The Islamic world came into contact with the great civilizations of antiquity during the eighth and ninth centuries. The scientific knowledge of not only Greece, but also India and Persia, was then assimilated by its scholars in a two-stage process. First, all the sources were translated into Arabic, and then, during the ninth and tenth centuries, their contents were held up to critical analysis.

The expansion of the Islamic Empire during this period had created great prosperity. Several social classes had appeared as a result, and it was these new classes, under the Abbasid caliphs, that patronized the translation movement. Nearly all the nonliterary and nonhistorical Greek texts available in the Byzantine Empire were subsequently translated into Arabic.

The process of assimilation was not merely an act of recapitulation. It was always imbued with a critical character. By the time of the caliph al-Ma’mūn (813–833), it was clear that the astronomical traditions of India and Greece, as reflected in the astronomical tables of the Sindhind and Ptolemy’s Almagest, were contradictory. To resolve this situation, the caliphate constructed the first Islamic observatories. Although they existed only briefly (ca. 828–833), the results obtained from the observatories in Baghdad and Damascus were used to make corrections to Ptolemaic dogmas, such as the fixity of the solar apogee. These changes were no small matter. According to Ptolemy, the angle between the equator and the ecliptic was 23° 51' 20". From the Ma’munian observations, the angle was calculated as 23° 33' or 23° 35'.

The first Arabic critiques of ancient Greek classics appeared during the tenth and eleventh centuries. Among them were Al-Shukūk ‘alā Jālīnūs (Doubts concerning Galen) by al-Rāzī, Al-Shukūk ‘alā Batlamyūs (Doubts concerning Ptolemy) by Ibn al-Haytham, and the list of disagreements with Aristotelian ideas that can be found in the al-Hikma al-mashriqiyya (Eastern Philosophy) of Ibn Sīnā. Arabic science was now beginning to move beyond classical science.

The initial stages of this new era of Arabic science first emerged in Baghdad, the center of Abbasid power. It soon spread to other parts of the empire, as the result of a political crisis that unfolded in the second half of the tenth century. The Abbasid caliphate lost its sovereignty and power passed into the hands of regional rulers. The latter were, in theory, representatives of the caliph, but his influence declined to merely that of a religious authority. In the Islamic West, even this limited influence receded and disappeared.

The decline of Abbasid power was not accompanied by any form of cultural or scientific decline. Many regional rulers sought to adorn their courts with men of letters and scientists, whose mere presence enhanced their standing and prestige. As a result, there was a considerable increase in the number of wealthy men prepared to encourage and fund scientific activity. In common with most of his scholarly contemporaries, the scholar and polymath al-Bīrūnī was consumed throughout his life by an unending search for patrons that might provide for his material necessities and allow him to continue his work.

A Learned Life

On September 4, 973, Abū al-Rayhān Muhammad ibn Ahmad al-Bīrūnī was born in a neighborhood, or bīrūn, on the outskirts of Kāth, the capital of the central Asian region of Khwārizm. In Persian, Bīrūnī means, “from an outer district.” The area is now known as Bīrūnī, in honor of its most illustrious son.

Kāth was ruled by the Banū ‘Irāq dynasty. Al-Bīrūnī was a disciple of a prince, Abū Nasr Mansūr ibn ‘Alī ibn ‘Irāq (ca. 965 – before 1036), who himself was a brilliant mathematician. Al-Bīrūnī acted as his secretary and was responsible for sending copies of the prince’s works to other leading scientific figures, such as the mathematician and astronomer Abū l-Wafā’ al-Buzjānī (940–998). This arrangement was not to last. A civil war broke out in the region that ended with the triumph of the Ma’mūnī dynasty and the death in 995 of the monarch Abū ‘Abd Allāh Muhammad b. Ahmad.

Al-Bīrūnī was forced into exile. He spent some time in Rayy, near Tehran, where he was unable to find financial support, but made contact with the great astronomer al-Khujandī (d. ca. 1000). He later surfaced in Jīlān, a region between the Caspian Sea and the Elburz mountains, where he dedicated a book to the ruler Marzubān b. Rustam, and also in Bukhāra, in the service of Mansūr II (997–999), a monarch of the Sāmāni dynasty. According to al-Bīrūnī, Mansūr II was his first patron.

