Article

How trisections of the angle were transmitted from Greek to islamic geometry

Authors:
To read the full-text of this research, you can request a copy directly from the author.

Abstract

Two trisections of the angle were transmitted from Greek to Islamic geometry, one in the Arabic translation of the Lemmata of (pseudo-?) Archimedes, and the other in a hitherto unpublished 9th-century treatise by Ahmad ibn , which contains a translation from another Greek source. This paper presents an edition of the Arabic text of the latter treatise, as well as an English translation and a commentary, in which the text is compared with Propositions 36–42 of Book 4 of the Collection of Pappus of Alexandria (4th century). Following this is a summary of the work of ibn Qurra on this trisection and an attempt to explain why some 10th-century Islamic geometers thought that the ancients had not been successful in trisecting the angle.

No full-text available

Request Full-text Paper PDF

To read the full-text of this research,
you can request a copy directly from the author.

Article
There was a widespread belief among historians of science of my generation that high competence with regard to content and languages alone can guarantee better, more reliable results than can good philology combined with high competence in history or the other human sciences. In my casestudy of Wilbur R. Knorr’s analysis of several medieval Arabic and Latin texts on the balance, or steelyard, I highlight a variety of factors that compromised time and again his understanding and interpretation of his chosen texts. I conclude that a greater openness to more complex historiographical assumptions and more sophisticated methodological approaches as well as a greater willingness to contextualize documents in numerous dimensions before coming to conclusions about their specific meaning is crucial if we are to correct and improve upon work such as Knorr’s analysis of the Kitab al-qarastun, ascribed to Thābit ibn Qurra, and the Liber de canonio. The way forward is to enhance and temper philological analysis with solid analysis of scientific content within its relevant contexts. Published Online (2021-04-30)Copyright © 2021 by Sonja Brentjes Article PDF Link: https://jps.library.utoronto.ca/index.php/aestimatio/article/view/37604/28609 Corresponding Author: Sonja BrentjesMPIWG, BerlinE-Mail: brentjes@mpiwg-berlin.mpg.de
Chapter
From the advent in the Islamic milieu of falsafa as a widely recognised intellectual discipline in the third/ninth century, kalam and falsafa existed as parallel discourses on issues of physics, metaphysics and ethics. The mutakallimun are generally divided into two camps, the Mutazilites and the Asharis, although reasoned theological disputes antedate these groupings. Philosophy in the Islamic milieu followed upon the availability of texts of the Aristotelian and Platonic traditions. Islamic philosophy was appeared in many thinkers such as the widely travelled Andalusian philosopher Ibn Sabin, who reflected this movement with a philosophical approach learned in Aristotelianism and Neoplatonism but constituted as Sufi mystical wisdom. The prolific and influential theologian Fakhr al-Din al-Razi in his early years became well versed in philosophy and read Avicenna broadly. Adud al-Din al-Iji rejects the notion of knowledge by presence and prefers to understand knowledge as a created attribute.
Article
In recent years, many discoveries in the history of Islamic mathematics have not been reported outside the specialist literature, even though they raise issues of interest to a larger audience. Thus, our aim in writing this survey is to provide to scholars of Islamic culture an account of the major themes and discoveries of the last decade of research on the history of mathematics in the Islamic world. However, the subject of mathematics comprised much more than what a modern mathematician might think of as belonging to mathematics, so our survey is an overview of what may best be called the “mathematical sciences” in Islam; that is, in addition to such topics as arithmetic, algebra, and geometry we will also be interested in mechanics, optics, and mathematical instruments. We must, however, mention two limitations on our survey. The first is that for astronomy we simply refer the reader to the recent surveys in King 1980, a general survey published in this journal, and King 1983a, restricted to the Mamluk period. The other is that our survey is limited to those works published in English, French, and German, so that the reader must look elsewhere for an account of the large number of sources and studies published in Russian.
Article
This article presents an account of my attempt to follow Thomas Hood’s instructions for the inscription of two scales on the sector, as laid out in his The Making and Use of the Geometricall Instrument, Called a Sector (1598). This will allow us to identify those aspects of the work that he expected readers to infer for themselves. I will then piece together the ways in which readers might have inscribed the scales and build up a picture of the kind of background knowledge that the author expected of his readers. It was this knowledge that made it possible to communicate the tacit skills involved in instrument making through a book. We will find that much of the background knowledge required involved experience of manual techniques, rather than abstract principles. Thus, we will develop an understanding of the thoroughly practical nature of vernacular mathematics.
