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500 600 700 800 900 1000 1100 1200 1300 1400 Charlemagne  The Crusades  Black Death Oxford Cambridge Thomas Bologna Aquinas Paris Medieval Times

• Early Middle Ages –Dark ages (c.450–750): – Medieval Europe was a large geographical region divided into smaller and culturally diverse political units that were never totally dominated by any one authority. – Feudalism – No traveling – No new math to get excited about – Gregorian Chants! Boethius (480 – 524)

• Boethius became an orphan when he was seven years old. • He was extremely well educated. • Boethius was a philosopher, poet, mathematician, statesman, and (perhaps) martyr. Boethius (480 – 524)

• He is best known as a translator of and commentator on Greek writings on logic and mathematics (Plato, Aristotle, Nichomachus). • His mathematics texts were the best available and were used for many centuries at a time when mathematical achievement in Europe was at a low point. • Boethius’ Arithmetic taught medieval scholars about Pythagorean number theory. Boethius (480 – 524)

• His writings and translations were the main works on logic in Europe becoming known collectively as Logica vetus. • One of the first musical works to be printed was Boethius's De institutione musica, written in the early sixth century. • It was for medieval authors, from around the ninth century on, the authoritative document on Greek music‐theoretical thought and systems. For example, Franchino Gaffurio in Theorica musica (1492) acknowledged Boethius as the authoritative source on music theory. Medieval Times

• (Middle) Middle Ages (750 ‐ 1000 CE) • The Church became the standard bearer of civilization and education • The Papacy became the most important secular power. • Monasteries became places where ancient learning was preserved. • Carolingian Empire (Charlemagne) Names from Carolingian Empire

• Pippin III • Carloman • Charles the Great –Charlemagne – Alcuin • Louis the Pious • Charles the Bald • Charles the Fat Medieval Times

• High Middle Ages (1000‐ 1300 CE) • The Church was the unifying institution • The Crusades • Church developed universities – Universities of Paris, Oxford, and Cambridge were founded • Scholasticism (Thomas Aquinas) Medieval Times

• Crusades – Brought Europe into contact with the Arabic world and the Greek writings which had been preserved, as well as the work of the Arabic mathematicians and astronomers. • Scholasticism – incorporated the Greek philosophies into the Church. Thomas Aquinas particularly liked Aristotle. Thomas Aquinas (1225 – 1272)

• St. Thomas Aquinas was an Italian philosopher and theologian, Doctor of the Church, known as the Angelic Doctor. • He is the greatest figure of scholasticism ‐ philosophical study as practiced by Christian thinkers in medieval universities. Thomas Aquinas (1225 – 1272)

• He is one of the principal saints of the Roman Catholic Church, and founder of the system declared by Pope Leo XIII to be the official Catholic philosophy. • St. Thomas Aquinas held that reason and faith constitute two harmonious realms in which the truths of faith complement those of reason; both are gifts of God, but reason has an autonomy of its own. Leonardo of Pisa (1170‐1250)

• Also known as Fibbonaci (from filius Bonaccia, “son of Bonnaccio”) • The greatest mathematician of the middle ages. • He traveled widely in the Mediterranean with his father while young. • Studied under a Muslim teacher. Fibonacci

• Published Liber abaci (Book of the abacus) in 1202. But it isn’t about abaci. • Summarizes Arabic arithmetic knowledge, and explains the merits of the “Hindu‐Arabic” number system, the “nine Indian figures” together with a 0, or zephirum*, in Arabic.

* Root of both zero and cipher. Liber abaci

• After summarizing arithmetic knowledge, lays out a series of problems, including commercial transactions, exchanges of currency, etc. • Uses fractions, but curiously does not use decimal fractions. Instead, he uses common fractions, sexagesimal fractions, and unit fractions, particularly sums of unit fractions in the Egyptian style. Liber abaci

• Has tables to convert common fractions to unit fractions, e.g. , (the +’s are implied by juxtaposition here). • He also used another strange notation for fractions: . ∙∙ ∙ • This led to the following kinds of headache‐ inducing reading: Liber abaci

• If of a rotulus is worth of a bizantium, then of a bizantium is worth of a rotulus.

