Definitions
from The Century Dictionary.
 Not comprehended within or not in conformity with the system of geometry established by Euclid; pertaining to or designating any system of geometry which, while differing in essential particulars from that of Euclid, is nevertheless in accord with the facts of experience within the limits of the errors of observation.
Etymologies
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Examples

I have to figure out how to fit "noneuclidean" in there someplace and I've got an H.P. Lovecraft story on my hands.

February 14, 2010 at 9:51 am a weird packing foam structure that comes in the box for the new laptop you ordered from Best Buy, but which appears on closer inspection to be designed to fit and cushion not an ordinary laptop, but an unknowable item of noneuclidean dimensions.

At the same time I was just getting into reading stories by HP Lovecraft, all that barelyrestrained hysteria and cosmic horror, mindcracking abominations bubbling up from beneath the floorboards of noneuclidean, ratinfested cellars, or leering from the dark, unbalanced vistas painted by former inmates of some creaking Providence institution whose walls are stained with the shrieks of vanished patients….
The Blue Oyster Cult – A Lifelong Fascination With High Weirdness « INTERSTELLAR TACTICS

Another example to show this fascinating elevation to noneuclidean, gravitational lensing, could be seen in this same light.

It turns out that nowadays, teachers are using landscape theory coupled with noneuclidean geometry to get far more accurate results.
John West's attempt at "swiftboating" sinks  The Panda's Thumb

Thus, the theory of general relativity, on adoption of the coordinative definition of rigid rods ( "universal forces = 0"), affirms that the geometry of spacetime in this region is of a noneuclidean kind.

Thus whether general relativity affirms a Euclidean or a noneuclidean metric in the solar gravitational field rests upon a conventional choice regarding the existence of "universal forces".

Why did Gauss wait and publish, seeing that Riemann would take noneuclidean geometries in a positive direction?

So such a challenge to society and the rise to noneuclidean perspective had to be quite a achievement.

Easter is just around the corner, and we all need to load up on antioxidants, especially the noneuclidean kind.
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