MATHEMATICS. The mathematical sciences occupy a prominent place in Islamic intellectual history. Historically called `ulum riyadiyah or ta’limiyah (pedagogic mathematics), they comprised the four main branches-arithmetic, geometry, astronomy, and music- of the quadrivium of the ancient schools. The establishment of the classical scientific heritage in the first few centuries of the spread of Islam further brought such fields as algebra, trigonometry, mechanics, and optics-together with their practical, even experimental aspects-under this domain. Although fields considered mathematical did not have comparable status or existence throughout the Islamic world, mathematical subjects in all their manifestations drew the attention of a large number of Islamic scholars who produced impressive, often historically significant, works.

There is no shortage of sources in Islamic or European languages about the rich history of mathematics in the premodern Islamic world. The Annotated Bibliography of Islamic Science (ABIS) lists in its third volume devoted to mathematical sciences nearly eighteen hundred printed sources on the subject published before 197° alone. Since that date, additional works equal to at least half of that number have appeared. Nonetheless, a comprehensive study on the history of mathematics in Islamic societies remains a distant dream. The current state of research is such that it is not possible to reconstruct such a history beyond the stage of theoretical and partial studies of the first half of the Islamic era, and the coverage of such crucial subjects as the social context of mathematics and the history of mathematics in the modern Islamic world is still limited and fragmentary. Attention to social context has been occasional (Hoyrup, 1987, 1990; Heinen, 1978; King, 1980, 1990; Berggren, history is just beginning (Ihsa1992). Such isolated studies, espehematics in modern Turkey, Iran, or countries of the region, are bound to ronology and scope of our understanding atics in the Islamic world.

Historical Meaning. Mathematics in the premodern Islamic world differed in its meaning and domain from that of the modern era (to which it is unquestionably bound), as well as that of the ancient world (from which it initially arose). Its disciplinary boundaries were changing even during the period in Islamic intellectual history that is commonly considered its peak. As the collective form `ulum al-riyadiyah indicates, these sciences existed as composite mathematical disciplines which had themselves evolved as a branch of the socalled “sciences of the ancients” (`ulum al-awd’il), that is, the original, pre-Islamic sciences in contrast to Islamic sciences (`ulum Isldmiyah). The affiliation of these “rational” (‘aqli)-in contrast to “traditional” (naqli)-sciences with the ancient sciences of Greece, India, or Persia was itself of a varied nature, and the practicing mathematical disciplines each inevitably had different elements of the classical heritage, different disciplinary encounters and boundaries, and naturally, different historical fates.

The science of arithmetic was not a single science. Of its main two divisions, the science of numbers (`ilm to al-adad), which was at the head of the seven divisions of mathematical sciences (`ulum ta`alim) according to alFarabi’s (d. AH 339/950 CE) Ihsa’ al-`ulum, was more theoretical. It was cast in the tradition of the arithmetic books (vii-ix) of Euclid’s Elements as well as Nichomachus’s Introduction to Arithmetic. The science of reckoning (`ilm al-hisdb) on the other hand, dealt more with arithmetical operations, and had its own distinct intellectual currents and systems of numerical calculation. One of unknown origin employed the fingers in the calculation process, and so was often termed “finger reckoning” (hisdb al-`uqud)”, or else “hand reckoning” (hisdb al -yad)” or “mental reckoning” (al-hisdb alhawd’i); it had a rhetorical mode of expression for numbers. In contrast, the so-called “Indian system of reckoning” (hisdb al-Hindi), also known as “board and dust calculation” (hisdb al-takht wa-al-turab) was based on the place-value concept and expressed numbers in terms of ten figures including zero (sifr) as the empty place. In addition to the first system, which became the arithmetic of the scribes or secretaries, and the second system from which the medieval European “Arabic numerals” are supposed to have been derived, there was another system in which numbers were represented neither by fingers nor by figures but by letters; this third system was linked to the old Babyonian astronomical tradition according to which computations were performed in sexagesimals indicated by alphabetical symbols. This kind of treatment was known as the “arithmetic of astronomers or astrologers” (hisab al-munajjim), “arithmetic of astronomical tables” (hisdb al-zij), or “arithmetic of degrees and minutes” (hisdb al-dard’ij wa al-daqa’iq). Besides the abjad system of ciphered numeration with twenty-eight Arabic letters, there was also a siyaq style of representation used until very recent times in Iran and Turkey, according to which forty-five Pahlavi-style characters where employed for commercial purposes. Finally, books on reckoning included an algebraic section for the determination of unknown quantities from known ones, where the expression “algebra” (al -jabr), meant an operation, not the distinctiot discipline it eventually became.

