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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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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• The Mathematical Cultures of Medieval Europe - Introduction

Mathematics in medieval Europe was not just the purview of scholars who wrote in Latin, although certainly the most familiar of the mathematicians of that period did write in that language, including Leonardo of Pisa, Thomas Bradwardine, and Nicole Oresme. These authors – and many others – were part of the Latin Catholic culture that was dominant in Western Europe during the Middle Ages. Yet there were two other European cultures that produced mathematics in that time period, the Hebrew culture found mostly in Spain, southern France, and parts of Italy, and the Islamic culture that predominated in Spain through the thirteenth century and, in a smaller geographic area, until its ultimate demise at the end of the fifteenth century. These two cultures had many relationships with the dominant Latin Catholic culture, but also had numerous distinct features. In fact, in many areas of mathematics, Hebrew and Arabic speaking mathematicians outshone their Latin counterparts. In what follows, we will consider several mathematicians from each of these three mathematical cultures and consider how the culture in which each lived influenced the mathematics they studied.

We begin by clarifying the words “medieval Europe”, because the dates for the activities of these three cultures vary considerably. Catholic Europe, from the fall of the Western Roman Empire up until the mid-twelfth century, had very little mathematical activity, in large measure because most of the heritage of ancient Greece had been lost. True, there was some education in mathematics in the monasteries and associated schools – as Charlemagne, first Holy Roman Emperor, had insisted – but the mathematical level was very low, consisting mainly of arithmetic and very elementary geometry. Even Euclid’s Elements were essentially unknown. About the only mathematics that was carried out was that necessary for the computation of the date of Easter.

Recall that Spain had been conquered by Islamic forces starting in 711, with their northward push being halted in southern France in 732. Beginning in 750, Spain (or al-Andalus) was ruled by an offshoot of the Umayyad Dynasty from Damascus. The most famous ruler of this transplanted Umayyad Dynasty, with its capital in Cordova, was ‘Abd al-Raḥmān III , who proclaimed himself Caliph early in the tenth century, cutting off all governmental ties with Islamic governments in North Africa. He ruled for a half century, from 912 to 961, and his reign was known as “the golden age” of al-Andalus. His son, and successor, al-Ḥakam II , who reigned from 961 to 977, was, like his father, a firm supporter of the sciences who brought to Spain the best scientific works from Baghdad, Egypt, and other eastern countries. And it is from this time that we first have mathematical works written in Spain that are still extant.

Al-Ḥakam’s son, Hishām, was very young when he inherited the throne on the death of his father. He was effectively deposed by a coup led by his chamberlain, who soon instituted a reign of intellectual terror that lasted until the end of the Umayyad Caliphate in 1031. At that point, al-Andalus broke up into many small Islamic kingdoms, several of which actively encouraged the study of sciences. In fact, Sā‘id al-Andalusī, writing in 1068, noted that “The present state, thanks to Allah, the Highest, is better than what al-Andalus has experienced in the past; there is freedom for acquiring and cultivating the ancient sciences and all past restrictions have been removed” [Sā‘id, 1991, p. 62].

Figure 1. Maps of Spain in 910 (upper left), 1037 (upper right), 1150 (lower left), and 1212-1492 (lower right)

Meanwhile, of course, the Catholic “Reconquista” was well underway, with a critical date being the reconquest of Toledo in 1085. Toledo had been one of the richest of the Islamic kingdoms, but was conquered in that year by Alfonso VI of Castile. Fortunately, Alfonso was happy to leave intact the intellectual riches that had accumulated in the city, and so in the following century, Toledo became the center of the massive transfer of intellectual property undertaken by the translators of Arabic material, including previously translated Greek material, into Latin. In fact, Archbishop Raymond of Toledo strongly encouraged this effort. It was only after this translation activity took place, that Latin Christendom began to develop its own scientific and mathematical capabilities.

But what of the Jews? There was a Jewish presence in Spain from antiquity, and certainly during the time of the Umayyad Caliphate, there was a strong Jewish community living in al-Andalus. During the eleventh century, however, with the breakup of al-Andalus and the return of Catholic rule in parts of the peninsula, Jews were often forced to make choices of where to live. Some of the small Islamic kingdoms welcomed Jews, while others were not so friendly. And once the Berber dynasties of the Almoravids (1086-1145) and the Almohads (1147-1238) from North Africa took over al-Andalus, Jews were frequently forced to leave parts of Muslim Spain. On the other hand, the Catholic monarchs at the time often welcomed them, because they provided a literate and numerate class – fluent in Arabic – who could help the emerging Spanish kingdoms prosper. By the middle of the twelfth century, most Jews in Spain lived under Catholic rule. However, once the Catholic kingdoms were well-established, the Jews were often persecuted, so that in the thirteenth century, Jews started to leave Spain, often moving to Provence. There, the Popes, in residence at Avignon, protected them. By the end of the fifteenth century, the Spanish Inquisition had forced all Jews to convert or leave Spain.

Figure 2. Papal territories in Provence

It was in Provence, and later in Italy, that Jews began to fully develop their interest in science and mathematics. They also began to write in Hebrew rather than in Arabic, their intellectual language back in Muslim Spain.  

Victor J. Katz (University of the District of Columbia), "The Mathematical Cultures of Medieval Europe - Introduction," Convergence (December 2017)

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The Mathematical Cultures of Medieval Europe

• The Mathematical Cultures of Medieval Europe - The Mathematics of the Muslims
• The Mathematical Cultures of Medieval Europe - The Mathematics of the Jews
• The Mathematical Cultures of Medieval Europe - Mathematics in Catholic Europe
• The Mathematical Cultures of Medieval Europe - Conclusions
• The Mathematical Cultures of Medieval Europe - References

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Medieval and Early Modern Science and Medicine: Mathematics

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• Occultism and Esotericism
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• Transmission and Reception
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Intended as an overview of available material on mathematics. Grouped into the following categories:

Medieval Mathematics

Early modern mathematics.

• "Maria Gaetana Agnesi: mathematics and the making of Catholic Enlightenment" by Massimo Mazzotti Isis 92 (2001): 657-683
• "Science and Humanism in the Renaissance: Regiomontanus's Oration of the Dignity and Utility of the Mathematical Sciences." by Noel Swerdlow Call Number: Q175.3 .W6 1993 in World Changes: Thomas Kuhn and the Nature of Science, edited by P. Horwich (Cambridge: MIT Press, 1993): 131-168.

Euclid Vatican City, Biblioteca apostolica vaticana, Cod. Vat. graec. 190, 288v-289r

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• Last Updated: Apr 18, 2024 11:38 AM
• URL: https://libguides.slu.edu/medren_science