GREEK MATHEMATICS. Greek was very clumsy in writing down the numbers. They didnt like algebra. They found it very hard to write down equations or number.

GREEK GREEK MATHEMATICSMATHEMATICS

Greek was very clumsy in writing down the Greek was very clumsy in writing down the numbers.numbers.

They didn’t like algebra.They didn’t like algebra.

They found it very hard to write down They found it very hard to write down equations or number problems.equations or number problems.

Instead Greek mathematicians were focused Instead Greek mathematicians were focused on geometry, and used geometric methods on geometry, and used geometric methods to solve problems that you might use to solve problems that you might use algebra for.algebra for.

Greek mathematicians were very Greek mathematicians were very interested in proving that certain interested in proving that certain mathematical ideas were true.mathematical ideas were true.

They spent a lot of time using They spent a lot of time using geometry to prove that things were geometry to prove that things were always true,even thoughpeople like always true,even thoughpeople like Egyptians and Babylonians already Egyptians and Babylonians already knew that they were true most of the knew that they were true most of the time away.time away.

The Greeks in general were very interesed The Greeks in general were very interesed in rationality, in things making sense and in rationality, in things making sense and

handing together.handing together.

Music were very important for them, Music were very important for them, because it followed strict rules to produce because it followed strict rules to produce

beauty.beauty.

Some famous Greek mathematicians were:Some famous Greek mathematicians were:

Pythagoras, Aristotle, Anaxagoras, Thales, Pythagoras, Aristotle, Anaxagoras, Thales, Antiphon, Archytas, Democritus, Euclid, Antiphon, Archytas, Democritus, Euclid,

Hipocrates, Plato, Xenocrates, Hipocrates, Plato, Xenocrates, Zeno,Socrates...Zeno,Socrates...

Early Greek Apppreciaton of Early Greek Apppreciaton of Geometric FormsGeometric Forms

-use of crude parallels-use of crude parallels

-less crude and more elaborate forms-less crude and more elaborate forms

-more delicate forms-more delicate forms

Greek AlgebraGreek Algebra

The Greeks proved thatThe Greeks proved that(a+b)(a+b) ²² == a²+2ab+b²a²+2ab+b²

They had no algebraic shorthand and consider They had no algebraic shorthand and consider only lines and rectangles instead of numbers only lines and rectangles instead of numbers

and products.and products.They knew such other identities asThey knew such other identities as

(a+b) (a-b) = a(a+b) (a-b) = a²²-b-b²²a(x+y+z) = ax+ay+aza(x+y+z) = ax+ay+az

(a-b) (a-b) ²² = a = a²²-2ab+b-2ab+b²²They could complete the square of binomial They could complete the square of binomial

expressionexpressionaa²±²±2ab2ab

Origin of Greek MathematicsOrigin of Greek Mathematics

-three important periods in the development of -three important periods in the development of Greek mathematicsGreek mathematics

The periods may be characterized as:The periods may be characterized as:

First - the one subject to the inluence of PythagorasFirst - the one subject to the inluence of Pythagoras

Second - the one dominated by Plato and his schoolSecond - the one dominated by Plato and his school

Third- the one in which the Alexandrian School Third- the one in which the Alexandrian School flourished in Grecian Egypt and extended its flourished in Grecian Egypt and extended its

influence to Sicily, the influence to Sicily, the ǼǼgean Islands and gean Islands and PalestinePalestine

ThalesThales

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

-merchant, statesman, mathematician, -merchant, statesman, mathematician, astronomer, philopherastronomer, philopher

Arithmetic of ThalesArithmetic of Thales

-he knew many number relations-he knew many number relations

-in his work is founding deductive -in his work is founding deductive geometrygeometry

Geometry of ThalesGeometry of Thales

-he is credited with a few of the simplest -he is credited with a few of the simplest propositions relating to the plane figurespropositions relating to the plane figures

1. Any circle is bisected by its diameter.1. Any circle is bisected by its diameter.2.The angles at the base of an isosceles triangle are 2.The angles at the base of an isosceles triangle are

equal.equal.3. When two lines intersect, the vertical angles are 3. When two lines intersect, the vertical angles are

equal.equal.4. An angle in a semicircle is a right angle.4. An angle in a semicircle is a right angle.

5. The sides of similar triangles are proportional.5. The sides of similar triangles are proportional.6. Two triangles are congruent if they have two 6. Two triangles are congruent if they have two

angles and a side respectively equal.angles and a side respectively equal.