Chapter
This chapter presents the art and architecture of the Islamic world from the first/seventh century through to the end of the twelfth/ eighteenth. It isolates consistent preoccupations found in the art and architecture produced in the Islamic regions of Europe, Africa and Asia. The chapter considers the cultural background of the Hijaz in the first/ seventh century and the formative phase of Islamic art. It explores the period from the migration to Medina in 1/622 through to 84/703. It looks at the varied ways in which key issues such as sacred space, the role of writing, ornamental traditions, commemorative art and architecture, and modes of artistic interaction with other cultures, are interpreted in different regions and cultural contexts. The chapter finally concerns the creation of art and architecture in different types of human environment: urban life, palatial culture and rural areas.
Chapter
Throughout the last three or four centuries European modernity has produced legal systems and legal doctrines that are almost exclusively the preserve of the equally modern nation-state. The sources of authority that governed the emerging Islamic law were three, including the Quran, the sunan and considered opinion. The activities of the legal specialists initiated what was to become a fundamental feature of Islamic law: that legal knowledge as an epistemic quality was to be the final arbiter in law making. By the beginning of the third/ninth century the sharia courts and a corpus of positive law had fully developed. Legal theory and the doctrinal legal schools, however, were to emerge much later, reaching their apex in the middle of the fourth/tenth century. The product of juristic activity was the fiqh work that continued to gauge and be gauged by legal practice. By every indication the sharia served Muslim societies well for centuries.
Chapter
The coexistence of written and spoken forms of the language appears to be as old as Arabic itself, and has led to the emergence and perpetuation of distinct literary and oral cultures. Of the several forms of oral literature thought to have existed in pre-Islamic Arabia, only some poems and stories have survived. The earliest Arabic writings are simple graffiti, funerary inscriptions and the like. The language of the Quran appears to lie somewhere between the standard poetical idiom of pre-Islamic poetry and the spoken dialect of the Hijaz. Memorisation of the Quran also created an environment in which orality and memorisation played a critical role in the creation and transmission of knowledge. The prevalence of writing is yet another remarkable feature of medieval Islamic culture, for words were routinely inscribed on buildings, textiles and objects of daily use. Some medieval Arabic documents even confirm the persistence of orality.
Chapter
The systematic crystallisation of a newly sophisticated art after the advent of Islam may be described as the forging of a great musical tradition. The mystical movement developed a different approach to music. The encyclopaedic and literary works are primarily conceived along the lines of the adab literature. Abu Yusuf Yaqqub ibn Ishaq al-Kindi is the first author in the field of the science of music whose achievements are known to people. This chapter provides a systematic classification of all the sounds that exist in nature and mankind. Ii explains a definition of the science of music, its principles, methodology, musical parameters and instruments, rules of rhythm and composition. The chapter also deals with genres and systems that foreshadow the systematic modal presentation of Safi al-Din. Al-Maqqari, in his Nafh al-tib, depicted the Baghdadi musician Ziryab, who the chief minstrel at the Cordoban court and the innovator of Andalusian music in toto.
Chapter
In many ways Islamic history writing began with a ‘clean slate’. While pre- Islamic Arabian inscriptions, poetry and the ayyam al-arab folklore reflect a nostalgic curiosity about the past, the rise of Arabic-Islamic historiography stemmed from a more practical and immediate motivation. Al-Tabari’s prose is simple and straightforward, with little rhetorical embellishment. The richness, and diversity, of post-classical Muslim historiography is also seen in yet another significant frontier: North Africa and Andalusia. Arab chroniclers’ interest in Iran goes back to the beginning of Islamic historiography. Persian Islamic historical writing, however, came much later. The rise of Persian Islamic historical writing coincided with historical developments in the fourth/tenth century, when breakaway dynasties began to carve out territory. The wide horizons were explored under Mongol patronage in what was the golden age of Persian historiography. Medieval Indian history writing effectively began with the Muslims.
