Muhammad Al-Karaji: A Mathematician Engineer from the Early 11th Century

Abu Bakr Muhammed Al-Karaji is a Muslim mathematician and engineer from the late 10th century-early 11th century. Of Persian origin, he spent an important part of his scientific life in Baghdad where he composed ground breaking mathematical books. Al-Karaji is also the author of Inbat al-miyah al-khafiya (The Extraction of Hidden Waters), a technical treatise that reveals such a profound knowledge of hydrology that it should be celebrated as the oldest text of its kind in this field. The book provides an outstanding study on the different kinds of waters, the methods to find the water level, the description of instruments for surveying, the construction of the conduits, their lining, protection against decay, and their cleaning and maintenance. In this article the scientific work of Al-Karaji is characterized, details of his biography are surveyed and a special attention is paid to expound the contents of his treatise of hydology.

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Mohammed Abattouy*

Table of contents

1. The career of mathematician-engineer
2. Al-Karaji: His life and works
3. Al-Karaji's book on the extraction of groundwater
4. Contents of the book
5. A subterranean conduit for water supply: the Qanat
6. References and further reading


From the 9th through the 16th centuries, Islamic societies experienced a "golden age" of science and technology. One of the most important fields in which they applied their knowledge and practical experience is the vast area of hydrology, in the sense of the various means of water supply, the control of the movement of water, and the different devices invented and applied therein. The rise of cities like Baghdad, Cairo, Cordoba, Damascus, Fez and Marrakech required increasingly sophisticated methods of water management to supply rapidly growing populations. Integrating, adapting and refining irrigation techniques and water distribution methods inherited from local expertise or borrowed from ancient civilisations, the water engineers of Islam started as early as the 8th century to build a real agricultural revolution based in great part on their mastery of hydrology [1].

Figure 1: Diagrams from the original manuscript of Al-Karaji's Inbat al-miyah al-khafiya, from Transformation of Knowledge: Early Manuscripts from the Schoenberg Collection (edited by Crofton Black, Paul Holberton publishing, 2007, p. 115). (Source: section 7 "Technology").

1. The career of mathematician-engineer

One of the earliest Arabic texts explaining how to locate aquifers, dig survey wells and build underground canals, is the treatise of the mathematician Muhammad Al-Karaji Inbat al-miyah al-khafiya (Book of the extraction of hidden waters), written about 1000 C.E in Iraq or Iran. The book is a technical treatise which gives good details on the finding of the water level, instruments for surveying, construction of the conduits, their lining, protection against decay, and their cleaning and maintenance. Before Al-Karaji and after him, such as it is explicitly stated by Ibn Sina (980-1037) in his Risala fi aqsam al-'ulum al-'aqliya (Treatise on the divisions of the rational sciences), hydraulics was established as an independent discipline on a par with geometry and astronomy [2]. It is not surprising then to find a gifted mathematician interested in such a practical area.

Al-Karaji was not the only one who joined to scientific expertise interest in engineering. Several of his immediate predecessors and contemporaries did the same: Al-Farghani (fl. ca. 860), Thabit ibn Qurra (d. 901), Al-Kuhi (d. ca 1000), to name just a few. Al-Farghani illustrates well this double interest in pure and applied science in the Islamic tradition, although in his case things did not turn out very successfully as is shown through the following story.

Abu ‘l-'Abbas Ahmad ibn Muhammad ibn Kathir Al-Farghani (d. ca. 865), known as Alfraganus, born in Farghana, Transoxiana, was one of the most distinguished astronomers in the service of Al-Mamun and his successors. His book Kitab fi al-harakat al-samawiya wa-jawami' ‘ilm al-nujum (The Book on celestial motion and thorough science of the stars) was translated into Latin in the 12th century and exerted great influence upon European astronomy before Regiomontanus.

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Figure 2: Page from Al-Kitâb al-Fakhrî by Al-Karaji. (Source).

Al-Farghani's activities extended to engineering. According to Ibn Tughri Birdi, he supervised the construction of the Great Nilometer at Al-Fustat (old Cairo). It was completed in 861 CE. But engineering was not Al-Farghani's forte, as transpires from the following story narrated by Ibn Abi Usaybi'a.

