Middle Age in Europe

1
1
Middle Ages in Europe
The Dark and Middle Ages in Europe cover the period from 476 to 1400 CE. After
the collapse of the western Roman Empire in 476 CE, the Dark Ages set in, and
cultural studies in all areas disappeared. Europe was divided between the
Visigoths from north Germany, the Vandals from north Africa, the Franks who
ruled France and part of Germany as well as the Saxons from Germany who ruled
Britain. Central governments were replaced by small feudal districts and city life
deteriorated. Pope Leo III separated the Roman Church from the Byzantine
Church. He also made Charlemagne Holy Roman Emperor in 800, temporarily
unifying Europe. From 814 to 1042 there were Viking invasions of coastal Europe,
England and Russia.
European society was organized under the Feudal System. There were three
estates, or classes: nobles, priests and serfs. The economy was based on
agriculture which was run through small manors. The serfs did the work and were
protected by a hierarchy of nobles who did the fighting. Crusades from 1096 to
1260 resulted in the capture of Israel, Lebanon and Syria from the Moslems who
recaptured this territory and drove out the European invaders. The Crusades led
to the growth of trade. Towns developed through fairs, and trades were organized
into guilds. Their inhabitants were free men. Plagues were frequent. The worst
one, the Black Death, killed over half of Europe’s population in the 1340s.
2
The Magyar invasion of western Europe ended in 955 with their defeat by Otto the
Great at the Battle of Lechfeld. In 1241 the Mongols conquered eastern Europe but
withdrew to fight over the succession when Ogedei Khan died. The most notable
example of the constant fighting in Europe began with the conquest of England by
William of Normandy in 1066. By 1152, Henry III controlled England and 2/3 of
France. However, in the Hundred Years War (1362 – 1453) England was driven out
of France.
The only vestiges of intellectual life in the Dark Ages were in Church monastery
schools based on the quadrivium, four elementary texts written by the Roman
Boethius ca 500 CE on arithmetic, geometry, astronomy and music. For example,
there is a ten times table in the arithmetic text and the geometry text does not
include the Pythagorean theorem which became unknown in Europe. Basically, no
European knew any significant mathematics and there were no copies of Greek
mathematical works in any language. Arithmetic only involved integers and the four
basic operations. Division was done ad hoc, there were no algorithms. Fractions
were rarely used, and irrational numbers were never mentioned. Good calculators
were called practitioners of the black art (magic). The lack of interest in mathematics
until 1100 resulted from a religious emphasis on the spiritual with no interest in the
physical world. This originated with St. Augustine (Italy, 4th century CE):
“Whatever knowledge man has acquired outside of Holy Writ, if it be harmful
it is therefore condemned; if it be wholesome it is there contained.”
Moslems had a similar attitude with regard to the Koran during their initial conquests,
but this attitude changed rapidly to encourage secular learning.
2
3
From 1100 CE onwards is called the Middle Ages. European society became stable,
towns grew and industry began. There were contacts with Moslems and the
Byzantine Empire from growing trade, the Crusades (1096 to 1260 CE) and the long
slow conquest of Spain. Europeans heard about Greek works and Arabic texts.
Over a 200 year period many of these texts were translated into Latin. (See the table
on page 327 of the text.) Many Arabic texts were translated by Jews in Spain from
Arabic to Spanish and then by Christians from Spanish to Latin. However, it took
hundreds of years to assimilate the information in these difficult texts since there
were no competent European teachers. In particular, the Indian-Islam base ten
decimal number system was slow to replace Roman numerals.
European intellectual life eventually developed at universities after the Renaissance.
The university at Bologna was founded in 1088 and the universities at Paris, Salerno,
Oxford and Cambridge were established shortly before 1200. They were initially
dominated by the Church with no academic freedom. The curriculum consisted of the
trivium (logic, grammar, rhetoric) which focused on Aristotle and on Boethius’
quadrivium. Excerpts from Euclid, Ptolemy’s Almagest and methods for practical
calculations were also studied. The standard scholarly language in Europe was Latin
which allowed communication across the continent. From the Middle Ages through
the Renaissance this university learning, called scholasticsm, was a blend of
Christian theology and Greek ideas, based on the philosophy of St. Thomas Aquinas
(1225 – 1274). It focused on certain Greek texts and the Bible while rejecting
experimentation and observation. Church doctrine and Aristotle were accepted as
absolute truths.
4
Roger Bacon (1214 – 1294) argued against blindly accepting Aristotle’s theories in
science and was placed under house arrest for a couple of years. He was the
forerunner of empiricism and the scientific method, approaches which developed in
the renaissance.
The main deterrents to scholarly activity in the Middle Ages were:
1. the large number of small states;
2. the lack of availability and high cost of texts which were hand written in Latin;
3. Church control and censorship;
4. the large loss of life and assets in frequent wars and the Crusades;
5. the Black Death which killed over half the population of Europe in the 1340s.