An independent science of algebra (`ilm al -jabr), correctly associated with Muslim mathematicians, did in fact take form as such during the appropriation of ancient learning in Islamic lands. Its early character as an offshoot of applied arithmetic may explain why it was later classified under “the science of devices” (`ilm alhiyal) as an applied branch of mathematics. But algebra had an equally close association with geometry from the start, as the common method of supplying algebraic problems with geometrical demonstrations; this indicates the distinction between its methodology of proofs (barahin), and the method of checks (mawdzin) in arithmetic. As algebra seems to have been placed somewhere between geometry and arithmetic, a number of other related fields such as trigonometry and optics were considered intermediate between geometry and another of the four main mathematical disciplines, astronomy. Such intermediate fields were ultimately based on a series of Greek mathematical texts known as the “middle books” (al-mutawassitat), because they were studied intermediately between Euclid’s Elements and Ptolemy’s Almagest, the two chief authorities on geometry and astronomy, respectively.

Geometry (`ilm al-handasah) officially came to the Islamic intellectual world through Greek sources, and mainly through Euclid. Initially known as jumatriyah, Arabic geometry was predominantly Greek in origin as well as method, although it also reflects encounters with Indian works such as Siddhantas (Sind hind in Arabic) and with Persian sources-hence its designation handasah (geometry), from Persian andazah (measure).

Astronomy, by contrast, had more and stronger links to non-Greek ancient traditions, as the Babylonian heritage can now safely be added to Sanskrit, Pahlavi, and Syriac astronomical sources alongside Greek ones. Astronomy also started off as a much wider discipline than it later became, with a cluster of subfields including instruments and tables, star movements, chronology, and astrology. The existence of such designations as “the science of the figure of the heaven” (`ilm al-hay’ah) as well as “the science of the heavens” (`ilm al-aflak) or “the science of stars” (`ilm al-nujum) by itself points to a historical distinction between observational and theoretical astronomy, exemplified in the respective traditions of Ptolemy’s Almagest (al-Majisti) and Planetary Hypothesis (Iqtisas). More curious, however, is the absence of an explicit historical distinction between the traditions of Ptolemy’s Almagest and Tetrabiblos (Arba` maqalat), namely between the two fields of astronomy and astrology; not only were these designated by the common expression nujum, but they also shared such terms as al-hasib (computer) to refer to their practitioners. The historical affinity of astronomy and astrology to yet another of the mathematical propedeutical sciences, theoretical music (musiqa) in this same period is further indication of the invalidity of assuming fixed mathematical fields with distinct borders between them. There is another aspect to the historical meaning of mathematical sciences. Fields like astronomy and arithmetic acquired new meaning and domain as they continued to grow on Islamic soil, as the emergence of such categories as a muwaqqit (time-keeper) as an astronomer attached to the mosque, or a branch of arithmetic called fard’id (dealing with the division of legacies) as part of the equipment of Islamic law attests.

**Achievements**. The achievements of mathematicians in the early Islamic world were of quite a varied nature and their most significant achievements were not always the most lasting. In the field of arithmetic there were few breakthroughs. The appearance of decimal fractions occurred as early as the work of the Damascene arithmetician Abu al-Hasan al-Uqlidisi in his Kitdb al -fusul ft al-hisab al-Hindi, composed in Ax 341/952-953 CE. These were much later reintroduced as al-kusur a’shari-yah, together with the first appearance of a unified place-value system for both integers and fractions, in the work of the Persian mathematician Jamshid ibn Mas’ud al-Kashi in Miftah al-hisab (Key to Arithmetic), composed in Samarkand in 830/1427. Although the earlier work of al-Uglidisi was of less impact, credit goes to him for the use of strokes for decimal signs, for the first successful treatment of the cube root, and for the alteration of the dust-board method to suit ink and paper. Other breakthroughs include notable steps toward considering irrationals as numbers, as in the work of the famous mathematician and poet `Umar Khayyam, as opposed to treating them as incommensurable lines as did those in the tradition of Book X of Euclid’s Elements.

