Differences in Geometry

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Differences in Geometry Geometry is the branch of mathematics that deals with the properties of space. Geometry is classified between two separate branches, Euclidean and Non-Euclidean Geometry. Being based off different postulates, theorems, and proofs, Euclidean Geometry deals mostly with two-dimensional figures, while Demonstrative, Analytic, Descriptive, Conic, Spherical, Hyperbolic, are Non-Euclidean, dealing with figures containing more than two-dimensions. The main difference between Euclidean, and Non-Euclidean Geometry is the assumption of how many lines are parallel to another. In Euclidean Geometry it is stated that there is one unique parallel line to a point not on that line. Euclidean Geometry has been around for over thousands of years, and is studied the most in high school as well as college courses. In it's simplest form, Euclidean geometry, is concerned with problems such as determining the areas and diameters of two-dimensional figures and the surface areas and volumes of solids. Euclidean Geometry is based off of the parallel postulate, Postulate V in Euclid's elements, which states that, "If a straight line meets two other straight lines so as to make the two interior angles on one side of it together less than two right angles, the other straight lines, if extended indefinitely, will meet on that side on which the angles are less than two right angles." For centuries, mathematicians tried to contradict Euclid's Postulate V, and determine that there was more than one line parallel to that of another. It was declared impossible until the 19th century when Non-Euclidean Geometry was developed. Non-Euclidean geometry was classified as any geometry that differed from the standards of Euclidean geo... ... middle of paper ... ...is a theorem in Euclidean Geometry, yet in Hyperbolic Geometry it is proved false by the above counter example (Both BA and BC are parallel to DE, yet BA is not parallel to BC). However, you may not be convinced that BA and DE are parallel. Bibliography: Bibliography: Books: O'Reilly, Geometry in a Nutshell, O'Reilly & Associates, Inc. California, ©1996. Textbooks: Ray C. Jurgensen, Richard G. Brown, John W. Jurgensen, Geometry, Houghton Mifflin Company. Boston, ©1988. John C. Peterson, Technical Mathematics 2nd Edition, Delmar Publishers, Inc. Washington, ©1997. Reference: Leon L. Bram, Funk & Wagnalls New Encyclopedia, Funk & Wagnalls, Inc. ©1990. URL Reference: NonEuclid: http://math.rice.edu/~joel/NonEuclid/ The Geometry of a Sphere: http://math.rice.edu/~pcmi/sphere/

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