The Caliph Al-Mutawakkil had entrusted two brothers Banu of Musa, Muhammad and Ahmad, with supervising the digging of a canal named Al-Ja'fari. They delegated the work to Al-Farghani, thus deliberately ignoring a better engineer, Sind ibn ‘Ali, whom, out of professional jealousy, they had caused to be sent to Baghdad, away from Al-Mutawakkil's court in Samarra. The canal was to run through the new city, Al-Ja'fariya, which Al-Mutawakkil had built near Samarra on the Tigris and named after himself. Al-Farghani committed a grave error, making the beginning of the canal deeper than the rest, so that not enough water would run through the length of the canal except when the Tigris was high. News of this angered the Caliph, and the two brothers were saved from severe punishment only by the gracious willingness of Sind ibn ‘Ali to vouch for the correctness of Al-Farghani's calculations, thus risking his own welfare and possibly his life. However, Al-Mutawakkil died shortly before the error became apparent. The explanation given for Al-Farghani's mistake is that, being a theoretician rather than a practical engineer, he never successfully completed a construction [3].

Whether true or not, this story informs that the active and dynamic milieu of competition that reigned in Baghdad in 9th-10th centuries was a powerful motive force behind the high degree of creativity and inventiveness that distringuished Islamic science of this period.

2. Al-Karaji: His life and works

Al-Karaji (spelled Al-Karadji), Abu Bakr Muhammad b. al-Hasan (or al-Husayn in some sources) was a 10th-century mathematician and engineer (4th century H). He is known as Al-Hasib (the calculator, meaning the mathematician). According to Girogio Levi Della Vida [4], he is a native of Karadj (in Iran) and not from Al-Karkh district of Baghad, as it is claimed in certain modern writings.

Figure 3: Diagram of a qanat, developed in Islamic lands as a water management system used to provide a reliable supply of water to human settlements or for irrigation in hot, arid and semi-arid climates. (Source).

While still young, he went to Baghdad where he held high positions in the administration and composed, towards 402 H/1011-12 CE, his known works in mathematics Al-Fakhri, Al-Kafi and Al-Badi', in which he attempted to free algebra from geometry. He returned afterwards to his native land, where he must have died after 406 H/1015 CE, the probable date of the composition of his Inbat al-miyah al-Khafiya. His sojourn in Baghdad was during the Buyahid era (which lasted between 334 and 447 H/945-1055 CE). No ancient source mentions the dates of his birth or death, but certain details of his biography have been reconstructed from the events of his time and from very few sentences in his books, from which we deduce that he died after 406 H/1015 CE [5].

The name of our scholar appears in modern scholarship both as Al-Karaji and Al-Karakhi (or Al-Karkhi). Karaj is a city in Iran (region of Tehran) and as the mathematician's name is Al-Karaji then certainly his family were from that city. On the other hand, Karkh is a district in the suburb of Baghdad; in this case the name Al-Karkhi would indicate that the mathematician originated from Baghdad. The historians of his work today use most often Al-Karaji to refer to him. As formulated by Roshdi Rashed, in the present state of our knowledge, it is far from easy to decide in favour of either name, at least because of the scarcity of the biographical information available about him in the classical Arabic sources (he is not mentioned by Ibn al-Nadim nor Ibn abi Usaybi'a in their major books of bio-bibliography of the Islamic scientific tradition). At any rate, he certainly lived and worked for the most fruitful period of his life in Baghdad and his chief mathematical works were written during the time when he lived in that city. His important treatise on algebra Al-Fakhri was dedicated to the vizier Fakhr al-Mulk, minister of Baha' al-Dawla, the Buyahid ruler of Baghdad (d. 406H/1015 CE). However, at some later point in his career, Al-Karaji left the Abbasid capital to live in what are described as the "mountain countries". He seems to have given up mathematics at this time and concentrated on engineering topics such as hydrology and hydraulics. Plausibly, therefore, his book Kitab inbat al-miyah al-khafiya (Book on the extraction of hidden waters) belongs to this latter period [6].

Besides his books mentioned above, he is credited with several other titles, some of which are still lost whilst others are extant and some of them were edited. These include several titles of classical subjects in mathematics and astronomy: a book of mathematics of inheritance (Al-Dawr wa-'l-wasaya), a book of mathematics of which we know only the title, Nawadir al-ashkal (Rare theorems), a treatise on The Reasons of the calculation of algebra (‘ilal hisab al-jabr wa-'I-muqabala), a book on buildings contracts (‘Uqud al-abniya), Kitab fi hisab al-hind, Kitab fi al-'istiqra' bi-'l-takht, Al-Madkhal ila ‘ilm al-nujum, Kitab al-muhit fi ‘l-hisab, Kitab al-ajdhar, Kitab hawla tasnif al-judhur, and Risalat al-khta'ayn.

