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MATH 5400, History of MathematicsLecture 2: Early origins and the Greek era

Professor: Peter Gibson

pcgibson@yorku.ca

http://people.math.yorku.ca/pcgibson/math5400

September 22, 2016

Early civilization

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The emergence of mathematics

It is noteworthy that mathematics has a long history, longer than that ofany other scientific discipline. Why is this?

One reason is: proof.

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The emergence of mathematics

It is noteworthy that mathematics has a long history, longer than that ofany other scientific discipline. Why is this?

One reason is: proof.

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Like language and literature, or the biological evolution of life, the originsof mathematics are obscure, and the vast majority of mathematics hasemerged only recently in its long history.

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Civic administrationAn example of early record keeping, reflecting the need for accounting inurban society.

Writing tabletMesopotamia∼3.1K-3K BCE

significance?

beer

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Civic administrationAn example of early record keeping, reflecting the need for accounting inurban society.

Writing tabletMesopotamia∼3.1K-3K BCE

significance?

beer

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Early mathematicsPlimpton 322

Mesopotamia∼1.8K BCE

significance?

Pythagorean triples

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Early mathematicsPlimpton 322

Mesopotamia∼1.8K BCE

significance?

Pythagorean triples

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The Rhind papyrus

Egypt∼1.65K BCE

significance?

ArithmeticAreasProblems

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The Rhind papyrus

Egypt∼1.65K BCE

significance?

ArithmeticAreasProblems

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The Greek period (∼800 BCE - 700 AD)

There was a remarkable flowering of intellectual culture in ancient Greece,including theatre, literature, philosophy and mathematics.

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Some historical remarks

The Dark Ages (1100-800BC), The Golden Age of Greece (∼ 500-300BC)

Homer is alleged to have lived c. 750BC

Pythagorus lived c. 530BC, founding a religious community in Croton

The Peleponnesian War took place 431-404BC

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Some historical remarks

The Dark Ages (1100-800BC), The Golden Age of Greece (∼ 500-300BC)

Homer is alleged to have lived c. 750BC

Pythagorus lived c. 530BC, founding a religious community in Croton

The Peleponnesian War took place 431-404BC

P. Gibson (YorkU) Math 5400 22.9.2016 10 / 26

Some historical remarks

The Dark Ages (1100-800BC), The Golden Age of Greece (∼ 500-300BC)

Homer is alleged to have lived c. 750BC

Pythagorus lived c. 530BC, founding a religious community in Croton

The Peleponnesian War took place 431-404BC

P. Gibson (YorkU) Math 5400 22.9.2016 10 / 26

Some historical remarks

The Dark Ages (1100-800BC), The Golden Age of Greece (∼ 500-300BC)

Homer is alleged to have lived c. 750BC

Pythagorus lived c. 530BC, founding a religious community in Croton

The Peleponnesian War took place 431-404BC

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The Peleponnesian War undermined the political institutions of the day,although high culture continued to flourish...

A famous succession of Golden age scholars:

Socrates (469-399BC)

Plato (427-347BC)

Aristotle (384-322BC)

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The Peleponnesian War undermined the political institutions of the day,although high culture continued to flourish...

A famous succession of Golden age scholars:

Socrates (469-399BC)

Plato (427-347BC)

Aristotle (384-322BC)

P. Gibson (YorkU) Math 5400 22.9.2016 11 / 26

The Peleponnesian War undermined the political institutions of the day,although high culture continued to flourish...

A famous succession of Golden age scholars:

Socrates (469-399BC)

Plato (427-347BC)

Aristotle (384-322BC)

P. Gibson (YorkU) Math 5400 22.9.2016 11 / 26

The Peleponnesian War undermined the political institutions of the day,although high culture continued to flourish...

A famous succession of Golden age scholars:

Socrates (469-399BC)

Plato (427-347BC)

Aristotle (384-322BC)

P. Gibson (YorkU) Math 5400 22.9.2016 11 / 26

The Peleponnesian War undermined the political institutions of the day,although high culture continued to flourish...

A famous succession of Golden age scholars:

Socrates (469-399BC)

Plato (427-347BC)

Aristotle (384-322BC)

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The dodecahedron:

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The icosahedron:

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The political authority of the Golden Age was fragile, however, and in338BC Philip of Macedon conquered much of Greece.

His son, Alexander, succeeded him in 336BC.

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The political authority of the Golden Age was fragile, however, and in338BC Philip of Macedon conquered much of Greece.

His son, Alexander, succeeded him in 336BC.

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What is the difference between

Hellenic and

Hellenistic?

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What is the difference between

Hellenic and

Hellenistic?

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What is the difference between

Hellenic and

Hellenistic?

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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)

P. Gibson (YorkU) Math 5400 22.9.2016 16 / 26

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)

P. Gibson (YorkU) Math 5400 22.9.2016 16 / 26

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)

P. Gibson (YorkU) Math 5400 22.9.2016 16 / 26

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)

P. Gibson (YorkU) Math 5400 22.9.2016 16 / 26

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.

P. Gibson (YorkU) Math 5400 22.9.2016 20 / 26

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.

P. Gibson (YorkU) Math 5400 22.9.2016 23 / 26

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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Some Euclidean mathematics

Onto the blackboard...

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