Euclids geometry


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Euclids geometry

  1. 1. Geometry (geo-earth, metron-measurement) is a branch of mathematics which deals with questions related to shape, size, relative position of figures, and the properties of space. Geometry is the study of angles and triangles, length, perimeter, area and volume. A mathematician who works in the field of geometry is called a geometer. Few of the known geometers are Euclid, Archimedes, René Descartes, Euler and Gauss. Euclid
  2. 2. Early geometry was a collection of principles concerning lengths, angles, areas, and volumes, which were developed to meet some practical need in surveying, construction, astronomy, and various crafts. In the 7th century BC, a Greek mathematician used to solve problems such as calculating the height of pyramids andThe oldest surviving Latin translationthe shore. the distance of ships from of the Elements Around 300 BC, geometry was revolutionized by Euclid, whose Elements, widely considered the most successful and influential textbook of all time, introduced the axiomatic method and is the earliest example of the format still used in mathematics today, that of definition, axiom, theorem, and proof. A European and an Arab practicing geometry in the 15th
  3. 3. Euclid was a Greek mathematician, who is known as the "Father of Geometry". His Elements is one of the most influential works in the history of mathematics, In the Elements, Euclid deduced the principles of what is now called Euclidean Euclid in Raphael's School of Athens geometry from a small set of axioms. Euclid’s life is not known. Nothing is known about his birth or death. Even description of Euclid's physical appearance made during his lifetime survived antiquity. Therefore, Euclid's depiction in works of art is the product of the artist's imagination. It is believed that Euclid may have studied at Plato's Academy in Athens.
  4. 4. WHAT IS THE BASIS OF EUCLID’S GEOMETRY??? • The Elements is based on theorems proved by other mathematics supplemented by some original work. • Euclid put together many of Eudoxus' theorems, many of Theaetetus', and also bringing to irrefragable demonstration the things which were only somewhat loosely proved by his predecessors. • Most of books 1 and 2 were based on Pythagoras, book 3 on Hippocrates of Chios, and book 5 on Eudoxus , while books 4, 6, 11 and 12 probably came from other Pythagorean or Athenian mathematicians. • Euclid often replaced misleading proofs with his own. • The use of definitions, postulates, and axioms dated back to Plato.
  5. 5. WHAT IS THE CONTENT IN THE BOOK??? 5 and 7–10 deal with number theory. Books 1–4 and 6 discuss plane geometry. •It deals with numbers treated geometrically • Many results about plane figures are proved. Books 11–13 concern solid geometry. through their representation as line segments with • Pons Asinorum i.e. If a ratio between two volume of triangle has the equal •A typical result is the 1:3 various lengths. angles, a cylinder with subtended by the base. a cone andthen the sides the same height andangles •Prime Numbers and Rational and Irrational are equal is proved. numbers are introduced. • The Pythagorean theorem is proved. A •The infinitude of prime numbers is proved. 2, 3, 5, 7, 9 etc. are PrimeABC, Angle B = Angle C In Triangle Nos.. -1/2, 5/7, 11/16 etc.Therefore, by Nos. Asinorum, are Rational Pons √2, 1.74653875…… , √71 etc. are AB=AC. Irrational Nos. B C
  6. 6. • The Elements begins with a list of definitions. • Euclid deduced a total of 131 definitions. • Though Euclid defined a point, a line, and a plane, the definitions are not accepted by mathematicians. Therefore, these terms are now taken as undefined. • Some of the Definitions by Euclid are :A 1. A point is that which has no part. 2. A line is breathless length. l 3. An even number is that which is divisible into two equal parts. 4. A solid is that which has length, breadth, and depth. 5. Parallel planes are those which do not meet.
  7. 7. • Euclidean geometry is an axiomatic system, in which all theorems ("true statements") are derived from a small number of axioms. • These are ‘self-evident truths’ which we take to be true without proof. • Some of the Axioms stated by Euclid are :1. Things which equal the same thing also equal one another. – If A=B and C=B, then A=C 2. If equals are added to equals, then the wholes are equal. – If A=B, then A+C=B+C 3. If equals are subtracted from equals, then the remainders are equal. – If A=B, then A-C=B-C 4. The whole is greater than the part. – 1 > 1/2 5. Things which are halves or double of the same things are equal to one another. – If A= ½ B and C= ½ B or A=2B and A=2C, then A=C
  8. 8. • Each postulate is an axiom—which means a statement which is accepted without proof— specific to the subject matter. Most of them are constructions. • Some of the Postulates by Euclid are :1. A straight line may be drawn from any one point to any other point. 2. A terminated line can be produced indefinitely. 3. A circle can be drawn with any centre and any radius. 4. All right angles are equal to one another. 5. If a straight line falling on two straight lines makes the interior angles on the same side of it taken together less than two right angles, then the two straight lines, if produced indefinitely, meet on that side on which the sum of angles is less than two right angles.
  9. 9. • After Euclid stated his postulates and axioms, he used them to prove other results. Then using these results, he proved some more results by applying deductive reasoning. The statements that were proved are called propositions or theorems. • Euclid deduced 465 propositions using his axioms, postulates, definitions and theorems proved earlier in the chain. • Some of the Propositions stated by Euclid are :1. In isosceles triangles the angles at the base equal one another, and, if the equal straight lines are produced further, then the angles under the base equal one another. 2. If magnitudes are proportional taken separately, then they are also proportional taken jointly. 3. If two triangles have their sides proportional, then the triangles are equiangular with the equal angles opposite the corresponding sides. 4. Any cone is a third part of the cylinder with the same base and