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Astronomy was one of the oldest, most developed, and most esteemed exact sciences of antiquity. Many of the mathematical sciences were originally developed to facilitate astronomical research. Initial interest in astronomy had its roots in astrology and the fascination with the powers and mysteries of the heavens. Practical considerations, such as finding one's direction during night travel or understanding the correlation between the seasons of the year and the positions of the planets, provided additional incentives for the study of astronomy. The Babylonians, Greeks, and Indians had devised elaborate systems for the study of astronomy that went beyond simple empirical observation and were characterized by various degrees of mathematical rigor and sophistication. Before Islam, however, the Arabs had no scientific astronomy. Their knowledge was empirical, and it was limited to the division of the year into precise periods on the basis of the rising and setting of certain stars. This area of astronomical knowledge was known as anwa; it continued to attract attention under later Arab astronomers after the rise of Islam, and its study gained much from the mathematical methods employed by these astronomers.

From its beginnings in the ninth century through its maturity in the sixteenth century, astronomical activity was widespread and intensive. This activity is reflected in the large number of scientists working in practical and theoretical astronomy, the number of books written, the active observatories, and the new observations. Astronomy, it should be noted, was unambiguously differentiated from astrology. Astrology continued to be practiced and to draw on and encourage astronomical knowledge. In fact, a good portion of the funding for astronomical research was motivated by the desire to make astrological predictions. Nevertheless, a clear line was drawn between the two disciplines. The vast majority of the thousands of written works are on astronomy, whereas only a handful deal with astrology. Many astronomers served as court astrologers, but many more condemned astrology and distanced themselves from it. Distinct terms were also used to refer to either field: ilm ahkam al-nujum or simply tanjim referred to astrology, whereas ilm al-falak, ilm al-haya, or ilm al-azyaj referred to the science of the celestial orb, the science of the configuration of the heavens, and to major astronomical treatises containing tables for the motion of the stars and instructions on using these tables.

The first astronomical texts that were translated into Arabic in the eighth century were of Indian and Persian origin. The earliest extant Islamic astronomical texts date to the second half of the eighth century. Two astronomers, Muhammad ibn Ibrahim al-Fazari (d. a. 777) and Yaqub ibn Tariq (eighth century), translated an eighth-century Indian astronomical work known as Zij al-Sindhind (a zij being an astronomical handbook with tables). Sources indicate that they produced this translation after 770, under the supervision of an Indian astronomer visiting the court of the Abbasid caliph al-Mansur (r. 754–75). Extant fragments of the works of these two astronomers also reveal a somewhat eclectic mixing of Indian parameters with elements of Persian origin as well as some from the Hellenistic pre-Ptolemaic period. These fragments also reflect the use of Indian calculation methods and the use of the Indian sine function in trigonometry, in place of the cumbersome chords of arc used in Greek astronomy. Late Islamic sources also contain references to Zij al-Shah, a collection of astronomical tables based on Indian parameters, which was compiled in Sasanid Persia over a period of two centuries.

Arab astronomers were first exposed to Persian and Indian astronomy, and they continued to use some of the parameters and methods of these two traditions, yet the greatest formative influence on Islamic astronomy was undoubtedly Greek. In the early ninth century astronomers realized that the Greek astronomical tradition was far superior to that of Persia or India, in both its comprehensiveness and its use of effective geometrical representations. One particular second-century Greek author, Ptolemy, and more specifically one work by this author, the Almagest, exerted a disproportionate influence on all of medieval astronomy through the Islamic period and until the eventual demise of the geocentric astronomical system. That this text exerted so much influence is neither accidental nor surprising, for it is the highest achievement in Hellenistic mathematical astronomy and one of the greatest achievements of all of Hellenistic science. Other works by Ptolemy, commentaries on his works, and several treatises by other authors were also used in conjunction with the Almagest and as introductions to it. These include eleven short treatises in Greek, by different authors, called the “Small Astronomy Collection,” which were all translated into Arabic during the ninth century.

In the Almagest, Ptolemy synthesized the earlier knowledge of Hellenistic astronomy in light of his own new observations. The book's main purpose was to establish the geometric models that would accurately account for observational phenomena. A large part of the work is dedicated to the methods for constructing various models and for calculating their parameters. Ptolemy also provided tables for planetary motions to be used in conjunction with these models. Of all the books of antiquity, the Almagest represents the most successful work of mathematical astronomy: its geometric representations of the universe provided the most accurate and best predictive accounts for the celestial phenomena. The Greek tradition of physical astronomy is reflected in the Almagest and in Ptolemy's other influential work, Planetary Hypothesis. According to this predominantly Aristotelian tradition, the universe is organized into a set of concentric spheres, each carrying a star and rotating around the stationary earth at the center of the universe. Ptolemy adopted, at least in theory, these two basic Aristotelian principles: that the earth is stationary at the center of the universe and that the motion of heavenly bodies ought to be represented by a set of perfectly uniform circular motions. In practice, however, mathematical considerations often forced Ptolemy to disregard these principles.

