MATH 4400, History of MathematicsHistorical Context

Peter Gibson

October 1, 2019

Some review of last lecture

The political authority of the Golden Age was fragile, and in 338BC Philipof Macedon conquered much of Greece.

His son, Alexander, succeeded him in 336BC.

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Some review of last lecture

The political authority of the Golden Age was fragile, and in 338BC Philipof Macedon conquered much of Greece.

His son, Alexander, succeeded him in 336BC.

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Upon Alexander’s death in 323BC, his empire was divided amongst hisgenerals,

leading to the Hellenistic period.

Some mathematicians from the period after Alexander:

Euclid of Alexandria (323-285BC)

Archimedes of Syracuse (287-212BC)

Appolonius of Perga (262-190BC)

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Upon Alexander’s death in 323BC, his empire was divided amongst hisgenerals, leading to the Hellenistic period.

Some mathematicians from the period after Alexander:

Euclid of Alexandria (323-285BC)

Archimedes of Syracuse (287-212BC)

Appolonius of Perga (262-190BC)

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Upon Alexander’s death in 323BC, his empire was divided amongst hisgenerals, leading to the Hellenistic period.

Some mathematicians from the period after Alexander:

Euclid of Alexandria (323-285BC)

Archimedes of Syracuse (287-212BC)

Appolonius of Perga (262-190BC)

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Upon Alexander’s death in 323BC, his empire was divided amongst hisgenerals, leading to the Hellenistic period.

Some mathematicians from the period after Alexander:

Euclid of Alexandria (323-285BC)

Archimedes of Syracuse (287-212BC)

Appolonius of Perga (262-190BC)

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Upon Alexander’s death in 323BC, his empire was divided amongst hisgenerals, leading to the Hellenistic period.

Some mathematicians from the period after Alexander:

Euclid of Alexandria (323-285BC)

Archimedes of Syracuse (287-212BC)

Appolonius of Perga (262-190BC)

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Upon Alexander’s death in 323BC, his empire was divided amongst hisgenerals, leading to the Hellenistic period.

Some mathematicians from the period after Alexander:

Euclid of Alexandria (323-285BC)

Archimedes of Syracuse (287-212BC)

Appolonius of Perga (262-190BC)

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Archimedes

Archimedes lived from 287-212 BC, during the Hellenistic period.

The age in which he lived was marked by the rise of Rome as a regionalpower, and by the first and second Punic Wars.

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Archimedes

Archimedes lived from 287-212 BC, during the Hellenistic period.

The age in which he lived was marked by the rise of Rome as a regionalpower, and by the first and second Punic Wars.

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The punic wars (264-146 BC) pitted Rome against Carthage.

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The punic wars (264-146 BC) pitted Rome against Carthage.

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During the Second Punic War Carthage was led by Hannibal.

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During the Second Punic War Carthage was led by Hannibal.

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Archimedes himself was killed by a Roman soldier during the seige ofSyracuse.

He left behind numerous works, including

On the Equilibrium of Planes

On the Measurement of a Circle

On Spirals

On Floating Bodies

The Method of Mechanical Theorems

He is also credited with numerous mechanical inventions.

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Archimedes himself was killed by a Roman soldier during the seige ofSyracuse.

He left behind numerous works, including

On the Equilibrium of Planes

On the Measurement of a Circle

On Spirals

On Floating Bodies

The Method of Mechanical Theorems

He is also credited with numerous mechanical inventions.

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Archimedes himself was killed by a Roman soldier during the seige ofSyracuse.

He left behind numerous works, including

On the Equilibrium of Planes

On the Measurement of a Circle

On Spirals

On Floating Bodies

The Method of Mechanical Theorems

He is also credited with numerous mechanical inventions.

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The screw of Archimedes:

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Transmission of Archimedes worksSome of Archimedes works have been rediscovered relatively recently.

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Transmission of Archimedes worksSome of Archimedes works have been rediscovered relatively recently.

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The Archimedes Palimpsest contains the only known version of his Methodof Mechanical Theorems, along with other previously known works.

The former is a letter, written to Erastosthenes of Alexandria, a famouscontemporary of Archimedes.

Translations of all Archimedes’ known works are freely available.

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The Archimedes Palimpsest contains the only known version of his Methodof Mechanical Theorems, along with other previously known works.

The former is a letter, written to Erastosthenes of Alexandria, a famouscontemporary of Archimedes.

Translations of all Archimedes’ known works are freely available.

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The Archimedes Palimpsest contains the only known version of his Methodof Mechanical Theorems, along with other previously known works.

The former is a letter, written to Erastosthenes of Alexandria, a famouscontemporary of Archimedes.

Translations of all Archimedes’ known works are freely available.

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ON THE EQUILIBKIUM OF PLANES

OR

THE CENTKES OF GEAVITY OF PLANES.

BOOK I.

"I POSTULATE the following:

1. Equal weights at equal distances are in equilibrium,and equal weights at unequal distances are not in equilibriumbut incline towards the weight which is at the greater distance.

2. If, when weights at certain distances are in equilibrium,

something be added to one of the weights, they are not in

equilibrium but incline towards that weight to which the

addition was made.

