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History of Non-EuclideanGeometryhttp://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Non-Euclidean_geometry.htmlhttp://en.wikipedia.org/wiki/Non-Euclidean_geometryEuclid’s Postulatesfrom Elements, 300BC1.To draw a straight line from any point to anyother.2.To produce a finite straight line continuously in astraight line.3.To describe a circle with any centre and distance.4.That all right angles are equal to each other.5.That, if a straight line falling on two straight linesmake the interior angles on the same side lessthan two right angles, if produced indefinitely,meet on that side on which are the angles lessthan the two right angles.What is up with #5? 5. That, if a straight line falling on two straight lines make theinterior angles on the same side less than two right angles, ifproduced indefinitely, meet on that side on which are theangles less than the two right angles. Equivalently, Playfair’s Axiom: Given a line and a point not on the line, it ispossible to draw exactly one line through the given point parallelto the line. To each triangle, there exists a similar triangle of arbitrarymagnitude. The sum of the angles of a triangle is equal to two right angles. Through any point in the interior of an angle it is always possibleto draw a line which meets both sides of the angle.Can the 5th Postulate be provenfrom the other 4? Ptolemy tried (~150 BC) Proclus tried (~450BC) Wallis tried (1663) Saccheri tried (1697) This attempt was important, he tried proof by contradiction Legendre tried… for 40 years (1800s) Others tried, making the 5th postulate the hot problem inelementary geometryD’Ambert cal ed it“the scandal of elementary geometry”Gauss and his breakthrough Started working on it at age 15 (1792) Still nothing by age 36 Decided the 5th postulate was independent of the other 4. Wondered, what if we allowed 2 linesthrough a single point to BOTH be parallel toa given line The Birth of non-Euclidean Geometry!!! Never published his work, he wanted to avoid controversy.Bolyai’s Strange New World Gauss talked with Farkas Bolyai about the 5thpostulate. Farkas told his son Janos, but said don’t “waste onehour's time on that problem”. Janos wrote daddy in 1823 saying“I have discovered things so wonderfulthat I was astounded ... out of nothingI have created a strange new world.”Bolyai’s Strange New World Bolyai took 2 years to write a 24 page appendixabout it. After reading it, Gauss told a friend,“I regard this young geometer Bolyai as agenius of the first order” Then wrecked Bolyai by telling him that hediscovered this all earlier.Lobachevsky Lobachevsky also published a work about replacing the 5thpostulate in 1829. Published in Russian in a local university publication, no oneknew about it. Wrote a book, Geometrical investigations on the theory ofparallels in 1840.Lobachevsky's Parallel Postulate. There exist two linesparallel to a given line through a given point not onthe line.5th postulate controversy Bolyai’s appendix Lobachevsky’s book the endorsement of Gauss… but the mathematical community wasn’t accepting it. WHY?5th postulate controversy Many had spent years trying to prove the 5thpostulate from the other 4. They still clung to thebelief that they could do it. Euclid was a god. To replace one of his postulateswas blasphemy. It still wasn’t clear that this new system wasconsistent.