The mathematical sciences occupy a prominent place in Islamic intellectual history. Historically called *ʿulūm riyāḍīyah* (mathematical sciences) or *ʿulūm taʿlīmīyah* (pedagogical sciences), 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 Islam further brought such fields as algebra, trigonometry, mechanics, and optics—together with their practical, even experimental aspects—into this domain. Although not all fields considered mathematical had comparable status—or even existed—in all parts of the Islamic world, mathematical subjects in all their manifestations drew the attention of many Islamic scholars who produced impressive, often historically significant, works.

There is an abundance of sources in Islamic and European languages for the rich premodern history of Islamic mathematics. 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 1970 alone. Since that date, at least half that number of additional works have appeared. Nonetheless, a comprehensive study on the history of mathematics in Islamic societies remains a distant goal. Given the current state of research, 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 (Høyrup, 1987, 1990; Heinen, 1978; King, 1980, 1990; Berggren, 1992), while that to modern history has only begun (İhsanoğlu, 1992; Rashed, 1992). Such isolated studies, especially those of mathematics in modern Turkey, Iran, and other non-Arab countries of the region, are bound to extend the chronology and scope of our understanding of mathematics 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 arose). Its disciplinary boundaries were changing even during the preeminent period of Islamic intellectual history. As the collective term *ʿulūm al-riyāḍīyah* (sciences of mathematics) indicates, these sciences existed as composite mathematical disciplines which had evolved as a branch of the so-called “sciences of the ancients” (*ʿulūm al-awāʿil*), that is, the original, pre-Islamic sciences as distinguished from Islamic sciences (*ʿulūm Islāmīyah*). The affiliation of these “rational” (*ʿaqlī*)—in contrast to “traditional” (*naqlī*)—sciences with the ancient sciences of Greece, India, or Persia was itself of a varied nature, and the various practicing mathematical disciplines inevitably had different elements of the classical heritage, different disciplinary encounters and boundaries, and naturally, different historical fates.

The science of arithmetic was not unitary. Of its main two divisions, the science of numbers (*ʿilm al-ʿadad*), which was at the head of the seven divisions of mathematical sciences (*ʿulūm taʿālīm*) according to al-Fārābī's (d. 950 C.E.) *Iḥṣāʿ al-ʿulūm* (The Enumeration of the Sciences), was more theoretical. It was cast in the tradition of the arithmetic books (vii–ix) of Euclid's *Elements* as well as Nicomachus's *Introduction to Arithmetic*. The science of calculation (*ʿilm al-ḥisāb*) on the other hand, dealt more with arithmetical operations and had its own distinct intellectual currents and systems of calculation. One system of unknown origin employed the fingers and so was often termed “reckoning by finger-joints (tens)” (*ḥisāb al-ʿuqūd*), “hand reckoning” (*ḥisāb al-yad*), or “mental reckoning” (*al-ḥisāb al-hawāʿī*); it had a rhetorical mode of expression for numbers. In contrast, the so-called “Indian reckoning” (*ḥisāb al-Hindī*), also known as “board and dust calculation” (*ḥisāb al-takht wa-al-turāb*) was based on the place-value concept and expressed numbers in terms of ten figures including zero (*ṣifr*). In addition to the first system, which became the arithmetic of scribes and 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 by letters rather than by fingers or figures; this third system was linked to the old Babyonian astronomical tradition in which computations were performed in sexagesimals represented by alphabetical symbols. This was known as the “arithmetic of astronomers/astrologers” (*ḥisāb al-munajjim*), “arithmetic of astronomical tables” (*ḥisāb al-zīj*), or “arithmetic of degrees and minutes” (*ḥisāb al-darāʿij wa-al-daqāʿiq*). In addition to the *abjad* (alphabetic) system of ciphered numeration with twenty-eight Arabic letters, there was the *siyāq* style of representation used until recently in Iran and Turkey, in which forty-five Pahlavi-style characters were employed for commercial purposes. Finally, books on reckoning included an algebraic section for the determination of unknown quantities from known ones, in which the expression “algebra” (*al-jabr*) meant an operation, not the entire discipline it eventually became.

An independent science of algebra (*ʿilm al-jabr*), correctly associated with Muslim mathematicians, did in fact take shape during the adoption of ancient learning into Islamic science. Its early classification as an offshoot of applied arithmetic may explain why it was later classified in “the science of devices/stratagems” (*ʿilm al-ḥiyal*) as an applied branch of mathematics. But algebra had an equally close association with geometry from the start, as geometrical demonstrations were used to elucidate algebraic problems; this illustrates the distinction between the method of proofs (*barāhīn*) used in algebra and that of checks (*mawāzīn*, literally “balances”) used in arithmetic. Just as algebra seems to have been placed somewhere between geometry and arithmetic, 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 based originally on Greek mathematical texts known as “the intermediate [books]” (*al-mutawassiṭāt*), because they were studied between Euclid's *Elements* and Ptolemy's *Almagest*, the chief authorities on geometry and astronomy, respectively.

Geometry (*ʿilm al-handasah*) officially came to the Islamic intellectual world through Greek sources, and mainly through Euclid. Initially known as *jūmāṭrīyah*, Arabic geometry was predominantly Greek in origin as well as method, although it also reflects encounters with Indian works such as the astronomical *siddhāntas* (literally “canons”; rendered *sind-hind* in Arabic) and with Persian sources; the Arabic term *handasah* (geometry) comes from the Persian *andāzah* (measure).

