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«m3hw5.tex Week 5. 10 – 12.2,2016 FROM THE ARABS TO GALILEO Edward GIBBON, Decline and fall of the Roman Empire, 1776 Boyer Ch. 14; Dreyer Ch. X, ...»

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m3hw5.tex Week 5. 10 – 12.2,2016


Edward GIBBON, Decline and fall of the Roman Empire, 1776

Boyer Ch. 14; Dreyer Ch. X, Mediaeval cosmology

Al-Kwarizmi’s Algebra (concluded). Its strengths include logical exposition

and proof (shared by the Greeks but not the Mesopotamian and Hindu in-

fluences), and facility in passing between geometric (Greek) and numeric

(Hindu) algebra. Its weaknesses include that only positive rots of equations were permitted, and that everything, even numerals, is written out in words.

Astronomy The Caliph al-Mamun founded an observatory in Baghdad in 829, ‘where continuous observations were made and tables of planetary motions con- structed, while an important attempt was made to determine the size of the Earth’ (Dreyer, 246). One of the leading astronomers here was al-Fargani (= Alfragamus), ‘whose Elements of Astronomy were translated into Latin in the 12th C., and contributed greatly to the revival of science in Europe.’ Hindu astronomy was derived from the Greek through the conquest of Alexander the Great (B 12.12). Hindu astronomical texts, the Siddhanthas, reached Baghdad during the reign of Caliph al-Mansur ‘in 773) (Dreyer, 244), or ‘by 766’ (B, 254). These were translated into Arabic as the Sindhind.

Note. One of the main spurs behind Arab interest in astronomy derives from the Muslim religious year.

Omar Khayyam (c. 1050-1123) (B 13.15).

Kayyam wrote a text Algebra. Like al-Khwarizmi, he restricted himself to positive roots, but went beyond degree 2. He gave algebraic and geometric solutions for quadratics, and geometric solutions (involving intersecting con- ics) for cubics. He was also a noted poet. His poem The Rubaiyat of Omar Khayyam was translated into English poetry by Fitzgerald in 1872/79/89.

Postscript to ‘pre-Europe’.

Mathematics is an intensely cultural activity – because unless one builds on the work of others, one will come much less far in one lifetime than hu- manity has come already! From now on, we enter the framework of European culture (apart from N. American in the 20th C.). We do this for brevity, not cultural bias – ancient and non-European cultures produced far more than we have time to touch on. But accusations of cultural bias do exist, and 1 ‘ethnomathematics’ is sometimes advocated as an antidote. See e.g.

G. C. JOSEPH, The crest of the peacock: Non-European roots of mathe- matics, I. B. Tauris, 1991. Rev.: Amer. Math. Monthly 99.7 (1992), 692-4;

Marcia ASCHER, Ethnomathematics: A multicultural view of mathematical ideas, Brooks Cole, 1991.

Background: pre-Europe to Europe Historically, one speaks of ancient, or classical, history, the Middle Ages or mediaeval period, and modern history. Customarily, the Middle Ages are taken from the fall of Rome in 4761 to the fall of Constantinople in 1553.

Mathematically, we take the end of the classical period as 529, when Justinian closed the Academy at Athens. The surviving scholars dispersed East to Syria and Persia, where they eventually encountered Arab scholarship.

The early mediaeval period is often called the Dark Ages by historians, and this is apt in mathematics too. In West Europe, the dominant institutions (which often quarreled) were the Roman Catholic Church, led by the Pope as Bishop of Rome, and the Holy Roman Empire (which lasted from Charlemagne in 800 to Napolean in 1804). Throughout the Middle Ages, the great majority of educated people were clerics, monks etc. (ordinary people, and many political leaders, were illiterate), and the language of learning was Latin. The subjects of learning were Christian theology (something of a minefield – one had to beware of being accused of heresy), works of the classical Latin authors, etc. (a revival of interest in classical Greek authors came later), rather than science. The Church remained suspicious of science as late as Galileo’s time, c. 1600 (and indeed, later).

In the East, the energies of the Byzantine Empire during its last eight centuries were absorbed by its struggles with Islam2, and with the West.

Greek Orthodox Christianity was considered partly heretical by the Roman Catholic Church3, and Constantinople was sacked by Crusaders during the Fourth Crusade (1202-04). It never recovered, though it survived till 1453.4 1 Recall that Rome was sacked by Alaric the Visigoth in 410; 476 marked the final dissolution of the Roman Empire in the West, when Emperor Romulus Augustus was deposed by Odoacer, a Germanic chieftain.

