History of Mathematics

Euclidean Geometry -

Controversial Parallel Postulate

Anisoara Preda

Geometry

A branch of mathematics dealing with the properties of geometric objects

Greek word geos- earth

metron- measure

Geometry in Ancient Society

In ancient society, geometry was used for:

Surveying

Astronomy

Navigation

Building

Geometry was initially the science of

measuring land

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

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

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

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

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

Book I

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

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 3To describe a circle with any centre and distance. Postulate 4That all right angles are equal to one another.

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.

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

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.

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

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.

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

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

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

János Bolyai(1802-1860)

Hungarian mathematicianThe 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

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.

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 .

Practical Application of Hyperbolic Geometry

Einstein stated that space is curved and his general theory of relativity uses hyperbolic geometry

Space travel and astronomy

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. 

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

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.

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

The Three Geometries