Definitions
from The American Heritage® Dictionary of the English Language, 4th Edition
 adj. Of or relating to Euclid's geometric principles.
from Wiktionary, Creative Commons Attribution/ShareAlike License
 adj. Adhering to the principles of traditional geometry, in which parallel lines are equidistant.
 adj. Of or relating to Euclid's Elements, especially to Euclidean geometry.
from The Century Dictionary and Cyclopedia
 Of or pertaining to Euclid, an illustrious Greek mathematician (who lived about 300 b. c.), the author of the “Elements of Geometry,” which has been the chief textbook of this subject down to recent times, and is still much used in England.
 Of or pertaining to Euclid, or Eukleides, Arch on Eponymos of Athens for the year 403 b. c.
 Also spelled Eukleidean.
from WordNet 3.0 Copyright 2006 by Princeton University. All rights reserved.
 adj. relating to geometry as developed by Euclid
Etymologies
Examples

The term Euclidean quantum gravity may make you want to nod off.

Metric tensors are used to define the angle between and length of tangent vectors (somewhat analogous to the dot product of vectors in Euclidean space)
Bad Language: Metric vs Metric Tensor vs Matrix Form vs Line Element

Just as it is unscientific to look for square circles in Euclidean geometry, so it is unscientific at this stage to look for purely chemical and physical explanations for life.

Not in Euclidean geometry, another abstract system.

For example, photons (which appear as particles in Euclidean space traveling at the speed of light) take advantage of the ultimate "shortcut" available in Minkowskian geometry.

Approximately equal to 3.14159, Pi represents the ratio of any circle’s circumference to its diameter in Euclidean geometry.

They were trying to use the particular technique called Euclidean Quantum Gravity (in which you temporarily forget that time is any different than space) to calculate rates at which different things could happen, when the stumbled across a puzzle.

To justify “2” rather than any other close number, we need only one assumption  that the space is Euclidean, which is eminently reasonable.
Should Science Pursue Methodological Supernaturalism?  The Panda's Thumb

(The theory of relativity showed what had been held to be an example of the synthetic a priori, namely Euclidean geometry, to be false as the geometry of physical space).

This restriction of the model is called the Euclidean “kernel” of the model.
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