MATH 4400, History of Mathematics

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1 MATH 4400, History of Mathematics Lecture 2: The Greek era (con t) Professor: Peter Gibson September 27, 2016

2 Recall from last lecture... The political authority of the Golden Age was fragile, however, and in 338BC Philip of Macedon conquered much of Greece. P. Gibson (YorkU) Math / 15

3 Recall from last lecture... The political authority of the Golden Age was fragile, however, and in 338BC Philip of Macedon conquered much of Greece. His son, Alexander, succeeded him in 336BC. P. Gibson (YorkU) Math / 15

4 Upon Alexander s death in 323BC, his empire was divided amongst his generals, P. Gibson (YorkU) Math / 15

5 Upon Alexander s death in 323BC, his empire was divided amongst his generals, leading to the Hellenistic period. P. Gibson (YorkU) Math / 15

6 Upon Alexander s death in 323BC, his empire was divided amongst his generals, leading to the Hellenistic period. Some mathematicians from the period after Alexander: P. Gibson (YorkU) Math / 15

7 Upon Alexander s death in 323BC, his empire was divided amongst his generals, leading to the Hellenistic period. Some mathematicians from the period after Alexander: Euclid of Alexandria ( BC) P. Gibson (YorkU) Math / 15

8 Upon Alexander s death in 323BC, his empire was divided amongst his generals, leading to the Hellenistic period. Some mathematicians from the period after Alexander: Euclid of Alexandria ( BC) Archimedes of Syracuse ( BC) P. Gibson (YorkU) Math / 15

9 Upon Alexander s death in 323BC, his empire was divided amongst his generals, leading to the Hellenistic period. Some mathematicians from the period after Alexander: Euclid of Alexandria ( BC) Archimedes of Syracuse ( BC) Appolonius of Perga ( BC) P. Gibson (YorkU) Math / 15

10 Archimedes Archimedes lived from BC, during the Hellenistic period. P. Gibson (YorkU) Math / 15

11 Archimedes Archimedes lived from BC, during the Hellenistic period. The age in which he lived was marked by the rise of Rome as a regional power, and by the first and second Punic Wars. P. Gibson (YorkU) Math / 15

12 The punic wars ( BC) pitted Rome against Carthage. P. Gibson (YorkU) Math / 15

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

13 The punic wars ( BC) pitted Rome against Carthage. P. Gibson (YorkU) Math / 15

14 During the Second Punic War Carthage was led by Hannibal. P. Gibson (YorkU) Math / 15

15 During the Second Punic War Carthage was led by Hannibal. P. Gibson (YorkU) Math / 15

16 Archimedes himself was killed by a Roman soldier during the seige of Syracuse. P. Gibson (YorkU) Math / 15

17 Archimedes himself was killed by a Roman soldier during the seige of Syracuse. 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 P. Gibson (YorkU) Math / 15

18 Archimedes himself was killed by a Roman soldier during the seige of Syracuse. 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 / 15

19 The screw of Archimedes: P. Gibson (YorkU) Math / 15

20 Transmission of Archimedes works Some of Archimedes works have been rediscovered relatively recently. P. Gibson (YorkU) Math / 15

21 Transmission of Archimedes works Some of Archimedes works have been rediscovered relatively recently. P. Gibson (YorkU) Math / 15

22 The Archimedes Palimpsest contains the only known version of his Method of Mechanical Theorems, along with other previously known works. P. Gibson (YorkU) Math / 15

23 The Archimedes Palimpsest contains the only known version of his Method of Mechanical Theorems, along with other previously known works. The former is a letter, written to Erastosthenes of Alexandria, a famous contemporary of Archimedes. P. Gibson (YorkU) Math / 15

24 The Archimedes Palimpsest contains the only known version of his Method of Mechanical Theorems, along with other previously known works. The former is a letter, written to Erastosthenes of Alexandria, a famous contemporary of Archimedes. Translations of all Archimedes known works are freely available. P. Gibson (YorkU) Math / 15

25 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 equilibrium but 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 applied to 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 straight lines be drawn from them to the equal angles, they make equal angles with the corresponding sides. P. Gibson (YorkU) Math / 15

26 Both Euclid of Alexandria and Appolonius of Perga are known for having each authored a compilation of mathematics. Euclid wrote the Elements, consisting of thirteen volumes mainly concerning geometry (but also other mathematics) Appolonius is famous for the eight volumes entitled Conics These works were highly influential in later eras. P. Gibson (YorkU) Math / 15

27 Ptolemy After Euclid s Elements the next important compilation of Greek mathematics from antiquity is Ptolemy s Almagest. P. Gibson (YorkU) Math / 15

28 Ptolemy After Euclid s Elements the next important compilation of Greek mathematics from antiquity is Ptolemy s Almagest. Ptolemy of Alexandria (85-165AD) was known for the Almagest, consisting of 13 books, covering geometrical theorems, trigonometric tables the motion of the sun the moon eclipses the fixed stars the motion of the planets Geography P. Gibson (YorkU) Math / 15

29 The geocentric theory of Ptolemy held sway for 14 centuries, until the time of Copernicus, who in 1543 proposed a heliocentric theory. (By contrast, Euclid s Elements endure.) P. Gibson (YorkU) Math / 15

30 Diophantus Diophantus of Alexandria (c. 3rd century AD) is known for is compilation, entitled the Arithmetica, of 130 algebraic problems. P. Gibson (YorkU) Math / 15

Credited with formulating the method of exhaustion for approximating a circle by polygons

Credited with formulating the method of exhaustion for approximating a circle by polygons MATH 300 History of Mathematics Figures in Greek Mathematics Sixth Century BCE Thales of Miletus May have formulated earliest theorems in geometry (e.g., ASA) Predicted an eclipse in 585 BCE Pythagoras