GREEK MATHEMATICS by 24FZkW

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									   GREEK
MATHEMATICS
Greek was very clumsy in writing down the numbers.
They didn’t like algebra.
They found it very hard to write down equations or
  number problems.
Instead Greek mathematicians were focused on
  geometry, and used geometric methods to solve
  problems that you might use algebra for.
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.
 The Greeks in general were very interesed in rationality, in
        things making sense and handing together.
Music were very important for them, because it followed strict
                  rules to produce beauty.
     Some famous Greek mathematicians were:
Pythagoras, Aristotle, Anaxagoras, Thales, Antiphon,
   Archytas, Democritus, Euclid, Hipocrates, Plato,
             Xenocrates, Zeno,Socrates...
Early Greek Apppreciaton of Geometric Forms


              -use of crude parallels
      -less crude and more elaborate forms
               -more delicate forms
                     Greek Algebra

                  The Greeks proved that
                    (a+b) ² = a²+2ab+b²
They had no algebraic shorthand and consider only lines and
         rectangles instead of numbers and products.
             They knew such other identities as
                     (a+b) (a-b) = a²-b²
                   a(x+y+z) = ax+ay+az
                     (a-b) ² = a²-2ab+b²
   They could complete the square of binomial expression
                            a²±2ab
             Origin of Greek Mathematics

    -three important periods in the development of Greek
                           mathematics
              The periods may be characterized as:
     First - the one subject to the inluence of Pythagoras
     Second - the one dominated by Plato and his school
Third- the one in which the Alexandrian School flourished in
    Grecian Egypt and extended its influence to Sicily, the
                   Ǽgean Islands and Palestine
                          Thales

 -the first of the Greeks who took any scientific interest in
                     mathematics in general
-merchant, statesman, mathematician, astronomer, philopher
               Arithmetic of Thales

          -he knew many number relations
    -in his work is founding deductive geometry


                Geometry of Thales

-he is credited with a few of the simplest propositions
                relating to the plane figures
           1. Any circle is bisected by its diameter.
  2.The angles at the base of an isosceles triangle are equal.
  3. When two lines intersect, the vertical angles are equal.
         4. An angle in a semicircle is a right angle.
      5. The sides of similar triangles are proportional.
6. Two triangles are congruent if they have two angles and a
                      side respectively equal.

-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
               Anaximander




      -the leadership of the Jonian School
-he brought the gnomon in Greece, and used for
                 determining noon
  Pythagoras

      -he had been Thales pupil
-the familiar proposition in geometry
  that bears his name was known, as
    already started, in India, China
               and Egypt
 -he had two groups of the disciples:
  the hearers and the methematicians
-he asserted that unity is the essence
               of number
              Geometry of Pythagoras


 -he investigated his theorems from the immaterial and
                 intellectual point of view
-he discovered the theory of irational quantities and the
           construction of the mundane figures
      -he defined a point as unity having position
                        Zeno

           It is important Zeno’s paradoks.
It talks about that Achilles cuold not pass a tortoise,
        even thogh he went faster than tortoise.

                    Agatharchus

He showed how to make use of notion of projection
             upon a plane surface.
                      Socrates

He should be mentioned in connection with the early
         development of a logical geometry.

                     Enopides

He discovered two problems of Euclid, one referring to
  the drawing of a perpendicular to a given line from
    an external point,and the other referring to the
       making of an angle equal to a given anlge.
                      Democritus

   He was the first to show the relation between the
   volume of a cone and that of a cylinder of equal base
   and equal height, and similarly for the pyramid and
                          prism.

                    Hippias of Elis

He invented a simple device for trisecting any angle, this
         device being known as the quadratrix.
                     Hippocrates

He arranged the propositions of geometry in a scientific
                        fashion.
    He discovered the first case of quadrature of a
  curvilinear figure, namely, the proof that the sum of
     the two shaded lines here shown is equal to the
                     shaded triangle.
           The Method of Exhaustion

 -the area between a
curvilinear figure (e.g. a
 circle) and a rectilinear
figure (e.g. an inscribed
 regular polygon) could
     be aproximately
exhausted by increasing
 the number of sides of
         the latter
                      Antiphon

He inscribed a regular polygon in a circle,doubled the
   number of sides, and continued doubling until the
         sides finally coincided with the circle.
 Since he could construct a square equivalent to any
 polygon, he could then construct a square equivalent
    to the circle; that is, he could “square the circle”.
                      Archytas

1. If a perpendicular is drawn to the hypotenuse from
       the vertex of the right angle of a right angled
    triangle, each side is the mean proportional betwen
          the hipotenuse and its adjecent segment.
2. The perpendicular is the mean proportional between
               the segments of the hypotenuse.
3. If the perpendicular from the vertex of a triangle is
    the mean proportional between the segments of the
       opposite side, the angle at the vertex is a right
                             angle.
4. If two chords intersect, the rectangle of the segments
    of one is equivalent to the rectangle of the segments
                          of the other.
   5. Angles in the same segment of a circle are equal.
6. If two planes are perpendicular to a third plane their
   line of intersection is perpendicular to that plane and
     also to their lines of intersection with that plane.
                       Theæte’tus

He discovered a cosiderable part of elementary geometry and
                      wrote upon solids.

                           Plato

                  -the method of analysis
                  -interested in arithmetic
                    -mystcism of numbers
                   -60 - Platonic number
   -accurate definitions, clear assumptions, logical proof
                    Speusip’pus
          -wrote upon Pythagorean numbers
                -wrote upon proportion
-rare elegance the subjects of linear, polygonal, plane,
                    and solid numbers


                     Xenoc’rates

             -deified unity and duality
      -assumed the existence of indivisible lines
Ar’istotle

    He wrote two works of a
       mathematical nature.
           Continuity:
 “A thing is continuous when of
   any two successive parts the
    limits at which they touch
   are one and the same and are,
     as the word implies, held
             together.”

								
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