Alexander the Great (356–323 BC) Born in Macedonia, son of Philip II At 13, Aristotle became his...

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Alexander the Great (356– 323 BC) • Born in Macedonia, son of Philip II • At 13, Aristotle became his tutor • At 16, as regent in his father’s absence he put down an insurgency

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Transcript of Alexander the Great (356–323 BC) Born in Macedonia, son of Philip II At 13, Aristotle became his...

Page 1: Alexander the Great (356–323 BC) Born in Macedonia, son of Philip II At 13, Aristotle became his tutor At 16, as regent in his father’s absence he put.

Alexander the Great (356–323 BC)

• Born in Macedonia, son of Philip II

• At 13, Aristotle became his tutor

• At 16, as regent in his father’s absence he put down an insurgency

Page 2: Alexander the Great (356–323 BC) Born in Macedonia, son of Philip II At 13, Aristotle became his tutor At 16, as regent in his father’s absence he put.

Alexander the Great (356–323 BC)

• 336 BC, Philip II is assassinated

• Alexander is king at 20, puts down revolts, secures northern borders, defeats or co-opts other Greek States

• 334 BC, he crosses to Asia Minor with an army of 40,000

• 331 BC, conquers Egypt and founds Alexandria

Page 3: Alexander the Great (356–323 BC) Born in Macedonia, son of Philip II At 13, Aristotle became his tutor At 16, as regent in his father’s absence he put.

Alexander conquers the Persian Empire

• 336–323 BC, King of Macedon• 332–323 BC, King of Egypt• 330–323 BC, Great King of Persia• 323 BC, Alexander dies in Babylon

Page 4: Alexander the Great (356–323 BC) Born in Macedonia, son of Philip II At 13, Aristotle became his tutor At 16, as regent in his father’s absence he put.

After Alexander’s Death

• Ptolemy Soter, king of Egypt 323–283 BC• Founds the great library of Alexandria• First chief librarian, Demetrios Phalerus

(c.350–283), exiled and poisoned by a snake

Page 5: Alexander the Great (356–323 BC) Born in Macedonia, son of Philip II At 13, Aristotle became his tutor At 16, as regent in his father’s absence he put.

Summoned from the Void

Euclid• Birth: place and date

unknown• Death: circumstances and

date unknown• “Oh, King, for travelling the

country there are royal roads and roads for common citizens, but in geometry there is but one road for all”.

Page 6: Alexander the Great (356–323 BC) Born in Macedonia, son of Philip II At 13, Aristotle became his tutor At 16, as regent in his father’s absence he put.

The Elements

• composed by Euclid in 13 books• starts with 23 definitions and 5 postulates for

plane geometry• written using Lemmas, Theorems and proofs

Page 7: Alexander the Great (356–323 BC) Born in Macedonia, son of Philip II At 13, Aristotle became his tutor At 16, as regent in his father’s absence he put.

Transmission of the Elements

• 6th century, Boethius translates parts into Latin

• 800, translation into Arabic• 1120 translation from Arabic to

Latin• 1260 new edition in Latin, cited

by Doctor Mirabilis (a.k.a. Roger Bacon)

• 1505 translation from Greek directly to Latin

• 1570 English edition

Page 8: Alexander the Great (356–323 BC) Born in Macedonia, son of Philip II At 13, Aristotle became his tutor At 16, as regent in his father’s absence he put.

Euclid’s Five Postulates

• There is a line between any two points• Any finite line can be infinitely extended• There is a circle with any centre and radius• All right angles are equal• If a line crosses two lines, and the sum of

the interior angles on the same side is less than 180˚, then the two lines intersect on that side

Page 9: Alexander the Great (356–323 BC) Born in Macedonia, son of Philip II At 13, Aristotle became his tutor At 16, as regent in his father’s absence he put.

