Transcript The Word Geometry

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 3
To describe a circle with any centre and distance.
Postulate 4
That 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 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
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