Islamic sources report at least four Arabic translations of the Almagest, of which two are extant. The first is a translation by al-Hajjaj ibn Matar in the first half of the ninth century. The second is a translation by Ishaq, the son of the famous translator Hunayn; this second translation was revised by Thabit ibn Qurra toward the end of the ninth century. Separated by more than fifty years, the second translation reflected the maturity of Arabic technical terminology; whereas certain parts of the first translation lacked full clarity, the second translation provided a coherent text that eliminated any need for further reference to the Greek original.

The first extant original work of Islamic astronomy is al-Khwarizmi's (fl. 830) Zij al-Sindhind (which is unrelated to the translation of the Indian text mentioned earlier with same name). This work contains tables for the movements of the sun, the moon, and five planets, with explanatory remarks on how to use these tables. Most of the parameters used by al-Khwarizmi are of Indian origin, but some are derived from Ptolemy's Handy Tables, and no attempt is made to harmonize the two sources. This work is significant not only for its content but also because it was written simultaneously with the earliest translations of the Almagest. The first introduction of Ptolemaic astronomy into Islamic science thus occurred in the context of two significant trends. First, research in Islamic astronomy went hand in hand with translation; despite its manifest superiority, Ptolemaic astronomy did not exclusively set the agenda for future research in Islamic astronomy. The second trend was the selective use of parameters, sources, and methods of calculation from different scientific traditions. As a result, the Ptolemaic tradition was rendered receptive from the beginning to the possibility of observational refinement and mathematical restructuring. These revisionist tendencies characterize the first period of Islamic astronomy.

A significant part of the intensive ninth-century astronomical research was dedicated to the dissemination of Ptolemy's astronomy, not just by translating parts or all of his work into Arabic, but also by composing summaries and commentaries on it. Ptolemy's work was thus made available and accessible to a large audience among the educated classes. In the first half of the ninth century, al-Farghani (d. ca. 850), for example, wrote Kitab fi Jawami Ilm al-Nujum (A compendium of the science of the stars). This book was widely circulated in the Arabic version and also in later Latin translations. This work provided a brief and simplified descriptive overview of Ptolemaic cosmography, without mathematical computations. Unlike the Almagest, however, it started with a discussion of calendar computations and conversions between different eras. Although its primary purpose was to introduce Ptolemaic astronomy in a simplified way, it also corrected Ptolemy based on findings of earlier Arab astronomers. Al-Farghani gave revised values for the obliquity of the ecliptic, the precessional movement of the apogees of the sun and the moon, and the circumference of the earth. This critical approach, thus far restricted to the correction of constants and parameters, had already been set by earlier astronomers at the beginning of the ninth century.

Under the Abbasid caliph al-Mamun, a program of astronomical observations was organized in Baghdad and Damascus. Like any organized research project, this program endowed astronomical activity in the Islamic world with formal prestige. It also set a precedent for future support of scientific activity by other rulers and established patronage as one of the modes of supporting such activities. The professed purpose of this program was to verify the Ptolemaic observations by comparing the results derived by calculation, based on Ptolemaic models, with actual observations conducted in Baghdad and Damascus some seven hundred years after Ptolemy. The results were compiled in al-Zij al-Mumtahan (The verified tables), which is no longer extant in its entirety but is widely quoted by later astronomers. The most important correction introduced was to show that the apogee of the solar orb moves with the precession of the fixed stars. On a more general note, this program stressed the need for continuing verification of astronomical observations and for the use of more precise instruments. The program also represented the first recorded instance in history of a collective scientific undertaking.

From its beginnings, Islamic astronomy set out to rectify and complement Ptolemaic astronomy. Having noted several discrepancies between new observations and Ptolemaic calculations, Arab astronomers then proceeded to reexamine the theoretical basis of Ptolemy's results. This critical reexamination took several forms. One example of the critical works of the ninth century is Fi Sanat al-Shams (The book on the solar year), which was wrongly attributed to the mathematician Thabit ibn Qurrah, but was produced around his time. This work corrected some of Ptolemy's constants, and although it retained Ptolemy's geometrical representations, it questioned his observations and calculations. Other astronomers devised enhanced methods of calculation. New mathematical tools were introduced to modernize the computational procedures. For example, in his al-Zij al-Dimashqi (The Damascene zij) written around the middle of the ninth century, the mathematician Habash al-Hasib (d. between 864 and 874) introduced the trigonometric functions of sine, cosine, and tangent, which were at that time unknown to the Greeks. Habash also worked on a problem that was not treated in the Greek sources: he examined the visibility of the crescent moon and produced the first detailed discussion of this complicated astronomical problem. Habash is an example of an astronomer who undertook his study to verify the results of the Almagest, but in the process he expanded these results and applied them to new problems. Although the general astronomical research of this period was largely conducted within the framework of Ptolemaic astronomy, this research reworked and critically examined the observations and the computational methods of Ptolemaic astronomy and in a limited way was able to explore problems outside its framework.