If a line crosses two lines, and the sum of the interior angles on the same side is less than 180˚, then the two lines intersect on that side

Page 10: Alexander the Great (356–323 BC) Born in Macedonia, son of Philip II At 13, Aristotle became his tutor At 16, as regent in his father’s absence he put.

Al-Haytham (965 Basra–c.1040 Cairo)

• Famous work on optics, 1011–21

• used motion to prove Euclid’s Fifth

• his work led to a connection between the parallel postulate and the sum of the angles in a quadrilateral

Page 11: Alexander the Great (356–323 BC) Born in Macedonia, son of Philip II At 13, Aristotle became his tutor At 16, as regent in his father’s absence he put.

Omar Khayyam (1048-1131, Persia)

• Refuted earlier work on parallels• There are many things wrong [with Al-Haytham’s proof]• How could a line move, remaining normal to a given line?• How could a proof be based on this idea?• How could geometry and motion be connected?

Page 12: Alexander the Great (356–323 BC) Born in Macedonia, son of Philip II At 13, Aristotle became his tutor At 16, as regent in his father’s absence he put.

Girolamo Saccheri (1667–1733)

• Thought he had proved Euclid’s Fifth

• His methods used three cases for the sum of angles in a triangle:

• (1) less than 180˚• (2) exactly 180˚• (3) greater than 180˚

Page 13: Alexander the Great (356–323 BC) Born in Macedonia, son of Philip II At 13, Aristotle became his tutor At 16, as regent in his father’s absence he put.

Any three points lie on a straight line or a circle

QuickTime™ and aMPEG-4 Video decompressor

are needed to see this picture.

Page 14: Alexander the Great (356–323 BC) Born in Macedonia, son of Philip II At 13, Aristotle became his tutor At 16, as regent in his father’s absence he put.

Any three points lie on a straight line or a circle

QuickTime™ and aMPEG-4 Video decompressor

are needed to see this picture.

Page 15: Alexander the Great (356–323 BC) Born in Macedonia, son of Philip II At 13, Aristotle became his tutor At 16, as regent in his father’s absence he put.

Any three points lie on a straight line or a circle

QuickTime™ and aMPEG-4 Video decompressor

are needed to see this picture.

Page 16: Alexander the Great (356–323 BC) Born in Macedonia, son of Philip II At 13, Aristotle became his tutor At 16, as regent in his father’s absence he put.
Page 17: Alexander the Great (356–323 BC) Born in Macedonia, son of Philip II At 13, Aristotle became his tutor At 16, as regent in his father’s absence he put.

Proof that L and M intersect

QuickTime™ and aMPEG-4 Video decompressor

are needed to see this picture.

Page 18: Alexander the Great (356–323 BC) Born in Macedonia, son of Philip II At 13, Aristotle became his tutor At 16, as regent in his father’s absence he put.

Proof that L and M intersect

QuickTime™ and aMPEG-4 Video decompressor

are needed to see this picture.

Page 19: Alexander the Great (356–323 BC) Born in Macedonia, son of Philip II At 13, Aristotle became his tutor At 16, as regent in his father’s absence he put.

Proof that L and M intersect

QuickTime™ and aMPEG-4 Video decompressor

are needed to see this picture.

Page 20: Alexander the Great (356–323 BC) Born in Macedonia, son of Philip II At 13, Aristotle became his tutor At 16, as regent in his father’s absence he put.

Proof that L and M intersect

QuickTime™ and aMPEG-4 Video decompressor

are needed to see this picture.

Page 21: Alexander the Great (356–323 BC) Born in Macedonia, son of Philip II At 13, Aristotle became his tutor At 16, as regent in his father’s absence he put.

Johann Heinrich Lambert (1728–77)

• Proved is not a fraction a/b• Introduced hyperbolic

functions• Showed that in the area of a

triangle in the hyperbolic plane is proportional to 180˚– the sum of its angles

• Saw the promised land, but never entered it

Page 22: Alexander the Great (356–323 BC) Born in Macedonia, son of Philip II At 13, Aristotle became his tutor At 16, as regent in his father’s absence he put.