Sometime before 1008, al-Bīrūnī returned to his homeland. An extraordinary group of scientists and philosophers had gathered in the court of Khwārizm-shāh Ma’mūn II (1009–1017). Among them, Ibn Sīnā was preeminent. Al-Bīrūnī’s teacher, Abū Nasr, was there, too, despite being a prominent member of the Banū ‘Irāq dynasty. Al-Bīrūnī was in Ma’mūn’s service for some seven years. He was engaged not only in scientific investigation, but also in political and diplomatic work.

Around 1014, the situation changed again. The sultan Mahmūd of Ghazna, which is in Afghanistan, presided over an immense empire that, at the time of his death in 1030, extended from the west of Iran to the Ganges valley. Nizāmī ‘Arūdī Samarqandī, a Persian writer in the mid-twelfth century, suggested that Mahmūd coveted the scientists and philosophers of Khwārizm for his own court. Other more reliable sources note that Mahmūd had ordered that his name be mentioned as that of the region’s true monarch in the sermon, or khutba, during the Friday noon prayer in Khwārizm’s mosques. Fearful of Mahmūd’s power, Ma’mūn II obeyed, but only half-heartedly. The result was a rebellion during which Ma’mūn II was assassinated. Mahmūd’s army captured Kāth in 1017, installing one of his associates as Khwārizm-shāh.

Al-Bīrūnī, his teacher Abū Nasr, and the physician Abū l-Khayr Husayn al-Khammār were all deported to Ghazna. A year later, al-Bīrūnī found himself living in a village near Kabul. In desperate circumstances, he continued working on a book about mathematical geography and attempting to carry out astronomical observations without the necessary instruments. Al-Bīrūnī’s relationship with Mahmūd was always cold and none of his writings were dedicated to the sultan. He does mention sultan Mas‘ūd, Mahmūd’s son and successor, to whom he dedicated his most important astronomical work, al-Qānūn al-Masū‘dī (The Mas‘ūdic Canon). Al-Bīrūnī reports that he was able to determine the latitude of Ghazna from observations made between 1018 and 1020 using a graduated astronomical ring, which al-Bīrūnī called the Yamīnī ring in honor of Mahmūd.

Mahmūd’s bloody conquest of northern India enabled al-Bīrūnī to travel extensively in the region. His assessment of the sultan’s military campaign was highly critical. “[Mahmūd] utterly ruined the prosperity of the country,” al-Bīrūnī remarked, “and performed there wonderful exploits, by which the Hindus became like atoms of dust scattered in all directions, and like a tale of old in the mouth of the people.” The annexation of the region led to an exodus of Hindu scientists, which, in turn, interfered with al-Bīrūnī’s work. Throughout his travels, al-Bīrūnī sought, above all else, to gather information from these sages. In his own book on India, he even goes so far as to implicitly criticize the patronage of Mahmūd, complaining that monarchs did not honor the sciences and those who cultivated them.

It seems likely that Al-Bīrūnī remained in Ghazna until the end of his life, although he may have returned to Khwārizm at some point. In the end, Al-Bīrūnī outlived both Mahmūd’s son and successor, Mas‘ūd (d. 1040), and his son and successor, Mawdūd (d. 1048). Al-Bīrūnī remarked in one of his later works that he had, at that point, lived for eighty years. He died sometime after 1053.

Al-Bīrūnī exhibited an almost universal, though broadly scientific, sense of curiosity throughout his life. Among his works, astronomy and astronomical geography predominate, but there are also discussions of history, pharmacology, mineralogy, and mechanics, amongst many other topics. At the age of sixty-three, al-Bīrūnī compiled a catalogue of his writings: one hundred and thirteen titles, plus another twenty-five written in his name by friends and collaborators, such as his teacher Abū Nasr, Abū Sahl ‘īsā b. Yahyā al-Masīhī, and Abū ‘Alī l-Hasan al-Jīlī. “As for what others have done in my name,” al-Bīrūnī remarked, “it is at the level of stepchildren in laps and necklaces [that adorn] throats: I do not distinguish between them and my own children.” In the end, he wrote a total of 146 or 180 works—the precise figure is still debated by scholars. Only twenty-three have been preserved and only thirteen have been edited.

Chronology and Trigonometry

Written after 998, the al-Athār al-bāqiya (Chronology of Ancient Nations) is among al-Bīrūnī’s early works. It begins with an analysis of the day as a fundamental chronological unit and goes on to describe solar, lunar, and lunisolar years, the different eras used by various cultures, and the names of the months, before concluding with a detailed description of the Jewish calendar. Al-Bīrūnī’s Maqālīd ‘ilm al-hay’a (The Keys to Astronomy), the first known treatise on spherical trigonometry, also dates from this early period.

