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 co- opts other Greek States 334 BC, he crosses to Asia Minor with an army of 40, BC, conquers Egypt and founds Alexandria

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

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 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

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 ( , 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

Proof that L and M intersect

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, 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

The fault in my ‘proof’