GREEK MATHEMATICS

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 μάθημα (mathema), meaning "subject of instruction“

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

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)

GREEK NUMBERS

Some other numbers in greek symbols

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.

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.

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.

MOST FAMOUS GREEK MATHEMATICIANS

Thales Pythagoras Anaxagoras Democritus

Aristotle Hipocrates Euclid Archimedes

THALES (grč. Θαλής)

Born 624. BC in Miletus

the first of the Greeks who took any scientific interest in mathematics in general

Improved Egyptian mathematics

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

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.

INTERCEPT THEOREM

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

THALES THEOREM

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

PHYTAGORAS (grč. Πυθαγόρας)

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

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

TV screen size is measured diagonally across the screen. A widescreen TV has an aspect ratio of 16:9, 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?

SQUARE NUMBERS

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

PYTHAGORAS AND MUSIC

musical notes could be translated into mathematical equations

DEMOCRITUS (grč. Δημόκριτος )

Born 460. BC, died 370.BC

Famous atomist introduced the

idea of an infinite number of points that make up the line

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

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.

Plato’s Cave Analogy

In Plato’s 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

ARISTOTLE (grč. Ἀριστοτέλης )

Born 384. BC, died 322. BC

Greek philosopher, a student of Plato and teacher of Alexander the Great

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

HIPPOCRATES (grč. Ἱπποκράτης )

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

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

EUCLID (grč. Εὐκλείδης )

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

ELEMENTS

written about 300 B.C. textbook that includes number

theory the Euclidean algorithm for

finding the greatest common divisor of two numbers

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

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.””

Euclid's fifth postulate cannot be proven from others, though attempted by many people.

Euclid used only 1—4 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.

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).

ARCHIMEDES (grč. Ἀρχιμήδης)

mathematician and inventor born 287. BC in Syracuse

founder of quantitative physics

as a mathematician, advocate of logical processes

He determined approximate values of some irrational numbers

1351/780> >265/153

28/7> π >223/71

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

LITERATURA Vladimir Devide: “Na izvorima

matematike” Dadić 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

Authors: Ivana Pušić Dajana Rudić Ines Malić