弗. The Greek Mathematics : Demonstrative Geometry

Ⅱ Characteristic of Greek Mathematics

Ⅱ Pythagorean Mathematics

Ⅱ The Three Famous Problems

Ⅱ Euclid's<Elements>

Ⅱ 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 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.

﹣ 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.
≡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 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.

≡ 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.

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≡ 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.

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﹣ 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.

≡ 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.

≡ A History of 休: '休' is used to calculate the area of a circle which is called ratio of circumference of circle to its diameter.

休 : the ratio of the circumference of a circle to its diameter
l : pheriphery of a circle
2r : diameter of a circle.

    '休'is fixed to any circles.
    The man who used '休' for the first time was Euler, Leonhard.     If we, actually, want to calculate the area of a circle, we should know the value of '休'.
    Unable to reckon the accurate value of '休' (nobody can do that), Archimedes got the approximate value of '休'.
  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休r (circumference of a circumscribed 96 - sided polygons)
3﹞1/7 < 休 < 3﹞10/71
This value is quite accurate ⊥ 3.14084 < 休 <3.142858
Archimedes used the approximate value of '休' as 3.14.

Approximate value of '休'.

Ahmes'(a.1650 B.C) Papyrus 休＞ 3.16

Arithmetic in Nine section 休＞ 3

Archimedes 休＞ 3.14

Tsu Chung - chih(430-501) 休＞ 3.1415929

﹣ 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 弘,必,戊,扔,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.

﹣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 parobola(curve) and chord (straight line).
    This way of reckoning provide the base of modern integral calcus.
    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 <The theory of conic sections> which made him a great geometrician.
    He stated conic sections as cut stains from circular corns.
    These parts were omitted in <Elements> but Apollonius compiled many fields called ☆the theory of quadratic curve★.  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 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.

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