“The search for the curvature K indicates that, after making all known corrections, the number N seems to increase faster with d than the third power, which would be expected in a Euclidean space, hence K is positive.”

— Howard P. Robertson

The space implied thereby is therefore bounded, of finite total volume, and of a present "radius of curvature" <math>R = \frac{1}{K^\frac{1}{2}}</math> which is found to be of the order of 500 million light years. Other observations, on the "red shift" of light from these distant objects, enable us to conclude with perhaps more assurance that this radius is increasing...
Geometry as a Branch of Physics (1949)

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American mathematician and physicist 1903–1961

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“The search for the curvature K indicates that, after making all known corrections, the number N seems to increase faster with d than the third power, which would be expected in a Euclidean space, hence K is positive. The space implied thereby is therefore bounded, of finite total volume, and of a present "radius of curvature" R = \frac{1}{K^\frac{1}{2}} which is found to be of the order of 500 million light years. Other observations, on the "red shift" of light from these distant objects, enable us to conclude with perhaps more assurance that this radius is increasing…”

Howard P. Robertson (1903–1961) American mathematician and physicist

Geometry as a Branch of Physics (1949)

“Euclidean geometry is only one of several congruence geometries… Each of these geometries is characterized by a real number K, which for Euclidean geometry is 0, for the hyperbolic negative, and for the spherical and elliptic geometries, positive. In the case of 2-dimensional congruence spaces… K may be interpreted as the curvature of the surface into the third dimension—whence it derives its name…”

Howard P. Robertson (1903–1961) American mathematician and physicist

Geometry as a Branch of Physics (1949)

“[T]he astronomical data give the number N of nebulae counted out to a given inferred "distance" d, and in order to determine the curvature… we must express N, or equivalently V, to which it is assumed proportional, in terms of d. …from the second of formulae (3) and… (4)… to the approximation here adopted, 5)V = \frac{4}{3} \pi d^2 (1 + \frac{3}{10} K d^2 + …);…plotting N against… d and comparing… with the formula (5), it should be possible operationally to determine the "curvature" K.”

Howard P. Robertson (1903–1961) American mathematician and physicist

Geometry as a Branch of Physics (1949)

“In all these congruence geometries, except the Euclidean, there is at hand a natural unit of length R = \frac{1}{K^\frac{1}{2}}; this length we shall, without prejudice, call the "radius of curvature" of the space.”

Howard P. Robertson (1903–1961) American mathematician and physicist

Geometry as a Branch of Physics (1949)

“[T]he space constant K… "curvature" may in principle at least be determined by measurement on the surface, without recourse to its embodiment in a higher dimensional space.”

Howard P. Robertson (1903–1961) American mathematician and physicist

Geometry as a Branch of Physics (1949)

“After succeeding his mother, Rajiv Gandhi asked K. K. Tewary, a Congress MP, to make the reckless charge on the floor of the Rashtrapati Bhawan.”

Zail Singh (1916–1994) Indian politician and former President of India

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“Measurements which may be made on the surface of the earth… is an example of a 2-dimensional congruence space of positive curvature K = \frac{1}{R^2}… [C]onsider… a "small circle" of radius r (measured on the surface!)… its perimeter L and area A… are clearly less than the corresponding measures 2\pi r and \pi r^2… in the Euclidean plane. …for sufficiently small r (i. e., small compared with R) these quantities on the sphere are given by 1):L = 2 \pi r (1 - \frac{Kr^2}{6} + …),
A = \pi r^2”

Howard P. Robertson (1903–1961) American mathematician and physicist

1 - \frac{Kr^2}{12} + …
Geometry as a Branch of Physics (1949)

“Let a thin, flat metal plate be heated… so that the temperature T is not uniform… clamp or otherwise constrain the plate to keep it from buckling… [and] remain [reasonably] flat… Make simple geometric measurements… with a short metal rule, which has a certain coefficient of expansion c… What is the geometry of the plate as revealed by the results of those measurements? …[T]he geometry will not turn out to be Euclidean, for the rule will expand more in the hotter regions… [T]he plate will seem to have a negative curvature K… the kind of structure exhibited… in the neighborhood of a "saddle point."”

Howard P. Robertson (1903–1961) American mathematician and physicist

Geometry as a Branch of Physics (1949)

“The Greeks encountered the confusion of tongues when numbers invaded Euclidean space.”

Marshall McLuhan (1911–1980) Canadian educator, philosopher, and scholar-- a professor of English literature, a literary critic, and a …

Source: 1960s, The Gutenberg Galaxy (1962), p. 203

“Is the position tenable, that certain phenomena, possible in Euclidean space, would be impossible in non-Euclidean space, so that experience, in establishing these phenomena, would directly contradict the non-Euclidean hypothesis? For my part I think no such question can be put. To my mind it is precisely equivalent to the following, whose absurdity is patent to all eyes: are there lengths expressible in meters and centimeters, but which can not be measured in fathoms, feet, and inches, so that experience, in ascertaining the existence of these lengths, would directly contradict the hypothesis that there are fathoms divided into six feet?”

Henri Poincaré book Science and Hypothesis

Source: Science and Hypothesis (1901), Ch. V: Experiment and Geometry (1905) Tr. https://books.google.com/books?id=5nQSAAAAYAAJ George Bruce Halstead

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