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Foundations of geometry—Euclidean and non-Euclidean geometry
1975Euclidean geometry is the oldest and historically most important example of a deductive scientific discipline. Down to modern times it has been a model of an exact science and it became the starting point for a systematic development of the foundations of geometry.
W. Gellert +4 more
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, 2021 Geometry is one of the first concepts that a child learns. At an early age itbecomes a game to put the circle in the round hole, the triangle in the triangular hole, andthe rectangle in the rectangular hole.
Wiese, Timothy P.
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1994
Abstract It will be noted that the ratio of the (essentially positive) Euclidean distances only gives the absolute value of the invariant (δac/δab). It is this fact which gives rise to some tiresome technical difficulties in classical Euclidean geometry, which relies principally on the notion of distance.
Peter M Neumann +2 more
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Abstract It will be noted that the ratio of the (essentially positive) Euclidean distances only gives the absolute value of the invariant (δac/δab). It is this fact which gives rise to some tiresome technical difficulties in classical Euclidean geometry, which relies principally on the notion of distance.
Peter M Neumann +2 more
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Euclidean and Non-Euclidean Geometries
2014Ancient mathematics was motivated by very practical reasoning. What we now call land management and commerce were the overriding considerations, and calculational questions grew out of those transactions. As a result, many of the ideas considered involved meshing rectangles and triangles, their areas, and their relative proportions.
Steven G. Krantz, Harold R. Parks
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2006
Abstract Euclidean geometry was the first branch of mathematics to be treated in anything like the modern fashion (with postulates, definitions, theorems, and so forth); and even now geometry is conducted in a very logical, tightly knit fashion.
Valerio Scarani, Rachael Thew
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Abstract Euclidean geometry was the first branch of mathematics to be treated in anything like the modern fashion (with postulates, definitions, theorems, and so forth); and even now geometry is conducted in a very logical, tightly knit fashion.
Valerio Scarani, Rachael Thew
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1993
Abstract Let S be an affine space. We saw in the last chapter that if we also assume that S has the extra structure of an absolute distance |AB| between two points A and B, then it is possible to define a dot product u • v between any two vectors u and V in S.
H. S. M. Coxeter, George Beck
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Abstract Let S be an affine space. We saw in the last chapter that if we also assume that S has the extra structure of an absolute distance |AB| between two points A and B, then it is possible to define a dot product u • v between any two vectors u and V in S.
H. S. M. Coxeter, George Beck
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1989
Surprisingly, the geometry of curved surfaces throws light on the geometry of the plane. More than 2000 years after Euclid formulated axioms for plane geometry, differential geometry showed that the parallel axiom does not follow from the other axioms of Euclid.
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Surprisingly, the geometry of curved surfaces throws light on the geometry of the plane. More than 2000 years after Euclid formulated axioms for plane geometry, differential geometry showed that the parallel axiom does not follow from the other axioms of Euclid.
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Euclidean and Non-Euclidean Geometry
1986This book gives a rigorous treatment of the fundamentals of plane geometry: Euclidean, spherical, elliptical and hyperbolic. The primary purpose is to acquaint the reader with the classical results of plane Euclidean and nonEuclidean geometry, congruence theorems, concurrence theorems, classification of isometries, angle addition and trigonometrical ...
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A variation of Hilbert’s axioms for euclidean geometry
Mathematische Semesterberichte, 2022Hermann Hahl, Hahl Hermann
exaly