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" "The value of the intrinsic approach is especially apparent in considering 3-dimensional congruence spaces... The intrinsic geometry of such a space of curvature <math>K</math> provides formulae for the surface area <math>S</math> and the volume <math>V</math> of a "small sphere" of radius <math>r</math>, whose leading terms are 3)<math>S = 4 \pi r^2 (1 - \frac{Kr^2}{3} + ...)</math>,
<math>V = \frac{4}{3} \pi r^3 (1 - \frac{Kr^2}{5} + ...)</math>.
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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What is needed is a homely experiment which could be carried out in the basement with parts from an old sewing machine and an Ingersoll watch, with an old file of Popular Mechanics standing by for reference! This I am, alas, afraid we have not achieved, but I do believe that the following example... is adequate to expose the principles...
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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.