2 Arabs initially, later the Ottoman Turks 3 The differences centred on the divinity of Christ, and came to a head over the presence or absence of the word filioque – and of the Son – in liturgy.

4 330-1453 – 1,123 years – is not bad for a human institution. When Hitler began WWII by invading Poland, he announced ‘The fighting that begins today decides the future of the German-speaking people for the next thousand years’. His ‘thousand-year Reich’ did not happen – but Constantinople did even better.

2 Byzantium contributed nothing original to mathematics (B 14.2); its only importance lay in its preservation of Greek texts, and commentaries on these, such as those of Proclus (B 11.13).

The ‘Golden Age of Islamic Mathematics’ resulted in the spread of Arab mathematic – algebra, trigonometry, astronomy etc. – from Baghdad and Persia in the East, along N. Africa, to Spain in the West. A particularly important centre of learning was Cordoba, which produced several noted astronomers (Dreyer, 262) and the Jewish scholar Maimonides (1135-1204 – see W for details). Contact between the Christian and Muslim worlds were mainly through Spain, Venice, Constantinople and Sicily (a Muslim province at that time).

Translation Toledo had been for centuries part of Muslim (Moorish) Spain (al-Andalus, hence Andalucia), but was reconquered by Christendom. The Archbishop was enlightened enough to approeciate the Islamic cultural heritage, and encouraged the translation of Arabic works into Latin. Toledo, which had excellent libraries and scholars of all religions and languages, became a centre for translation. In particular, the extensive mingling over long periods of time between Muslim, Jewish and Christian scholars did much to prepare the ground for the great flowering that marked the take-off point for European culture and learning – including mathematics and science – the Renaissance (below).

Adelard of Bath (c. 1075-1160).

Adelard (Athelhard) produced the first translation of Euclid’s Elements from Arabic to Latin in 1142 (Heath I, 362-3). A passage of Old English verse quoted by Heath puts the introduction of Euclid into England as far back as King Athelstan’s reign (924-939).5 Fibonacci, Leonardo of Pisa (c.1180-1250) (B 14.6-10) Leonardo of Pisa, son of Bonaccio (hence ‘Fibonacci’) wrote the Liber Abaci (Book of the Abacus) in 1202. This was the most influential European mathematical work before the Renaissance, and was the first such book to stress the value of the (Hindu-)Arabic numerals (Fibonacci had studied in the Muslim world and travelled widely in it).

The Fibonacci sequence u = (un )∞, 1,1,2,3,5,8,13,21,... (B 14.8) is genn=0 5 Athelstan (c.894-939) was the first King of England, bringing Northumbria, Mercia and Wessex together politically for the first time with his conquest of Viking York in

927. Offa, Alfred the Great and Athelstan are regarded as the three greatest Anglo-Saxon kings.

–  –  –

It occurs naturally in various problems of growth.

The Rise of Universities The academies (Pythagorean, Athenian, Alexandrian) of the ancient world played the role of universities in their day, as did their Arab counterparts in Baghdad, Cordoba etc. By the 12th C., the modern concept of a university as an autonomous academic institution awarding degrees, and as a centre of learning, teaching and research, began to emerge. This was a gradual process. The earliest continental universities are Bologna (founded 1088; Royal Charter 1130), Salamanca (founded 1134, from a Cathedral School, 1130;

Royal Charter 1218) and Paris (mid-12th C.). In Britain, the universities of Oxford and Cambridge simply describe themselves as ‘founded in the 12th C.’ and ‘founded in the 13th C.’ respectively. Scotland has four ancient universities: St. Andrews, 1410; Glasgow, 1451; Aberdeen, 1495; Edinburgh,

1583. Ireland has Trinity College, Dublin, 1592. London has UCL, 1826, and KCL, 1828-9; the University of Durham was founded in 1832 and the University of London (incorporating UCL and KCL) in 1836. Imperial College London was founded in 1907 (and left U. London in 2007).

Latin We have commented repeatedly on the adverse effect on mathematics of the Roman influence. Here at last we see the positive side of the Roman bequest to mathematics, and to science and learning generally: Latin. The Latin language was the common medium of communication among scholars from the Dark Ages on, through the rise of the universities, for several centuries,6 till the gradual replacement by (mainly) French, German and English. Thus Newton and Euler wrote in Latin, while the mathematicians of the French Revolution used French. Gauss began with Latin and ended with German. Scientific Latin survives today in biology (names of species and genera, e.g. in the classification by Carl Linnaeus (1707-1778), medicine (names of diseases), etc.7 6 Queen Elizabeth I of England once conducted a spirited political argument with the Polish Ambassador – in fluent Latin.