-his great contribution lay in suggesting a geometry -his great contribution lay in suggesting a geometry of lines and in making the subject abstractof lines and in making the subject abstract

-he gave the idea of a logical proof as applied to -he gave the idea of a logical proof as applied to geometrygeometry

AnaximanderAnaximander

-the leadership of the Jonian School-the leadership of the Jonian School

-he brought the gnomon in Greece, -he brought the gnomon in Greece, and used for determining noonand used for determining noon

PythagorasPythagoras

-he had been Thales pupil-he had been Thales pupil-the familiar proposition in -the familiar proposition in

geometry that bears his geometry that bears his name was known, as name was known, as

already started, in India, already started, in India, China and EgyptChina and Egypt

-he had two groups of the -he had two groups of the disciples: the hearers and disciples: the hearers and

the methematiciansthe methematicians-he asserted that unity is the -he asserted that unity is the

essence of numberessence of number

Geometry of PythagorasGeometry of Pythagoras

-he investigated his theorems from the -he investigated his theorems from the immaterial and intellectual point of viewimmaterial and intellectual point of view

-he discovered the theory of irational -he discovered the theory of irational quantities and the construction of the quantities and the construction of the

mundane figuresmundane figures

-he defined a point as unity having position-he defined a point as unity having position

ZenoZeno

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

faster than tortoise.faster than tortoise.

AgatharchusAgatharchus

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

SocratesSocrates

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

geometry.geometry.

EnopidesEnopides

He discovered two problems of Euclid, one He discovered two problems of Euclid, one referring to the drawing of a perpendicular referring to the drawing of a perpendicular to a given line from an external point,and to a given line from an external point,and

the other referring to the making of an the other referring to the making of an angle equal to a given anlge.angle equal to a given anlge.

DemocritusDemocritus

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

Hippias of ElisHippias of Elis

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

quadratrix.quadratrix.

HippocratesHippocrates

He arranged the propositions of He arranged the propositions of geometry in a scientific fashion.geometry in a scientific fashion.

He discovered the first case of He discovered the first case of quadrature of a curvilinear figure, quadrature of a curvilinear figure,

namely, the proof that the sum of the namely, the proof that the sum of the two shaded lines here shown is equal two shaded lines here shown is equal

to the shaded triangle.to the shaded triangle.

The Method of ExhaustionThe Method of Exhaustion

-the area between a -the area between a curvilinear figure curvilinear figure

(e.g. a circle) and a (e.g. a circle) and a rectilinear figure (e.g. rectilinear figure (e.g. an inscribed regular an inscribed regular polygon) could be polygon) could be

aproximately aproximately exhausted by exhausted by increasing the increasing the

number of sides of number of sides of the latterthe latter

AntiphonAntiphon

He inscribed a regular polygon in a He inscribed a regular polygon in a circle,doubled the number of sides, circle,doubled the number of sides,

and continued doubling until the and continued doubling until the sides finally coincided with the circle.sides finally coincided with the circle.

Since he could construct a square Since he could construct a square equivalent to any polygon, he could equivalent to any polygon, he could

then construct a square equivalent to then construct a square equivalent to the circle; that is, he could “square the circle; that is, he could “square

the circle”.the circle”.

ArchytasArchytas

1. If a perpendicular is drawn to the 1. If a perpendicular is drawn to the hypotenuse from the vertex of the right hypotenuse from the vertex of the right

angle of a right angled triangle, each side angle of a right angled triangle, each side is the mean proportional betwen the is the mean proportional betwen the hipotenuse and its adjecent segment.hipotenuse and its adjecent segment.2. The perpendicular is the mean 2. The perpendicular is the mean

proportional between the segments of the proportional between the segments of the hypotenuse.hypotenuse.

3. If the perpendicular from the vertex of a 3. If the perpendicular from the vertex of a triangle is the mean proportional between triangle is the mean proportional between

the segments of the opposite side, the the segments of the opposite side, the angle at the vertex is a right angle.angle at the vertex is a right angle.

4. If two chords intersect, the rectangle of 4. If two chords intersect, the rectangle of the segments of one is equivalent to the the segments of one is equivalent to the rectangle of the segments of the other.rectangle of the segments of the other.

5. Angles in the same segment of a circle 5. Angles in the same segment of a circle are equal.are equal.

6. If two planes are perpendicular to a third 6. If two planes are perpendicular to a third plane their line of intersection is plane their line of intersection is

perpendicular to that plane and also to perpendicular to that plane and also to their lines of intersection with that plane.their lines of intersection with that plane.

TheTheææte’tuste’tus

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

PlatoPlato

-the method of analysis-the method of analysis-interested in arithmetic-interested in arithmetic-mystcism of numbers-mystcism of numbers-60 - Platonic number-60 - Platonic number

-accurate definitions, clear assumptions, -accurate definitions, clear assumptions, logical prooflogical proof

Speusip’pusSpeusip’pus

-wrote upon Pythagorean numbers-wrote upon Pythagorean numbers

-wrote upon proportion-wrote upon proportion

-rare elegance the subjects of linear, -rare elegance the subjects of linear, polygonal, plane, and solid numberspolygonal, plane, and solid numbers

Xenoc’ratesXenoc’rates

-deified unity and duality-deified unity and duality

-assumed the existence of indivisible -assumed the existence of indivisible lineslines

Ar’istotleAr’istotle

He wrote two works of a He wrote two works of a mathematical nature.mathematical nature.

Continuity:Continuity:

““A thing is continuous A thing is continuous when of any two when of any two

successive parts the successive parts the limits at which they limits at which they

touch are one and the touch are one and the same and are, as the same and are, as the

word implies, held word implies, held together.”together.”