The exact sciences

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In Byzantium

The transmission of the exact sciences, based on principles established in Late Antiquity, continued throughout the existence of the Byzantine Empire. The four scientific disciplines of the quadrivium—geometry, mathematics, astronomy, and musical theory—were taught in the average school. Learning aids consisted of introductory manuals and commentaries on Classical works in the form of treatises or scholia.

During Late Antiquity, Alexandrian scholars produced commentaries on the great scientific texts from Antiquity, which formed the basis of teaching in later periods. In geometry, Heron (first century) and Theon Alexandricus (fifth century) produced commentaries on the Elements of Euclid. The treatises of Archimedes on the relation between the surface and volume of a sphere and its circumscribing cylinder (Eutokios of Askalon wrote a commentary on the treatise in the fifth century) were used by Isidore of Miletus and Anthemios of Tralles, the architects of Hagia Sophia in Constantinople (sixth century). In mathematics, Hypatia of Alexandria, Theon's daughter, produced a commentary on the treatises of Diophantus of Alexandria, a famous mathematician from the third century. In astronomy, Eutokios, Theon, and Stephan of Alexandria wrote a commentary on the Grande Syntaxe by Ptolemy, which was also known under the name of Almagest.
In addition to these commentaries, a series of new treatises and manuals were produced: manuals on applied geometry by Heron of Alexandria, treatises on cylinders and cones by Serenus of Antinopolis (fourth century), an introduction to mathematics by Domninus of Larissa (fifth century), and a treatise on burning-glasses by Anthemius of Tralles  (sixth century).
With the loss of Alexandria, conquered by the Arabs, Stephan brought Alexandrian science—he took his books and teaching—to Constantinople. And then the scholastic tradition died out and went underground, only concerning itself with the preservation of the manuscripts.

Science became more prominent again in the ninth century, with Leon the Mathematician. After learning mathematics and geometry on the Island of Andros, he taught these disciplines in Constantinople. News of his teachings reached Baghdad where, in the presence of the caliph al-Ma'mūn (reigned 813–817), one of his students was able to resolve problems, which none of the court scholars were able to solve. The caliph asked the Byzantine emperor to send Leon to his court, but the emperor Leon V the Armenian (reigned 813–820) decided instead to give the mathematician the task of overhauling the further education system (Magnaur school in Constantinople). Leon wrote commentaries on the works of Euclid, Archimedes, Apollonius of Perga, and the Neoplatonists. During this period, copies were made of manuscripts on astronomy: tables indicating the position of the stars were updated. From the eleventh century, Arab treatises on mathematics and astronomy were translated into Greek, introducing Arab science to the Byzantines.

During the Fourth Crusade and the conquest of Constantinople by the Latin peoples, the Greek scholars sought refuge in the empire of Nicaea (now İznik, Tur.), where they gradually rebuilt a cultural environment that was favourable to scientific development. After the reconquest of the city, these scholars contributed to the emergence of a renaissance movement in the arts and sciences. This is known as the Renaissance of the Palaiologoi.
In mathematics, scholars wrote teaching manuals (e.g. the Quadrivium of Pachymeres) and new commentaries on Classical works (aided by the rediscovery of forgotten manuscripts). Examples include the scholia on Diophantus by Maximus Planudes and Artabasdos Rhabdas, and the commentaries on Euclid by Pachymeres, Barlaam, and Isaac Argyros (who also wrote treatises on triangles and square roots). Maximus Planudes (end of the thirteenth century), in his The Great Calculation According to the Indians, introduced Arab numerals to Byzantium (more suitable for calculation than the Greek letters used up until then) and the practice of placing a zero next to the figures, which facilitated ‘astronomical’ calculations. Correspondence between scholars reflects the intellectual effervescence during this period: Manuel Moschopoulos sent a treatise on magic squares to Artabasdos Rhabdas, who sent his correspondents short treatises on calculation.

A knowledge of Ptolemaic astronomy was restored by Manuel Bryennios, Theodore Metochites, Nicephorus Gregoras, and Isaac Argyros, and took the form of an updated version (geographical and chronological) of the Tables of Ptolemy. Nicephorus Gregoras proposed certain reforms in the Julian calendar, which prefigured Pope Gregory's reforms. At the same time, eastern astronomy came to Byzantium thanks to scholars like Gregory Chioniades and his disciple George Chrysokokkes. They were introduced to the science in Trabzon and translated the Arab and Persian tables of astronomy into Greek. Ignoring the debate between the two schools of astronomy, some scholars, like Theodore Meliteniotes and John Abramius, reconciled the two schools of thought. Other influences emerged, as shown by the translation of Toledan and Jewish tables (in the fifteenth century, Michael Chrysokokkes translated the tables of Immanuel ben Jacob Bonfils of Tarascon from Hebrew into Greek). The need for exact calculations, especially for the purposes of liturgical festivals, favoured the acceptance of non-Greek science.

The empire was experiencing a complete scientific revival when it collapsed in the middle of the fifteenth century with the Fall of Constantinople. Some of the scholars, who emigrated with their manuscripts, contributed to the Renaissance in the West.
M.-H. C.