7 Latin was compulsory for entrance to Oxford and Cambridge in my day. I studied classical Latin for O Level (GCSE came later), and remember studying Newtonian Latin in the Scholarship Sixth (in those days, Oxbridge entrance was held at Christmas after A Level, so applicants had to at least begin a third year in the Sixth, now almost unknown).


Background The Renaissance, or re-birth, marks the end of the Dark Ages and the re-emergence of European culture dormant since the classical period. The history of mathematics, and science, will be primarily concerned with Europe (and later, its offshoots in America) from now on.

The Renaissance (the term dates back only to 1855, in Michelet’s Histoire de France) is a broad term covering the 14th-17th centuries, but is regarded as having begun in Florence in the 14th C. One factor here was the role of the Medici family, who were originally bankers before going into politics.

Banking and finance (of which more below) flourished on contact between civilisations, which tended to have a cross-fertilising effect; also, the Medici’s money enabled them to become great patrons of the arts. Later, the influx of Greek scholars after the Fall of Constantinople in 1453, bringing with them many texts and much learning, was also an important factor.

Perspective As we have seen, perspective was known (at least in part) in the ancient world, but was then lost.

Filippo Brunelleschi (1377-1446) discovered the main principle of perspective – the use of vanishing points – and convinced his fellow-artists of this in a famous experiament of 1420 involving the chapel outside Florence Cathedral.

Leon Battista Alberti (1404-72), Della pictura (1435, printed 1511) gave the first written account of perspective.

Piero della Francesca (1410-92), De prospectivo pingendi (c. 1478). In his book, and in his painting, Piero della Francesca did much to popularise perspective, which spread throughout the Western art world.

Leonardo da Vinci (1452-1519); Trattato della pittura. Leonardo is usually regarded as the personification of Renaissance genius. He was a prolific inventor, an artist who wrote on perspective, and a mathematician.

Albrecht D¨rer (1471-1528) of Nuremburg; Investigations of the measurement u with circles and straight lines of plane and solid figures (1525-1538, German and Latin). Like Leonardo, D¨rer was both a mathematician and an artist.

u He adopted perspective after visiting Italy.

Source: E. C. ZEEMAN, The discovery of perspective in the Renaissance (LMS Popular Lecture, 1983).

Printing and books An important turning-point was the invention of the printing-press. Movable type was introduced by Johannes Gutenberg (1395-1468) of Mainz in Germany (where the University is named after him), in his Bible of 1455. The first printed book in English (on the Trojan War) was produced by William Caxton in Bruges in 1476 (he then moved to London and established the first printing press in Britain). Printing and books enormously increased the scope for the rapid dissemination of knowledge.

Universities Learning was now dominated by the universities, focussing on mediaeval Latin for theology etc. and Latin translations from the Arabic in science.

Printing and books led to more emphasis on Greek culture, both in literature and in science. The Greek classics in both could now be read in the original, translated directly (from the Arabic), printed, and distributed widely.

Regiomontanus (1436-76) (B 15.2-5) Born Johann M¨ller of K¨nigsberg (the german city in East Prussia u o founded by the Teutonic Knights, now Kaliningrad in Russia), he was known as Regiomontanus (= king’s mountain in Latin, = K¨nigsberg in German), o Almagest. Regiomontanus completed a new Latin translation of Ptolemy’s Almagest (begun by Peuerbach), which – with its commentary – was mathematically superior to previous versions.

De triangulis omnimodis (1464). Probably influenced by the work of the Arab mathematician Nasir Eddin, this was the first major European work on trigonometry, and assisted in the evolution of trigonometry as a subject in its own right, independent of astronomy.

Book I: Solution of triangles; Book II: Sine rule; Book IV: Sine rule for spherical triangles.

Nicholas Chuquet (fl. c. 1500) Triparty en la science des nombres, 1484 (B 15.6): the most important European mathematical text since the Liber Abaci.

Part I: Hindu-Arabic numerals; addition, subtraction; multiplication; division; Part II: Surds; Part III: Algebra; laws of exponents; solution of equations.

Luca Pacioli (1445-1514) Summa de arithmetica, geometrica, proportioni et proportionalita, 1494 (B 15.7). This was an elementary text (more influential than the Triparty) on arithmetic, algebra and geometry. It is notable for double-entry book-keeping, and use of the decimal point.

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