When Ptolemy wrote the Almagest, he had only one trigonometric function at his disposal—the chord—and a single tool for solving spherical triangles. A theorem attributed to Menelaus relates the sides of two intersecting spherical triangles as a composite ratio: a/b = c/d · e/f. Indian astronomy had long employed sines and cosines, and these were adopted by astronomers in the Islamic world, along with tangents, cotangents, secants, and cosecants. Around the turn of the eleventh century, Abū Nasr Mansūr, Abū l-Wafā‘ al-Buzjānī, and Abū Mahmūd al-Khujandī developed theorems that became the laws of sines, cosines, and tangents. These new theorems had an obvious advantage. They establish relationships between the sides and angles of a single triangle: a/b = c/d. In the Maqālīd, al-Bīrūnī systematized this new trigonometry. He also insisted on the predominant role of his teacher, Abu Nasr, in its creation. The new trigonometry reached the Islamic West, and was systematized again by Ibn Mu‘ādh of Jaen (d. 1093), and then by Jābir ibn Aflah of Seville during the first half of the twelfth century.

Another important work dedicated to trigonometry is Kitāb fi ifrād al-maqāl fī amr al-zilāl (The Exhaustive Treatise on Shadows). The book begins by classifying trigonometric functions into two groups: circular functions, comprised of sines and cosines; and shadows, comprised of tangents and cotangents. The cotangent is the shadow projected in a horizontal plane by a sundial’s gnomon that is itself perpendicular to the plane. The tangent is the shadow projected in a vertical plane by a gnomon that is parallel to the horizontal. These shadows are dependent on the length of the gnomon, which is usually estimated as 12 digits (or 6.5 or 7 feet). A tangent here is 12 (or 6.5 or 7) times greater than the value that we now use. This was inconvenient when tangents and cotangents were used simultaneously with sines and cosines, which were calculated as a function of the radius. Sines and cosines were 60 times greater than their present value calculated for r = 1. Al-Bīrūnī was one of the first mathematicians to calculate tangent and cotangent tables, as well as sines and cosines, in the Qanun, for r = 1.

Latitude, Longitude, and India

In his Tahqīq mā li-l-hind min maqūlah maqbūlah fī al-‘aql aw mardhūlah (Verifying All That the Indians Recount, the Reasonable and the Unreasonable), al-Bīrūnī provided a contemporary description of India that covered a wide range of topics, including religion, philosophy, the caste system, marriage customs, units of measurement, geography, the time cycles used by Indian astronomy, calendars, rites, pilgrimages, and diet. Al-Bīrūnī’s account is remarkable both for its scope and the level of detail provided. He described things he had seen, but did not offer any criticism of the local inhabitants, even when their beliefs were clearly at odds with Islam. Instead, he drew analogies with similar ideas from the ancient Greeks, quoting from the Platonic dialogues or the works of Aristotle, which he knew in their Arabic translations. Al-Bīrūnī had also acquired Sanskrit by this time, since he translated passages from the books and sources at hand. Hindu scientific texts of the era were often composed in Sanskrit verses, or sloka, to facilitate memorization. Al-Bīrūnī remarked that he was planning to translate the works of Euclid and Ptolemy into Sanskrit verse for dissemination among Hindu scientists.

Al-Bīrūnī’s Tahdīd nihāyāt al-amākin li-tashīh masāfāt al-masākin (Determination of the Coordinates of Places for the Correction of Distances Between Cities) was another major work conceived during his stay in Ghazna. These coordinates are needed to calculate the sacred direction, or qibla, toward Mecca. Al-Bīrūnī began with a consideration of latitude, which is obtained by observing the meridian height of the sun, or a star, if its declination is known, or by resorting to the semi-sum of the maximum and minimum altitude of a circumpolar star. He then determined the obliquity of the ecliptic—the angle between the ecliptic and equator, which he approximated as 23° 35'—because calculating the sun’s declination depends on this angle.

