EUCLIDS GEOMETRY

MATHS PROJECT WORK

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EUCLID’S

GEOMETRY









INTRODUCTION Euclidean geometry is a mathematical system attributed to the Greek mathematician Euclid Euclid of of Alexandria. Alexandria. Euclid's Elements is the earliest known systematic discussion of geometry of geometry . It has been one of the most influential books in history, as much for its method as for its mathematical content. The method consists of assuming a small set of intuitively appealing axioms axioms,, and then proving many other propositions propositions ( theorems theorems ) from those axioms. Although many of Euclid's results had been









Much of the Elements states results of what is is now called number called number theory , proved using geometrical methods. For over two thousand years, the adjective "Euclidean" was unnecessary because no other sort of geometry had been conceived. Euclid's axioms seemed so intuitively obvious that any theorem proved from them was deemed true in an absolute sense. Today, however, many other self-consistent non-Euclidean self-consistent non-Euclidean geometries are known, the first ones having been discovered in the early 19th century. It also is no longer taken .









AXIOMATIC APPROACH Euclidean geometry is an axiomatic system , in which all theorems ("true statements") are derived from a finite number of axioms. Near the beginning of the first book of the Elements, Euclid gives five postulates five postulates (axioms): Any two points two points can be joined by a straight line. line. Any straight Any straight line segment can segment can be extended indefinitely in a straight line. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center. All right All right angles are congruent . Parallel postulate. postulate. If two lines intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must









These axioms invoke the following concepts: point, straight line segment and line, side of a line, circle with radius and center, right angle, congruence, inner and right angles, sum. The following verbs appear: join, extend, draw, intersect. The circle described in postulate 3 is tacitly unique. Postulates 3 and 5 hold only for plane geometry; in three dimensions, postulate 3 defines a sphere. Postulate 5 leads to the same geometry as the following statement, known as Playfair's axiom , which also holds only in the plane: Through a point not on a given straight line, one and only one line can be drawn that never meets the given line. Postulates 1, 2, 3, and 5 assert the existence and uniqueness of certain geometric figures, and these assertions are of a constructive nature: that is, we are not only told that certain things exist, but are also given methods for creating them with no more than a compass and an unmarked straightedge. straightedge. In this sense, Euclidean geometry is more concrete than many modern axiomatic systems











Things that equal the same thing also equal one another. If equals are added to equals, then the wholes are equal. If equals are subtracted from equals, then the remainders are equal. Things that coincide with one another equal one another. The whole is greater than the part. p art. Euclid also invoked other properties pertaining to magnitudes magnitudes.. 1 is the only part of the underlying logic that Euclid explicitly









The very first geometric proof in the Elements, shown in the figure on the right, is that any line segment is part of a triangle; Euclid constructs this in the usual way, by drawing circles around both endpoints and taking their intersectio intersection n as the third vertex third vertex . His axioms, however, do not guarantee that the circles actually intersect, because they are consistent with discrete, rather than continuous, space. Starting with Moritz Pasch in 1882, many improved axiomatic systems for









THE PARALLEL POSTULATE To the ancients, the parallel postulate seemed less obvious than the others; verifying it physically would require us to inspect two lines to check that they never intersected, even at some very distant point, and this inspection could potentially take an infinite amount of time. [1] [1] Euclid himself seems to have considered it as being qualitatively different from the others, as evidenced by the









TREATMENT USING ANALYTIC GEOMETRY The development of analytic of analytic geometry provided geometry provided an alternative method for formalizing geometry. In this approach, a point is represented by its









AS A DESCRIPTION OF PHYSICAL REALITY











A disproof of Euclidean geometry as a description of physical space. In a









They were later verified by observations such as the observation of the slight bending of starlight by the Sun during a solar eclipse in 1919, and









While Euclidean geometry, the Standard Model and general relativity are all in principle compatible with any number of spatial dimensions and any specification









CONIC SECTIONS AND









Given his timing and measuring apparatus, this was an excellent approximation. Over such small distances that the acceleration of gravity





















EUCLIDS GEOMETRY

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