In Islam

Scientific activities and intercultural relations in Muslim lands (from the eighth to the fifteenth centuries)
Scientific activities increased in the Islamic world between the eighth and fifteenth centuries (continued in Europe, India and China) and this resulted in a multitude of contacts and exchanges between communities with different languages, religions and cultures. This was made possible by individuals and groups sharing the same scientific goals, who overstepped ideological, cultural and language boundaries.
There were three pivotal moments during this long phase of production and exchange: the time when scientific knowledge was “borrowed” from civilisations prior to Islam (mid-eighth to mid-tenth centuries), then the time of assimilation of ancient science and finally the time of original work (from the “Golden Age” – the ninth to twelfth or fifteenth centuries), when medieval Europe entered the scene (end of ninth to second half of fifteenth centuries).

The first appropriations
The first wave of appropriation, while knowledge circulated in a now opened-up area, began with the translation of Indian scientific literature in Sanskrit, which was probably known by the Persian elite since the fourth century at least, sometimes in a Pahlavi version dating from before Islam. The availability of these texts allowed pioneers to participate in the intellectual movement of the Umayyad period. This movement was amplified by the advent of the Abbasids and their Persian allies, who – with the Christian, Jewish and pagan communities – played an important role in the appropriation of scholarly knowledge from earlier civilizations. This first period was a time of exchanges between the elites of different communities in the Islamic world.
In this context, the Syriac-speaking citizens (whether Christian or pagan) who mastered Greek and Arabic played a decisive role in the translation of ancient knowledge preserved in the libraries of well-read families or monasteries. From the time of the first caliphs (632–61) and until the early tenth century, their translations of scientific and philosophical texts from Syriac to Arabic then directly from Greek to Arabic, served as intermediaries.
Euclid’s renowned Elements (third century BC) illustrates the cultural, religious and linguistic diversity characteristic of the first great period of scientific scholarship in Islamic countries: the book was translated twice within a few decades by al-Hajjāj Ibn Yūsuf, a Muslim; a third version, by the Christian Hunayn Ibn Ishāq, was then completely revised by pagan mathematician Thābit Ibn Qurra (d. 901). It is this version, considered the best, that has come down to us.
During this translation phase, unique in its duration and prolificacy, the diversity of participants was a decisive factor in rediscovering and reactivating ancient knowledge and using it to create a new scientific tradition. In so doing, patterns and exchange networks that played an important role in the second phase – the assimilation of ancient knowledge and its enrichment through new inputs – were created or developed.

Intercultural exchanges and the Golden Age
From the early ninth century to the late eleventh century or, for certain disciplines, the late fifteenth century, a scientific tradition took shape, initially mainly in Baghdad and in towns of the Fertile Crescent, as well as in many cities of the Islamic East and West. The ancient disciplines (mathematics, physics, medicine, philosophy, music, engineering, geography) and newer subjects, such as algebra, meteorology, trigonometry, and combinatorial analysis were nurtured. Technological knowledge, already sustained by local knowledge and Greek treaties (Archimedes, Hero of Alexandria and Philo of Byzantium), saw many innovations in the fields of hydraulic and military engineering, watch-making and even games.
All these advances took place in the context of lay scientific practices which transcended denominational particularities. They played a significant role in the establishment and preservation of a secular space in the Islamic world, but these practices were also made possible through some specific features of Islamic societies between the ninth and fifteenth centuries (cultural and religious diversity, secular administrative systems, and an absence of clergy).
In reconquered Spain and Norman Sicily, the secular nature of Arab scientific practice was sometimes preserved in a quite different political and ideological context. Thus, the geographer al-Idrīsī (d. after 1157) worked in Palermo with several collaborators to create a map of the world commissioned by Roger II of Sicily. This would not have been possible without the interculturality of newly reconquered territories. Similarly, Alfonso X the Wise (1252–82), king of Castile, recreated for scholarly reasons, a space that possessed the same characteristics as scholarly communities in the Islamic world.

The second wave of appropriations
From the eleventh century, science literature produced in the Islamic world, especially astronomical treatises, were translated from Arabic to Greek. After the fall of Byzantium to the Turks in 1453, these texts then circulated around Western Europe. It is by this means that scientific elements were brought to the attention of Copernicus who was thought to have used them in his description of the model of the Moon.
From late eleventh century, thanks to Constantine the African, a Carthaginian scientist who worked in southern Italy, Greek and Arabic medicine books became known to European practitioners. This individual undertaking was prolonged in early twelfth century by a major project of translation of dozens of science and philosophy books from Arabic to Hebrew and Latin in Toledo (newly reconquered by the Castilians) and Palermo (capital of Norman Sicily). Again, these activities bringing together people of diverse cultures and different faiths could only take place with the support of patrons interested in the knowledge produced in Muslim lands.
Meanwhile, often overlooked individual initiatives enabled the sharing of some of the knowledge and know-how of Islamic countries, including the circulation of astronomical instruments. In technology, the innovators were all anonymous, and most did not claim to be scientists. On the contrary, theoretical knowledge was transmitted by scientists trained in Arabic. Some lived in the Islamic world, for example Abraham Ibn Ezra and Abraham Bar Hiya (late eleventh century, Spain). Others, from southern Europe, had learned Arabic and specialised in a scientific area, and published books directly in Latin. This is the case in the twelfth century of the author of the Liber Mahamaleth and, in the thirteenth century of Leonardo Pisano, whose books were key in transmitting – from Italy – some aspects of the Arab tradition of calculation and algebra.

A. D.