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Euclidean geometry is a mathematical system attributed to the Greek Mathematician Euclid of Alexandria . Euclid's text '' Elements '' is the earliest known systematic discussion 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 Axiom s, and then proving many other Proposition s ( Theorem s) from those axioms. Although many of Euclid's results had been stated by earlier Greek mathematicians, Euclid was the first to show how these propositions could be fitted together into a comprehensive deductive and logical system. The ''Elements'' begin with Plane Geometry , still often taught in Secondary School as the first Axiomatic System and the first examples of Formal Proof . The ''Elements'' goes on to the Solid Geometry of three Dimension s, and Euclidean geometry was subsequently extended to any finite number of Dimension s. Much of the ''Elements'' states results of what is now 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 Geometries are known, the first ones having been discovered in the early 19th century. It also is no longer taken for granted that Euclidean geometry describes physical space. An implication of Einstein 's theory of General Relativity is that Euclidean geometry is only a good approximation to the properties of physical space if the Gravitational Field is not too strong. 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 Postulate s (axioms): # Any two Point s can be joined by a Straight Line . # Any Straight Line 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 Angle s are Congruent . # Parallel Postulate . If two lines intersect a third in such a way that the sum of the Inner Angle s on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough. 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: 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 . In this sense, Euclidean geometry is more concrete than many modern axiomatic systems such as Set Theory , which often assert the existence of objects without saying how to construct them, or even assert the existence of objects that cannot be constructed within the theory. Strictly speaking, the constructs of lines on paper etc are '' Models '' of the objects defined within the formal system, rather than instances of those objects. For example a Euclidean straight line has no width, but any real drawn line will. The ''Elements'' also include the following five "common notions": # 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. Euclid also invoked other properties pertaining to Magnitude s. 1 is the only part of the underlying logic that Euclid explicitly articulated. 2 and 3 are "arithmetical" principles; note that the meanings of "add" and "subtract" in this purely geometric context are taken as given. 1 through 4 operationally define Equality , which can also be taken as part of the underlying logic or as an Equivalence Relation requiring, like "coincide," careful prior definition. 5 is a principle of Mereology . "Whole," "part," and "remainder" beg for precise definitions. In the 19th century, it was realized that Euclid's ten axioms and common notions do not suffice to prove all of theorems stated in the ''Elements''. For example, Euclid assumed implicitly that any line contains at least two points, but this assumption cannot be proved from the other axioms, and therefore needs to be an axiom itself. 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 intersection as the third . 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 geometry have been proposed, the best known being those of Hilbert , George Birkhoff , and Tarski . To be fair to Euclid, the first Formal Logic capable of supporting his geometry was that of Frege 's 1879 '' Begriffsschrift '', little read until the 1950s. We now see that Euclidean geometry should be embedded in First-order Logic with Identity , a formal system first set out in Hilbert and Wilhelm Ackermann 's 1928 '' Principles Of Theoretical Logic ''. Formal Mereology began only in 1916, with the work of Lesniewski and A. N. Whitehead . Tarski and his students did major work on the Foundations Of Elementary Geometry as recently as between 1959 and his death in 1983. The parallel postulate See Also: 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.For the assertion that this was the historical reason for the ancients considering the parallel postulate less obvious than the others, see p. 9 of E. Nagel and J.R. Newman, ''Godel's Proof'', New York University Press, 1958. Euclid himself seems to have considered it as being qualitatively different from the others, as evidenced by the organization of the ''Elements'': the first 28 propositions he presents are those that can be proved without it. Many geometers tried in vain to prove the fifth postulate from the first four. By 1763 at least 28 different proofs had been published, but all were found to be incorrect.Douglas R. Hofstadter, ''Godel, Escher, Bach: An Eternal Golden Braid'', New York, Basic Books, 1979, p. 91. In fact the parallel postulate cannot be proved from the other four: this was shown in the 19th Century by the construction of alternative ( Non-Euclidean ) systems of geometry where the other axioms are still true but the parallel postulate is replaced by a conflicting axiom. One distinguishing aspect of these systems is that the three angles of a the sum of the three angles is always less than 180° and can approach zero, while in Elliptic Geometry it is greater than 180°. If the parallel postulate is dropped from the list of axioms without replacement, the result is the more general geometry called Absolute Geometry . TREATMENT USING ANALYTIC GEOMETRY The development of Analytic Geometry provided an alternative method for formalizing geometry. In this approach, a point is represented by its Cartesian (x,y) coordinates, a line is represented by its equation, and so on. In the 20th century, this fit into David Hilbert 's program of reducing all of mathematics to arithmetic, and then proving the consistency of arithmetic using finitistic reasoning. In Euclid's original approach, the Pythagorean Theorem follows from Euclid's axioms. In the Cartesian approach, the axioms are the axioms of algebra, and the equation expressing the Pythagorean theorem is then a definition of one of the terms in Euclid's axioms, which are now considered to be theorems. The equation |
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