Chapter
The origins of Islam’s divisions into Sunnism and Shiism may be broadly traced to the crisis of succession to the Prophet Muhammad, who died after a brief illness in 11/632. The Imamiyya now splits into six groups, one of which eventually acquired the designation of the Ithna Ashariyya or Twelvers, recognising a line of twelve imams. Representing the most important Shiite community, the Ismailis have had their own complex history. The Ismailis of southern Iraq became generally known as the Qaramita, after their first local leader. Considering al-Hakim as the last maqam of the Creator, the Druzes await his reappearance together with Hamza, who is considered an imam. The Zaydi branch of Shiism developed out of Zayd ibn Ali’s abortive revolt in 122/740. A Shiite community with syncretic doctrines, the Nusayriyya, who were initially also called the Namiriyya, retained the traditions of the early Shiite ghulat. The Kharijites were originally concentrated in Kufa, where they survived until early Abbasid times.
Chapter
The languages and literatures of Turkic peoples, spread over a large geography on the Silk Route, have been written with a variety of scripts over the centuries. From the seventh/thirteenth century onwards various regional written languages and literatures begin to appear among the Turkic-speaking peoples. This chapter discusses the linguistic variation between Eastern and Western Turkic languages in Central Asia. The Eastern Turkic language developed into a literary language in Turkistan, Khwarazm and the Golden Horde. Ottoman poets and intellectuals took great interest in Chaghatay and Persian literature. During Turkish rule of Egypt and Syria Mamluks played a crucial role in enabling the preservation and dissemination of important works of Turkic literature. For example, two of the oldest and most important works of Islamic Turkish literature, the Kutadgu bilig and the Diwan lughat al-Turk. The printing-press had been introduced into the empire during the reign of Bayazid II by non-Muslim subjects, Christians, and Jews after their expulsion from Spain.
Chapter
Much of the so-called biographical literature in classical Arabic, Persian and Turkic has little in common with modern biography. Biography served as a documentary archive and a token of authority rather than a literary genre. Nevertheless, Muslim scholars of the seventh/thirteenth century and afterwards speak of such texts as a source of readerly pleasure. In Arabic, names conventionally take the form ‘so and so the son or daughter of so and so’: that is, they contain a genealogical component. The biographies of the Prophet Muhammad, for example, begin with genealogies that trace his lineage through the ancient prophets all the way back to Adam. Like all constructed histories, the charter myths of the various tawaif were subject to change over time. The expansion of the charter myth provoked an angry reaction and a new round of biography writing. Two sub-genres of biography appear anomalous, at least from the perspective of the genealogical model, including single-subject biography, and spiritual autobiography.
Chapter
The surviving translations from Arabic are in prose, the influence of Arabic poetic models on Persian prosody was also clearly extensive. However, alongside Arabic literary production as a significant model, elements derived from pre-Islamic Persian literature are also discernible in New Persian works. The most famous ethical prose work of Persian literature is Sadi’s Gulistan, a collection of anecdotes arranged into eight books according to their subject matter. The development of Persian verse has traditionally been divided according to three historically successive styles: the Khurasani; the Iraqi; and the Hindi. The Khurasani style is dominated, in aesthetic achievement as well as in sheer bulk, by a number of narrative mathnawis. The major epic is the Shahnama of Firdawsi, which recounts the pre-Islamic myths and romanticised history of Iran, from the creation of the world until the Arab conquest of the seventh century CE. The most significant romance of the Khurasani style is the Wis wa Ramin by Gurgani.
Chapter
The central role of knowledge flows from the importance of making Islamic civilisation’s greatest treasures, the Quran and the hadith, live and work in each day, each year and each generation of Muslim life. Throughout the middle period of Islamic history knowledge tended to be divided into two broad fields, including the ulum naqliyya, and ulum aqliyya. The pattern for transmitting knowledge with the emphasis on person to person transmission and the involvement on occasion of madrasas, which had developed in Iran, Iraq and Egypt, spread through the rest of the Muslim world. To begin the process of spiritual education a man, although it could also be a woman, would have to be accepted by a master. Approaches to the early education of children were for the most part driven by the need to save their souls. The school, in fact, was at the apex of a cluster of schools in which slaves were trained.
