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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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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
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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
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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
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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”.
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The Elements
• composed by Euclid in 13 books• starts with 23 definitions and 5 postulates for
plane geometry• written using Lemmas, Theorems and proofs
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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
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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
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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
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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
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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?
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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˚
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Any three points lie on a straight line or a circle
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Any three points lie on a straight line or a circle
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Any three points lie on a straight line or a circle
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Proof that L and M intersect
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Proof that L and M intersect
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Proof that L and M intersect
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Proof that L and M intersect
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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
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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
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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
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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
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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
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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
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Back to Saccheri’s Triangles
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The fault in my ‘proof’