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

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 9^th through the 16^th 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 8^th century to build a real agricultural revolution based in great part on their mastery of hydrology [1].

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 12^th century and exerted great influence upon European astronomy before Regiomontanus.

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 9^th-10^th 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 10^th-century mathematician and engineer (4^th 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. 

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. 

‘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 19^th 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].

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 18^th-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.

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].

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].

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:

The title of the book contains a word worth of some comments. The inbat, like istikhraj, means precisely "extraction" of underground water, to show what is hidden and to extract ground and hidden waters for economic and social benefit. The term may have to do with the mathematical concept of istinbat, meaning ‘deduction by reasoning' [12]. If this is verified, the link between the two is natural, as Al-Karaji would have coined the term in the aftermath of his long experience as a mathematician. 

4. Contents of the book

The Inbat al-miyah al-khafiya is an excellent manual on hydraulic water supplies; besides its main interest in hydrology, it contains some auto-biographical notes, as well as a discussion of a series of conceptions relative to the geography of the globe, various remarks on natural phenomena, and pays a great attention to surveying techniques, whether in general or in what regards hydrology. The author describes a certain number of surveying instruments, the geometrical bases of which he demonstrates, and ends with very concrete details on the construction and servicing of qanats, subterranean tunnels (he makes an express allusion to those of Isfahan) for providing water in arid places. He likewise discusses the basis of the legality of the construction of wells and hydraulic conduits and in what circumstances these might be prejudicial to the people. Here he refers to the schools of fiqh (Islamic law) and shows that he is aware of the legal dimensions of hydrology, as a techno-scientific discipline closely related to society and economy.

As a scientific treatise, the book is an original contribution in hydrology, surveying and other aspects of geology, and testifies to the advanced knowledge concerning groundwater around the 10^th century in the Islamic lands. Relying on the knowledge of his time and on his owns researches, Al-Karaji reveals a profound and exacting technical understanding of groundwater theory and as such his contribution in this field is the oldest known text on the subject. His knowledge of groundwater is in general agreement with modern understanding of the subject. For example, our scholar was familiar with the general concepts of the hydrological cycle. While he never featured the whole cycle as we know it, he records in different passages of his book each individual phase.

The contents of the book can be summarized roughly as follows. The treatise is divided into 25 chapters that may be grouped in seven sections or parts:

• The first part is an introduction to the treatise: it begins with the basmala and the dedication of the book to the minister Abu Ghanim Ma'ruf b. Muhammad, and the assertion of the subject of the book.

• The second part (chapters 2-11): the author expounds here various considerations of natural philosophy on geology, the aspect of the earth, ground water, water sources, the mountains, the various kinds of waters, the methods to distinguish between them, and numerous considerations on the soils. This part is the basis of the treatise in terms of scientific knowledge as it contextualises hydrology in the larger field of natural science and geology.

• The third part (chapters 12-14): this is a treatise of legal content based on the arguments of the various Islamic schools of law concerning qanats, their digging, characteristics and use. This part is the social background of the work.

• The fourth part (chapters 15-17): centred on themes of engineering in hydrology, mainly relevant to the transport of water, digging qanats, water aqueducts and the description of the required techniques for their maintenance.

• The fifth part (chapters 18-19): on surveying and its instruments; description of the latter and demonstration of theorems for their use; a special interest is paid in this section and in the next one to the implementation of surveying techniques in hydorology.

• The sixth part (chapters 20-24): further analysis of surveying methods and instruments; the author describes traditional surveying instruments and procedures with them. In chapter 23, he describes ‘some instruments I have invented', namely a calibrated plate for surveying and its chains and staves; it continues with the description of a dioptre-like instrument [13].

• The seventh part (chapter 25): this is the concluding chapter of the treatise, it contains some practical advices to the surveyor.

