medieval mathematics essay

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MEDIEVAL MATHEMATICS

During the centuries in which the Chinese , Indian and Islamic mathematicians had been in the ascendancy, Europe had fallen into the Dark Ages, in which science, mathematics and almost all intellectual endeavour stagnated.

Scholastic scholars only valued studies in the humanities, such as philosophy and literature, and spent much of their energies quarrelling over subtle subjects in metaphysics and theology, such as “ How many angels can stand on the point of a needle? “

From the 4th to 12th Centuries, European knowledge and study of arithmetic, geometry, astronomy and music was limited mainly to Boethius’ translations of some of the works of ancient Greek masters such as Nicomachus and Euclid . All trade and calculation was made using the clumsy and inefficient Roman numeral system, and with an abacus based on Greek and Roman models.

By the 12th Century , though, Europe , and particularly Italy, was beginning to trade with the East, and Eastern knowledge gradually began to spread to the West. Robert of Chester translated Al-Khwarizmi ‘s important book on algebra into Latin in the 12th Century, and the complete text of Euclid ‘s “Elements” was translated in various versions by Adelard of Bath, Herman of Carinthia and Gerard of Cremona. The great expansion of trade and commerce in general created a growing practical need for mathematics, and arithmetic entered much more into the lives of common people and was no longer limited to the academic realm.

The advent of the printing press in the mid-15th Century also had a huge impact. Numerous books on arithmetic were published for the purpose of teaching business people computational methods for their commercial needs and mathematics gradually began to acquire a more important position in education.

Europe’s first great medieval mathematician was the Italian Leonardo of Pisa , better known by his nickname Fibonacci . Although best known for the so-called Fibonacci Sequence of numbers, perhaps his most important contribution to European mathematics was his role in spreading the use of the Hindu-Arabic numeral system throughout Europe early in the 13th Century, which soon made the Roman numeral system obsolete, and opened the way for great advances in European mathematics.

An important (but largely unknown and underrated) mathematician and scholar of the 14th Century was the Frenchman Nicole Oresme. He used a system of rectangular coordinates centuries before his countryman René Descartes popularized the idea, as well as perhaps the first time-speed-distance graph. Also, leading from his research into musicology, he was the first to use fractional exponents, and also worked on infinite series, being the first to prove that the harmonic series 1 ⁄ 1 + 1 ⁄ 2 + 1 ⁄ 3 + 1 ⁄ 4 + 1 ⁄ 5 … is a divergent infinite series (i.e. not tending to a limit, other than infinity).

The German scholar Regiomontatus was perhaps the most capable mathematician of the 15th Century , his main contribution to mathematics being in the area of trigonometry. He helped separate trigonometry from astronomy, and it was largely through his efforts that trigonometry came to be considered an independent branch of mathematics. His book “ De Triangulis “, in which he described much of the basic trigonometric knowledge which is now taught in high school and college, was the first great book on trigonometry to appear in print.

Mention should also be made of Nicholas of Cusa (or Nicolaus Cusanus), a 15th Century German philosopher, mathematician and astronomer, whose prescient ideas on the infinite and the infinitesimal directly influenced later mathematicians like Gottfried Leibniz and Georg Cantor . He also held some distinctly non-standard intuitive ideas about the universe and the Earth’s position in it, and about the elliptical orbits of the planets and relative motion, which foreshadowed the later discoveries of Copernicus and Kepler.

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Medieval and early modern mathematics

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Jacqueline Stedall

There is much fascinating material to be explored in the history of medieval and early modern mathematics, but perhaps the first thing to be aware of it is that, particularly for the earlier centuries, almost all original sources are in Latin. Mathematical Latin uses a specialised but relatively limited vocabulary, and if you know some Latin already (A-level or a good GCSE, say) you can build on it, but if you have none you will need to devote time to learning it.

Medieval mathematics (roughly 1100–1500)

Medieval mathematics was on the whole far removed from anything that we think of as mathematics today. Indeed to study this period at all you need to be prepared to enter a world whose preconceptions, political, religious, or mathematical, were very different from our own. There are texts that are recognisably devoted to arithmetic, geometry, or occasionally algebra, but most of the writings that were later described as 'mathematical' were concerned with astrology and astronomy (the distinction between the two was often blurred). Others border on philosophy, natural philosophy (or early science), and even theology.

The main centres for such studies in northern Europe were Paris and Oxford, and Oxford still holds major collections of medieval manuscripts, many with mathematical content. This has not been a major area of study in the last few years, but for useful introductory material and overviews see the following.