The problem of calculating longitude, on the other hand, was far more challenging and was not fully resolved until the invention of the chronometer in the eighteenth century. In al-Bīrūnī’s time, scholars relied on observations of a lunar eclipse from two locations as a basis for determining the difference in time between them. In view of the practical difficulties involved, al-Bīrūnī refined the method used by Ptolemy in his Geography. If the number of days that a caravan takes to go from one city to another is known, as well as its daily route, it becomes possible to calculate the distance between the cities and convert this figure to degrees. To work out the coordinates of Ghazna, al-Bīrūnī made a series of calculations, obtaining, in turn, the differences in longitude between Baghdad, Rayy, Jurjāniya (Urgench, south of the Aral Sea), Balkh (a town in Afghanistan), and Ghazna. Al-Bīrūnī’s final result is impressive. He calculated the difference in longitude between Baghdad and Ghazna as 24° 22'. The modern value is 24° 20'.

The al-Qānūn al-Mas‘ūdī was al-Bīrūnī’s astronomical masterpiece. Dedicated to the sultan Mas‘ūd and written during his reign over Ghazna between 1030 and 1040, the Qānūn was the largest astronomical encyclopedia compiled during the Middle Ages. The title is a clear allusion to Ptolemy’s Manual Tables, known in the Arab world as al-Qānūn, and the work itself follows the model established by the Almagest. The Qānūn is concerned with all manner of astronomical matters, but also contains a large number of numerical tables, many of which were original. The Qānūn can, in fact, be considered a revised and updated version of the Almagest. While the basic principles remain Ptolemaic, the Qānūn contains a great deal of information gleaned from scientific cultures outside the Islamic world, notably that of India.

The Qānūn comprises eleven books. Books I and II deal with chronology and include material already covered by al-Bīrūnī in his Chronology, to which it adds information on the time-measurement systems used in India. Book III is concerned with trigonometry, begins with a detailed analysis of the chord function, a subject to which al-Bīrūnī had dedicated a book, and incorporates the new trigonometric material from the Maqālīd. Books IV and V are focused on spherical astronomy and mathematical geography, respectively. Book VI begins by addressing differences of longitude between localities and continues with the study of the mean motion of the sun, based on observations by Hipparchus, Ptolemy, and a multitude of Islamic astronomers of the ninth and tenth centuries, including al-Bīrūnī himself. Books VII and VIII deal with the motion of the moon, solar and lunar eclipses, and the visibility of the new moon. Book IX is concerned with the fixed stars and includes a catalogue of 1,029 stars, while Book X examines planetary motion, in longitude and in latitude. The final volume, Book XI, deals with mathematical astrology.

Astrology

It seems that al-Bīrūnī may have served as an astrologer to Mahmūd of Ghazna at one time. Writing in the mid-twelfth century, Nizāmī ‘Arūdī Samarqandī offered the following anecdote:

Mahmūd was seated in a room with four doors on the second floor of his palace. He asked al-Bīrūnī to predict, in writing, by which door he was going to leave. Abū l-Rayhān asked for an astrolabe, with which he took the sun’s height (to determine the time), cast the horoscope, and, after thinking for a while, wrote his answer on a piece of paper and hid it under the carpet. Mahmud called for a mason, who opened a fifth door on the eastern wall. The monarch went out by this door and ordered that the opinion of the astrologer be brought to him. In it he read: “The exit will not take place by any of the four doors, but another one will open in the eastern wall, which will be the exit door.”

Nizāmī’s account appears to demonstrate both al-Bīrūnī’s gift for psychology and his understanding of Mahmud’s personality. Yet a strikingly similar encounter between the Cordoban emir ‘Abd al-Rahmān II (821–852) and the poet-astrologer Ibn al-Shamir had been described some two centuries earlier. Clearly, the story itself is apocryphal. The next passage reveals both Mahmūd’s violent nature and the nature of his relationship with al-Bīrūnī:

When the monarch saw the failure of the joke he had prepared for his astrologer, he was outraged and ordered him thrown out the window. This was carried out by the servants, but al-Bīrūnī fell into a net hanging at the first floor, so he emerged unscathed. Mahmūd ordered him to appear again before him. Al-Bīrūnī showed him a taqwīm [a book containing planetary ephemeris for the current year, as well as the corresponding astrological predictions] in which everything that had happened was written. Mahmūd’s indignation then overflowed and he ordered that the astrologer be locked up in a fortress, where he remained for six months.

Nizāmī’s account, although embellished and at least partially apocryphal, nonetheless helps convey some sense of al-Bīrūnī’s ambiguous attitude towards astrology. Al-Bīrūnī was largely concerned with the technical aspects of astrology, or mathematical astrology. While there has always been a popular branch of astrology that offers predictions based on simple rules, such as the zodiacal sign of the sun at the moment of birth, there was also a more serious version reserved for the upper classes. This type of astrology involves astronomy and mathematical problems of increasing complexity, such as the division of the houses, the projection of planetary rays, and the progression of an ecliptical degree (tasyīr).