• I hope that clears up the rotulus – bizantium question. Liber abaci

• There were some interesting problems included as well: – Seven old women went to Rome; each woman had seven mules; each mule carried seven sacks; each sack contained seven loaves; and with each loaf there were seven knives; each knife was put up in seven sheathes. . . . – Ahmes would have been proud. Liber abaci

• The most famous problem in the book: – A certain man put a pair of rabbits in a place surrounded on all sides by a wall. How many pairs of rabbits can be produced from that pair in a year if it is supposed that every month each pair begets a new pair which from the second month on becomes productive? • The solution gives rise to the celebrated Fibonacci sequence. Liber abaci

• Other types of problems in the third section of Liber abaci include: – A spider climbs so many feet up a wall each day and slips back a fixed number each night; how many days does it take him to climb the wall? – A hound whose speed increases arithmetically chases a hare whose speed also increases arithmetically. How far do they travel before the hound catches the hare? – Calculate the amount of money two people have after a certain amount changes hands and the proportional increase and decrease are given. Liber abaci

• There are also problems involving perfect numbers, the Chinese remainder theorem and problems involving the summing of arithmetic and geometric series. • In the fourth section, he deals with irrational numbers both with rational approximations and with geometric constructions. Fibonacci

• Also wrote other books, including • Practica geometriae, • Flos, and • Liber quadratorum. Practica geometriae

• Contains a large collection of geometry problems arranged into eight chapters with theorems based on Euclid's Elements and On Divisions. • Includes practical information for surveyors, including a chapter on how to calculate the height of tall objects using similar triangles. • Included is the calculation of the sides of the pentagon and the decagon from the diameter of circumscribed and inscribed circles, as well as the inverse calculation. Flos

• Johannes of Palermo, a member of the Holy Roman emperor Frederick II's court, presented a number of problems as challenges to Fibonacci. • Fibonacci solved three of them and put his solutions in Flos. Flos

• One of these problems Johannes of Palermo took from Omar Khayyam's algebra book where it is solved by means of the intersection a circle and a hyperbola. • It is to give an accurate approximation to a root of 10x + 2x2 + x3 = 20. • Fibonacci proves that the root of the equation is neither an integer nor a fraction, nor the square root of a fraction. Flos

• Without explaining his methods, Fibonacci then gives the approximate solution in sexagesimal notation as 1;22,7,42,33,4,40.

• This converts to the decimal 1.3688081075 which is correct to nine decimal places, a remarkable achievement. Liber Quadratorum

• Liber quadratorum, written in 1225, is Fibonacci's most impressive piece of work, although not the work for which he is most famous. • The book's name means the book of squares and it is a number theory book which, among other things, examines methods to find Pythogorean triples. Liber Quadratorum

• In Liber Quadratorum Fibonacci also gave examples of cubic equations whose solutions could not be rational numbers. Liber Quadratorum

• Example: Find rational numbers x, u, v satisfying: . • Fibonacci’s solution involved finding three squares that form an arithmetic sequence, say so d is the common difference. Then let . • One example would be Then . Liber Quadratorum

• Finally, Fibonacci proved that all Pythagorean triples a, b, c with can be obtained by letting , for s and t positive integers. • It was known in Euclid’s time that this method would always general Pythagorean triples; Fibonacci showed that it in fact produced all Pythagorean triples. Liber Quadratorum

• You can now create Pythagorean Triples to amaze your friends and confuse your enemies: stabc 12435 1 4 8 15 17 1 5 10 24 26 1 6 12 35 37 2312513 2 4 16 12 20 2 5 20 21 29 2 6 24 32 40 3424725 3 5 30 16 34 3 6 36 27 45 3 7 42 40 58 4540941 4 6 48 20 52 18 174 6264 29952 30600 Fibonacci Sequence

• Although Fibonacci is justifiably famous for this sequence, his Liber abaci was not its first appearance. The earliest known appearance of this sequence was in the work of the Sanskrit grammarian Pingala, sometime between 450 and 200 BCE. • He was studying the number of meters of a given overall length could be made with the Long (L) and short (S) vowels used in syllables (with the long vowel twice as long as the short). Pingala

• To get a meter of length n, you could add a short syllable S to a meter of length n‐1, or a long syllable L to a meter of length n‐2. Adding these two gave you the total meters of length n¸ and since it started out with: S SS, L SSS, LS, SL, it is easy to see that the recurrence relations produced what we know as the Fibonacci sequence.