GREEK MATHEMATICS

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GREEK MATHEMATICS
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INTRODUCTION

• The beginnings of Greek mathematics originated
  from the 6th century BC to the 5th century AD
• The word mathematics comes from the Greek word
  µ???µa (mathema), meaning "subject of
  instruction

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PERIODS IN GREEK MATHEMATICS

• FIRST influenced by Pythagoras
• SECOND Plato and his school
• THIRD Alexandrian School flourished in Grecian
  Egypt and extended its influence to Sicily and
  Palestine

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GREEK NUMBERS

• Greeks had a variety of different ways of writing
  down numbers
• Some Greeks used a system based on writing the
  first letter of the word for that number
• For number ten Deka, they would draw a D to
  mean 10. (a delta, in the Greek alphabet)

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Some other numbers in greek symbols
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• Because the Greeks had very clumsy ways of
  writing down numbers, they didn't like algebra
• They were more focused on geometry, and used
  geometric methods to solve problems that you
  might use algebra for
• They found it very hard to write down equations
  or number problems.

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• Greek mathematicians were very interested in
  proving that certain mathematical ideas were
  true.
• They spent a lot of time using geometry to prove
  that things were always true,even thoughpeople
  like Egyptians and Babylonians already knew that
  they were true most of the time away.

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• Because the Greeks had very clumsy ways of
  writing down numbers, they didn't like algebra
• They were more focused on geometry, and used
  geometric methods to solve problems that you
  might use algebra for
• They found it very hard to write down equations
  or number problems.

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MOST FAMOUS GREEK MATHEMATICIANS

• Thales
• Pythagoras
• Anaxagoras
• Democritus

• Aristotle
• Hipocrates
• Euclid
• Archimedes

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THALES (grc. Ta???)

• Born 624. BC in Miletus
• the first of the Greeks who took any scientific
  interest in mathematics in general
• Improved Egyptian mathematics

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THALES

• He knew many number relations
• In his work is the foundation of deductive
  geometry
• He is credited with a few of the simplest
  propositions relating to the plane figures
• His great contribution lay in suggesting a
  geometry of lines and in making the subject
  abstract
• He gave the idea of a logical proof as applied to
  geometry

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PROPOSITION RELATING PLANE FIGURES

• a circle is bisected by its diameter,
• the angles at the bases of any isosceles triangle
  are equal
• if two straight lines cut one another, the
  opposite angles are equal.
• if two triangles have two angles and a side in
  common, the triangles are identical.

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INTERCEPT THEOREM

• The ratios of any 2 segments on the first line
  equals the ratios of the according segments on
  the second line

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THALES THEOREM

• If AC is a diameter, then the angle at B is a
  right angle

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PHYTAGORAS (grc. ???a???a?)

• Born 570. BC in Samos
• Died 495. BC
• worked with abstract geometric objects and
  numbers
• gathered his school as a sort of mathematician
  secret brotherhood

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PHYTAGORAS THEOREM

• in a right triangle, the sum of the squares of
  the two right-angle sides will always be the same
  as the square of the hypotenuse

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TV screen size is measured diagonally across the
screen. A widescreen TV has an aspect ratio of
169, meaning the ratio of its width to its
height is 16/9. Suppose that a TV has a one inch
boundary one each side of the screen. If Joe has
a cabinet that is 34 inches wide, what is the
largest size wide screen TV that he can fit in
the cabinet?
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SQUARE NUMBERS

• These numbers are clearly the squares of the
  integers 1, 4, 9, 16, and so on. Represented by a
  square of dots

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PYTHAGORAS AND MUSIC

• musical notes could be translated into
  mathematical equations

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DEMOCRITUS (grc. ??µ????t?? )

• Born 460. BC, died 370.BC
• Famous atomist
• introduced the idea of an infinite number of
  points that make up the line

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• He observed that a cone or pyramid has one-third
  the volume of a cylinder or prism respectively
  with the same base and height

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Plato (428 BC 348 BC),
Philosopher, mathematician, student of Socrates,
writer of philosophical dialogues, and founder of
the Academy in Athens, the first institution of
higher learning in the Western World.
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Platos Cave Analogy
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• In Platos Divided Line, Mathematics falls under
  the following category
• Highest form of true knowledge
• Second highest form of true knowledge
• A form of belief, but not true knowledge
• A form of perception

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ARISTOTLE (grc. ???st?t???? )

• Born 384. BC, died 322. BC
• Greek philosopher, a student of Plato and teacher
  of Alexander the Great

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• For him the base of mathematics is logic, but the
  nature of mathematical relations is completely
  specified by postulates that dictates the
  physical experience

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HIPPOCRATES (grc. ?pp????t?? )

• Lived from 460. BC to 377. BC
• an ancient Greek physician and was considered one
  of the most outstanding figures in the history of
  medicine

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HIPPOCRATUS PROBLEM

• He proved that the lune bounded by the arcs
  labeled E and F in the figure has the same area
  as does triangle ABO

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EUCLID (grc. ????e?d?? )

• Born 300. BC
• pioneer of axiomatics in geometry
• His work Elements fundamental work in the field
  of Greek mathematics
• influenced the development of mathematics in the
  next 20 centuries

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ELEMENTS

• written about 300 B.C.
• textbook that includes number theory
• the Euclidean algorithm for finding the greatest
  common divisor of two numbers

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• the first edition of the translation from Arabic
  into Latin 1482.

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The axiomatic method
The Elements begins with definitions and five
postulates. There are also axioms which Euclid
calls 'common notions'. These are not specific
geometrical properties but rather general
assumptions which allow mathematics to proceed as
a deductive science. For example Things which
are equal to the same thing are equal to each
other.
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Euclid's fifth postulate cannot be proven from
others, though attempted by many people. Euclid
used only 14 for the first 28 propositions of
the Elements, but was forced to invoke the
parallel postulate on the 29th. In 1823,Bolyai
and Lobachevsky independently realized that
entirely self-consistent "non-Euclidean
geometries" could be created in which the
parallel postulate did not hold.
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Our world is non Euclidean Restate the fifth
postulate Given a line and a point not on the
line, it is possible to draw exactly one line
through the given point parallel to the line.
Spherical geometry is just as real as Euclidean
geometry, but the theorems and general results
are very different. There are quite a few results
from Euclidean geometry that are completely false
in spherical geometry (and vice versa).
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ARCHIMEDES (grc. ????µ?d??)

• mathematician and inventor born 287. BC in
  Syracuse
• founder of quantitative physics
• as a mathematician, advocate of logical processes

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• He determined approximate values of some
  irrational numbers
• 1351/780gt gt265/153
• 28/7gt p gt223/71

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• A sphere has 2/3 the volume and surface area of
  its circumscribing cylinder
• A sphere and cylinder were placed on the tomb of
  Archimedes at his request

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LITERATURA

• Vladimir Devide Na izvorima matematike
• Dadic Žarko Povijest ideja i metoda u
  matematici i fizici ŠK, 1992.
• http//www.ibilio.org/expo/vatican.exhibit/exhibit/
  d-mathematics/Greek_math.html
• http//www.historyforkids.org

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Authors Ivana Pušic Dajana
Rudic Ines Malic