The Axioms of Non-Metrical Geometry

1963 ◽  
Vol 47 (361) ◽  
pp. 210
Author(s):  
S. N. Collings
Keyword(s):  
Metrical Geometry

scholarly journals Metrical Geometry Classification of Conic Section in Hyperbolic Space Form

2012 ◽  
Vol 02 (02) ◽  
pp. 97-102
Author(s):  
幼宁 王
Keyword(s):  
Hyperbolic Space ◽  
Space Form ◽  
Conic Section ◽  
Metrical Geometry ◽  
Hyperbolic Space Form

Metrical Geometry

2012 ◽  
pp. 49-78
Author(s):  
Sébastien Gandon
Keyword(s):  
Metrical Geometry

Metrical geometry of the classical gauge fields

1985 ◽  
Vol 86 (3) ◽  
pp. 259-271 ◽  
Author(s):  
A. Loinger
Keyword(s):  
Gauge Fields ◽  
Metrical Geometry

scholarly journals ON ELEMENTS OF CONTENT IN METRICAL GEOMETRY

1929 ◽  
Vol 15 (2) ◽  
pp. 99-102
Author(s):  
F. D. Murnaghan
Keyword(s):  
Metrical Geometry

Metrical Geometry.

2020 ◽  
pp. 404-418
Author(s):  
Bertrand Russell
Keyword(s):  
Metrical Geometry

scholarly journals One dimensional metrical geometry

2007 ◽  
Vol 128 (1) ◽  
pp. 145-166 ◽  
Author(s):  
N. J. Wildberger
Keyword(s):  
One Dimensional ◽  
Metrical Geometry

Metrical geometry of Galilean inertial structure. III

1984 ◽  
Vol 16 (10) ◽  
pp. 901-908 ◽  
Author(s):  
D. P. L. Castrigiano
Keyword(s):  
Metrical Geometry

3260. Orthogonal Matrices in Non-Metrical Geometry

1970 ◽  
Vol 54 (388) ◽  
pp. 151
Author(s):  
S. N. Collings
Keyword(s):  
Orthogonal Matrices ◽  
Metrical Geometry

II. On the non-euclidian plane geometry

1884 ◽  
Vol 37 (232-234) ◽  
pp. 82-102
Keyword(s):  
Dimensional Space ◽  
Spherical Surface ◽  
Presidential Address ◽  
Physical Space ◽  
Real Surface ◽  
Two Dimensional ◽  
Straight Line ◽  
Third Dimension ◽  
Metrical Geometry ◽  
Pseudospherical Surface

1. I consider the hyperbolic or Lobatschewskian geometry: this is a geometry such as that of the imaginary spherical surface x 2 + y 2 + z 2 = —1; and the imaginary surface may be bent (without extension or Contraction) into the real surface considered by Beltrami, and which I will call the Pseudosphere, viz., this is the surface of revolution defined by the equations x = log cot ½θ—cosθ, √y 2 + z 2 = sinθ. We have on the imaginary spherical surface imaginary points corresponding to real points of the pseudosphere, and imaginary lines (arcs of great circle) corresponding to real lines (geodesics) of the pseudosphere, and, moreover, any two such imaginary points or lines of the imaginary spherical surface have a real distance or inclination equal to the corresponding distance or inclination on the pseudosphere. Thus the geometry of the pseudo­sphere, using the expression straight line to denote a geodesic of the surface, is the Lobatschewskian geometry; or rather I would say this in regard to the metrical geometry, or trigonometry, of the surface; for in regard to the descriptive geometry, the statement requires (as will presently appear) some qualification. 2. I would remark that this realisation of the Lobatschewskian geometry sustains the opinion that Euclid’s twelfth axiom is undemonstrable. We may imagine rational beings living in a two-dimensional space, and conceiving of space accordingly, that is having no conception of a third dimension of space; this two-dimensional space need not however be a plane, and taking it to be the pseudospherical surface, the geometry to which their experience would lead them would be the geometry of this surface, that is, the Lobatschewskian geometry. With regard to our own two-dimensional space, the plane, I have, in my Presidential Address (B. A., Southport, 1883) expressed the opinion that Euclid’s twelfth axiom in Playfair’s form of it does not need demonstration, but is part of our notion of space, of the physical space of our experience; the space, that is, which we become acquainted with by experience, but which is the representation lying at the foundation of all physical experience.

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