Thursday, July 26, 2012

MODULE 6 - EUCLIDE (325BC – 265BC)

EUCLIDE (325BC – 265BC)

325 BC (1707-04-15)
265 BC
Egypt, Athens, Alexandria,
Mathematics (geometry),
Royal school at Alexandria in Egypt
Plato academy in Athens

Eulcid, best known for his treatise on mathematics, ‘Elements’ was born about 325 BC. Little is known about his life except that he taught mathematics about 300 BC in Royal School at Alexandria in Eqypt that has been founded by Ptolemy, the successor of Alexander the great. He must have studies in Plato’s academy in Athens and learnt the geometry of Eudoxus and theaetetus of which he was familiar. He is considered a one of the great Greek mathematicians.

Euclid Elements / BOOKS:
Books are given the content of subject matter of the thirteen books of the elements.
Ü   Book-1 : Proves elementary theorems about triangles and parallelograms and ends with  Pythagorean theorem.
Ü   Book –2 : has been known as “geometric algebra” because it states algebraic identities  as theorems about equivalent geometric  figures. This book contains a construction of “the section”, the division of a line into two parts such that the ratio of the original line to the larger segment. It also generalizes the Pythagrorean theorem to arbitrary triangles, a result that is equivalent to the law of cosines.
Ü   Book –3 : gives details of properties of circles.
Ü   Book –4 : deals with the construction of regular polygons, in particular the pentagon.
Ü   Book –5 : develops a general theory of ratios and proportions. It formed the foundation for a geometric theory of numbers the foundation for a geometric theory of numbers until an analytic theory developed in the late 19th century.
Ü   Book-6 : applies the theory of ratios to plane geometry, mainly triangles and parallelograms, culminating in the “application of areas” a procedure for solving Quadratic problems by geometric means.
Ü   Book-7 : describes a method, antanaresis (now known as Euclidean algorithms), for finding the greatest common divisor of two or more numbers.
Ü   Book-8 : examines numbers in continued proportions, now known as geometric sequences (such as ax, ax2, ax3, ax4).
Ü   Book-9 : proves that there are infinite numbers.
Ü   Book-10 : which comprises roughly one-fourth of the Elements seems disproportionate  to the importance of its classification of incommensurable lines and areas.
Ü   Book-11 : deals with the intersections of planes, lines and parallelepipeds (solids with parallel parallelograms as opposite faces).
Ü   Book-12 : provides Eudoxus’ method of exhaustion to prove that areas of a circle are to one another as the squares of their diameters and the  volumes of spheres are to one another as the cubes of their diameters.
Ü   Book-13 : culminates with the construction of the five regular Platonic solids (pyramid, cube, octahedron, dodecahedron, icosahedron) in a given spheres.
Book 1.       Triangles, parallelogram, Pythagorean theorem
Book 2.       Geometric algebra : Algebraic identities.
Book 3.       Properties of Circles
Book 4.       Construction of regular polygons
Book 5.       Geometrical Theory : Theory of ratios and Proportion
Book 6.       Idea of proportion applied to similar figure
Book 7.       Even, odd numbers, numerical theory of proportion
Book 8.       Continued proportion
Book 9.       Number theory
Book 10.     Irrational
Book 11.     Solid geometry
Book 12.     Method of exhaustion used to show that circles are
Proportional to their diameters etc.
Book 13.     Regular solids.
Euclid also wrote other books in mathematics and a few in physics. The important of them are ‘data’, ‘on Division of Figures’, ‘Phenomena’, ‘optics’, etc.
@    Data:  Data deals with the nature and implications of "given" information in geometrical problems; the subject matter is closely related to the first four books of the Elements. It comprising supplementary geometrical material concerned with algebraic problems is important in the development of algebra.
@    Division of figures: consists of 36 propositions concerning the division of various figures into two or more equal parts or parts in given ratios.
@    Phenomena: Phenomena, a treatise on spherical astronomy survives in Greek; it deals with the celestial sphere and contains 25 geometrical propositions.
@    Optics: apparent shapes of cylinder and cones with viewed from different angles.

Euclid Contributions:
Ø  Euclid proved what is generally known as Euclid’s second theorem that prime numbers are infinite.
Ø  He thought about the three current problems of time namely
-Dividing an angle into three equal parts
-Making double of a cube
-Obtaining square from a cube
Ø  He solved unresolved problems related to irrational numbers
Ø  He also discussed the so called Euclidean algorithm for finding the greatest common divisor of two numbers
Ø  He is created with the well-known proof of the Pythagorean Theorem.
Ø  He also formulated few mathematical riddles.
Ø  Euclid axioms:
Ø   We connect two points in a straight line
Ø   We extend the line in both sides
Ø   Right angle triangle consists of 90 degree
Ø   We can draw the circle from the center point.