Given its date of composition, Al-Karaji's book seems to be the oldest extant manuscript on ground water science. Its content is startling, revealing that Al-Karaji was familiar with present-day concepts and principles inherent with the hydrological cycle, the classification of soils, the description of aquifers, and the search for ground water. Emphasis in the manuscript centers on the illustration of effective water search techniques, such as the study of the colour of rock formations and the examination of roots of phreatophyte types of plants. The treatise also describes instruments and devices employed in surveying and in the construction of qanats or underground conduits.

Inbat al-miyah al-khafiya is a manual on hydraulics and water supply, including practical information on the construction of irrigation systems in the form of subterranean tunnels (qanat; pl. qanawat). According to Al-Karaji, this was the most beneficial of crafts, since it helped the earth to flourish and enables men to attain order in their lives. The work begins with a general description of the earth and the waters that are to be found in it—how to find them, what types and tastes there are, and how to clean contaminated water. It goes on to discuss springs and wells, drilling, the measurement of water and the construction and upkeep of qanats, including dealing with blockages.

When we read the book, we have the clear impression that Al-Karaji was quite familiar with the basic hydrologic, geologic, and engineering principles associated with groundwater. Al-Karaji himself exhibited extensive skills and expertise regarding: (1) the classification of soils, (2) the search for fresh water, and (3) the different types and hydraulic characteristics of aquifers. He pioneered work on the use of plant growth as an indicator of groundwater aquifers, and invented ingenious devices employed in surveying and tunneling. Much of Al-Karaji's book deals with the techniques of exploring for groundwater, mainly how to dig wells and qanats. The methods he describes are still used in many parts of the Middle East and Asia [14].

5. A subterranean conduit for water supply: the Qanat

Qanat is a type of underground irrigation canal between an aquifer on the piedmont to a garden on an arid plain. The word is Arabic, but the system is best known from ancient Iran. To make a qanat, one needs a source of water, which may be a real well, but can also be an underground reservoir or a water-bearing geological layer. When this source is identified, a tunnel is cut to the farm or village that needs the water.

Shafts are as air supply, to remove sand and dirt, and to prevent the tunnels from becoming dangerously long. The shafts are not very far apart, and as a result, a qanat seen from the air gives the impression of a long, straight line of holes in the ground - as if the land has been subjected to a bombing run. Typically, the qanat becomes a ditch near its destination; in other words, the water is brought to the surface by leading it out of the slope. In fact, one creates an artificial artesian well and an oasis. In Islamic times, the qanat became one of the most effective methods for providing water in regions without perennial streams. The technique probably originated in northern Iran in ancient times, but progressively this system of supplying water over a long distance was in widespread use in the Muslim world in the medieval period and up to modern times. Indeed, recent estimates have shown that 75 per cent of all water used in Iran comes from qanats and that their total length exceeds 100 000 miles. Outside Iran, qanats are still in use in parts of the Arab world, notably in the south-east of the Arabian Peninsula and in North Africa.

The qanat system was used by the Umayyad and the Abbasid caliphs. The Caliph Al-Mutawakkil (847-866) constructed a qanat system for the supply of water to his new palace at Samarra. Recent excavations there showed that the water was obtained from ground water of the upper Tigris and conveyed to Samarra in qanat conduits totaling 300 miles in length.

In Al-Karaji's Inbat al-miyah al-khafiyya we find good details on the construction of the qanat-conduits, their lining, protection against decay, and their cleaning and maintenance.

A part of the book is devoted to techniques of exploring ground water, mainly how to dig wellsand qanats. For example, he describes how to survey the slope of qanats and how to work under difficult circumstances.

As an example, wells to be dug through unconsolidated rocks have to be timbered by means of centerings. Bricks are laid behind this stage. The same procedure can be applied if a qanat in development hits a lens or layer of loam or clay saturated with water. In such a case, Al-Karaji's advice is to stop the project because of the risk of collapse of the under-ground construction [15].