Figure 4: A pool at Aqiq, Saudi Arabia, one of dozens of rest and water stations on the pilgrim road from Iraq to Makkah. It still holds water more than a thousand years after it was constructed under the patronage of Zubaydah, the wife of caliph Harun al-Rashid. (Source).

‘Adil Anbuba, in the introduction of his edition of the Badi' (Beirut 1964), lists 12 works of Al-Karaji, most of which have been lost. The following four titles of mathematics and hydraulic engineering are of interest: (1). Al-Fakhri fi ‘l-jabr wa ‘l-muqabala (The Fakhri [or ‘The Glorious'] book of algebra;); (2). Al-Badi' fi ‘l-hisab (The Marvellous book in arithmetic), (3). Al-Kafi fi ‘l-hisab (The Sufficient book in arithmetic); and (4). Inbat al-miyah al-khafiya, which will be surveyed below.

Al-Fakhri fi ‘l-jabr wa ‘l-muqabala was studied by Franz Woepcke since the middle of the 19th century. Woepcke demonstrates the agreements between this work and the Arithmetica of Diophantus, which Al-Karaji must have known through the partial translation (the first three books and a part of the fourth) by Qusta ibn Luqa (d. 296/912), and concludes that "more than a third of the problems of the first book of Diophantus, the problems of the second book beginning with the eighth one, and almost all the problems of the third, have been inserted by Alqarkhi into his collection" [7].

Figure 5: The Albolafia noria, or waterwheel, is the last vestige of an array of mills and dams built on the Guadalquivir River in Cordoba between the 8th and 10th centuries as it appears in its present condition. (Source).

In the Fakhri, Al-Karaji studied the successive powers of a binomial, developed it further in the Badi', and concluded his analysis in a work now lost but preserved in fragments in the Bahir of Al-Samaw'al b. Yahya Al-Maghribi (d. ca. 570/1174), through the discovery of the generation of the coefficients of (a-b) n by means of the triangle which is now named after Pascal or Tartaglia.

Al-Karaji's other mathematical book Al-Badi' fi ‘l-hisab is a systematic treatise in which are developed the chapters treated by Euclid and Nicomachus and in which an important place is accorded to algebraic operations. Al-Karaji expounds for the first time the theory of the extraction of the square root of a polynomial with an unknown and resolves the equations of the type x2 + 5, x2 - 5, x2 + y and y2 + x. Studied later on by Fibonacci (Leonardo of Pisa, (ca. 1170-ca. 1250) in his Liber Quadratorum, those equations attracted much more later the attention of Euler and others in 18th-century European mathematics. In his study of these problems, Al-Karaji often utilizes the expedient of changing the variable, the auxiliary variables or the process through substitution.

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Figure 6: The shaduf was known in ancient times in Egypt and Assyria. It consists of a long beam supported between two pillars by a wooden horizontal bar. A counterweight was attached to the short arm of the beam. A bucket suspended by a rope or a pole was attached to the long arm of the beam. The bucket was lowered into the water by bearing down on the rope/pole and the counterweight raised the full bucket. The shaduf is still used in Egypt. See: Sandra Postel, "Egypt's Nile Valley Basin Irrigation". (Source).

His other book Al-Kafi fi ‘l-hisab, written on the use of functions, was a summary of arithmetic, algebra, geometry and the processes of mental calculus (called hawa'i, "aerial") as opposed to "Indian" calculus).

The examination of these text books by modern historians shows Al-Karaji as a mathematician of the highest calibre. His enduring contributions to the field of mathematics are still recognized today, as the canonical form of the table of binomial coefficients (in its formation law and its expansion form), for integer n, are named after Al-Karaji.