One of the main ninth-century scientists from whom several extant astronomical manuscripts exist today is Thabit ibn Qurra (ca. 836–901). Thabit was a pagan from Harran (in southeast Turkey); his native language was Syriac, but he was fluent in Greek and his working language was Arabic. Thabit joined the Banu Musa circle in Baghdad, and produced numerous works on several scientific disciplines. Of about forty treatises on astronomy, only eight are extant. All the treatises reflect Thabit's full command of Ptolemaic astronomy and illustrate the level to which this astronomy was thoroughly absorbed by Arab astronomers. A few of these are of particular interest. In one treatise, for example, Thabit analyzed the motion of a heavenly body on an eccentric, and the model he used was Ptolemaic. In contrast to Ptolemy's description, which was stated without proof, Thabit provided a rigorous and systematic mathematical proof with the aid of the theorems of Euclid's Elements. In the course of this proof, Thabit introduced the first known mathematical analysis of motion. For the first time in history, he also referred to the speed of a moving body at a particular point. In another work, Thabit provided general and exhaustive proofs for problems that Ptolemy examined only for special cases or for boundary conditions. Another work is exclusively devoted to lunar visibility. Thabit's solution, which was far more complex than that of Habash, exhibited the same mathematical rigor apparent everywhere in his work: he proved the general law that applies to the visibility of any heavenly body, then he applied this law to the special case of the crescent moon. Thabit's work is significant because it illustrates the high creativity of Islamic astronomy in its earliest periods. The roots for this creativity lie in the application of diverse mathematical disciplines to each other. This application had the immediate effect of expanding the frontiers of various disciplines and introducing new scientific concepts and ideas. The use of systematic mathematization transformed the methods of reasoning and enabled further creative developments in the diverse branches of science.

Another famous astronomer of this early period is Abu Abd Allah Muhammad ibn Jabir al-Battani (ca. 858–929), who originally came from Harran but lived in Raqqa in northern Syria. At Raqqa, al-Battani conducted observations for more than thirty years. The results of his research were recorded in al-Zij al-Sabi (The Sabian tables), which was translated into Latin in the twelfth century and into Spanish in the thirteenth. Although al-Battani did not contribute significantly to theoretical astronomy, his meticulous observations enabled him to make some important discoveries. For example, he noted the variations in the apparent diameters of the sun and the moon and deduced, for the first time in the history of astronomy, the possibility of an annular eclipse of the sun.

In the ninth century, then, Islamic astronomy had already struck deep roots. It integrated all the knowledge there was to integrate from earlier traditions and was justly positioned to surpass this knowledge. The achievements of the ninth century laid the foundation for the high-quality work in the following two centuries. The tenth and eleventh centuries witnessed important developments in trigonometry, with dramatic effects on the accuracy and facility of astronomical calculations. In this period steps were taken toward the formal establishment of large-scale observatories. Although the information from these two centuries is spotty and fragmented, several extant sources provide evidence for significant attempts to reevaluate Ptolemaic astronomy. In the tenth and eleventh centuries the earlier examinations of Ptolemaic astronomy led to systematic projects that rather than addressing the field in its totality, focused on specific aspects of astronomy. The work of Abd al-Rahman al-Sufi (who was born in Rayy and worked in the Iranian centers of Shiraz and Isfahan, 903–86) illustrates this tendency. In his famous book, Kitab Suwar al-Kawakib al-Thabita (Book on the constellations), al-Sufi reworked the star catalog of the Almagest on the basis of a corrected value of 1°/66 years for the precessional movement (in the place of Ptolemy's 1°/100 years), as well as several other new observations and verifications. Al-Sufi produced an accurate representation of the constellations and their coordinates and magnitudes. His work was translated into Latin and is the source of many Latin star names of Arabic origin. Another example of the tendency to synthesize is Abu al-Hasan Ali Ibn Yunus’ (Cairo, d. 1009) al-Zij al-Hakimi al-Kabir (The Hakimi zij), a monumental work in eighty-one chapters, of which only about one-half is preserved. The book is a complete treatise on astronomy, which contains tables for the movement of the heavenly bodies, their various parameters, and instructions on the use of these tables. Here, too, the objective of the work was to provide an exhaustive documentation of previous observations, subsequent verifications or corrections of these, and new observations recorded by the author.