Al-Bīrūnī had little time for professional astrologers. In his treatise on astrological transits, he described al-Hasan ibn ‘Alī ibn ‘Abdūs as, “a non-studious listener,” noting that, “this is the case with most of the class of the astrologers; they bubble proudly about things they barely hear, without verifying them, and they are satisfied by associating fancies with them.” An even less favorable assessment appears in the Tahdīd:

As the profession of astrology is based on weak foundations, as its subsidiary branches are defective, as the measurements made are inaccurate, as the predictions made are based on probability and not on certainty … so their predictions can never be valid unless the data of their object are accurate.

Around 1029, al-Bīrūnī wrote a book of astrological questions and answers that was fundamentally didactic in nature. It was composed at the request of Rayhanat bint al-Hasan, a member of the ruling family in Khwārizm who was also forced into exile in Ghazna. Only a third of the book deals with astrology. The remainder is concerned with geometry, arithmetic, algebra, and astronomy. “Nobody deserves the title of astrologer,” Al-Bīrūnī remarked, “if he does not have full competence in these four subjects.”

However much he may have been required to practice it, there is little evidence that al-Bīrūnī believed in astrology. He never wrote about its predictive applications. He explained how to cast a horoscope, but not how to interpret it.

The Pinnacle

Throughout his professional life, al-Bīrūnī was obsessed with observations. Impressive instruments were available to him, from the fakhrī sextant of al-Khujandī, which had a radius of sixty meters, to the horizontal astronomical ring, with a diameter of 8.1 meters. In 1016, al-Bīrūnī used a large meridian ring in Jurjāniyya to make around fifteen observations of meridian transits of the sun. Between 1018 and 1019, he determined the latitude of Ghazna using the yamīnī ring, which, in common with large instruments, was divided into degrees and minutes.

When a suitable instrument was not available, al-Bīrūnī improvised. In Jayfūr, near Kabul, he took a tablet that was used to record calculations and drew an arc of a circle on the back, which he divided into degrees and arcs of ten minutes. By suspending the tablet from a plumb line, al-Bīrūnī was able to measure the meridian height of the sun and determine its declination. The latitude he calculated for Kabul at the end of 1018, 34° 41', compares favorably with the modern value, 34° 30'.

Al-Bīrūnī always sought to expand his knowledge using all the sources available to him. New findings were compared and examined in the light of his own observations, and subjected to rigorous analysis. He was a prolific author, penning great syntheses, which included much information gathered from outside the Islamic world, and an enormous quantity of his own work. His books about India and geography opened new fields of research.

Translated and adapted from the Spanish by the editors.

