Creation of NonEuclidean Geometry

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Creation of Non-Euclidean Geometry
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The 17th Century Scientific Revolution

• Virtually all science is overturned
• Euclidean geometry survives untouched
• It is admired as a
• model of perfect science.

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Immanuel Kant (1724-1804)

• The last major philosopher of the Enlightenment
  and one of the most influential thinkers of
  modern Europe.

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Immanuel Kant (1724-1804)

• He credited David Hume with awakening him from
  "dogmatic slumber" (circa 1770). Kant would not
  publish another work in philosophy for the next
  eleven years. Kant spent his silent decade
  working on a solution to the problems posed. When
  he emerged from his silence in 1781, the result
  was the Critique of Pure Reason, now recognized
  as one of the greatest works in the history of
  philosophy.

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Immanuel Kant (1724-1804)

• Kant regarded the truths of Euclidean geometry as
  a priori and certain.
• The sum of the angles in a triangle is 180
  degrees,
• the square on the hypotenuse of a right triangle
  is the sum of the squares on the other sides,
• through a point not on a line exactly one line
  can be drawn parallel to the first line.
• a) They are known prior to any experience of
  space.
• b) They pre-configure how we experience space.

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Euclids Axiomatics

• Undefined terms point, line, plane
• Common Notions
• Things equal to the same thing are equal.
• If equals are added to equals, the results are
  equal.
• If equals are subtracted from equals, the results
  are equal.
• Things that coincide with one another are equal
  to one another.
• The whole is greater than the part.

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Euclids Axiomatics

• Undefined terms point, line, plane
• Postulates
• Exactly one straight can be drawn from any point
  to any other point.
• A finite straight line can be extended
  continuously in a straight line.
• A circle can be formed with any center and
  distance (radius).
• All right angles are equal to one another.
• If a straight line falling on two straight lines
  makes the sum of the interior angles on the same
  side less than two right angles, then the two
  straight lines, if extended indefinitely, meet on
  that side on which the angle sum is less than two
  right angles.

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• Postulates 1-4 are
• intuitive, trivial, self-evident.
• The 5th postulate, the parallel postulate
• spoils that.

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For 2000 years mathematicians tried to prove the
parallel postulate from the earlier postulates

• Saccheri
• Lambert
• Legendre
• and many others.

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Carl Friedrich Gauss(1777 1855)Prince of
Mathematicians
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Carl Friedrich Gauss

• Began attempting to prove the parallels
  postulate from the other four.
• 1817 Is convinced that the fifth postulate
  was independent of the other four postulates.
  Began to work out the consequences of a geometry
  in which more than one line can be drawn through
  a given point parallel to a given line. Kept his
  work secret.

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Theorema Egregium ('Remarkable Theorem')

• Informally, the theorem says that the curvature
  of a surface can be determined entirely by
  measuring angles and distances on the surface,
  that is, it does not depend on how the surface
  might be embedded in (3-dimensional) space.

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Gauss Non-Euclidean Experiment

• Gauss measured a large triangle formed by the
  mountain peaks of Brocken, Hohehagen, and
  Inselsberg. The sides of the triangle were about
  43, 53, and 67 miles (69, 85, and 109 km).
• The results? 180 degrees and 14.85 seconds
• This excess (less than a quarter of a thousandth
  of a percent) was far smaller than the margin of
  error for the measurement. So the result was
  inconclusive.

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Nikolai Lobachevsky (1792 - 1856)
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Nikolai Lobachevsky

• 1829 Published in local university paper,
  Kazan Messenger , in Russian.
• 1837 Published an account in French in
  Crelles Journal, gets wide audience
• Published Geometrical investigations on the
  theory of parallels, 61 pages.
• Lobachevsky's Parallel Postulate. There exist two
  lines parallel to a given line through a given
  point not on the line.

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János Bolyai (1802 1860)
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János Bolyai

• 1823 I have discovered things so wonderful
  that I was astounded ... out of nothing I have
  created a strange new world.
• 1825 Published as a 24 page appendix to his
  fathers book.

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Bernhard Riemann(1826 1866)
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Bernhard Riemann

• Submitted 3 topics to Gauss to select from
  for his Habilitation lecture. To his surprise
  Gauss selects Foundations of Geometry.
• 1854 Monastyrsky writes-
• Among Riemann's audience, only Gauss was able to
  appreciate the depth of Riemann's thoughts. ...
  The lecture exceeded all his expectations and
  greatly surprised him. Returning to the faculty
  meeting, he spoke with the greatest praise and
  rare enthusiasm to Wilhelm Weber about the depth
  of the thoughts that Riemann had presented.

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Gauss-Lobachevsky-Bolyai Geometry

• straight lines have two infinitely distant points
• no lines parallel
• hyperbolic geometry geometry of surfaces of
  negative curvature
• sum of the angles in a triangle is less than 180
  degrees

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Riemann Geometry

• lines have no (or more precisely two imaginary)
  infinitely distant points
• spherical geometry
• elliptic geometry
• more than one line parallel
• geometry of space-time used in relativity
• sum of the angles in a triangle is more than 180
  degrees

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Non-Euclidean GeometryWho Dunnit?
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Did Gauss aid the development and going
public with Non-Euclidean Geometry?
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