In conclusion, it appears evident that Al-Karaji was familiar with the basic hydrologic, geologic, and engineering principles associated with ground water, as known today. Al-Karaji himself exhibited extensive skills and expertise in the discussion of the construction of qanats, in the classification of soils, in the search for fresh water, and in his knowledge of different types of aquifers and their hydraulic characteristics. Al-Karaji pioneered work with geological structures in his use of plant growths as indicators and locators of ground water reservoirs (aquifers), and invented ingenious devices employed in surveying and tunneling. The reception and authentication of Al-Karaji's treatise in history of science should strengthen the valuedness of this millennium work.

6. References and further reading

• Al-Karaji, Kitab inbat al-miyah al-khafiya. Haydarabd: Da'irat al-ma'arif al-'uthmaniya, 1359/1940, 75 pp. Reprinted in: Water-Lifting Devices in the Islamic World: Texts and Studies. Collected and reprinted by F. Sezgin et al.. Frankfurt: Institute for the History of Arabic-Islamic Sciences, 2001, pp. 302-398.

• Al-Karaji, Kitab al-badi' fi ‘l-hisab. L'algèbre al-Badi' d'Al-Karagî. Edition, introduction et notes par Adel Anbouba. Beirut: Université libanaise, 1964. [Edition of the manuscript of the Vatican Library, MS Barberini 36.1].

• [Al-Karaji], Estexra-e abha-ye penhami. [Translation From Arabic to Persian of Al-Karaji's Extraction of Hidden Water by Husayn Khadiv-Djam]. Tehran: Iranian Culture Foundation, 1345 H (1966-67), 127 pp.

• [Al-Karaji], La civilisation des eaux cachées: Traité de l'exploitation des eaux souterraines composé en 1017 a.d. Texte établi, traduit et commenté par Ali Mazahéri. Nice : Universite´ de Nice, Institut d'e´tudes et de recherches interethniques et interculturelles (IDERIC), 1973, II+187 pp.

• Al-Karaji, Al-Kafi fi ‘l-hisab (Genügendes über Arithmetik) von (4.-5. Jhd/10-11.Jhd.u.) Ediert und kommentiert von Sami Chalhoub. Al-Kafi fi ‘l-hisab li-Abi Bakr Muhammad b. al-Hassan al-Karaji, darasahu wa haqqaqahu wa-sharahahu Sami Shalhub. Aleppo: Manshurat jami'at Halab/Institute for the History of Arabic Science, 1986, 320 pp.

• Al-Karaji, Kitab inbat al-miyah al-khafiya. Tahqiq wa-dirasa [edition and analysis by] Baghdad Abdul-Mun'im. Cairo: Institute of Arabic Manuscripts, 1997, 283 pp.

• [Al-Karaji], L'Estrazione delle acque nascoste: Trattato tecnico-scientifico di Karaji Matematico-ingegnere persiano vissuto nel Mille by Giuseppina Ferriello. Con una Prefazione di Romano Gatto. Turin: Kim Williams Books, 2007, 232 pp.

• Amir-Moéz, A. R., "Comparison of the Methods of Ibn Ezra and Karkhi", Scripta Mathematica vol. 23 (1957), pp. 173-178.

• Bruin, Fr., Surveying and Surveying Instruments being chapters 26, 27, 29 and 30 of the Book on Finding Hidden Water by Abu Bakr Muhammad al-Karaji. Beyrouth, 1970.

• Covington, Richard, The Art and Science of Water, in Saudi Aramco World, May/June 2006, vol. 57, Number 3, pp. 14-23.

• Hill, Donald R., Islamic Science and Engineering. Edinburgh: Edinburgh University Press, 1993.

• Hubbard, Matthew and Tom Roby, "The History of the Binomial Coefficients in the Middle East". (Retrieved 14.01.2009).

• O'Connor, John J., and Robertson, Edmund F., "Abu Bekr ibn Muhammad ibn al-Husayn Al-Karaji" (July 1999 ). In: MacTutor History of Mathematics archive.

• Krenkow F., 1947-9, "The Construction of Subterranean Water Supplies during the Abbaside [sic] Caliphate", Transactions of the Glascow University Oriental Society, vol. 13, pp. 23-32. Reprinted in Water-Lifting Devices in the Islamic World. Texts and Studies. Collected and Reprinted by F. Sezgin et al., Frankfurt, 2001.

