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In the sum <math>\sigma</math> of the three angles of a triangle (whose sides are arcs of s) is greater than two right angles [180°]; it can... be sh… - Howard P. Robertson
" "In the sum <math>\sigma</math> of the three angles of a triangle (whose sides are arcs of s) is greater than two right angles [180°]; it can... be shown that this "spherical excess" is given by 2)<math>\sigma - \pi = K \delta</math>where <math>\delta</math> is the area of the spherical triangle and the angles are measured in s (in which 180° = <math>\pi</math> [radians]). Further, each full line (great circle) is of finite length <math>2 \pi R</math>, and any two full lines meet in two points—there are no parallels!
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About Howard P. Robertson
Howard Percy Robertson (January 27, 1903 – August 26, 1961) was an American mathematician and physicist known for contributions related to physical cosmology and the uncertainty principle. He was Professor of Mathematical Physics at the California Institute of Technology and Princeton University.
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Howard Percy Robertson
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H. P. Robertson
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In considerations involving the nature of the world as a whole the irregularities caused by the aggregation of matter into stars and stellar systems may be ignored; and if we further assume that the total matter in the world has but little effect on its macroscopic properties, we may consider them as being determined by the solution of an empty world.
Now it is the practice of astronomers to assume that brightness falls off inversely with the square of the "distance" of an object—as it would do in Euclidean space, if there were no absorption... We must therefore examine the relation between this astronomer's "distance" <math>d</math>... and the distance <math>r</math> which appears as an element of the geometry.
is a congruence geometry, or equivalently the space comprising its elements is homogeneous and isotropic; the intrinsic relations between... elements of a configuration are unaffected by the position or orientation of the configuration. ...[M]otions of are the familiar translations and rotations... made in proving the theorems of Euclid.
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