History of Mathematics

1
History of Mathematics

• Euclidean Geometry -
• Controversial Parallel Postulate
• Anisoara Preda

2
Geometry

• A branch of mathematics dealing with the
  properties of geometric objects
• Greek word
• geos- earth
• metron- measure

3
Geometry in Ancient Society

• In ancient society, geometry was used for
• Surveying
• Astronomy
• Navigation
• Building
• Geometry was initially the science of
• measuring land

4
Alexandria, Egypt

• Alexander the Great conquered Egypt
• The city Alexandria was founded in his honour
• Ptolemy, one of Alexanders generals, founded the
  Library and the Museum of Alexandria
• The Library- contained about 600,000 papyrus
  rolls
• The Museum - important center of learning,
  similar to Platos academy

5
Euclid of Alexandria

• He lived in Alexandria, Egypt between 325-265BC
• Euclid is the most prominent mathematician of
  antiquity
• Little is known about his life
• He taught and wrote at the Museum and Library of
  Alexandria

6
The Three Theories

• We can read this about Euclid
• Euclid was a historical character who wrote the
  Elements and the other works attributed to him
• Euclid was the leader of a team of mathematicians
  working at Alexandria. They all contributed to
  writing the 'complete works of Euclid', even
  continuing to write books under Euclid's name
  after his death
• Euclid was not an historical character.The
  'complete works of Euclid' were written by a team
  of mathematicians at Alexandria who took the name
  Euclid from the historical character Euclid of
  Megara who had lived about 100 years earlier

7
The Elements

• It is the second most widely published book in
  the world, after the Bible
• A cornerstone of mathematics, used in schools as
  a mathematics textbook up to the early 20th
  century
• The Elements is actually not a book at all, it
  has 13 volumes

8
The Elements- Structure

• Thirteen Books
• Books I-IV ? Plane geometry
• Books V-IX ?Theory of Numbers
• Book X ? Incommensurables
• Books XI-XIII ?Solid Geometry
• Each books structure consists of definitions,
  postulates, theorems

9
Book I

• Definitions (23)
• Postulates (5)
• Common Notations (5)
• Propositions (48)

10
The Four Postulates

• Postulate 1
• To draw a straight line from any point to any
  point.
• Postulate 2
• To produce a finite straight line continuously in
  a straight line.
• Postulate 3
• To describe a circle with any centre and
  distance.
• Postulate 4
• That all right angles are equal to one another.

11
The Fifth Postulate

• That, if a straight line falling on two straight
  lines makes the interior angles on the same side
  less than two right angles, the two straight
  lines, if produced indefinitely, meet on that
  side on which are the angles less than the two
  right angles.

12
Troubles with the Fifth Postulate

• It was one of the most disputable topics in the
  history of mathematics
• Many mathematicians considered that this
  postulate is in fact a theorem
• Tried to prove it from the first four - and failed

13
Some Attempts to Prove the Fifth Postulate

• John Playfair (1748 1819)
• Given a line and a point not on the line, there
  is a line through the point parallel to the given
  line
• John Wallis (1616-1703)
• To each triangle, there exists a similar triangle
  of arbitrary magnitude.

14
Girolamo Saccheri (16671733)

• Proposed a radically new approach to the problem
• Using the first 28 propositions, he assumed that
  the fifth postulate was false and then tried to
  derive a contradiction from this assumption
• In 1733, he published his collection of theorems
  in the book Euclid Freed of All the Imperfections
• He had developed a body of theorems about a new
  geometry

15
Theorems Equivalent to the Parallel Postulate

• In any triangle, the three angles sum to two
  right angles.
• In any triangle, each exterior angle equals the
  sum of the two remote interior angles.
• If two parallel lines are cut by a transversal,
  the alternate interior angles are equal, and the
  corresponding angles are equal.

16
Euclidian Geometry

• The geometry in which the fifth postulate is true
• The interior angles of a triangle add up to 180º
• The circumference of a circle is equal to 2?R,
  where R is the radius
• Space is flat

17
Discovery of Hyperbolic Geometry

• Made independently by Carl Friedrich Gauss in
  Germany, Janos Bolyai in Hungary, and Nikolai
  Ivanovich Lobachevsky in Russia
• A geometry where the first four postulates are
  true, but the fifth one is denied
• Known initially as non-Euclidian geometry

18
Carl Friedrich Gauss (1777-1855)

• Sometimes known as "the prince of mathematicians"
  and "greatest mathematician since antiquity",
• Dominant figure in the mathematical world
• He claimed to have discovered the possibility of
  non-Euclidian geometry, but never published it

19
János Bolyai(1802-1860)

• Hungarian mathematician
• The son of a well-known mathematician, Farkas
  Bolyai
• In 1823, Janos Bolyai wrote to his father saying
  I have now resolved to publish a work on
  parallels I have created a new universe from
  nothing
• In 1829 his father published Jonos findings, the
  Tentamen, in an appendix of one of his books

20
Nikolai Ivanovich Lobachevsky(1792-1856)

• Russian university professor
• In 1829 he published in the Kazan Messenger, a
  local publication, a paper on non-Euclidian
  geometry called Principles of Geometry.
• In 1840 he published Geometrical researches on
  the theory of parallels in German
• In 1855 Gauss recognized the merits of this
  theory, and recommended him to the Gottingen
  Society, where he became a member.

21
Hyperbolic Geometry

• Uses as its parallel postulate any statement
  equivalent to the following
• If  l is any line and P is any point not on l ,
  then there exists at least two lines through P
  that are parallel to l .

22
Practical Application of Hyperbolic Geometry

• Einstein stated that space is curved and his
  general theory of relativity uses hyperbolic
  geometry
• Space travel and astronomy

23
Differences Between Euclidian and Hyperbolic
Geometry

• In hyperbolic geometry, the sum of the angles of
  a triangle is less than 180
• In hyperbolic geometry, triangles with the same
  angles have the same areas
• There are no similar triangles in hyperbolic
  geometry
• Many lines can be drawn parallel to a given line
  through a given point. 

24
Georg Friedrich Bernhard Riemann

• His teachers were amazed by his genius and by his
  ability to solve extremely complicated
  mathematical operations
• Some of his teachers were Gauss,Jacobi,
  Dirichlet, and Steiner
• Riemannian geometry

25
Elliptic Geometry (Spherical)

• All four postulates are true
• Uses as its parallel postulate any statement
  equivalent to the following
• If  l is any line and P is any point not on
• l then there are no lines through P that are
  parallel to l.

26
Specific to Spherical Geometry

• The sum of the angles of any triangle is always
  greater than 180
• There are no straight lines. The shortest
  distance between two points on the sphere is
  along the segment of the great circle joining
  them

27
The Three Geometries