• Levi della Vida, Giorgio, "Appunti e quesiti di storia letteraria araba. 4. Due nuove opere del matematico al-Karagi (al-Karkhi)", Rivista degli Studi Orientali (Roma) vol. 14, 1934, pp. 249-264. Reprinted in Al-Karaji Abu Baker Muhammad ibn al-Hasan (400 H.) : Texts and Studies. Frankfurt: Institute for the History of Arabic-Islamic Sciences, 1998, pp. 247-262.

• Lewis, Michael J. T., Surveying Instruments of Greece and Rome. Cambridge: Cambridge University Press, 2001.

• Lightfoot, Dale R., "The Origin and Diffusion of Qanats in Arabia: New Evidence from the northern and southern Peninsula", Geographical Journal (Royal Geographical Society, UK), vol. 166, Issue 3, July 2005, pp. 215-226.

• Parshall, Karen H., "The Art of Algebra from al-Khwarizmi to Viète: A Study in the Natural Selection of Ideas", History of Science vol. 26, 1988, pp. 129-164.

• Rashed, R., "L'induction mathématique: al-Karaji et As-Samaw'al", Archive for the History of Exact Sciences vol. 9, 1972, pp. 1-21.

• Rashed, Roshdi, "Al-Karaji", Dictionary of Scientific Biography, New York: Charles Scribner's Sons, 1973, vol. 7, pp. 240-246.

• Rashed, R., "Al-Karkhi", in Dictionary of the Middle Ages, edited by J.R. Strayer. New York: Charles Scribner's Sons, vol 7, 1986, pp. 211-212.

• Seaquist, Carl R., Padmanabhan Seshaiyer, and Dianne Crowley, "Calculation across Cultures and History", Texas College Mathematics Journal vol. 1:1, 2005, pp. 15-31.

• Sesiano, Jacques, "Le traitement des équations indéterminées dans le Badi' fi al-Hisab d'Abu Bakr Al-Karaji, Archive for the History of Exact Sciences, vol. 17, 1977, pp. 297-379.

• Solignac, Marcel, "Mohamed al-Karagi, ingénieur hydrologue (m.410/1019)", Revue de l'Institut des Belles Lettres Arabes (Tunis), XXXVII/134 (1974), pp. 315-328.

• Vernet, J., and Catala, M. A., "Un ingeniero árabe de siglo XI: al-Karayi", Al-Andalus: revista de las Escuelas de Estudios Árabes de Madrid y Granada, vol. 35, Nº 1, 1970, pp. 69-92.

• Vernet, J. "Al- Karadji, Abu bakr Muhammad b. al-Hasan (and also al-Husayn)." Encyclopaedia of Islam, Second Edition. Brill, 2006, vol.4, p. 599.

• Wiedemann, Eilhard, and Hauser, Franz, "Über Vorrichtungen zum Heben von Wasser in der Islamischen Welt", Beitrage zür Geschichte der Tecknik und Industrie, vol. 8, 1921, pp. 121-154. Reprinted in: Water-Lifting Devices in the Islamic World. Texts and Studies. Frankfurt: Institute for the History of Arabic-Islamic Sciences, 2001.

• [Wikipedia], Al-Karaji (retrieved 14.01.2009).

• Wodzicki, Mariusz, Early History of Algebra: a Sketch, Math No. 160, Fall 2005.

• Woepcke, Franz, Kitab fi l-jabr wa-l-muqabala wahuwa l-ma'ruf bi l-Fakhri li l-shaykh Abi Bakr Mu?ammad ibn al-Hasan al-Karkhi. Extrait du Fakhri, traité d'algèbre par Aboù Bekr Mohammed ben Alhaçan Alkarkhî (manuscrit 952, supplément arabede la bibliothèque Impériale) ; précédé d'un mémoire sur l'algèbre indéterminée chez les Arabes. Paris 1853. Reprinted Hildesheim: Georg Olms Verlag, 1982. Reprinted in: Franz Woepcke, Études sur les mathématiques arabo-islamiques : Nachdruck von Schriften aus den Jahren 1842-1874. Frankfurt: Institute for the History of Arabic-Islamic Sciences, 1986, 2 vols. Reprinted in Al-Karaji Abu Baker Muhammad ibn al-Hasan (400 H.): Texts and Studies. Frankfurt: Institute for the History of Arabic-Islamic Sciences, 1998.

