The Elements of Non-Euclidean Geometry
Author | : Duncan M'Laren Young Sommerville |
Publisher | : |
Total Pages | : 291 |
Release | : 1914 |
Genre | : Geometry, Non-Euclidean |
ISBN | : |
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Author | : Duncan M'Laren Young Sommerville |
Publisher | : |
Total Pages | : 291 |
Release | : 1914 |
Genre | : Geometry, Non-Euclidean |
ISBN | : |
Author | : Julian Lowell Coolidge |
Publisher | : |
Total Pages | : 300 |
Release | : 1909 |
Genre | : Geometry, Non-Euclidean |
ISBN | : |
Author | : Julian Lowell Coolidge |
Publisher | : Createspace Independent Publishing Platform |
Total Pages | : 282 |
Release | : 2017-06-03 |
Genre | : |
ISBN | : 9781547058419 |
The Elements of non-Euclidean Geometry by Julian Lowell Coolidge
Author | : JULIAN LOWELL. COOLIDGE |
Publisher | : |
Total Pages | : 0 |
Release | : 2018 |
Genre | : |
ISBN | : 9781033719237 |
Author | : Julian Lowell Coolidge, PhD |
Publisher | : |
Total Pages | : 274 |
Release | : 2020-06-04 |
Genre | : |
ISBN | : |
In this book Dr. Coolidge explains non-Euclidean geometry which consists of two geometries based on axioms closely related to those specifying Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean geometry arises when either the metric requirement is relaxed, or the parallel postulate is replaced with an alternative one. In the latter case one obtains hyperbolic geometry and elliptic geometry, the traditional non-Euclidean geometries. When the metric requirement is relaxed, then there are affine planes associated with the planar algebras which give rise to kinematic geometries that have also been called non-Euclidean geometry. The essential difference between the metric geometries is the nature of parallel lines. Euclid's fifth postulate, the parallel postulate, is equivalent to Playfair's postulate, which states that, within a two-dimensional plane, for any given line l and a point A, which is not on l, there is exactly one line through A that does not intersect l. In hyperbolic geometry, by contrast, there are infinitely many lines through A not intersecting l, while in elliptic geometry, any line through A intersects l. Another way to describe the differences between these geometries is to consider two straight lines indefinitely extended in a two-dimensional plane that are both perpendicular to a third line: In Euclidean geometry, the lines remain at a constant distance from each other (meaning that a line drawn perpendicular to one line at any point will intersect the other line and the length of the line segment joining the points of intersection remains constant) and are known as parallels. In hyperbolic geometry, they "curve away" from each other, increasing in distance as one moves further from the points of intersection with the common perpendicular; these lines are often called ultraparallels. In elliptic geometry, the lines "curve toward" each other and intersect.
Author | : Harold E. Wolfe |
Publisher | : Courier Corporation |
Total Pages | : 274 |
Release | : 2013-09-26 |
Genre | : Mathematics |
ISBN | : 0486320375 |
College-level text for elementary courses covers the fifth postulate, hyperbolic plane geometry and trigonometry, and elliptic plane geometry and trigonometry. Appendixes offer background on Euclidean geometry. Numerous exercises. 1945 edition.
Author | : Horatio Scott Carslaw |
Publisher | : |
Total Pages | : 193 |
Release | : 1916 |
Genre | : Geometry |
ISBN | : |
Author | : Duncan M'Laren Young Sommerville |
Publisher | : |
Total Pages | : 274 |
Release | : 1958 |
Genre | : Geometry, Non-Euclidean |
ISBN | : |