Astronomy, by contrast, had more and stronger links to non-Greek ancient traditions; the Babylonian heritage can now safely be added to Sanskrit, Pahlavi, and Syriac astronomical sources alongside Greek ones. Astronomy also started as a much wider discipline than it later became, with several subfields including instruments and tables, star movements, chronology, and astrology. Such designations as “the science of the figure of heaven” (*ʿilm al-hayʿah*), “the science of the heavens” (*ʿilm al-aflāk*), and “the science of the stars” (*ʿilm al-nujūm*) suggest a historical distinction between observational and theoretical astronomy, exemplified in the respective traditions of Ptolemy's *Almagest* (*al-Majisṭī*) and *Planetary Hypotheses* (*Iqtiṣāṣ*; literally, exposition). Curious, however, is the absence of an explicit historical distinction between the traditions of Ptolemy's *Almagest* and his *Tetrabiblos* (*Arbaʿ maqālāt*, four treatises), that is, between the fields of astronomy and astrology; not only were these designated by the common expression *nujūm*, but they also shared such terms as *ḥāsib* (computer) referring to their practitioners. The historical affinity in this same period of astronomy and astrology to yet another of the mathematical propaedeutical sciences, theoretical music (*mūsīqā*), is further indication of the invalidity of fixed boundaries between mathematical fields of study. Mathematical sciences like astronomy and arithmetic acquired new meaning and domain as they continued to grow on Islamic soil, as attested, for example, by the emergence of new categories such as *muwaqqit* (time-keeper), an astronomer attached to the mosque, or by the appearance of a branch of arithmetic called *farāʿiḍ* (which dealt with the division of legacies) as part of the equipment of Islamic law.

#### Achievements.

The achievements of mathematicians in the early Islamic world were varied, and their most significant achievements were not always the longest lasting. In the field of arithmetic there were few breakthroughs. Decimal fractions appeared as early as the work of the Damascene arithmetician Abū al-Ḥasan al-Uqlīdisī in his *Kitāb al-fuṣūl fī al-ḥisāb al-Hindī* (Book of Chapters on the Indian Method of Calculation), composed in 952/953 C.E. These were much later reintroduced under the name *al-kusūr aʿshārī* (decimal fractions), together with the first appearance of a unified place-value system for both integers and fractions, in the *Miftāḥ al-ḥisāb* (Key to Arithmetic) of the Persian mathematician Jamshīd ibn Masʿūd al-Kāshī, composed in Samarkand in 1427. Although the earlier work of al-Uqlīdisī was of less impact, he is credited with 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 steps toward treating irrationals as numbers, as in the work of the famous mathematician and poet ʿUmar Khayyām (d. 1131), rather than treating them as incommensurable lines as did those following the tradition of the tenth book of Euclid's *Elements*.

On the whole, developments in theoretical arithmetic (*ʿilm al-ʿadad*) are of less historical significance, and despite much theoretical treatment—e.g., the arithmetic sections of *Rasāʿil* (Treatises) composed by the secret Ikhwān al-Ṣafā (Brotherhood of Purity), or works on “amicable numbers” (*aʿdād mutaḥabbah*) and pyramidal numbers)—it is in the area of computation that more important contributions were made. Especially important in this category are treatises devoted to algebra, often including in their titles the term *hiṣāb* linked with terms referring to practical aspects of arithmetic, as in *Kitāb mukhtaṣar fī al-ḥisāb al-jabr wa-al-muqābalah* (Brief Book on Calculation by Algebra) by al-Khwārizmī, *Ṭarāʿif al-ḥisāb* (Jewels of Calculation) by Abū Kāmil al-Shujāʿ (d. 900), *Al-kafī fī al-ḥisāb* (The Sufficient for Arithmetic), by Abū Bakr al-Karajī (d. 1000), and *Al-bāhir fī ʿilm al-ḥisāb* (The Dazzling Book of the Science of Computation) by al-Samawʿal al-Maghribī (d. ca. 1175). The algebra in these and similar texts is often associated with the successful treatment of problems involving quadratic—and occasionally cubic—equations, many of which combine the reduction of rhetorical algebraic problems into canonical form with geometrical proofs.

The most significant achievement of Islamic arithmetic may have been to fuse various methods into a unified system. But their achievements in other areas were not limited to such fusion. In geometry, for example, attempts by several mathematicians to prove Euclid's parallel postulate, culminating in the work of Naṣīr al-Dīn al-Ṭūsī (d. 1273), 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 influential scientific movement, theoretical 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 centuries later. What is remarkable about this astronomical movement, which ranged from eastern Persia, to Damascus, and on to Andalusia, is its preoccupation with philosophical rather than observational concerns. Nonetheless, a strong observational program did exist as astronomical records, in the form of *zīj* (tables, almanacs), were produced for many major cities from Baghdad, Damascus, and Cairo in the Arab world to Shīrāz, Khwārizm, and Marāgheh in Persia. In fact, this same observational tradition produced a system of testing, adopted and developed under the terms *miḥnah* (trial), *imtiḥān* (experiment) or *iʿtibār* (contemplation), which converted 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 Islamic science through Ibn al-Haytham's (d. 1040) *Kitab al-manāẓir* (Book of Optics), 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, 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 throughout the Islamic world that are so far available only in manuscript form should be used to construct a fuller and broader history of the exact sciences and achieve a deeper understanding of the history of mathematics in Islamic societies.

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*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 Sciences9, no. 3 (1992): 309–324.
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