Chapter
The history of women in Islamic societies has made steady progress over the last few decades, following the spectacular growth of the field in other historiographical arenas. Generally speaking, women occupied a gendered sector of the work space, as most of the tasks they performed were centred on the domestic area or answered needs caused by gender segregation. Gender segregation in public spaces hindered the presence of women in the madrasas, the most important institution of high learning in the Muslim world from the fifth/ eleventh century onwards. It has been asserted that religion and religious practices were the privileged field for women’s agency. A long-lived Western tradition characterises Islamic societies by an unbridled sensuality and a self-indulgent allowance of fleshly pleasures. There is, however, a difference of approach to sexuality in Christianity and Islam that has influenced Western as well as Muslim interpretations.
Chapter
This chapter recognizes economic aspects of life in the pre-modern Islamic world. 1974 also saw the publication of the second edition of The legacy of Islam, and this book made a step forward in terms of bringing economic history to a wider audience. The chapter concerns sweeping matters: questions of agricultural and commercial legacy, particularly Europe. At the heart of many commercial transactions is money. Perhaps the most well-known example of a money of account is the dinar jayshi. Systems and units of measurement varied tremendously across time and region, and source material has survived in differing amounts from those periods and places. In Islamic Egypt the terminology of weight units often overlapped with the terminology for coins and units of account, resulting in much confusion. People have multiple-qirat weights which survive from the early Islamic period in Egypt.
Chapter
The fourth/tenth-century traveller and geographer al-Muqaddasi observed laconically that the inhabitants of central Arabia were both ‘frugal and emaciated, so little nourished are they by food’. With the rise and expansion of Muslim rule beyond the confines of the Hijaz from the first/seventh through the fourth/tenth centuries, came the inheritance of the food cultures of the conquered lands. The earliest Arabic work on agronomy, written in the early fourth/tenth century and attributed to one Ibn Wahshiyya, contains information on the culinary and medical uses of the many plants covered in it. In Persian, two works describing Indian cookery of the sultan’s court belong to the tenth/sixteenth and eleventh/seventeenth centuries. Plain white rice made in this fashion is called chelow. The basic method is described in medieval Arabic cookbooks, and the more elaborate preparations, already widely used in Persia, found their way into Turkish and Mughal cooking as well.
Chapter
This chapter discusses the complex relationships between the work of scientists and physicians and the societies that they lived in. The translation movement was primarily caused by forces that opposed the Umayyad dynasty and sought to restore pre-Islamic Iranian rule and splendor. Court patronage was the major element that provided the necessary means to carry out the translations. The chapter places important developments in astronomy within the context of its relationship to astrology and to other applications and areas of religious scholarship. Cartography in Islamic societies shares common ground with geography without being part of it. The relationship between the arts and the sciences in Islamic societies was, however, more complex than these views suggest. Mediterranean pre-Islamic arts and sciences were the sources of inspiration. The permeation of kalam by philosophy led to a stable linkage between miracles, magic, the theory of prophecy and the theory of the rational soul.
Chapter
Oldest historical document, the so-called ‘Constitution of Medina’, did provide a basis for the organisation of authority in the nascent Islamic polity. The imported Persian literature on statecraft was easily absorbed into the public law of the caliphate and Muslim monarchies, and shaped the medieval Muslim conception of government. The privileged status of the Arab Muslims made the Umayyad polity very much an Arab empire. The differentiation and mutual articulation of the caliphate and the sultanate became clearer in the second half of the sixth/twelfth century. The central paradox of Islamic royalism is that, in order to protect and promote Islam, the kings of Islam have to commit what is forbidden by the sacred law. Thus, in contrast to the theory of Islamic monarchy under the ‘king of Islam’, Ibn Taymiyya’s juristic theory subordinated monarchy to the shari order. The Ottomans combined the Persianate and Turco-Mongolian traditions of kingship with the law.
Chapter
The built environment of the Islamic city appears to have certain definable characteristics. The Islamic city, in short, holds an immense attraction for historians of Islamic culture, primarily because it appears to present a particularly and distinctively Islamic contribution to civilization. Cities also expanded because they became the centres of expanding industries. The new Muslim government shaped the distribution of cities in many ways. The model of Baghdad had a lasting influence on the formation of new capitals in the Muslim world. The nomads who supported the Almoravids and the village dwellers who sustained the Almohads both used and created cities to exercise their power. True nomads have traditionally lived in three major geographical areas: the deserts of Arabia and Syria, the steppes of Central Asia and the Sahara desert in Africa. The impact of the nomads has been enormous, sometimes destructive, sometimes leading to the formation of major states and empires.
