Alternatives to Euclidean Geometry and Its Applications Negations to Euclid’s fifth postulate, known as the parallel postulate, give rise to the emergence of other types of geometries. Its existence stands in the respective models which their originators have imagined and designed them to be. The development of these geometries and its eventual recognition give humans some mathematical systems as alternative to Euclidean geometry. The controversial Euclid’s fifth postulate is phrased in this manner, to wit: “If a straight line crossing two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if extended indefinitely, meet on that side on which is the angles less than the two right angles.” which has been rephrased, and what is known as the parallel postulate as follows: “Given a line L and an external point P not on L, there exists a unique line m passing through P and parallel to L.” With the sphere as its model, is spherical (also called reimannian or elliptic) geometry being advanced by German mathematician, Bernhard Riemann who proposes the absence of a parallel line with Euclid’s fifth postulate as reference. His proposition is as follows: “ If L is any line and P is any point not on L, then there are no lines through P that are parallel to L” It contradicts Euclid’s fifth postulate mainly because no matter how careful one in constructing a line with a straightedge- as straight as it is- that line normally bends with the sphere. In such situation, there would be no straight lines and consequently, parallel lines cannot exist due to the intersection of these lines at the poles. Spherical Geometry has a direct connection to our daily lives for we live in an almost spherical world. Navigators find this kind of geometry useful in calculating the shortest distance en route to their destinations. So did the weather forecasters, in predicting paths of weather. Another type of geometry exists with three-dimensional curved spaces as its model, is the hyperbolic (also called saddle or Lobachevskian) geometry being named after the Russian mathematician, Nicholas Lobachevsky, who proposes the following:
“If L is any line and P is any point not on L , then there exists at least two lines through P that are parallel to L.” Euclid’s fifth postulate holds no water in hyperbolic geometry due to the presence of parallel lines upon construction. Hyperbolic geometry is used by astronomers in predicting behaviors of space objects’ orbits. It is also useful in space travel and computations involving gravitational pulls. Each of these geometries has defined terms and own independent system that normally posits different and varying logical equivalents and consequences which responds and answers to problems in the models that each of them stands for. The geometries bear distinct properties as summarized in the following table:

Properties | Types of Geometry | | Euclidean | Spherical | Hyperbolic | Curvature | | | | Given a line m and a point P not on m, the number of lines passing through P and parallel to m | 1 | 0 | many | Sum of interior angles of a triangle | 180° | > 180° | < 180° | Square of hypotenuse of a right triangle with sides a and b | a2 + b2 | < a2 + b2 | > a2 + b2 | Circumference of a circle with diameter 1 | π | < π | > π | These geometries (spherical and hyperbolic) give mankind useful tools, in addition to Euclidean geometry, in the quest to untangle the complexities of the world. The advent of spherical and hyperbolic geometries gives us alternatives to Euclidean geometry in dealing with geometrical problems. Assertions to abandon the latter- with regards to the ambiguities of the fifth postulate- by reason of antiquity or of logical consequences that break down upon a model different from the plane figure, is absurd. As mathematical systems, Euclid’s planar, spherical, and hyperbolic geometries are separate entities independent from each other having distinct properties and separate limitations on the applications being worked with. Being planar, Euclidean geometry and the first five of Euclid’s postulates are true to all respects in dealing with problems in plane figures and solid mensuration. The very task of determining which type of geometry is apt and applicable in the premises remains with us.

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