Grant, Edward, (ed), A source book in medieval science , Harvard University Press, 1974.
Molland, George, 'Mathematics', in David C Lindberg and Ronald L Numbers (eds), Cambridge history of science , II, Cambridge University Press, 1999.
North, J D, Chaucer's universe , Clarendon Press, 1988.
North, J D, 'Astronomy and mathematics' and 'Natural philosophy in late medieval Oxford', in J I Catto and T A Evans, The history of the University of Oxford , II, Clarendon Press, 1992.
North, J D, 'Medieval Oxford', in Fauvel, Flood and Wilson, Oxford figures: 800 years of the mathematical sciences , Oxford University Press, 1999.
Stedall, Jacqueline, 'How algebra was entertained and cultivated in Europe' in Stedall, A discourse concerning algebra , Oxford University Press, 2002, 19–54.

Early modern mathematics (roughly 1500–1700)

Like all other areas of intellectual activity in the sixteenth century, mathematics was revitalized by the translation of Classical texts from Greek to Latin. It was further stimulated by the absorption of ideas from Islamic sources, and by the new technical challenges posed by increased trade and navigation. During the seventeenth century in particular, mathematics in western Europe began to change rapidly and dramatically. At the beginning of that century, mathematicians looked back on ancient learning as something they could barely hope to emulate. By the end of it, they had far outstripped Classical achievements in both methods and results, and had developed their own tools and language, recognisably similar to those we use today. The most notable mathematical advances of the seventeenth century were the development of analytical geometry, the new acceptance of indivisibles, the discovery and use of infinite series, the discovery of the calculus, and the beginnings of a mathematical interpretation of nature. All of these changes were continued, consolidated, and argued about during the eighteenth century.

At the same time, mathematical learning was becoming more widespread, and the number of publications increased rapidly. In recent years the availability of electronic databases and searchable text has transformed research possibilities for this period. Every book published in England in the seventeenth century is now catalogued, and usually digitally available, on a database known as EEBO (Early English Books Online). Its eighteenth-century counterpart is ECCO (Eighteenth Century Collections Online). Access to these is available only through academic libraries, but they are an excellent way to begin to explore the literature.

Accounts of the early modern period are to be found in all general histories of mathematics, and you should read as many as you can. The following more specialised bibliography, by no means exhaustive, is intended to illustrate just some of the different ideas, approaches, and research methods that are to be found in recent literature.

Bos, Henk, Redefining geometrical exactness: Descartes' transformation of the early modern concept of construction , Springer, 2001.
Feingold, Mordechai, The mathematician's apprenticeship: science, universities and society in England 1560–1640 , Cambridge University Press, 1984.
Feingold Mordechai, 'Decline and fall: Arabic science in seventeenth-century England' in F Jamil Ragep and Sally P Ragep (eds), Tradition, transmission, transformation , Leiden: Brill, 1996, 441–469.
Feingold Mordechai, 'Gresham College and London practitioners: the nature of the English mathematical community', in Francis Ames-Lewis (ed), Sir Thomas Gresham and Gresham College: studies in the intellectual history of London in the sixteenth and seventeenth centuries , Ashgate, 1999, 174–188.
Guicciardini, Niccolo, '"Gigantic implements of war": images of Newton as a mathematician', in E Robson and J Stedall (eds), Oxford Handbook of the History of Mathematics , Oxford University Press, 2008.
Hill, Katherine, 'Juglers or Schollers?: negotiating the role of a mathematical practitioner', British Journal for the History of Science , 31 (1998), 253–274.
Mahoney, Michael Sean, 1973, The mathematical career of Pierre de Fermat 1601–1665 , Princeton University Press, 1973, reprinted, 1994.
Malcolm, Noel, and Stedall, Jacqueline, John Pell (1611–1685) and his correspondence with Sir Charles Cavendish: the mental world of an early modern mathematician , Oxford University Press, 2005.
Stedall, Jacqueline, A discourse concerning algebra: English algebra to 1685 , Oxford University Press, 2002.
Stedall, Jacqueline, 'Symbolism, combinations, and visual imagery in the mathematics of Thomas Harriot', Historia mathematica , 34 (2007).
Wardhaugh, Benjamin, 'Poor Robin and Merry Andrew: mathematical humour in Restoration England', BSHM Bulletin , 23 (2007).