• Zeuthen, Hieronymus Georg, "Sur l'arithmétique géométrique des Grecs et des Indiens", Bibliotheca mathematica (Leipzig), vol. 3F.5, 1904, pp. 97-112 [on the mathematics of al-Karaji].

Footnotes

[1] For bull bibliographical references and further reading on the different topics are aboarded in this article, see below the detailed reference list in section 6.

[2] On water technology in the Islamic East and West, see Donald R. Hill and Ahmad Y. Al-Hassan, Engineering in Arabic-Islamic Civilization. Part I (retrieved 14.01.2009); and Richard Covington, The Art and Science of Water, in Saudi Aramco World, vol. 57, 2006, pp. 14-23.

[3] On Al-Farghani, see Ibn Abi Usaybi'a, ‘Uyun al-anba' fi tabaqat al-atibba'. Edited by Nizar Ridha. Beirut: Maktabat al-hayat, 1965, pp. 286-287; A. I. Sabra, "Farghani", Dictionary of Scientific Biography. New York: Charles Scribner's Sons, vol. 4, 1971, pp. 541-545; and Al-Farghani (c. 860 C.E.), in [Ahmed Monzur], Muslim Scientists and Scholars (July 1998).

[4] Giorgio Levi della Vida, , "Appunti e quesiti di storia letteraria araba. 4. Due nuove opere del matematico al-Karagi (al-Karkhi)", Rivista degli Studi Orientali (Roma) vol. 14, 1934, pp. 249-264; p. 250.

[5] One of the names is that of the vizier Abu Ghanim Ma'ruf b. Muhammad ibn Ma'ruf who was active between 403-420 H/… CE: see Ibn al-Athir, Al-Kamil fi ‘l-tarikh (Cairo: Al-Matba'a al-Muniriya, 1353H, 9 vols; vol. 9, pp. 102, 394-395.

[6] On the life and works of al-Karaji, see the following reference works: Carl Brockelman, Geschichte der arabischen Litteratur, Leiden: Brill, 2^e edition, 1943-49, 3 vols. and 2 Supplements; vol. 1, p. 219; Suppl. 1, p. 389; George Sarton, Introduction to the History of Science. Baltimore, 1927-48, 3 vols.; vol. 1, p. 718; Khayr al-Din al-Zirikli, Al-A'lam, Beirut: Dar al-'ilm li-'l-malayin, 2002, 8 vols.; vol. 6, pp. 83, 99; ‘Umar Ridha Kahhala, Mu'jam al-mu'allifin, Damascus: Al-Maktaba al-'arabiya, 1957-1961, 15 vols.; vol. 9, p. 211; Juan Vernet and M. A. Catala, "Un ingeniero árabe de siglo XI: al-Karayi", Al-Andalus, vol. 35, 1970, pp. 69-92; Roshdi Rashed, "Al-Karaji", Dictionary of Scientific Biography, New York: Charles Scribner's Sons, 1973, vol. 7, pp. 240-246; Juan Vernet, "Al- Karadji", Encyclopaedia of Islam, Second Edition. Brill, 2006, vol. 4, p. 599; and Giuseppina Ferriello, L'Estrazione delle acque nascoste: Trattato tecnico-scientifico di Karaji Matematico-ingegnere persiano vissuto nel Mille. Turin: Kim Williams Books, 2007, pp. 51-57.

[7] Franz Woepcke, Extraits du Fakhri, traité d'algèbre, Paris, 1853, p. 21.

