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MATH 5400, History of MathematicsLecture 2: Early origins and the Greek era
Professor: Peter Gibson
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
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
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
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)
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)
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)
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.
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.
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.
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.
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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