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 Alexander’s 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 Plato’s 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 book’s 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 ( )
To each triangle, there exists a similar triangle of arbitrary magnitude.

14. Girolamo Saccheri (1667–1733)
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 2ΠR, 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