[8] "Another important idea introduced by al-Karaji and continued by al-Samaw'al and others was that of an inductive argument for dealing with certain arithmetic sequences. Thus al-Karaji used such an argument to prove the result on the sums of integral cubes already known to Aryabhata [...] Al-Karaji did not, however, state a general result for arbitrary n. He stated his theorem for the particular integer 10 [...] His proof, nevertheless, was clearly designed to be extendable to any other integer. [...] Al-Karaji's argument includes in essence the two basic components of a modern argument by induction, namely the truth of the statement for n = 1 (1 = 1³) and the deriving of the truth for n = k from that of n = k - 1. Of course, this second component is not explicit since, in some sense, al-Karaji's argument is in reverse; this is, he starts from n = 10 and goes down to 1 rather than proceeding upward. Nevertheless, his argument in al-Fakhri is the earliest extant proof of the sum formula for integral cubes", Victor J. Katz, History of Mathematics: An Introduction, Reading, MA: Addison Wesley, 2^nd edition, 1998, p. 255-259; p. 255.

[9] Quoted in J. J. O'Connor and E. F. Robertson, "Abu Bekr ibn Muhammad ibn al-Husayn Al-Karaji" (July 1999 ).

[10] Al-Maqdisi, Ahsan al-taqasim fi ma'rifat al-aqalim, published by Ghazi Talimat. Damascus, 1980, p. 261.

[11] For the references of the various editions and translations of the book, see the bibliography below.

[12] See "inbat" in Lisan al-'Arab by Ibn Manzur.

[13] The sections on surveying in Al-Karaji's book are summarized in Donald R. Hill, Islamic Science and Engineering, Edinburgh University Press, 1993, pp. 187-205.

[14] On the contents of the book and its assessement in recent scholarshp, see Mehdi Nadji and Rudolf Voight, "Exploration for Hidden Water by M. Karaji: The Oldest Textbook on Hydrology?" Groundwater, September-October 1972, pp. 43-46; Hormoz Pazwash and Gus Mavrigian, "A Historical Jewelpiece-Discovery of the Millennium Hydrological Works of Karaji," Water Resources Bulletin, December 1980, pp. 1094-1096; Khalid ‘Azab, Kayfa wajahat al-hadhara al-islamiya mushkilat al-miyah [How islamic civilisation confronted the problem of water (supply)]. Rabat: ISESCO Publications, 2006, chap. 2; and Mohammed Karaji (retrieved 14.01.2009).

[15] Quoted in Mehdi Nadji and Rudolf Voight, "Exploration for Hidden Water by M. Karaji: The Oldest Textbook on Hydrology?" Groundwater, September-October 1972, p. 45. See also on the qanat, Salim Al-Hassani (chief editor), 1001 Inventions: Muslim Heritage in Our World, Manchester: FSTC, 2006, pp. 112-113. The construction of a qanat is described in: Donald R. Hill and Ahmad Y. Al-Hassan, "Engineering in Arabic-Islamic Civilization. Part One". For more details on the other devices used in the Islamic tradition for water management and supply, see Salim T. S. Al-Hassani, The Machines of Al-Jazari and Taqi Al-Din (published on www.MuslimHeritage.com 30 December, 2004), Salim T. S. Al-Hassani and Mohammed A. Al-Lawati, The Six-Cylinder Water Pump of Taqi al-Din: Its Mathematics, Operation and Virtual Design (published on www.MuslimHeritage.com 21 July, 2008), and Salim T. S. Al-Hassani and Colin Ong Pang Kiat, Al-Jazari's Third Water-Raising Device: Analysis of its Mathematical and Mechanical Principles (published on www.MuslimHeritage.com 24 April, 2008).

*Professor of History and Philosophy of Science, Mohammed Vth University, Rabat, Morocco, and Senior Research Fellow, Foundation for Science, Technology and Civilisation (FSTC), Manchester, UK. Chief Editor of FSTC academic web portal: http://www.MuslimHeritage.com.

by: FSTC Limited