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Numbers, in: Medieval Culture: A Compendium of Critical Topics (full text)

2015, Medieval Culture: A Compendium of Critical Topics Fundamental Aspects and Conditions of the European Middle Ages, ed. Albrecht Classen (Berlin and New York: De Gruyter 2015), pp. 1205–1260

This article outlines a history of numerical knowledge. It focuses on the forms of use that were developed for dealing with numbers. I begin with the act of counting and the notation of numbers (B); move from the role of numerals in medieval writing practices (C.) to the practices of measuring (D.) and calculating (E.); and close with comments on medieval theory building (F.), on speculative interpretations of numbers (G.), and with a short conclusion (H.). In doing so, the article focuses on the phenomenality of number—i.e., number as a spoken counting word (B.); as a graphic figure carved in wood or written on parchment (B.); as a formal pattern that organizes books and memory (C.); as a verbal expression for the documentation of quantities (D.); as a manual or symbolic instrument of calculation (E.); as a formally structured multitude determined by arithmetic properties (F.), which as symbolic carriers organize interpretation processes as well as the production of texts and works of art (G.). Thus, the approach allows for an exploration of how these varying phenomenons of number incorporate different kinds of semantic saturation that derive from the various fields of its use. As a result, this focus also contributes to dissolving traditional disciplinary gaps, especially the opposition between the history of science, which traditionally deals with the history of numerical knowledge, and historical branches of the humanities, which handle semantic and narrative phenomena.

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‘The Mysticism of Number in the Medieval Period before Eriugena’ in J.J. Cleary ed., The Perennial Tradition of Neoplatonism (Leuven University Press, Leuven 1997), pp. 397-416.

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The paper deals with early medieval mathematics (mainly arithmetic) and presents mathematical knowledge as an important tool for human way to God’s wisdom. The aim of this paper is focused on the definitions of the subject of arithmetic in early medieval texts created between the late 4th and early 7th century. The aim is to highlight the fact that the traditional definitions of a number (i.e., the subject of arithmetic) correspond with the appropriate topics which exist within arithmetic. If a number is characterised as a discrete quantity, it refl ects the classification and typological surveys of the mathematical properties of numbers. If a number is defined as a collection of units, this definition refers to the issue of fi gural numbers, whereas if the number is marked as the quantity that emerges and then returns to the unit, it is possible to detect the themes of numerical sequences and ratios, including their transfers.

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Christian number symbolism built upon the strong tradition of Platonic philosophy, Pythagorean lore and Babylonian astronomy. Numbers such as 1, 7, 8, 12, and 40 had magical and talismanic properties and were strongly represented in the Hebrew Scriptures and apocrypha. God was praised: 'you have ordered all things in measure, number and weight' (Wisdom 11:21). In the Christian Scriptures, when John the Divine was perplexed with the events happening around him, he was given a golden reed to measure the temple of God, the altar and those who worshipped within the temple (Revelation 11:1). Understanding would come from the process of measuring. Saint Augustine claimed that 'to ascend the path towards wisdom, we discover that numbers transcend our mind and remain unchangeable in their own'. Numbers had an ethereal existence. Augustine, Ambrose, Macrobius, Marcianus Capella, Isidore, Boethius, Thierry of Chartres, Abelard and many others praised the divine quality of particular numbers at length. Numbers were a model for theology and an analogy of creation. This paper examines the manifestation of this number symbolism in early medieval art, literature and architecture.

The volume gathers together a selection of eleven papers originally published by Burnett between 1996 and 2009. The papers deal with the various numeral forms used in the Middle Ages in mathematical and other contexts and show the complexity of the process of adoption of Hindu Arabic numerals over a period that extends from the tenth century, when the first reports of these numerals reached Europe, to their final adoption in the thirteenth century. Ordered according to the chronology of the subjects, this highly cohesive and stimulating set of essays will be essential reading for scholars studying the area, either thoroughly or tangentially.

Arts of Calculation: Numerical Thought in Early Modern Europe, ed. David Glimp and Michelle R. Warren (New York: Palgrave, 2004)

Signs of Writing: The Cultural, Social, and Linguistic Contexts of the World’s First Writing Systems, 2014

Across multiple disciplines, written numerical notation is a topic of keen interest, yet several unresolved issues in its analysis are either elided or taken as settled. Numerical notation is a complex phenomenon with multiple independent histories—more than 100 distinct systems used over the past 5,500 years, interweaving with, rather than strictly paralleling, the histories of writing systems. Social, semiotic, and cognitive approaches are brought to bear on six incompletely answered questions about numeration in relation to the earliest writing. Is numerical notation a necessary precursor to writing? Does the earliest numerical notation initially serve a bookkeeping function for early states? What is the relationship between tallying and numerical notation? Does the use of numerical notation change human cognition about the domain of number? How does the emergence of numerical notation relate to linguistic representations of number? Finally, among all domains of knowledge, why is number so widely represented using graphic notations? Recognizing that these issues are not resolved, and identifying different possible resolutions, must be preliminary to fully integrating numerical notation within the broader history of writing.

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