Greek Mathematics
The Greek Mathematics: Demonstrative Geometry
· Characteristic of Greek Mathematics
· Euclid's
· Greek Mathematics After Euclid
· Characteristic of Greek Mathematics
In the 600 B.C. Mathematics was focused as a study and a science in the ancient Greek as a matter of course in China, India and Babylonia and to learn Geometry in Egypt. Thales, Pythagoras and Plato in Greek studied in Egypt and joined with Egypt culture Greek produced achievements at mathematics formed a term of now civilization accepting the Egypt culture. That is "Elements" of Euclid, "The Theory of conic sections" of Apollonius, and "Arithmetical" of Diophantus and many research achievements of Archimedes. Many scholar represented as Aristotle. Plato focused only philosophy and mathematics. The story, Plato wrote "NO one knows Geometry, No admission" at the entrance to a hall, is famous. Euclid is known affected by Aristotle and Plato. His "Elements" is the first arranged and systematized book logically and had been used as a textbook toward the end of the 1800's in Europe. This book showed the closed to the present mathematics toward 300 B.C. demonstrating a proposition from the axiom in the view of today, this had many defects, but this had affected after that time. However, Greek mathematics was remarkable theoretically, but unremarkable in the field of number and calculus. The research in Algebra of Diophantus was remarkable. After that time, Europe had accepted arithmetic and Algebra from India and east countries until 900's.
Pythagorean mathematics
· The Pythagorean philosophy rested on the assumption
what whole number is the cause of the various qualities of man and
matter. This led to an exaltation and study of number properties,
and arithmetic (considered as the theory of numbers), along with geometry,
music, and spheres (astronomy), constituted the fundamental liberal arts of the
Pythagorean program of study. Because Pythagoras' teaching was entirely oral, and because
of the brotherhood's custom of referring all discoveries back to the revered
founder, it is now difficult to know just which mathematical findings should be
credited to Pythagoras himself and which to other members of the fraternity.
Pythagorean
Arithmetic: Pythagoras and his followers, in
conjunction with the fraternity's philosophy, took the first steps in the
development of number theory, and at the same time laid much of the basis of
future number mysticism. Amicable, or friendly, numbers.
Two numbers are amicable number if
each is the sum of the proper divisors of the other. For example,
284 and 220, constituting the pair ascribed to Pythagoras, are
amicable. They are the perfect, deficient, and abundant
numbers. A number is perfect if it is the sum of its
proper divisors, deficient if it exceeds the sum of its proper divisors,
and abundant if it is less than the sum of its proper
divisors. So God created the world in six days, a perfect number, Since 6= 1 + 2 + 3.So
people those times told fortunes with that number and they used an amulet to
avert evils, the figurate numbers were found by the
Pythagorean. These numbers, considered as the number of dots in certain geometrical configurations,
represent a link between geometry and arithmetic.
· As a last and very remarkable discovery about numbers, made by the Pythagoreans, we might mention the dependence of musical intervals upon numerical ratios. The Pythagoreans found that for strings under the same tension, the lengths should be 2 to 1 for the octave 3 to 2 for the fifth, and 4 to 3 for the fourth. These results, the first recorded facts in mathematical physics, led the Pythagoreans to initiate the scientific study of musical scales.
· Pythagorean Theorem and Discovery of Irrational Magnitudes: Pythagoras says that the square on the hypotenuse of a right triangle is equal to the sum of the squares on the two legs. Since Pythagoras' time, many different proofs of the Pythagorean Theorem have been supplied. In the second edition of his book, The Pythagorean Proposition, E.S. Loomis has collected and classified 370 demonstrations of this famous theorem.
Roughly saying the Pythagorean Theorem
is about width but actually about the length of three sides to make a right
triangle. The
problem of finding integers a, b, c that can represent the legs and
hypotenuse of a right triangle. A triple of numbers of this sort is
known as a Pythagorean triple.
By this theorem there exist
incommensurable line segments  that is, line segments having no common unit of
measure. The discovery of irrational number is the milestone in mathematics history. But the discovery ran
counter to the Pythagorean philosophy  'everything is decided by integer.'
The discovery
of the existence of irrational numbers was surprising and disturbing to the
Pythagoreans.
· The Regular Solids: A polyhedron is said to be regular if its
faces are congruent regular polygons and if its polyhedral angles are all
congruent.
