MATH 4400, History of Mathematics
Lecture 5: From the Roman Empire to the first universities
Professor: Peter Gibson
[email protected]
http://people.math.yorku.ca/pcgibson/math4400
October 11, 2016
The Roman Empire
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Roman Empire 117AD
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Roman Empire 271AD
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German invasions
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The Byzantine empire
In the fourth century AD, the Roman emperor Constantine moved the
capital from Rome to Constantinople and legalized Christianity, which, by
the end of the century became the official religion of the Empire.
In the 7th century AD the official language of the imperial administration
changed from Latin to Greek.
The continuation of the Roman Empire in the east, centred in
Constantinople, is usually referred to as the Byzantine Empire, which
lasted until the 15th century. It is characterized by use of the Greek
languange and by Orthodox Christianity.
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Byzantine Empire 555 AD
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The Islamic era
The Islamic conquest
Expansion under the Prophet Mohammad, 622-632
Expansion during the Patriarchal Caliphate, 632-661
Expansion during the Umayyad Caliphate, 661-750
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The principal cities of the Islamic world were:
Baghdad
Cairo
These were centres of political administration and intellectual high culture.
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The re-emergence of Western Europe
We shall focus on Western Europe in the 12th and 13th centuries, the
High Middle Ages. This is the era during which the first universities were
established. The main centers of culture and learning lay elsewhere, under
Arab and Byzantine dominion. By the standards of other eras, very little
new mathematics was developed at this time. The works of Leonardo of
Pisa stand out; but even they are reflective of a world influenced by the
east.
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Historical context
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Europe was an agglomeration of small principalities and city states, with
loose larger-scale alliances.
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Europe was an agglomeration of small principalities and city states, with
loose larger-scale alliances.
The Western European economy was mainly feudal or manorial.
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Europe was an agglomeration of small principalities and city states, with
loose larger-scale alliances.
The Western European economy was mainly feudal or manorial.
The most sophisticated European cities, such as Venice, thrived on trade
with the Arab world, and central and East Asia.
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Europe was an agglomeration of small principalities and city states, with
loose larger-scale alliances.
The Western European economy was mainly feudal or manorial.
The most sophisticated European cities, such as Venice, thrived on trade
with the Arab world, and central and East Asia.
The Arab and Byzantine worlds held the main cultural centers.
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Demographics circa 1200 reflect a starkly different world from today.
(Figures are approximate!)
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Demographics circa 1200 reflect a starkly different world from today.
(Figures are approximate!)
Place
Germany
France
British Isles
Italy
Spain & Portugal
Europe
World
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Population
7.3M
12M
3.2M
8M
7M
68M
360M
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More demographics...
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More demographics...
Place
London
Paris
Rome
Constantinople
Cairo
Baghdad
Fez
Beijing
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Population
22K
110K
20K
200K
225K
250K
200K
130K
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The first universities
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The first universities
Place
Bologna
Oxford
Salamanca
Paris
Cambridge
Padua
∼Year
1088
1096
1134
1150
1209
1222
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The first universities
Place
Bologna
Oxford
Salamanca
Paris
Cambridge
Padua
∼Year
1088
1096
1134
1150
1209
1222
Cambridge and Padua were offshoots of Oxford and Bologna, respectively.
Later, in the 15th century, there was a profusion of new universities.
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Later, in the 15th century, there was a profusion of new universities.
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Some charateristic features of early universities...
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Some charateristic features of early universities...
independent legal status (town vs. gown)
latin language
scholasticism
the trivium and quadrivium
law (civil law, canon law), theology, arts and medicine
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The following is attributed to Boethius (c. 600)...
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The following is attributed to Boethius (c. 600)...
trivium
I
I
I
grammar
logic
rhetoric
quadrivium
I
I
I
I
arithmetic
geometry
music
astronomy
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Leonardo of Pisa (1180-1250)
Wrote Liber abaci (1202), Flos (1225).
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Leonardo of Pisa (1180-1250)
Wrote Liber abaci (1202), Flos (1225).
Leonardo—also known as Fibonacci—had a Muslim teacher, and had
travelled to Egypt, Syria and Greece. His father was a merchant.
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Leonardo of Pisa (1180-1250)
Wrote Liber abaci (1202), Flos (1225).
Leonardo—also known as Fibonacci—had a Muslim teacher, and had
travelled to Egypt, Syria and Greece. His father was a merchant.
The mathematics of Fibonacci’s works is awkward. There are complicated
methods to convert currencies, and illustrations of decomposition into unit
fractions.
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Leonardo of Pisa (1180-1250)
Wrote Liber abaci (1202), Flos (1225).
Leonardo—also known as Fibonacci—had a Muslim teacher, and had
travelled to Egypt, Syria and Greece. His father was a merchant.
The mathematics of Fibonacci’s works is awkward. There are complicated
methods to convert currencies, and illustrations of decomposition into unit
fractions.
99
1 1 1
1
= + + +
100
2 4 5 25
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Leonardo of Pisa (1180-1250)
Wrote Liber abaci (1202), Flos (1225).
Leonardo—also known as Fibonacci—had a Muslim teacher, and had
travelled to Egypt, Syria and Greece. His father was a merchant.
The mathematics of Fibonacci’s works is awkward. There are complicated
methods to convert currencies, and illustrations of decomposition into unit
fractions.
99
1 1 1
1
= + + +
100
2 4 5 25
Fibonacci is credited with promoting the use of Hindu-Arabic numerals.
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Fibonacci’s most famous problem...
How many pairs of rabbits will be produced in a year, beginning
with a single pair, if in every month each pair bears a new pair
which becomes productive from the second month on?
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