Greek mathematics

Greek mathematics By: Ines Burazin TomislavHaršanji Igor Sušić MateaUgrica

Introduction: • from the 7th century BC to the 4th century AD • around the shores of the Mediterranean • word "mathematics" – from the ancient Greek μάθημα (mathema) = "subject of instruction“ • influences: Babylonian and Egyptian civilizations • differences: respect of geometry and the idea of formal proof

Introduction: • achievements: • ideas on number theory • mathematical analysis • applied mathematics • approached close to integral calculus • two periods: • Classical • Hellenistic (4th century BC)

Greek numeral system • known as Attic or Herodianic numerals • fully developed by about 450 BC • in regular use possibly as early as the 7th Century BC • it is a base 10 system

The Three Classical Problems

Squaring the circle It is the challenge of constructing a square with the same area as a given circle by using only a finite number of steps with compass and straightedge.

Doubling the cube • also known as the Delian problem • impossible to solve, because ≈ 1.25992105 is not a constructible number • double the size of the altar for Apollo, which was a regular cube

The trisection of an angle • involves the solving of a cubic equation • something that cannot be done • 90° can be trisected • a general angle cannot (example 60°)

Famous Greek Mathematicians • THALES • PYTHAGORAS • ZENO OF ELEA • EUCLID • ARCHIMEDES • ERATOSTHENES

Thales of Miletus “The most difficult thing in life is to know yourself.”

Thales of Miletus • the first of the seven wise men of Greece • used geometry to solve problems such as calculating the height of pyramids and the distance of ships from the shore • first person known to have studied electricity • died at 78 in the 58th Olympiad (548–545)

Propositions • A circle is bisected by any diameter. • The base angles of an isosceles triangle are equal. • The angles between two intersecting straight lines are equal. • Two triangles are congruent if they have two angles and the included side equal. • An angle in a semicircle is a right angle.

Pythagoras “We ought so to behave to one another as to avoid making enemies of our friends, and at the same time to make friends of our enemies.”

Pythagoras • called Pytha-goras because the Pythian oracle predicted his birth • known as “the father of numbers” • Pythagoras and his students believed that everything was related to mathematics • everything could be predicted and measured in rhythmic patterns or cycles.

Figurate Numbers • Numbers geometrically constructed had a particular importance to the Pythagoreans. • Triangular numbers. These numbers are 1, 3, 6, 10, ... . The general form is the familiar 1+2+3+....+n=n(n+1)/2 • Square numbers These numbers are clearly the squares of the integers 1, 4, 9, 16, and so on. Represented by a square of dots • the well known formula 1 + 3 + 5 + . . . + (2n − 1) = n2.

Pythagorean spiral

Zeno of Elea • Zeno's paradoxes: • "Achilles and the Tortoise" • "Arrow paradox"

Zeno's paradoxes • paradoxes of Achilles and the Tortoise

Euclid “There is no royal road to geometry.”

Euclid • nobody knows much about Euclid's life anymore • often referred to as the "Father of Geometry“

Euclid • Five works by Euclid have survived to our day: • 1. The Elements - The Elements can be topically divided into four sections: • Books I-VI — Plane geometry • Books VII-IX — Theory of Numbers • Book X — Incommensurables • Book XI-XIII — Solid Geometry • 2. Data -includes geometric methods for the solution of quadratics. • 3. Division of Figures • 4. Phaenomena - on spherical geometry, • 5. Optics

Euclid’s theorem • b² = a · m • c² = a · n

Euclid

Archimedes “Eureka!” “Give me the place to stand, and I shall move the earth.” “Do not disturb my circles!”

Archimedes • from Syracuse • the most famous and probably the best mathematician of antiquity • made many discoveries in mathematics and physics

The Golden Crown • used his principle of buoyancy to determine whether the golden crown was less dense than solid gold

The Burning Mirror (Glass)

Eratosthenes • born in Cyrene • invented the armillary sphere

Eratosthenes' other contributions include: • The measurement of the Sun-Earth distance, now called the astronomical unit. • The measurement of the distance to the Moon. • The measurement of the inclination of the ecliptic with an angle error of 7'. • He compiled a star catalogue containing 675 stars, which was not preserved. • A map of the Nile's route as far as Khartoum. • A map of the entire known world, from the British Isles to Ceylon, and from the Caspian Sea to Ethiopia.

Thank you for your time!

Greek mathematics