3. Similarly, if anything be taken away from one of the

weights, they are not in equilibrium but incline towards the

weight from which nothing was taken.

4. When equal and similar plane figures coincide if appliedto one another, their centres of gravity similarly coincide.

5. In figures which are unequal but similar the centres of

gravity will be similarly situated. By points similarly situated

in relation to similar figures I mean points such that, if straightlines be drawn from them to the equal angles, they make equal

angles with the corresponding sides.

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Both Euclid of Alexandria and Appolonius of Perga are known for havingeach authored a compilation of mathematics.

Euclid wrote the Elements, consisting of thirteen volumes mainlyconcerning geometry (but also other mathematics)

Appolonius is famous for the eight volumes entitled Conics

These works were highly influential in later eras.

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Ptolemy

After Euclid’s Elements the next important compilation of Greekmathematics from antiquity is Ptolemy’s Almagest.

Ptolemy of Alexandria (85-165AD) was known for

the Almagest, consisting of 13 books, coveringI geometrical theorems, trigonometric tablesI the motion of the sunI the moonI eclipsesI the fixed starsI the motion of the planets

Geography

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Ptolemy

After Euclid’s Elements the next important compilation of Greekmathematics from antiquity is Ptolemy’s Almagest.Ptolemy of Alexandria (85-165AD) was known for

the Almagest, consisting of 13 books, coveringI geometrical theorems, trigonometric tablesI the motion of the sunI the moonI eclipsesI the fixed starsI the motion of the planets

Geography

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The geocentric theory of Ptolemy held sway for 14 centuries, until thetime of Copernicus, who in 1543 proposed a heliocentric theory.(By contrast, Euclid’s Elements endure.)

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DiophantusDiophantus of Alexandria (c. 3rd century AD) is known for is compilation,entitled the Arithmetica, of 130 algebraic problems.

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Diophantine equations refers to polynomial equations with integercoefficients, where one seeks integer solutions.

For example,

xn + yn = zn where n ≥ 2 is and integer and x , y , z are unknown

ax + by = n where a, b, n ∈ Z+ and x , y are unknonwn

The first equation with n = 2 concerns Pythagorean triples. The casesn ≥ 3 have a long history.

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Diophantine equations refers to polynomial equations with integercoefficients, where one seeks integer solutions.

For example,

xn + yn = zn where n ≥ 2 is and integer and x , y , z are unknown

ax + by = n where a, b, n ∈ Z+ and x , y are unknonwn

The first equation with n = 2 concerns Pythagorean triples. The casesn ≥ 3 have a long history.

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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 thecapital from Rome to Constantinople and legalized Christianity, which, bythe end of the century became the official religion of the Empire.

In the 7th century AD the official language of the imperial administrationchanged from Latin to Greek.

The continuation of the Roman Empire in the east, centred inConstantinople, is usually referred to as the Byzantine Empire, whichlasted until the 15th century. It is characterized by use of the Greeklanguange 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, theHigh Middle Ages. This is the era during which the first universities wereestablished. The main centers of culture and learning lay elsewhere, underArab and Byzantine dominion. By the standards of other eras, very littlenew mathematics was developed at this time. The works of Leonardo ofPisa stand out; but even they are reflective of a world influenced by theeast.

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Historical context

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Europe was an agglomeration of small principalities and city states, withloose larger-scale alliances.

The Western European economy was mainly feudal or manorial.

The most sophisticated European cities, such as Venice, thrived on tradewith the Arab world, and central and East Asia.

The Arab and Byzantine worlds held the main cultural centers.

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Europe was an agglomeration of small principalities and city states, withloose larger-scale alliances.

The Western European economy was mainly feudal or manorial.

The most sophisticated European cities, such as Venice, thrived on tradewith the Arab world, and central and East Asia.

The Arab and Byzantine worlds held the main cultural centers.

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Europe was an agglomeration of small principalities and city states, withloose larger-scale alliances.

The Western European economy was mainly feudal or manorial.

The most sophisticated European cities, such as Venice, thrived on tradewith the Arab world, and central and East Asia.

The Arab and Byzantine worlds held the main cultural centers.

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Europe was an agglomeration of small principalities and city states, withloose larger-scale alliances.

The Western European economy was mainly feudal or manorial.

The most sophisticated European cities, such as Venice, thrived on tradewith 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!)

Place Population

Germany 7.3MFrance 12M

British Isles 3.2MItaly 8M

Spain & Portugal 7MEurope 68MWorld 360M

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Demographics circa 1200 reflect a starkly different world from today.(Figures are approximate!)

Place Population

Germany 7.3MFrance 12M

British Isles 3.2MItaly 8M

Spain & Portugal 7MEurope 68MWorld 360M

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More demographics...

Place Population

London 22KParis 110KRome 20K

Constantinople 200KCairo 225K

Baghdad 250KFez 200K

Beijing 130K

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More demographics...

Place Population

London 22KParis 110KRome 20K

Constantinople 200KCairo 225K

Baghdad 250KFez 200K

Beijing 130K

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