On the whole, developments in theoretical arithmetic (`ilm al-`adad) are of less historical significance, and despite much theoretical treatment, e.g. arithmetic sections of Ikhwan al-Safa’s Rasa’il, or works on “amicable numbers” (a’dad mutahabbah) or pyramidal numbers, it is in the area of computation that more important contributions seem to have been made. Especially important in this category are treatises devoted to algebra, often including the term hisab associated with practical aspects of arithmetic in their titles: Kitab mukhtasar ft al-hisab al -jabr), wa-al-muqabalah by al-Khuwarizmi and Tara’if al-hisab by Abu Kamil al-Shuja’ (d. 287/900), Al-kafi fi al-hisab, by Abu Bakr al-Karaji (d. 390/1000), or Al-bahir ft `ilm al-hisab by al-Samawa’l al-Maghribi (d. c.570/1175). Arabic algebra as contained in these and similar texts is often associated with the successful treatment of problems corresponding to quadratic-and occasionally third degree–equations, many of which combine the act of reducing rhetorical algebraic problems into canonical form with that of providing geometrical proofs.

The most significant achievement of Muslims in arithmetic may be characterized as the fusing together of various methods into a single unified system. But fusing by no means describes their achievements in other areas. In geometry, for example, attempts to prove Euclid’s Parallel postulate by a number of mathematicians, culminating in the work of Nasir al-Din al-Tfisi, resulted in the formulation and proof of some non-Euclidean theorems assumed to have been known to European founders of non-Euclidean geometry. In another impressive and influential scientific movement a theoretical program of reaction against certain inconsistencies in Ptolemaic astronomy resulted in a series of complex non-Ptolemaic models that have been compared to those proposed in Europe by Copernicus a few centuries later. What is particularly remarkable about this astronomical movement, which ranged from Eastern Persia, to Damascus, and all the way to Andalusia, is its preoccupation with philosophical, rather than observational, concerns. Nonetheless, a strong observational program did exist as astronomical records were produced for many major Islamic cities from Baghdad, Damascus, and Cairo to Shiraz, Khwarazm, and Maragha, as reflected in several tables (zij). In fact, this same observational tradition produced a concept of testing, adopted and developed under the terms mihnah, imtihan or i`tibar, which converted the methodology of the neighboring science of optics from a theoretical to an experimental science.

As it turned out, the most significant contribution of the mathematical sciences of the Islamic world to modern science was not in the field of mathematics proper, but in optics. Belonging not to physics, but to mathematics-more specifically a field intermediate between geometry and astronomy-optics, was to be revolutionized in the hands of Muslims. This revolution occurred during the golden era of sciences in Islamic civilization through Ibn al-Haytham’s (d. 432/1040) Kitab almanazir, which put this science on a new foundation. His seven books on optics were translated into Latin and Italian, and being among the first scientific books to be printed, strongly influenced the works of medieval Latin, Renaissance, and seventeenth-century thinkers.

Finally, it should be remembered that the history of optics-like the history of other mathematical disciplines in this period-includes significant developments that were not transmitted to Europe. This is particularly true of the history of astronomy, where a large body of nontransmitted literature survives from late periods in Islamic intellectual history. A study of this late mathematical tradition-now beginning to emerge from Turkey, Iran, and India-is both needed and promising. But much more is needed, and many important mathematical texts and treatises in Islamic languages now only available in manuscript form throughout the world must be brought to the aid of constructing a history of the exact sciences with a fuller context and a broader time line in the hope of approaching a deeper understanding of the history of mathematics in Islamic societies.

[See also Astronomy; Numerology.]

BIBLIOGRAPHY

Berggren, J. L. “History of Mathematics in the Islamic World: The Present State of the Art.” Bulletin of the Middle East Studies Association 19.r (1985): 9-33.

Berggren, J. L. Episodes in the Mathematics of Medieval Islam. New York, 1986. Contains sections on arithmetic, geometry, algebra, trigonometry, and spherics.

Berggren, J. L. “Islamic Acquisition of the Foreign Sciences: A Cultural Perspective.” American Journal of Islamic Social Sciences 9.3 (1992): 309-324.

Heinen, Anton M. “Mutakallimfin and Mathematicians: Traces of a Controversy with Lasting Consequences.” Islam 55 (1978): 57-73. Considers the relationship between religious thought and mathematics, particularly the attitudes of al-Jahiz and al-Biruni.

