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The Saga of Mathematics A Brief History th Century The 19 th The 19 Century Critical examination of Euclidean Critical examination of Euclidean geometry. geometry. A Century of Surprises A Century of Surprises Especially the

  1. The Saga of Mathematics A Brief History th Century The 19 th The 19 Century � Critical examination of Euclidean Critical examination of Euclidean � geometry. geometry. A Century of Surprises A Century of Surprises � Especially the Especially the parallel postulate parallel postulate � which Euclid took for granted. which Euclid took for granted. Chapter 11 Chapter 11 � Euclidean versus Non Euclidean versus Non- -Euclidean Euclidean � geometry. geometry. Lewinter & Widulski Lewinter & Widulski The Saga of Mathematics The Saga of Mathematics 1 1 Lewinter & Widulski Lewinter & Widulski The Saga of Mathematics The Saga of Mathematics 2 2 The Parallel Postulate The Parallel Postulate The Parallel Postulate The Parallel Postulate � If a straight line falling on two straight If a straight line falling on two straight � lines makes the interior angles on the L 1 lines makes the interior angles on the same side less than two right angles, same side less than two right angles, the two straight lines, if produced the two straight lines, if produced indefinitely, meet on that side on indefinitely, meet on that side on which are the angles less than the two which are the angles less than the two L 2 right angles. right angles. L Lewinter & Widulski Lewinter & Widulski The Saga of Mathematics The Saga of Mathematics 3 3 Lewinter & Widulski Lewinter & Widulski The Saga of Mathematics The Saga of Mathematics 4 4 The Parallel Postulate Alternatives The Parallel Postulate Alternatives � Poseidonius Poseidonius (c. 135 (c. 135- -51 BC): Two 51 BC): Two � The most convincing evidence of The most convincing evidence of � � parallel lines are equidistant. parallel lines are equidistant. Euclid’s mathematical genius. Euclid’s mathematical genius. � Proclus Proclus (c. 500 AD): If a line (c. 500 AD): If a line � Euclid had no proof. Euclid had no proof. � � intersects one of two parallel lines, intersects one of two parallel lines, � In fact, no proof is possible, but he In fact, no proof is possible, but he � then it also intersects the other. then it also intersects the other. couldn’t go further without this couldn’t go further without this � Saccheri Saccheri (c. 1700): The sum of the (c. 1700): The sum of the � statement. statement. interior angles of a triangle is two right interior angles of a triangle is two right � Many have tried to prove it but failed. Many have tried to prove it but failed. angles. angles. � Lewinter & Widulski Lewinter & Widulski The Saga of Mathematics The Saga of Mathematics 5 5 Lewinter & Widulski Lewinter & Widulski The Saga of Mathematics The Saga of Mathematics 6 6 Lewinter & Widulski 1

  2. The Saga of Mathematics A Brief History Alternatives Alternatives Alternatives Alternatives � Legendre Legendre (1752 (1752- -1833): A line 1833): A line � Farkas Farkas B Bó ólyai lyai (1775 (1775- -1856): There is 1856): There is � � through a point in the interior of an through a point in the interior of an a circle through every set of three a circle through every set of three angle other than a straight angle angle other than a straight angle non- non -collinear points. collinear points. B intersects at least one of the arms of intersects at least one of the arms of the angle. the angle. A P C Lewinter & Widulski Lewinter & Widulski The Saga of Mathematics The Saga of Mathematics 7 7 Lewinter & Widulski Lewinter & Widulski The Saga of Mathematics The Saga of Mathematics 8 8 Playfair’s Axiom Playfair’s Axiom Non- Non -Euclidean Geometry Euclidean Geometry � Given a line and a point not on the Given a line and a point not on the � In the beginning of the last century, some In the beginning of the last century, some � � line, it is possible to draw exactly one line, it is possible to draw exactly one mathematicians began to think along more mathematicians began to think along more line through the given point parallel to line through the given point parallel to radical lines. radical lines. the line. th Postulate is not true! the line. � Suppose the 5 Suppose the 5 th Postulate is not true! � – – The Axiom is not The Axiom is not Playfair’s Playfair’s own invention. own invention. � “Through a given point in a plane, “Through a given point in a plane, two two lines, lines, � He proposed it about 200 years ago, but He proposed it about 200 years ago, but parallel to a given straight line, can be parallel to a given straight line, can be Proclus Proclus stated it some 1300 years earlier. stated it some 1300 years earlier. drawn.” drawn.” – Often substituted for the fifth postulate Often substituted for the fifth postulate – � This would change proposition in Euclid. This would change proposition in Euclid. � because it is easier to remember. because it is easier to remember. Lewinter & Widulski Lewinter & Widulski The Saga of Mathematics The Saga of Mathematics 9 9 Lewinter & Widulski Lewinter & Widulski The Saga of Mathematics The Saga of Mathematics 10 10 Non- -Euclidean Geometry Euclidean Geometry Non- -Euclidean Geometry Euclidean Geometry Non Non � For example: The sum of the angles in a For example: The sum of the angles in a � After failing to prove the parallel postulate, After failing to prove the parallel postulate, � � triangle is less triangle is less than two right angles! than two right angles! mathematicians wondered if there was a mathematicians wondered if there was a � This of course led to Non This of course led to Non- -Euclidean Euclidean consistent “ “alternative alternative” ” geometry in which geometry in which consistent � geometry whose discovery was led by Gauss geometry whose discovery was led by Gauss the parallel postulate failed. the parallel postulate failed. (1777- -1855), 1855), Lobachevskii Lobachevskii (1792 (1792- -1856) and 1856) and (1777 � To their amazement, they found two! To their amazement, they found two! � Jonas B Bó ól lyai yai (1802 (1802- -1850). 1850). Jonas � The secret was to look at curved surfaces. The secret was to look at curved surfaces. � Later by Beltrami (1835 Later by Beltrami (1835- -1900), Hilbert 1900), Hilbert � � (1862- -1943) and Klein (1849 1943) and Klein (1849- -1945). 1945). – The plane is flat – The plane is flat – – it has no curvature or, as it has no curvature or, as (1862 mathematicians say, its curvature is 0. mathematicians say, its curvature is 0. � Fits Einstein’s Fits Einstein’s Theory of Relativity. Theory of Relativity. � Lewinter & Widulski Lewinter & Widulski The Saga of Mathematics The Saga of Mathematics 11 11 Lewinter & Widulski Lewinter & Widulski The Saga of Mathematics The Saga of Mathematics 12 12 Lewinter & Widulski 2

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