Transcript Any three points lie on a straight line or a circle

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
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 coopts other Greek States
• 334 BC, he crosses to
Asia Minor with an army
of 40,000
• 331 BC, conquers Egypt
and founds Alexandria
Alexander conquers the Persian Empire
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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
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
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”.
The Elements
• composed by Euclid in 13 books
• starts with 23 definitions and 5 postulates for
plane geometry
• written using Lemmas, Theorems and proofs
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
Euclid’s Five Postulates
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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
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
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
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?
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˚
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
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
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
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
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
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
Back to Saccheri’s Triangles
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The fault in my ‘proof’