There were two areas of progress in the Middle Ages. Jordanus de Nemore
(ca 1220) learned math by reading Latin translations of Arabic texts. He wrote the
Arithmetica in Latin using Euclid’s style. It included Euclid’s results on algebra and
number theory as well as more recent algebraic results and some original ones. He
ignored the Islam approach, rejecting irrational numbers and using Euclid’s
distinction between numbers and magnitudes. The Arithmetica also marked the first
appearance of Pascal’s triangle in Europe. Jordanus was the first person since
Diophantus to use symbols in algebraic computations. He was also the first to use
more than one variable but used words for operations and Roman numerals for
numbers. This notation developed in the Renaissance into modern mathematical
algebraic notation.
3
5
The interpretation of ratios as fractions was begun by Thomas Bradwardine (1295 –
1349) at Merton College of Oxford. However, he followed the Greek viewpoint of
only allowing fractions of quantities of the same kind. In particular, he would not
allow the statement “velocity equals distance divided by time”. He did, however,
present the modern procedures for multiplying and dividing fractions. Nicole
Oresme (1320 – 1382) of the University of Paris developed rules which we would
interpret as doing arithmetic with fractional exponents.
The only significant new mathematics of the Middle Ages was the study of
kinematics in the 14th century. It began in Merton College of Oxford in 1335 with
William Heytesbury’s definition of instantaneous velocity and his proof of the mean
speed rule for a body moving with constant acceleration:
s = ½ (vi + vf)(tf – ti ).
Following Aristotle, velocity, being a magnitude, was depicted by a line segment.
In the 1350s Nicole Oresme carried these ideas further. His advances include:
(1) basic formulas for motion under constant acceleration;
(2) the depiction of velocity as a function of time;
(3) the graph of the velocity function;
(4) the identification of the area under a linear velocity graph as the distance
traveled.
6
Example 1 (a) Oresme’s depiction of of v(t) the velocity of an object at time t.
(b) Oresme’s graph of an object moving with constant velocity.
(c) Oresme’s graph of an object moving with constant acceleration.
Oresme proves the mean speed rule from the velocity graph in (c). The area
under this graph is the area of the trapezoid with bases vi, vf and height tf – ti
which is ½ (vi + vf)(tf – ti). This area is the distance traveled under constant
acceleration.
Note Oreseme’s early use of functions and their graphs. Oresme’s ideas were
largely ignored until they reappeared 200 years later in the kinematics of
Galileo, 275 years later in the analytic geometry of Fermat, Descartes and
300 years later in Newton’s Fundamental Theorem of Calculus.
(a) (b) (c)
t
v(t) v
ti tf ti tf tf – ti
vi
vf
4
7
In addition, Oresme initiated the first advances in infinite series since Archimedes.
He summed geometric series and a few nongeometric series. He also showed that
the harmonic series diverges. This was the first example of a divergent series whose
terms have limit zero.
Example 2 The geometric series with initial term A/K and ratio 1 – 1/K, for K ≥ 1, has
sum A.
Example 3 The harmonic series diverges.
Solution Oresme observed that the 2nth partial sum is greater than ½ plus the 2n–1th
partial sum. Hence the limit of the increasing sequence of partial sums is infinite.
Example 4 (Richard Swineshead, Merton College,1350) The series has sum 2.
Conclusion
In 1300 C.E. China, India, the Islam world and Europe were at approximately the same
mathematical level. Why did modern mathematics develop in Europe? There were
inhibiting forces in all four cultures.
In China, mathematics was dominated by the government’s emphasis on training
bureaucrats to efficiently perform standard algorithms. Consequently there was no
incentive to innovate. Individuals with original ideas were isolated.

n1 2
n
n
8
In India, mathematical activity declined after 1200. They emphasized calculation,
rather than theory, calling mathematics ganita, the science of calculation. Kline,
citing the Persian historian al-Biruni ca. 1000, claims Indians lacked mathematical
values. They gave equal value to their own innovations and to crude outdated
Babylonian and Egyptian methods. A simpler and fairer explanation may be that
the violence of Mongol invasions and the conflicts between Hindus and Moslems
created a poor atmosphere for mathematical innovation.
In the Islam world, mathematics was encouraged, flourished and developed until
1300 through the encouragement of religious leaders and civil rulers. Then they
began to fear science as possibly subversive to Islam and distinguished between
“foreign sciences” and “religious sciences”. Unfortunately, mathematics was
classified as a foreign science. Consequently, government support and popular
interest fell dramatically.
In Europe, there was no tradition of mathematics and no government support for
its development. In addition, the Church discouraged new ideas as subversive.
However wealth, accumulated from trade, financed and encouraged an intellectual
revolution, called the Renaissance, which laid the basis for mathematical
breakthroughs. Simultaneously, the Catholic Church gradually lost its influence
culminating in the 16th century Reformation with the disintegration of the Church
into Protestant sects in northern Europe and Catholic countries in southern
Europe.