There is the tetrahedron with four
triangular faces, the hexahedron, or cube, with six square faces, the
octahedron with eight triangular faces, the dodecahedron with twelve pentagonal
faces, and the icosahedron with twenty triangular faces. Plato mystically
associates fire, earth, air, universe, and water to each regular solid.
The Three Famous Problems
· The
first three centuries of Greek mathematics, commencing with the initial efforts
at demonstrative geometry by Thales about 600 B.C. and culminating with the
remarkable Elements of Euclid about 300 B.C.
One can notice three important and
distinct lines of development during the first 300 years of Greek
mathematics. First, we have the development of the material that
ultimately was organized into the Elements.
There is the development of notions
connected with infinitesimals and with limit and summation processes. The third
line of development is that of higher geometry,
or the geometry of curves other than the circle and straight line, and of
surfaces other than the sphere and plane. Curiously enough, most of
this higher geometry originated in continued attempts to solve three now famous construction problems. By virtue of
this challenge, the development and creation of new mathematics were made.
· Duplication, Trisection, and Quadrature: The Greeks regarded logical thinking very highly. They considered height system of knowledge as important: not practical value. Unexpectedly they couldn't solve easy problems Typical examples were duplication, trisection and quadrature.
1. The duplication of the cube, or the problem of constructing the edge of a cube having twice the volume of a given cube.
2. The trisection of an angle, or the problem of dividing a given arbitrary angle into three equal parts.
3. The quadrature of the circle, or the problem of constructing a square having an area equal to that of a given circle.
People should solved these three problems by using unmarked straightedges and compasses. The impossibility of the three constructions, under the selfimposed limitation that only the straightedge and compasses could be used, was not established until the nineteenth century, more than 2000 years after the problems were first conceived. The energetic search for solutions to these three problems profoundly influenced Greek geometry and led to many fruitful discoveries, such as that of the conic sections, many cubic and quartic curves, and several transcendental curves. A much later outgrowth was the development of portions of the theory of equations concerning domains of rationality, algebraic numbers, and group theory.
· A History of π: 'π' is used to calculate the area of a circle which is called ratio of circumference of circle to its diameter.
π: the ratio of the circumference of a circle to its
diameter
l : periphery of a circle
2r : diameter of a circle.
'π'is fixed to any circles.
The man who used ' π ' for the first
time was Euler, Leonhard. If we, actually, want to
calculate the area of a circle, we should know the value of ' π '.
Unable to reckon the accurate value
of ' π ' (nobody can do that), Archimedes got
the approximate value of ' π '.
Starting from the regular inscribed
and circumscribed sixsided polygons, Archimedes drew regular inscribed
96sided polygons to the circle, and he drew regular circumscribed 96sided
polygons to it.
Then, the circumference of a circle
is longer than that of the regular inscribed 96  sided polygons and is smaller
than that of the regular circumscribed 96  sided
polygons. Thus,
(circumference of an inscribed 96side polygons) < 2 π r (circumference of a circumscribed 96  sided polygons)
3 1/7 < π < 3 10/71
This value is quite accurate 3.14084 <
π <3.142858
Archimedes used the approximate
value of ' π ' as 3.14.
Approximate value of ' π '.
Ahmes' (a.1650 B.C) Papyrus 
π 3.16 

Arithmetic in Nine section 
π 3 

Archimedes 
π 3.14 

Tsu Chung – chih (430501) 
π 3.1415929 




Euclid's
· Although
Euclid was the author of at least ten works (fairly complete texts of five of
these have come down to us), his reputation rests mainly on his Elements. It appears that this
remarkable work immediately and completely superseded all previous Elements;
in fact, no trace remains of the earlier efforts. As soon as the
work appeared, it was accorded the highest respect, and from Euclid's
successors on up to modern times, the mere citation of Euclid's book and
proposition numbers was regarded as sufficient to identify a particular theorem
or construction. No work, except the Bible, has been more widely
used, edited, or studied, and probably no work has exercised a greater influence
on scientific thinking. Over one thousand editions of Euclid's Elements have appeared since the first one printed in 1482; for more than two millennia,
this work has dominated all teaching of geometry.
Contrary to widespread impressions,
Euclid's Elements is not devoted to geometry alone, but contains much
number theory and elementary (geometric) algebra. The work is
composed of thirteen books with a total of 465 propositions.
American highschool plane and solid geometry texts contain much of
the material found in Books π °, π ², π ³, π µ, XI, and XII.
Certainly one of the greatest
achievements of the early Greek mathematicians was the creation of the postulation
form of thinking. In order to establish a statement in a deductive
system, one must show that the statement is a necessary logical consequence of
some previously established statements.
These, in their turn, must be
established from some still more previously established statements, and so
on. Since the chain cannot be continued backward indefinitely, one
must, at the start, accept some finite body of statements without proof or else
commit the unpardonable sin of circularity, by deducing statement A from
statement B and then later B from A. These initially assumed
statements are called the postulates, or axioms, of the discourse, and all other statements
of the discourse must be logically implied by them. Where the
statements of a discourse are so arranged, the discourse is said to be
presented in postulation form.
So great was the impression made by
the formal aspect of Euclid's Elements on following generations that the
work became a model for rigorous mathematical demonstration? It is not certain precisely what
statements Euclid assumed for his postulates and axioms, nor, for that matter,
exactly how many he had, for changes and additions were made by subsequent
editors. There is fair evidence, however, that he adhered to the
second distinction and that he probably assumed the equivalents of the
following ten statements, five "axioms," or common notions, and five
geometric "postulates":
A1 Things that are equal to the same thing are also equal to one another.
A2 If equals be added to equals, the wholes are equal.
A3 If equals be subtracted from equals, the remainders are equal
A4 Things that coincide with one another are equal to one another.
A5 The whole is greater than the part.
P1 It is possible to draw a straight line from any point to any other point.
P2 It is possible to produce a finite straight line indefinitely in that straight line.
P3 It is possible to describe a circle with any point as center and with a radius equal to any to finite straight line drawn from the center.
P4 All right angles are equal to one another.
P5 If a straight line intersects two straight lines so as to make the interior angles on one side of it together less than two right angles, these straight lines will intersect, if indefinitely produced, on the side on which are the angles which are together less than two right angles.
Greek Mathematics after Euclid
· One
of the greatest mathematicians of all time, and certainly the greatest of
antiquity, was Archimedes, showed his typical strict arguments in calculating
the area of a figure which was surrounded by parabola (curve) and chord (straight
line). This
way of reckoning provide the base of modern integral calculus. Great was his
Knowledge about a circular cylinder and a sphere with Euclid and Archimedes in
mathematics in 300 B.C. was a great mathematician Apollonius (ca. 200 B.C.)
argued about which made him a great geometrician. He stated conic sections as cut
stains from circular corns.
These parts were omitted in but
Apollonius compiled many fields called ،¸the theory of
quadratic curve،¹. This method reminds us of the
analytic geometry.
· Archimedes
was killed by a roman soldier in 212 B.C. The Roman Empire
conquered many city states in Greece and dominated the Mediterranean Sea.
But the flower of science that is
mathematics began to wither. Rome ruined Greek culture. In mathematics especially, Rome
didn't obtain good results except binary. The Roman Empire only assimilate and
copy the conquered culture of Greece, Egypt and Carthage. Although the pursuit of learning
weakened, Alexandrians was the center of learning and culture then.
As trade was frequent between the
West and the East, people came to need the art of navigation so they studied
astronomy and trigonometry.
Introduced was logistic system which
represent angle today. Representative astronomers at those times
were Aristarchus (280 B.C.) Eratosthenes and Hipparchus (150
B.C.) Eratosthenes, working at a library in Alexandria, computed the
size of earth by measuring altitude of the sun on summer solstice.
Maybe moose distinguished astronomer
in Ancient Age, Hipparchus drew up the logistic
system. He
made a kind of table and it is called trigonometric function today and also
studied spherical astronomy is maybe the best book about astronomy written by Claudius
Ptolemy in Alexandria about 150 A.D.
Arabians translated the book as which
was regarded as a criterial book of astronomy from Copernicus to Kepler. Theoretical
mathematics of Greece and practical mathematics of the orient coexisted at
those times.
The representative mathematicians
were Heron (250~150 B.C.) and Diophantus. The former is famous for
its 'Heron's formula' referring to the area of a
triangle. The
latter is 'the father of algebra' who studied 'theory of numbers' and equation
(primarily linear and quadratic)
Pappas wrote about Greek
geometry. Hepatica, daughter of annotator Theon was also famous
mathematician. As the Alexandrian School was burned by Arabians in
641. After this incident, the glorious and brilliant Greek mathematics
disappeared in the darkness.