Chapter
This chapter shows that literature on the dar al-harb, from the third/ ninth century until the thirteenth/nineteenth, was shaped not only by the juridical distinction between the two abodes, but also by a variety of cultural, religious, political, linguistic, geographical-astronomical and historical boundaries. Among the geographical traditions available to the early Muslims, that of the second-century Greek astronomer and geographer Ptolemy was the least concerned with political and cultural boundaries. Bureaucrats in the service of the Abbasid caliphate and its successor states composed comprehensive geographical works. In 309/921 the caliph al-Muqtadir sent an embassy to the Bulghars of the Volga, far to the north. Islamic travellers generally preferred to seek knowledge from established scholars, in their constant movements across the abode of Islam. During the many centuries of conflict between Muslim and non-Muslim states, countless individuals were taken captive. Non-Muslim sources regarding the dar al-harb were mostly avoided or ignored during the classical and postclassical periods.
Chapter
The lives of sedentary people in the Islamic countryside unfolded in a multifaceted context. Natural, technological, economic, political, cultural and religious factors all bore on rural life, and were in turn affected by it. The physical environment of the early Islamic world, though extremely varied, allowed agriculture in very limited areas. Rural people in the Islamic world had crucial relations with two non-agricultural elements of society: city-dwellers and nomadic pastoralists. Trade saw the cities exporting some industrial goods to the countryside, most notably textiles and tools, while the agricultural surplus of crops and animals was, in large measure, sold to the cities. The subsequent history of the rural economy in the Islamic world is the tale of periods of recovery, and even advance, followed by further decay, in which regional variation became more evident. Thus, in the early modern period, as the world economy tightened, agriculturists in many parts of the Islamic world had a foretaste of new vulnerabilities.
Chapter
Ibn Khaldun’s account constitutes two classic discourses on occult sciences in medieval Islam, and his report is, typically, of a very high historical and sociological value. This chapter shows that intricate backdrop provides a context to the occult sciences in Islam: their cultivation; their practice; and their legal and social status. The reality of magic is confirmed and reinforced in the Muslim tradition, particularly in the hadith, with the stories emerging that the Prophet Muhammad himself had fallen victim to magic. Reflecting a Quranic motif, Khaldun discusses magical powers in the context of prophecy. By the end of the Middle Ages Islamic culture had accumulated a fairly large body of magical literature, though much of it consisted of tantalising pseudotracts. Jabir ibn Hayyan and Abu Bakr al-Razi are the greatest names in the history of Arab alchemy, though for very different reasons. Abu Bakr al-Razi is one of the outstanding figures of the history of medicine in Islam.
Chapter
This chapter discusses Sufism’s evolution from a simple world-renouncing piety to the highly sophisticated doctrines and rituals practised primarily, within the institutional framework of the Sufi tariqa. Normative Sufi literature routinely portrays the Prophet and some of his ascetically minded Companions as ‘Sufis’ avant la letter. The ‘proto-Sufis’ strove to win God’s pleasure through self-imposed deprivations, self-effacing humility, supererogatory prayers, night vigils and meditation on the deeper implications of the Quranic revelation. In the western provinces of the caliphate people find a few ascetics who studied under al-Hasan al-Basri or his disciples, and who taught his ideas to their own students. Equally important for the self-identity of the Baghdad school of Sufism is Bishr ibn al-Harith. Rumi drew heavily on the Sufi tradition systematised by earlier Sufi writers. The chapter examines the rise and subsequent evolution of the major Sufi brotherhoods, and finally considers their respective roles in various geographical areas of the Muslim world over the last seven centuries.
Geschichte des arabischen Schrifttums, Band 5, ~at~emat~k. Leiden: Brill. ~ 1978. Geschichte des arabischen Schrifttums
  • F Sezgin
Sezgin, F. 1974 * Geschichte des arabischen Schrifttums, Band 5, ~at~emat~k. Leiden: Brill. ~ 1978. Geschichte des arabischen Schrifttums, Band 6, Astroi7omie. Leiden: Brill. Terzioqlu, N. 1974. Das vorwort des astronomen Bani Musa b.
On the trisection of the angle and the construction of a regular nonagon by means of conic sections in medieval Islamic geometry
  • Hogendijk
Das vorwort des astronomen Bani Musa b. Sakir zu den Conica des Apollonius von Perge
  • Terzioglu
Diocles on burning mirrors
  • Toomer