Maths in Europe by 1200. After the falling of the Roman Empire, Europe was immersed in the chaos and anarchy, mathematically speaking. Europeans only.

Maths in Europe by 1200

• 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.

And now, for something completely different…

…some of the most relevant mathematicians of this time…

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!

222 ayx bxy 2

1

Mathematical writings: Libellus de numerorum divisione,

De geometria,

Epistola ad Adelbodum,

De sphaerae constructione and

Libellus de rationali et ratione uti.

Contemporary with Gerbert was another mathematician and church-person, Hrotsvita of Saxony (940-1002),

who had an interest in perfect numbers.

Adelard of Bath (1075-1160)

• Was an expert in the Arabic language.

• Made three different latin translations of Euclid’s Elements from Arabic sources.

• Translated al-Khwarizmi’s 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.

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-Eḥad, on the peculiarities of the numbers 1-9.

Sefer ha-Mispar, explained the Hindu arithmetic, using Hebrew letters for numerals, with a zero added.

Luḥo, astronomical tables.

Sefer ha-'Ibbur, on the calendar.

Keli ha-Neḥoshet, on the astrolabe.

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 Ptolemy’s Almagest.

• Translated Al-Kwarizmi’s first book on algorithms: Hisab al-Yabrwa’l-Mukabala.

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

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.

• 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:

10 yx 5822 yx

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. “

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.

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 Morgan’s laws

Binary logic (Aristotle)

Ternary logic (Ockham)

Fuzzy logic (19-20th century)

The mathematician who became to change the situation of European mathematics was….

Fibonacci…but this is a theme for other exposition.