# „Nevertheless, it remains conceivable that the measure relations of space in the infinitely small are not in accordance with the assumptions of our geometry [Euclidean geometry], and, in fact, we should have to assume that they are not if, by doing so, we should ever be enabled to explain phenomena in a more simple way.“

Memoir (1854) Tr. William Kingdon Clifford, as quoted by A. D'Abro, The Evolution of Scientific Thought from Newton to Einstein https://archive.org/details/TheEvolutionOfScientificThought (1927) p. 55.

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### „…the differential element of non-Euclidean spaces is Euclidean. This fact, however, is analogous to the relations between a straight line and a curve, and cannot lead to an epistemological priority of Euclidean geometry, in contrast to the views of certain authors.“

— Hans Reichenbach American philosopher 1891 - 1953

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### „The concept of congruence in Euclidean geometry is not exactly the same as that in non-Euclidean geometry. …"Congruent" means in Euclidean geometry the same as "determining parallelism," a meaning which it does not have in non-Euclidean geometry.“

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### „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…“

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### „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" d… and the distance r which appears as an element of the geometry.“

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### „Geometry can in no way be viewed… as a branch of mathematics; instead, geometry relates to something already given in nature, namely, space. I… realized that there must be a branch of mathematics which yields in a purely abstract way laws similar to geometry.“

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### „Euclidean geometry can be easily visualized; this is the argument adduced for the unique position of Euclidean geometry in mathematics. It has been argued that mathematics is not only a science of implications but that it has to establish preference for one particular axiomatic system. Whereas physics bases this choice on observation and experimentation, i. e., on applicability to reality, mathematics bases it on visualization, the analogue to perception in a theoretical science. Accordingly, mathematicians may work with the non-Euclidean geometries, but in contrast to Euclidean geometry, which is said to be "intuitively understood," these systems consist of nothing but "logical relations" or "artificial manifolds". They belong to the field of analytic geometry, the study of manifolds and equations between variables, but not to geometry in the real sense which has a visual significance.“

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The Philosophy of Space and Time (1928, tr. 1957)

### „It is necessary to learn geometry only so far as might enable a man to measure land for the purposes of buying and selling.“

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### „What is the true geometry of the plate? …Anyone examining the situation will prefer Poincaré's common-sense solution… to attribute it Euclidean geometry, and to consider the measured deviations… as due to the actions of a force (thermal stresses in the rule). …On employing a brass rule in place of one of steel we would find that the local curvature is trebled—and an ideal rule (c = 0) would… lead to Euclidean geometry.“

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### „We should measure our wealth according to the means we have of satisfying our desires.“

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Il faut compter ses richesses par les moyens qu'on a de satisfaire ses désirs.

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### „In fact it is a stupidity, Maurice Princet told me in the presence of Juan Gris, to claim to be able to bring together in a single system of relations, colour, which is a sensation that only needs to be received, and form which is an organisation that has to be understood (14); and, introducing us to the non-Euclidean geometries, he urged us to create a geometry for painters.“

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Science, Vol. 18 (1903), p. 106, as reported in Memorabilia Mathematica; or, The Philomath's Quotation-Book https://archive.org/stream/memorabiliamathe00moriiala#page/81/mode/2up, (1914), by Robert Edouard Moritz, p. 352

### „I am coming more and more to the conviction that the necessity of our geometry cannot be demonstrated, at least neither by, nor for, the human intellect... Geometry should be ranked, not with arithmetic, which is purely aprioristic, but with mechanics.“

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### „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 American mathematician and physicist 1903 - 1961

Geometry as a Branch of Physics (1949)

### „Riemann has shewn that as there are different kinds of lines and surfaces, so there are different kinds of space of three dimensions; and that we can only find out by experience to which of these kinds the space in which we live belongs. In particular, the axioms of plane geometry are true within the limits of experiment on the surface of a sheet of paper, and yet we know that the sheet is really covered with a number of small ridges and furrows, upon which (the total curvature not being zero) these axioms are not true. Similarly, he says although the axioms of solid geometry are true within the limits of experiment for finite portions of our space, yet we have no reason to conclude that they are true for very small portions; and if any help can be got thereby for the explanation of physical phenomena, we may have reason to conclude that they are not true for very small portions of space.“

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Abstract

On the Space-Theory of Matter (read Feb 21, 1870)

### „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 American mathematician and physicist 1903 - 1961

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— Hans Reichenbach American philosopher 1891 - 1953

The Philosophy of Space and Time (1928, tr. 1957)

### „Although the Special Theory of Relativity does not account for electromagnetic phenomena, it explains many of their properties. General Relativity, however, tells us nothing about electromagnetism. In Einstein's space-time continuum gravitational forces are absorbed in the geometry, but the electromagnetic forces are quite unaffected. Various attempts have been made to generate the geometry of space-time so as to produce a unified field theory incorporating both gravitational and electromagnetic forces.“

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p, 125

The Structure of the Universe: An Introduction to Cosmology (1949)

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The Philosophy of Space and Time (1928, tr. 1957)