|¢º Characteristic of Greek Mathematics|
|¢º Pythagorean Mathematics|
|¢º The Three Famous Problems|
|¢º Greek Mathematics After Euclid|
¡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.
¡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.
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.
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.
¡Ý Duplication, Trisection, and Quadrature: The Greeks regarded logical thinking very highly.
They considered hight system of knowledge as important: not practical value.
1.The duplication of the cube, or the problem of constructing the edge of a cube having twice the volume of a given cube.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'¥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,
3¡¤1/7 < ¥ð < 3¡¤10/71
Archimedes used the approximate value of '¥ð' as 3.14.
¡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.
A1 Things that are equal to the same thing are also equal to one another.
¡ßGreek 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 parobola(curve) and chord (straight line).
But the flower of science that is mathematics began to wither. Rome ruined Greek culture.
In mathemtics especially, Rome didn't obtain good results except quinary.
The Roman Empire only assimilate and copy the conquered culture of greece, Eqypt and Carthage.
Althought the pursuit of learning weakened, Alexandrias 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 astronnomers at those times were Aristarchos (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 mose 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.
<Syntaxis Mathematica> is maybe the best book about astronomy written by Claudius Ptolemy in Alexandria about 150 A.D.
Arabians translated the book as <Almagest> which was regarded as a criterional 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 fater of algebra' who studied 'theory of numbers' and equation(primarily linear and quadratic)
Pappus wrote <Mathematical collection> about Greek geometry. Hypatia, doughter of annotator Theon was also famous mathematician. As the Alexandrian School was burned by Arabians in 641. After this incident, the glorous and brilliant Greek mathematics disappeared in the darkness.