Ferdinand Karl Schweikart (1780–1857)

• Letter to Gauss, 1818. This being assumed we can prove rigorously:

• a) that the sum of the angles in a triangle is less than 180˚

• b) that the sum becomes less as the area of the triangle becomes greater

• c) that the altitude of a right-angled isosceles triangle continually grows as the sides increase, but it can never become greater than a certain length, which I call the constant

Page 23: Alexander the Great (356–323 BC) Born in Macedonia, son of Philip II At 13, Aristotle became his tutor At 16, as regent in his father’s absence he put.

Carl Friedrich Gauss (1777–1855)

• with Archimedes and Newton one of the three greatest mathematicians of all time

• discovered the hyperbolic plane but did not publish anything about it

Page 24: Alexander the Great (356–323 BC) Born in Macedonia, son of Philip II At 13, Aristotle became his tutor At 16, as regent in his father’s absence he put.

Farkas Bolyai (1775–1856)

• Hungarian, fellow student of Gauss at Göttingen; they became lifelong friends

• poorly paid, wrote and published dramas, …

• Worked on his mathematical masterpiece

• Tried to dissuade his son Janos from wasting his life on the parallel postulate

Page 25: Alexander the Great (356–323 BC) Born in Macedonia, son of Philip II At 13, Aristotle became his tutor At 16, as regent in his father’s absence he put.

Janos Bolyai (1802–1860)

• 1820 studied in Vienna• 3 Nov 1823 wrote to his father, …

[I have] created a new, another world out of nothing …

• In 1832 he published a 24-page appendix to his father’s book

• Gauss, “I regard this young geometer Bolyai as a genius of the first order”.

• Never published again, but left 20,000 manuscript pages

Page 26: Alexander the Great (356–323 BC) Born in Macedonia, son of Philip II At 13, Aristotle became his tutor At 16, as regent in his father’s absence he put.

Nikolai Lobachevsky (1792–1856)

• strongly influenced by Martin Bartels at Kazan, previously Gauss’s tutor in Braunschweig

• 1827 Rector of Kazan University

• 1829 work on the hyperbolic plane published in the Kazan Messenger, rejected by St. Petersburg Academy of Sciences

• 1846 retired (dismissed)

• After 1846 his health deteriorates

Page 27: Alexander the Great (356–323 BC) Born in Macedonia, son of Philip II At 13, Aristotle became his tutor At 16, as regent in his father’s absence he put.

Back to Saccheri’s Triangles

Page 28: Alexander the Great (356–323 BC) Born in Macedonia, son of Philip II At 13, Aristotle became his tutor At 16, as regent in his father’s absence he put.

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Page 29: Alexander the Great (356–323 BC) Born in Macedonia, son of Philip II At 13, Aristotle became his tutor At 16, as regent in his father’s absence he put.

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Page 30: Alexander the Great (356–323 BC) Born in Macedonia, son of Philip II At 13, Aristotle became his tutor At 16, as regent in his father’s absence he put.

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Page 31: Alexander the Great (356–323 BC) Born in Macedonia, son of Philip II At 13, Aristotle became his tutor At 16, as regent in his father’s absence he put.

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Page 32: Alexander the Great (356–323 BC) Born in Macedonia, son of Philip II At 13, Aristotle became his tutor At 16, as regent in his father’s absence he put.

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Page 33: Alexander the Great (356–323 BC) Born in Macedonia, son of Philip II At 13, Aristotle became his tutor At 16, as regent in his father’s absence he put.
Page 34: Alexander the Great (356–323 BC) Born in Macedonia, son of Philip II At 13, Aristotle became his tutor At 16, as regent in his father’s absence he put.
Page 35: Alexander the Great (356–323 BC) Born in Macedonia, son of Philip II At 13, Aristotle became his tutor At 16, as regent in his father’s absence he put.

The fault in my ‘proof’


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