Maths in Europe by 1200

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Maths in Europe by 1200
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• After the falling of the Roman Empire, Europe was
  immersed in the chaos and anarchy, mathematically
  speaking.
• Europeans only began to wake up themselves from
  the intellectual slumber (sleep peacefully or
  comfortably) of the Dark Ages as they came in
  contact with Arab civilization, mostly in Spain.
• By 1200 the situation started to change.

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• And now, for something completely different
• some of the most relevant mathematicians of this
  time

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Gerbert of Aurillac (940-1003)

• Became Pope and was known as Silvester II.
• Studied in Spain, where he learned the Indian
  numerals (but no zero).
• He wrote on arithmetic and geometry.
• One of the most difficult problems in his
  Geometry was the following
• Find x, y such that
• and
• This would have been an easy exercise for
  Babylonian scribe!

Mathematical writings Libellus de numerorum
divisione,
De geometria,
Epistola ad Adelbodum,
De sphaerae
constructione and
Libellus de rationali et ratione
uti.
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• Contemporary with Gerbert was another
  mathematician and church-person,
• Hrotsvita of Saxony (940-1002),
• who had an interest in perfect numbers.

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Adelard of Bath (1075-1160)

• Was an expert in the Arabic language.
• Made three different latin translations of
  Euclids Elements from Arabic sources.
• Translated al-Khwarizmis astronomical and
  trigonometrical tables.
• Wrote arithmetic books.
• One of his most famous books is Quaestiones
  naturales, which consists of 76 scientific
  discussions based on Arabic science.

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Abraham Ben Ezra (1095-1160)

• was born at Toledo and was called The Wise, The
  Great and The Admirable Doctor.
• was one of the most distinguished Jewish men of
  letters and writers of the Middle Ages.
• excelled in philosophy, astronomy/astrology,
  medicine, poetry and linguistics.
• also wrote various works on mathematical
  subjects.

Main works about maths, astronomy and
astrology Sefer ha-E?ad, on the peculiarities of
the numbers 1-9. Sefer ha-Mispar, explained the
Hindu arithmetic, using Hebrew letters for
numerals, with a zero added. Lu?o, astronomical
tables. Sefer ha-'Ibbur, on the calendar. Keli
ha-Ne?oshet, on the astrolabe.
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Gherard of Cremona (1114-1187)

• Travelled to Toledo to learn Arabic. He became
  the most important translator of his time with
  more than 80 works.
• Was the first to translate Ptolemys Almagest.
• Translated Al-Kwarizmis first book on
  algorithms Hisab al-Yabrwal-Mukabala.

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Johannes de Sacrobosco (1195-1256)

• Was an English scholar who wrote an important
  text on astronomy and promoted Arabic methods of
  arithmetic and algebra.
• In De Algorismo, discussed calculating with
  positive integers.
• His most important book is Tractatus de Sphaera
  (the main book on astronomy until 17th century).
  There, he discussed
• a) The shape and place of the Earth within a
  spherical universe.
• b) Various circles on the sky.
• c) introduction to Ptolemy's theory of the
  planets and of eclipses

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Jordanus Nemorarius (1225-1260)

• His name does not appear in any list of clerics
  so it is generally assumed that he was a layman.
• wrote about triangles, circles, regular polygons,
  Arabic numerals, primes, perfect numbers,
  polygonal numbers, ratios, powers and
  progressions.
• was the first mathematician to correctly
  formulate the law of the inclined plane.
• His writings on geometry were important for
  explorers who relied on the astrolabe for
  navigation.

• used letters in place of numbers in his books on
  mathematics, and was able to articulate general
  algebraic theorems in this manner. However, his
  system of algebraic notation was only a distant
  antecedent to the algebra that is used today.

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• There are six mathematical treatises written by
  Jordanus
• - The Demonstratio de algorismo, which gives a
  practical explanation of the Arabic number
  system. It deals only with integers and their
  uses.
• - Demonstratio de minutiis deals with fractions.
• - De elementis arithmeticae artis , a theoretical
  work on arithmetic which became the standard
  source of Middle Ages texts.
• - Liber phylotegni de triangulis which is an
  excellent example of a Middle Ages Latin geometry
  text.
• - The Demonstratio de plana spera is a
  specialised work on geometry which studies
  stereographic projection.
• - Perhaps the most impressive of all is the De
  numeris datis which is the first advanced algebra
  to be written in Europe after Diophantus.
• In this treatise he discusses problems of
  the following sort
• find x and y such that
• 
  and
  
• And another of this problems

This is the sort of problem the ancient
Mesopotamians were goot at.
If a given number is separated into two parts
such that the product of the parts is known, then
each of the parts can be found.
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Ramon Llull (1235-1316)

• Llull used logic and mechanical methods involving
  symbolic notation and combinatorial diagrams to
  relate all forms of knowledge. This work makes
  him a precursor of combinatorics.

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William of Ockham
(1288-1348)

• Was an English mathematician who worked on logic.
  
• In Summa Logicae, he considered a three valued
  logic where propositions can take one of three
  truth values.
• He also came very close to stating De Morgans
  laws

Binary logic (Aristotle)
Ternary logic (Ockham)
Fuzzy logic (19-20th century)
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• The mathematician who became to change the
  situation of European mathematics was.
• Fibonacci
• but this is a theme for other exposition.