In the ninth century, the scientific and philosophical level of the Byzantines had sharply declined. It is curious to note the parallels between how official Abbasid ideology understood this decline and how Europeans in the nineteenth and twentieth centuries saw Islam as responsible for a similar scientific decline in the Arab world. For the thinkers of the caliphate, the cause of Byzantine decline rested in the irrational character of Christianity, with dogmas as absurd as those of the Trinity and the incarnation of God as man. Islam, a religion much more coherent from a logical point of view, could assimilate the entire Greek heritage without hesitating over the pagan context that so terrified the Byzantines. See Dimitri Gutas, Greek Thought, Arabic Culture (London: Routledge, 1998); George Saliba, Islamic Science and the Making of the European Renaissance (Cambridge, MA: MIT Press, 2007). ↩ Aydin Sayili, The Observatory in Islam (New York: Arno Press, 1981), 50–87. ↩ The scholars and scientists who worked in Baghdad during the ninth century and the beginning of the tenth century were not, for the most part, Arabs. They were from regions near Iraq, where there had been a dramatic increase in scientific activity among non-Arabs. Despite diverse geographical origins, these men of science were perfectly Arabized and consistently used Arabic as the language of scientific communication. ↩ The best biography of al-Bīrūnī is Edward Kennedy, “al-Bīrūnī,” in the Dictionary of Scientific Biography, vol. 2, ed. Charles Gillispie (New York: Scribner’s, 1970), 147–58. ↩ Muhammad ibn Ahmad Bīrūnī and Marie-Thérèse Debarnot, Kitāb Maqālīd ‘ilm al-hay’a: la trigonométrie sphérique chez les arabes de l’est à la fin du Xe siècle (Damascus: Institut Français de Damas, 1985), 96. ↩ Al-Bīrūnī described the great wall sextant, known as a fakhrī, which had been constructed on the outskirts of the city for the observation of meridian transits. ↩ It was in Bukhāra that al-Bīrūnī may have met the dethroned king of Jurjān, Qābūs (977–81 and 998–1013). When Qābūs regained power and returned to Jurjān in 998, al-Bīrūnī followed him. The Chronology, al-Bīrūnī’s first major work, is dedicated to Qābūs. ↩ Al-Nizāmī al-‘Arūdī al-Samarqandī, Chahār Maqāla, trans. ‘Abd al-Wahhāb ‘Azzām and Yahyā al-Khashshāb (Cairo, 1949), 81–82. ↩ The honorary title of Yamīn al-Dawla had been awarded to Mahmūd by the caliph. Edward Kennedy, “al-Bīrūnī,” in the Dictionary of Scientific Biography, vol. 2, ed. Charles Gillispie (New York: Scribner’s, 1970), 150. ↩ Edward Sachau, Alberuni’s India: An Account of the Religion, Philosophy, Literature, Geography, Chronology, Astronomy, Customs, Laws and Astrology of India about A.D. 1030, vol. 1 (London: Routledge, 1888), xi, 22. ↩ Edward Sachau, Alberuni’s India: An Account of the Religion, Philosophy, Literature, Geography, Chronology, Astronomy, Customs, Laws and Astrology of India about A.D. 1030, vol. 1 (London: Routledge, 1888), 152. ↩ Al-Bīrūnī might be interested in a subject but encounter difficulties, such as, for example, a lack of worked examples that allow one to be absolutely certain that there are no errors in the manuscript. Instead of doing this detailed work on his own, he would find a collaborator to do it, so that he could devote himself to more ambitious projects. See Julio Samsó, Estudios sobre Abū Nasr Mansūr b. ‘Alī b. ‘Irāq (Barcelona, 1969), 18–25. ↩ D. J. Boilot, “L’oeuvre d’al-Beruni: Essai bibliographique,” Mélanges de l’Institut Dominicain d’Études Orientales 2 (1955): 161–256 and 3 (1956): 391–96. ↩ In the end, he wrote a total of 146 (according to Edward Kennedy) or 180 works (according to Boilot). D. J. Boilot, “Al-Bīrūnī,” in Encyclopedia of Islam, vol. 1 (Leiden: Brill, 1986), 1,236–37. ↩ For an English translation, see Edward Sachau, trans., Chronology of Ancient Nations (London: W. H. Allen and Co., 1879). ↩ It was dedicated to a ruler of Jīlān, Abū l-‘Abbās Marzubān ibn Rustam ibn Sharwīn. Muhammad ibn Ahmad Bīrūnī and Marie-Thérèse Debarnot, Kitāb Maqālīd ‘ilm al-hay’a: la trigonométrie sphérique chez les arabes de l’est à la fin du Xe siécle (Damascus: Institut Français de Damas, 1985), 96. ↩ There were precedents in the work of Habash in the second half of the ninth century. ↩ The work of Jabir (Islāh al-Majistī) was translated into Latin and Hebrew and was used by Regiomontanus, in his De triangulis, as well as by Copernicus. ↩ Edward Kennedy, The Exhaustive Treatise on Shadows by Abū al-Rayhān Muhammad b. Ahmad al-Bīrūnī. Translation and Commentary, 2 vols. (Aleppo: Institute for the History of Arabic Science, 1976). Al-Bīrūnī