A Quote by Hans Reichenbach
...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.
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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.
The classical theorists resemble Euclidean geometers in a non-Euclidean world who, discovering that in experience straight lines apparently parallel often meet, rebuke the lines for not keeping straight as the only remedy for the unfortunate collisions which are occurring. Yet, in truth, there is no remedy except to throw over the axiom of parallels and to work out a non-Euclidean geometry.
It [ non-Euclidean geometry ] would be ranked among the most famous achievements of the entire [nineteenth] century, but up to 1860 the interest was rather slight.
At that stage of my youth, death remained as abstract a concept as non-Euclidean geometry or marriage. I didn't yet appreciate its terrible finality or the havoc it could wreak on those who'd entrusted the deceased with their hearts.
Not that the propositions of geometry are only approximately true, but that they remain absolutely true in regard to that Euclidean space which has been so long regarded as being the physical space of our experience.
I entered an omnibus to go to some place or other. At that moment when I put my foot on the step the idea came to me, without anything in my former thoughts seeming to have paved the way for it, that the transformations I had used to define the Fuchsian functions were identical with non-Euclidean geometry.
My family was well off but not rich. I spent the four years I was an undergraduate working on the beach. And it wasn't because I was lazy; it was because my freshman class would go to a hundred different employers and wouldn't get a nibble. That was a disequilibrium system. I realized that the ordinary old-fashioned Euclidean geometry didn't apply.
Development of Western science is based on two great achievements: the invention of the formal logical system (in Euclidean geometry) by the Greek philosophers, and the discovery of the possibility to find out causal relationships by systematic experiment (during the Renaissance). In my opinion, one has not to be astonished that the Chinese sages have not made these steps. The astonishing thing is that these discoveries were made at all.
The victory over Euclidean space was not achieved by isolated individuals, but by a field of young rebels opposed to all absolutes.
The once-surprising existence of non-Euclidean models of Euclid's first four axioms can be seen as a sort of mathematical joke.
[The Euclidean algorithm is] the granddaddy of all algorithms, because it is the oldest nontrivial algorithm that has survived to the present day.
But then, Cap'n Crunch in a flake form would be suicidal madness; it would last about as long, when immersed in milk, as snowflakes sifting down into a deep fryer. No, the cereal engineers at General Mills had to find a shape that would minimize surface area, and, as some sort of compromise between the sphere that is dictated by Euclidean geometry and whatever sunken treasure related shapes that the cereal aestheticians were probably clamoring for, they came up with this hard-to-pin-down striated pillow formation.
The world of shapes, lines, curves, and solids is as varied as the world of numbers, and it is only our long-satisfied possession of Euclidean geometry that offers us the impression, or the illusion, that it has, that world, already been encompassed in a manageable intellectual structure. The lineaments of that structure are well known: as in the rest of life, something is given and something is gotten; but the logic behind those lineaments is apt to pass unnoticed, and it is the logic that controls the system.
But I still read Shaw on a regular basis. What I love is the nakedness of the polemic and the irresistible good humour. For me, 'Major Barbara' is the greatest of all the plays in that it starts from the rational and proceeds to the ecstatic in a spectacular way, and leaves you very confused if you cling to Euclidean logic.
We find in the history of ideas mutations which do not seem to correspond to any obvious need, and at first sight appear as mere playful whimsies such as Apollonius' work on conic sections, or the non-Euclidean geometries, whose practical value became apparent only later.
Proofs of the Euclidean [parallel] postulate can be developed to such an extent that apparently a mere trifle remains. But a careful analysis shows that in this seeming trifle lies the crux of the matter; usually it contains either the proposition that is being proved or a postulate equivalent to it.
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