Some of the astronomers of this period were known as instrument builders and for their association with observatories. The astronomer Abu Mahmud Hamid al-Khujandi (d. a. 1000), for example, wrote several works on scientific instruments and built a large sextant at Rayy. The astronomer Abu al-Wafa al-Buzjani (940–98) worked in a large observatory built by the Buyid ruler Sharaf al-Dawla in the gardens of the royal palace in Baghdad. Like Abu Nasr Mansur ibn Iraq (d. a. 1036) of Ghazna, al-Buzjani was a mathematician-astronomer who made great contributions in the field of trigonometry. Although much of the trigonometric works of these early scientists is lost, ample information exists from the extensive discussion on these works by the illustrious scientist al-Biruni.

Al-Biruni was born in 973 in Khwarizm (modern-day Khorezm) and died in 1048 in Ghazna (in eastern Afghanistan). Among other places, he worked in Rayy, where he collaborated with al-Khujandi. He also studied with Abu Nasr Mansur ibn Iraq, who was a student of al-Buzjani. Al-Biruni considered these two scholars as his teachers, and with them he shared a focused interest in trigonometry and its application to astronomy. Al-Biruni's native language was Persian, but he composed the vast majority of his works in Arabic. He also knew Sanskrit, and as a result he had full command of Indian astronomy in addition to the well-established Greek and Islamic traditions. Al-Biruni wrote more than 150 works on most of the known sciences of his time, including astronomy, mathematics, mathematical geography, mineralogy, metallurgy, pharmacology, history, and philosophy. Although only a third of his works are extant, these contain a wealth of scientific and historical information. His al-Oanun al-Masudi (Canon Macudicus) is a veritable treasure, which, as a great synthesis of the Greek, Indian, and Islamic astronomical traditions, has been compared to the synthesis produced in the Almagest by Ptolemy. The book is also a history of Islamic astronomy through the early eleventh century, and it provides the only extant source of information on many of the contributions of earlier astronomers. The value of this and other historical works by al-Biruni is further enhanced by his keen historical consciousness and cultural sensitivity.

Advances in trigonometry resulting from the full integration of the Indian achievements in the field, as well as from new discoveries in the tenth and eleventh centuries, played a central role in the development of Islamic astronomy. This tendency is itself part of a larger phenomenon whereby the systematic mathematization of disciplines contributed to the expansion of their frontiers. Equipped with new and more rigorous mathematical tools, al-Biruni, like many of his predecessors and contemporaries, provided exhaustive studies of specialized topics within astronomy. His “exhaustive” treatises cover such topics as shadows; the theory, construction, and use of astrolabes; the coordinates of geographical locations; and many more. In most of these monographs, al-Biruni starts with a thorough critical overview of older theories and mathematical methods for solving the particular problems in question; he then proceeds either to choose one of these theories or to propose his own alternative theory. Al-Biruni's work as a whole represents a critical assessment of the state of mathematical astronomy through the early eleventh century. Such comprehensive surveys of earlier knowledge exhausted the possibilities of expanding the astronomical disciplines from within; to achieve further progress, scientists needed to move in new directions, devise new strategies, and explore new research programs.

Another characteristic of this period is the seemingly random use of old as well as new mathematical methods in the solution of astronomical problems. Thus the same author may have used an archaic method in one place and an advanced method in another. Al-Biruni, for example, used both the old, cumbersome Menelaus theorem as well as the new, elegant sine rule in several solutions to the problem of determining the qibla, the direction that Muslims have to face in prayers. This simultaneous use of different mathematical procedures cannot be attributed to the slow dissemination of scientific knowledge or to the limited circulation of this knowledge. There is ample evidence for a high level of mobility and of efficient and speedy communication among scientists working in various regions of the Muslim world. Al-Biruni himself did not travel to Baghdad, but he apparently corresponded with scientists there and was fully aware of scientific developments there and elsewhere. The use of different methods is likely a result of the increasing diffusion of scientific knowledge among large segments of the educated elites. Within the broad ranks of these elites, “full-time” scientists were expected to keep up with the latest research in their fields, while scholars with partial interest in science would be familiar only with older theories and methods. The use of a variety of mathematical methods is thus an indication of the degree to which scientific culture had filtered into society, and the extent to which it became available to average members of the educated class.

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