dedicated this work to an important citizen of Nīshāpūr in the Khurasan, Abū l-Hasan Musāfir b. al-Hasan. ↩ In the second half of the eighth century, at the time of the caliph al-Mansūr (754–775), an Indian delegation visited Baghdad. One of its members was an astronomer, who brought with him a set of astronomical tables in Sanskrit. By order of the caliph, the astronomer al-Fazārī, probably with the collaboration of Ya‘qūb b. Tāriq, translated these tables into Arabic. It was one of many examples of contact between India and the cultural world of the Abbasid caliphate, and the works of these two astronomers (especially Ya‘qūb b. Tāriq) are frequently cited by al-Bīrūnī in the astronomical sections of his work on India. ↩ Edward Sachau, Alberuni’s India: An Account of the Religion, Philosophy, Literature, Geography, Chronology, Astronomy, Customs, Laws and Astrology of India about A.D. 1030, vol. 1 (London: Routledge, 1888), 7. ↩ Edward Sachau, Alberuni’s India: An Account of the Religion, Philosophy, Literature, Geography, Chronology, Astronomy, Customs, Laws and Astrology of India about A.D. 1030, vol. 1 (London: Routledge, 1888), 137. ↩ The book includes a calculation of the qibla for Ghazna with a maximum error of 2' of arc. ↩ In passing, he mentions the observations of the obliquity he carried out in Būshkānz, a small town south of Kāth, for which he used a large horizontal ring with a diameter of 8.1 meters. He could not, unfortunately, finish his work after his observation of the summer solstice in the year 994; the civil war began. Jamil Ali, trans., The Determination of the Coordinates of Cities: Al-Bîrûnî’s Tahdîd al-Amâkin. (Beirut: American University of Beirut, 1967), 77; Edward Kennedy, A Commentary Upon Bîrûnî’s Kitâb Tahdîd al-Amâkin (Beirut: American University of Beirut, 1973), 49. ↩ This instrument records the local time of a particular place. Elsewhere, the passage of the sun through the meridian (local noon) was observed. The time difference made it possible to calculate, immediately and precisely, the difference in longitude between the two places. ↩ In his analysis, al-Bīrūnī establishes requirements for reliable observations. He himself observed in Kāth the lunar eclipse of May 24, 997, after having agreed with Abū l-Wafā‘ (940–998) that the latter would do the same in Baghdad. The result was a difference of one hour between Kāth and Bagdad, equivalent to a 15º difference in longitude. Jamil Ali, trans., The Determination of the Coordinates of Cities: Al-Bīrūnī’s Tahdīd al-Amākin (Beirut: American University of Beirut, 1967), 214–15; Edward Kennedy, A Commentary Upon Bīrūnī’s Kitāb Tahdīd al-Amākin (Beirut: American University of Beirut, 1973), 164–65. ↩ An edition of the Arabic text of al-Qānūn al-Mas‘ūdī was published in three volumes in Hyderabad, between 1954 and 1956. There is no translation available for any European language. A detailed summary of the contents can be read in Edward Kennedy, “Al-Bīrūnī’s Masudic Canon,” Al-Abhāth 24 (1971): 59–81. Reprinted in Edward Kennedy et al., Studies in the Islamic Exact Sciences (Beirut: American University of Beirut, 1983), 573–95. ↩ Istikhrāj al-awtār fī l-dā’ira (Obtaining the Chords of a Circle). ↩ The latter is important for Islamic worship since the lunar month begins when the moon, after its conjunction with the sun, starts to become visible. This determines the beginning and end of the fasting period during the month of Ramadan. ↩ Al-Nizāmī al-‘Arūdī al-Samarqandī, Chahār Maqāla (Four Discourses), trans. ‘Abd al-Wahhāb ‘Azzām and Yahyā al-Khashshāb (Cairo, 1949), 64–65. Translation from the Arabic by the author. ↩ Al-Nizāmī al-‘Arūdī al-Samarqandī, Chahār Maqāla (Four Discourses), trans. ‘Abd al-Wahhāb ‘Azzām and Yahyā al-Khashshāb (Cairo, 1949), 64–65. Translation from the Arabic by the author. ↩ For a general analysis of these problems, see Josep Casulleras and Jan Hogendijk, “Progressions, Rays and Houses in Medieval Islamic Astrology: A Mathematical Classification,” Suhayl 11 (2012): 33–102. ↩ Mohammad Saffouri, Adnan Ifram, and Edward Kennedy, Al-Bīrūnī on Transits (Beirut: American University of Beirut, 1959), 71. ↩ Abū al-Rayhān Muhammad ibn Ahmad al-Bīrūnī, The Book of Instruction in the Elements of the Art of Astrology, trans. R. Ramsay Wright, facsimile edition of Brit. Mus. MS. Or. 8349 (London: Luzac & Co., 1934), 1. ↩ Jamil Ali, trans., The Determination of the Coordinates of Cities: Al-Bīrūnī’s Tahdīd al-Amākin (Beirut: American University of Beirut, 1967), 86; Edward Kennedy, A Commentary Upon Bīrūnī’s Kitāb Tahdīd al-Amākin (Beirut: American University of Beirut, 1973), 54. ↩