Al-Karaji wrote about the work of earlier mathematicians, and he is now regarded as the first person to free algebra from geometrical operations, that were the product of Greek arithmetic, and replace them with the type of operations which are at the core of algebra today. His work on algebra and polynomials gave the rules for arithmetic operations to manipulate polynomials. In his pioneering work in French, the historian of mathematics Franz Woepcke (in Extrait du Fakhri, traité d'Algèbre par Abou Bekr Mohammed Ben Alhacan Alkarkhi, Paris, 1853), praised Al-Karaji for being "the first who introduced the theory of algebraic calculus". Stemming from this, Al-Karaji investigated binomial coefficients and Pascal's triangle. He was also the first to use the method of proof by mathematical induction to prove his results, which he also used to prove the sum formula for integral cubes, an important result in integral calculus. He also used a proof by mathematical induction to prove the binomial theorem and Pascal's triangle [8].

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Figure 7: The Saqiya machine of Al-Jazari, an animal powered device for raising water. Source: the original manuscript of Al-Jazari's treatise Al-Jami' bayna al-'ilm wa-'l-'amal al-nafi' fi sina'at al-hiyal held at Topkapi Palace Museum Library in Istanbul, MS Ahmet III 3472, p. 216. See S. Al-Hassani & C. Ong Pang Kiat, Al-Jazari's Third Water-Raising Device: Analysis of its Mathematical and Mechanical Principles.

In his book The Development of Arabic Mathematics: Between Arithmetic and Algebra (London, 1994), the historian Roshdi Rashed writes: "the more-or-less explicit aim of Al-Karaji's exposition was to find the means of realising the autonomy and specificity of algebra, so as to be in a position to reject, in particular, the geometric representation of algebraic operations… Al-Karaji's work holds an especially important place in the history of mathematics... The discovery and reading of the arithmetical work of Diophantus, in the light of the algebraic conceptions and methods of Al-Khwarizmi and other Arab algebraists, made possible a new departure in algebra by Al-Karaji" [9].

3. Al-Karaji's book on the extraction of groundwater

According to the resultst of recent scholarship, Al-Karaji wrote his mathematical works in Baghdad, but he composed his book on hidden waters in the Jabal region in Iran, where there were developed several hydraulic projects, including the qanats, an old Persian tradition, improved in Islamic times. Al-Maqdisi said in his geographical book Ahsan al-taqasim fi ma'rifat al-aqalim that iqlim al-jabal includes three districts or counties: Ray, Hamadhan and Isfahan, whilst Karaj with, which Al-Karaji's name is associated, is located between four mountains full of farms and villages, with several rivers and water sources [10].

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Figure 8: The largest norias or water wheels in the world, with a diameter of about 20 meters, exist on the Orontes River in Hama, Syria. Norias (na'ura in Arabic, pl. nawa'ir) are machines for lifting water into an aqueduct using energy derived from the water's flow. It consists of an undershot waterwheel to which are fixed a series of containers that lift water from the river to the aqueduct at a higher level. (Source).

The Inbat al-miyah al-khafiya is the only extant book of engineering of Al-Karaji. It was printed for the first time in Haydarabad in 1940 [11]. Another edition was issued in 1997 by the Institute of Arabic Manuscripts in Cairo. The book was translated into Persian by H. Khadiv-Djam in 1966 and into French by Aly Mazahéri in 1973. An Italian translation appeared recently in 2007.

There are no complete English and German translations, but some chapters of the book were rendered into these two languages in various works published respectively by Eilhard Wiedemann and Franz Hauser (German) and Fr. Bruin (English).

One of the manuscripts of the book surfaced recently. Dated 14 Dhu-'l-Qa'da 1084/20 February 1674, it was copied in Iran or Iraq. It was acquired by the library of the University of Pennsylvania from Sam Fogg, London, in December 2000. The manuscript is in paper (49 folios, 190x125 mm), in Arabic, nasta'liq script, black ink, rubrication and overlining in red; it contains 14 large diagrams in black and red.

Al-Karaji writes in the introduction of the book that when he arrived in Iraq, he saw that its people, young and older, love knowledge and value it highly, which drove him to compose books of mathemaics in arithmetics and geometry. Later on, he returned to the district of Jabal (ardh al-jabal). There he stopped writing books and making scientific research, until he was encouraged by the sponsorship and encouragement of the minister Abu Ghanim Ma'ruf b. Muhammad, and that this is the reason for which he wrote the book on hidden waters From this, we conclude that the book was written after the mathematical treatises mentioned above. From the mention of the minister Abu Ghanim, the date of the book must be after 406 H as asserted above: