¡ßCharacteristic 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 achivements at mathematics formed a term of now civilization
accepting the Egypt culture.
That is "Elements" of Euclid, "The Theory of conic sections" of Apollonius,
"Arithmetica" of
Diophantus and many reserch achivements 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
enterance 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 the that time.
However, Greek mathematics was remarkable theoretically, but unremarkable in the field of
number and calculus.
The reserch in Algebra of Diophantus was remarkable.
After that time, Europe had accepted arithmetic and Algebra from India and
east countries until 900's.
¡ß 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 spherics (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.
¡ÝPythagorean 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 theperfect
, 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,
Thefigulate 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.
¡Ý 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 acctually
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 aPythagorean 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 Pythogorean philosophy - 'everything is decided
by integer.'
The discovery of the existence of irrational numers was surprising and
disturbing to the Pythagoreans.
¡Ý 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.
¡ß 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 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.
¡Ý Duplication, Trisection, and Quadrature: The Greeks regarded logical thinking very highly.
They considered hight system of knowledge as important: not practical value.
Unexpectedly they couldn't solve easy problems Typical
exemples 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 impossibillity of the three constructions, under
the self-imposed 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 qroup theory.
¡Ý A History of ¥ð:
'¥ð' is used to calculate the area of a circle which is called ratio of
circumference of circle to its diameter.
¥ð : the ratio of the circumference of a circle to its diameter
l : pheriphery of a circle
2r : diameter of a circle.
'¥ð'is fixed to any circles.
The man who used '¥ð' for the first time was Euler, Leonhard.
If we, actually, want to calculate the area of a circle, we should know
the value of '¥ð'.
Unable to reckon the accurate value of '¥ð' (nobody can do that),
Archimedes got the approximate value of '¥ð'.
Starting from the regular inscribed and circumscribed six-sided polygons,
Archimedes drew regular inscribed 96-sided polygons to the circle, and he drew
regular circumscribed 96-sided polygons to it.
Then, the circumference of a circle is longer than that of the regular
inscribed 96 - sided plygons and is smaller than that of the regular
circumscribed 96 - sided polygons. Thus,
(circumference of an inscribed 96-side polygons) < 2¥ðr
(circumference of a circumscribed 96 - sided polygons)
3¡¤1/7 < ¥ð < 3¡¤10/71
This value is quite accurate ¡æ 3.14084 < ¥ð <3.142858
Archimedes used the approximate value of '¥ð' as 3.14.
Approximate value of '¥ð'.
| Ahmes'(a.1650 B.C) Papyrus | ¥ð ¡Ö 3.16 |
| Arithmetic in Nine section | ¥ð ¡Ö 3 |
| Archimedes | ¥ð ¡Ö 3.14 |
| Tsu Chung - chih(430-501) | ¥ð ¡Ö 3.1415929 |
¡ß Eicliod's <Elements>
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 influcnce 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 high-school plane and solid geometry texts contain much of
the material found in Books ¥°,¥²,¥³,¥µ,XI, and XII.
Certainly one of the greatest achievements of the early Greek mathematicians
was the creation of the postulational 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
postulational 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 equl 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.
