Nikolai Lobachevsky (1792-1856)

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Nikolai Lobachevsky (1792-1856)

• Born 1 Dec 1792 in Nizhny Novgorod, RussiaDied
  24 Feb 1856 in Kazan, Russia
• Went to school and studied in Kazan under Martin
  Bartels (1769 - 1833), who was a friend of Gauss.
• 1811 Master's Degree in physics and mathematics.
• 1814 he was appointed to a lectureship
• 1816 he became an extraordinary professor.
• 1822 he was appointed as a full professor
• 1820 -1825 dean of the Math. and Physics
  Department
• 1827 he became rector of Kazan University

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Nikolai Lobachevsky (1792-1856)

• Accepting the possibility of HAA Lobachevsky
  redefines parallels in the following way
• 16. All straight lines which in a plane go out
  from a point can, with reference to a given
  straight line in the same plane, be divided into
  two classes - into cutting and non-cutting. The
  boundary lines of the one and the other class of
  those lines will be called parallel to the given
  line.

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Nikolai Lobachevsky (1792-1856)
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Nikolai Lobachevsky (1792-1856)

• Depending on the angle HAD which is called ?(p)
  there are two possibilities.
• Either
• ?(p)p/2 and there is only one parallel to CB
  through A,
• or
• ?(p)ltp/2 there are exactly two Lobachevsky
  parallels.
• This means that there are infinitely many lines
  through A which do not intersect CB.

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Nikolai Lobachevsky (1792-1856)

• In the case that there are two parallels there
  are also two sides of parallelism.

l
The side of parallelism of l
?(p)
p
?(p)
The side of parallelism of l
l
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Nikolai Lobachevsky (1792-1856)

• Lobachevsky needs to prove the following.
• His definition of parallelism uses the point A,
  he has to show that in fact parallelism is a
  symmetric property of two lines
• 17. A straight line maintains the characteristic
  of parallelism at all its points.
• 18. Two lines are always mutually parallel.

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Nikolai Lobachevsky (1792-1856)

• He shows that HOA is not possible with the first
  four of Euclids axioms (which he made more
  precise e.g. 3).
• 19. In a rectilineal triangle the sum of the
  three angles can not be greater than two right
  angles.

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Nikolai Lobachevsky (1792-1856)

• He also shows that if one triangle has the
  property that the sum of the interior angels is
  180, then all triangles have this property.
• This is the dichotomy between Euclidean geometry
  (HRA) and non-Euclidian geometry (HAA).
• 20. If in any rectilineal triangle the sum of
  the three angles is equal to two right angles, so
  is this also the case for every other triangle.

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Nikolai Lobachevsky (1792-1856)

• Finally he shows that the dichotomy extends to
  parallels.
• In Euclidean geometry there is exactly one
  parallel line to a given line through a given
  point not on that line.
• In non-Euclidean geometry there are exactly two
  parallel lines, in Lobachevskys sense, which
  implies that there are infinitely many lines
  through the point that do not intersect the given
  line.
• 22. If two perpendiculars to the same straight
  line are parallel to each other then the sum of
  the three angles in a rectilineal triangle is
  equal to two right angels.

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Lobachevskys Euclidean and non-Euclidean geometry

• Euclidean or plane geometry is the geometry in
  which the equivalent assumptions
• For all lines and points p ?(p) p/2 and
  equivalently
• The sum of the interior angles of any triangle
  p
• hold.
• Non-Euclidean (imaginary) geometry is the
  geometry in which the equivalent assumptions
• For all lines and points p ?(p)lt p/2 and
  equivalently
• The sum of the interior angles of any triangle lt
  p
• hold.

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Nikolai Lobachevskys Non-Euclidean geometry

• Now assume that for all lines and points p ?(p)lt
  p/2.
• In this case parallels can have any angle which
  is less than p/2 and moreover for any given angle
  there is always a pair of parallels whose angle
  is that angle
• 23. For every given angle ? there is a line p
  such that ?(p) ?.

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Nikolai Lobachevskys Non-Euclidean geometry

• Parallels cease to have the same distance to
  each other at all points rather
• 24. The farther parallel lines are prolonged on
  the side of their parallelism, the more they
  approach on another.

B
B
x
s
s
A
A
x
Moreover in the above figure sse-x.