Hill, Donald R. “Mathematics and Applied Science.” In Religion, Learning, and Science in the ‘Abbasid Period, edited by M. J. L. Young et al., pp. 248-273. Cambridge, 1990. A more updated work on the early history of mathematics in Islamic civilization which includes the practical as well as theoretical aspects of the discipline.

Hoyrup, Jens. “The Formation of `Islamic Mathematics’: Sources and Conditions.” Science in Context 1 (1987): 281-329. “Subscientific Mathematics: Observations on a Premodern Phenomena.” History of Science 28.1 (1990): 63-81. Particularly valuable pieces for the often ignored subject of the encounter between mathematics and Islamic society.

Ihsanoglu, Ekmeleddin, ed. Transfer of Modern Science and Technology to the Muslim World. Istanbul, 1992. Includes articles related to mathematics in fourteenth- to eighteenth-century Turkey (Ihsanoglu, pp. 1-120), and nineteenth-century Iran (Roshdi Rashed, PP. 393-404).

Iushkevich, Adol’f Pavlovich. Les mathimatiques arabes, VIIIe-XVe siecles. Translated by M. Cazenave and K. Jaouiche. Paris, 1976. Reviewed in Journal of the History of Arabic Science 1 (1977): 111. Kennedy, E. S. Studies in the Islamic Exact Sciences. Beirut, 1983. By the author of several earlier articles on the subject, including “The Arabic Heritage in the Exact Sciences,” Al-Abhath 23 (1970); “The Exact Sciences,” in The Cambridge History of Iran, vol. 4, The Period from the Arab Invasion to the Saljuqs, edited by Richard N. Frye, pp. 378-395 (Cambridge, 1975); and “The Exact Sciences in Timurid Iran,” in The Cambridge History of Iran, vol. 6, The Timurid and Safavid Periods, edited by Peter Jackson and Laurence Lockhart, pp. 568-58o (Cambridge, 1986).

King, David A. “The Exact Sciences in Medieval Islam: Some Remarks on the Present State of Research.” Bulletin of the Middle East Studies Association 4 (1980): 10-26.

King, David A. “The Sacred Direction in Islam: A Study of the Interaction of Religion and Science in the Middle Ages.” Interdisciplinary Science Review to (1985): 315-328.

King, David A. Islamic Mathematical Astronomy. London, 1986. King, David A. Islamic Astronomical Instruments. London, 1987. King, David A. “Science in the Service of Religion.” UNESCO; Im pact of Science on Society 159 (1990): 245-262. This article along with “The Sacred Direction in Islam” (above) are valuable sources on this rarely studied interaction.

King, David A., and George Saliba, eds. From Deferent to Equant: A Volume of Studies in the History of Science in the Ancient and Medieval Near East in Honor of E. S. Kennedy. New York, 1987. Indispensable source that contains, in addition to a list of Kennedy’s own publications, many specialized articles and an extensive bibliography.

Nasr, Seyyed Hossein. An Annotated Bibliography of Islamic Science, vol. 3, Mathematical Sciences. Tehran, 1991. Valuable volume containing 1,831 sources printed before 1970, arranged according to subject, some with annotations in English and Persian.

Rashed, Roshdi. Optique et mathematiques: Recherches sur Phistoire de la pensee scientifique en Arabe. London, 1992. Collection of previously published articles on the history of mathematics.

Sabra, A. I. “The Scientific Enterprise: Islamic Contributions to the Development of Science.” In The World of Islam, edited by Bernard Lewis, pp. 181-199. London, 1976. Contains informative sections on various branches of mathematics: arithmetic, algebra, and geometry; applied mathematics (mechanics); and those involving testing and experimentation (astronomy, light, and vision).

Sezgin, Fuat. Geschichte der arabischen Schrifitums, vol. 5, Mathematik; vol. 6, Astronomie; vol. 7, Astrologie, Meteorologie, and Verwandts. Leiden, 1970-1979. Fundamental reference works for manuscript sources and bibliography of the exact sciences during the classical period of Islamic civilization. For a review of volume 5, see David A. King, “Notes on the Sources for the History of Early Islamic Mathematics,” Journal of the American Oriental Society 99 (1979) 450-459, which lists sources for later periods.

See also articles “`Ilm al-hisab,” “`Ilm al-handasa,” “`Ilm al-hay’a,” “al-Djabr wa al-muqabalah,” and “Musiqa” in Encyclopaedia of Islam, new ed., Leiden, 1960-.

ELAHEH KHEIRANDISH