Early Middle Ages
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- Early Middle Ages
- Great Schism
- Trivium
- Anthemius of Tralles
- Leonardo Pisano (of Pisa) called Fibonacci
- (chaplain to Pope): astronomy, Latin edition of Euclid’s Elements that became standard for 200 years 1274 Thomas Aquinas’ work “
- Lecture 2. Culture and Math in 14 th -15 th centuries
- Little Ice Age
- Religious Wars
- Math Ideas in 14
- Quattrocento: Italian Renaissance of 15 th cent.
Lecture 1. Education and Math in the Medieval Europe      
 
Boethius (480-524) noble Roman, Christian philosopher, “last of the Romans and the first of Scholastics” Magister officiorum (head of government and court services) of Theodoric, king of Italy and of Goths, who later imprisoned and executed Boethius in charges of conspiracy 
Consolation of Philosophy philosophical treatise composed in jail: on Weal of fortune, evil and death, etc., one of the most popular and influential works of the Middle Ages 
Translated many books from Greek to Latin: philosophy (Aristotle, Plato), Math (“Arithmetic” of Nicomachus, geometry books, etc.) were textbooks for quadrivium in the Middle Ages. Cassiodorus (485-585) Roman statesman and writer, was Magister officiorum after Boethius 
Established a system of monastery education (School at Vivarium) based on 7 Liberal Arts Through Early Middle Ages, schools at some monasteries and cathedrals were the only centers to get a minimal education based on Trivium and aimed for religious needs. There existed only a few places in Europe to get a higher education based on Quadrivium.  
Anthemius of Tralles (474-534) and Isidorus of Miletus mathematicians and architects hired by Emperor Justinian to construct a new church of Hagia Sophia at Constantinople in 532-37, after it was destroyed by the Nika Riot. Isidorus taught at the universities of Alexandria and Constantinople, produced the first comprehensive compilation of Archimedes' work. Arthemius studied the focal properties of conics.  
Alcuin of York (735-804) made York Cathedral School one of the most important European center of leaning, with the best library in Europe, in 775 writes elementary texts in arithmetic, geometry and astronomy, in 781 appointed head of Charlemagne’s Palace School at Aachen, developed Carolingian miniscule that replaced less readable unspaced capital script. Being a personal friend of Charlemagne became a teacher of his two sons.    
        
Peter Abelard 1079–1142 philosopher and the first major logician of the Middle Ages "the keenest thinker and boldest theologian of the 12th Century". His lectures in Paris were extremely popular and crowded. He was condemned by Church and the king of France forbade to teach on his land, so he was teaching sitting on a tree and then from a boat. His affair with and love for Héloïse d'Argenteuil has become legendary.
 
Leonardo Pisano (of Pisa) called Fibonacci (son of Bonacci family) 1170-1250 Liber Abaci (Book of Calculation) 1202: Hindo-Arabic numerals Modus Indorum (Method of Indians) Many arithmetical and algebraic examples: a problem about rabbits leaded to the Fibonacci sequence 1,2,3,5,8,13,… problems about perfect numbers, Chineese remainder theorem, sum of arith/geom series Practica geometriae 1220: geometry problems based on Euclid’s Elements Flos 1220: approximation to a root of 10x+2x2+x3=20 (asked by Johannes of Palermo, problem from Omar Khayyam’s book) Liber Quadratorium 1225 the most impressive work: 1+3+5+…; on Pythagorean triples; both x2+y2 and x2-y2 cannot be squares at the same time; x4-y4 cannot be a square; (a2+b2)(c2+d2)=(ab+cd)2+(ad-bc)2 
  
      
  
Lecture 2. Culture and Math in 14th-15th centuries   
    
  
Nicole Oresme (1320-1382) most original thinker of XIV: philosopher, economist, astronomer, mechanic and math Introduced an idea of graph of function, named coordinates lattitudo and longitude; for the graph of speed as function of time realized that the distance is the area; introduced an average speed. Proved divergence of a harmonic series; used fractional powers of numbers. Discussed possibility of rotation of Earth; incommensurability of lengths of day/month/year in a book of 1382 Le Livre du ciel et du monde (The Book of Heaven and Earth) he translated it from Latin to French by request of king Charles V
      
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