# 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
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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