Euclid Geometry And Non-Euclid Geometry. Have you ever asked yourself why is it that if you walk to a specific place from

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1 Hu1 Haotian Hu Dr. Boman Math 475W 9 November 2016 Euclid Geometry And Non-Euclid Geometry Have you ever asked yourself why is it that if you walk to a specific place from somewhere, you will always find your way back to the same place by same or different direction? ("Euclidean Geometry") Geometry as a branch of mathematics which tends to explain in depth the going off from one place which is based on the definition of an object then coming back to the point of origin. As a mathematician, I came to learn that any object in mathematics is always based on common outside experience which depends mostly on the definition of the objects which is the formulae. If we don t have clear definition of the objects then our results will automatically be wrong. Mathematics is not only a most demanding discipline but also full of imaginative and liberal of all human activities. Every terminologies used in mathematics must clearly be defined and finally proved how they came to be. Despite the clarity that we get from mathematics based on the process of definition till the final stage of proof, the most important thing is how we master the actual definition and formulas so that we don t come up with wrong result. This is clearly illustrated in geometry using Euclid s Elements theory. He was the first Mathematician who came around 300 BCE even before Pythagoras. He came up with many insertions in other mathematician s works. Euclid in his study about geometry said that geometry is divided into three categories; the plane geometry, the solid geometry and finally the number geometry. The plane of geometry

2 Hu2 comprises of the first principles of mathematics which are; the definition, common notions and finally the postulates. The main objective of the definition is to name the basic object and mathematical concepts, the common notion are the accepted rules which are usually accompanied with reasoning and show the relationship of the defined objects and finally the proof of the postulates. The above three are the starting point or the proposition in geometry which are at the end proved through logical rules. The proof of mathematical proposition is explained through deductive arguments which at the end we come up with statements that is finally proved. Some of the geometrical definitions are the definition of an angle. An angle is normally known to result if and only if two lines meet on a plane but most importantly is that they never lie on a straight line. Another example is the parallel line. They are known to be straight lines which will never meet despite the fact that they begin in the same plane. In conjunction to these we also have some geometrical notions some of which are like there is always a straight line which must have its point of origin to any point. Another geometrical notion is that whenever we are given a circle there must always be is center radius and finally its always assumed though never proved that all the right angles are always and with forever be equal to one another. Euclid used the three (definition, common notions and postulates) to give clear meaning of plane geometry. Some of the geometrical proposition that we are familiar with are like; angles in a straight line are add up to 180 degrees, the sum of all interior angles in triangle, square and rectangle are add up to 180 degrees. Euclid came up with the first principle when he came up with proofs and discussion on the Greek geometry and number theory. Euclid Elements is also rigor. His proofs were not only based on the postulates and the definition but a concentrated on the properties of what was being defined. He gave an example of this by saying that given two end points say P and Q and we draw a straight line joining the two points, then if we draw two circles centered at P and Q respectively, then

3 Hu3 eventually the two circle will meet at a point. Now if we draw any straight line to any of the points where the two circles meet, then the two segments will be equal meaning that their base angles are also equal. This is an argument that proofs Euclid proposition that given any line segment, we can have can create equilateral triangle which has the same segment as its sides. Some of the students turns away from mathematics due to the flawlessness of geometrical Elements in line with its apex and rigorous. Most of the Greek have failed to proof some of the Euclid s mathematical statements over a long period of time such as the parallel line postulate has not only led to his fame but also the invention of the non-euclidean geometries. ("Noneuclid: 1: Non-Euclidean Geometry") This is always any geometrical proof which opposes Euclidean geometry. They are normally divided into two categories, the spherical and hyperbolic geometry. The difference between the two geometries is based on parallel line nature or properties. But Proclus came in in the fifth century to prove Euclid s postulate on parallel lines. He said that suppose we have two parallel lines say A and B and C being distinct and it intersect with say line B then line C will also intersect with line A. He used the assumption that parallel lines are normally equidistant from each other. Another scholar who tried to proof Euclid error was Saccheri on the construction of quadrilateral where he discovered the new world of the absolute geometry and the theorems which were concern with the congruent triangles. Majority of the mathematicians who tried to come up with solution to some of the Euclid s postulates only worked which were logically equivalent to Euclid s fifth postulate. (p.667) Example when Legendre ( ) tried to prove that there always exist a triangle in which total value of angles of the three angles are always equal to the sum of measures of two right angles (right angle are always 90 degrees each hence their sum which is equivalent to 180 is equal to the sum of all angles in a triangle).despite his proof that sum of angle in a triangle adds up to 180 degrees, he never gave a proof the sum of this angles can never be less than 180 degrees. Friedrich Thibaut

4 Hu4 also tried to demonstrate property that sum of all angles adds up to 180 degrees. He assumed that every inflexible motion can be resolved into a rotation and independent translation.karl Gauss through his study to proof the theory behind the parallel line for a period of thirty years ended up with formulation of non- Euclidean geometry. He stated that the sum of angle sin a triangle being less than 180 degrees results to a questioning geometry since it appears to be paradoxical.( p.668) Gauss never bothered on trying to proof on Euclid s parallel postulate but rather he replaced it with contradicting with main aim of coming up with (inventing) new geometry. The first period when non-euclidean geometry was used according to Felix Klein was characterized with the use of synthetic method. Another mathematical scholar who also developed non-euclidean geometry was G.F.B Riemann when he said that any straight line always has an infinite length according to Euclid s postulate that a straight line is always restricted in length and has no specific end point and parallel lines postulate was also replaced by the statement that; through a point in a plane there can be drawn in the plane no line which doesn t intersect a given line not passing through the given point. In conclusion, non-euclidean geometry not only widened the knowledge used in geometry but most of all it stimulated leaners to want to find more about geometry and the truth behind mathematical knowledge. In regard to Euclid s view on geometry, it s evident that geometrical statements were only considered to be true if and only if they could describe nature correctly but was considered to be false if it gave wrong information about the nature. Hence geometrical truth involved description of nature. WORK CITATION "Euclidean Geometry". Encyclopedia Britannica, 2016, "Noneuclid: 1: Non-Euclidean Geometry".

5 Hu5 The parallel Postulates, Raymond H. Rolwing and Maita Levine The Poincare Conjecture, Donal O shea

Class IX Chapter 5 Introduction to Euclid's Geometry Maths

Class IX Chapter 5 Introduction to Euclid's Geometry Maths Exercise 5.1 Question 1: Which of the following statements are true and which are false? Give reasons for your answers. (i) Only one line can