ibn Sinān, Abu Sahl al-Kūhī, and Ibn al-Haytham used the ancient technique of analysis to reduce the solution of problems to constructions involving conic sections. (Ibn al-Haytham, for example, used this method to find the point on a convex spherical mirror at which a given object is seen by a given observer.) Thābit and Ibrāhīm showed how to design the curves needed for sundials. Abūʾl-Wafāʾ, whose book on the arithmetic of the scribes is mentioned above, also wrote on geometric methods needed by artisans.

In addition, in the late 10th century Abūʾl-Wafāʾ and the prince Abū Naṣr Manṣurstated and proved theorems of plane and spherical geometry that could be applied by astronomers and geographers, including the laws of sines and tangents. Abū Naṣr’s pupil al-Bīrūnī (973–1048), who produced a vast amount of high-quality work, was one of the masters in applying these theorems to astronomy and to such problems in mathematical geography as the determination of latitudes and longitudes, the distances between cities, and the direction from one city to another.

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.