Some helpful discussion with Dana Scott and Fred Linton got me to the point where I felt I could say reasonably succinctly exactly what a Euclidean space is (or ought to be). The following summarizes my present understanding, which is a lot clearer than 24 hours ago. I've seen only two essentially different approaches so far, bottom-up (adding structure) and top-down (forgetting structure). Bottom-up: A Euclidean space is an affine space transformable only by similarities. Since affine spaces are already understood to be transformable only by affinities, the further restriction to similarities is equivalent to the preservation of circles, or of right isosceles triangles. Affine spaces are in turn definable as an affine-closed subspace of a projective space transformable only by affinities. Constraining the morphisms in this way is morally equivalent to adding structure (of some unspecified kind), whence "bottom-up". Top-down: A Euclidean space is a torsor for a Euclidean inner product space E. A torsor, or principal homogeneous space, is (in this instance) a generalized metric space with vector distances in place of real distances so as to make the triangle inequality an equality, with d(x,y) = -d(y,x) and d(x,y) = 0 iff x = y. Like metric spaces torsors have no origin. E supplies the distances. I put "Euclidean" as a modifier for "inner product space" to connote the liberalization of morphisms thereof to preserving inner product only up to a constant factor (as opposed to the presumed default of on the nose). This liberalization accommodates scaling, and can be considered as forgetting the scale. That and torsors for forgetting the origin makes this approach "top-down." I've used both approaches in software for things like surveying and computer graphics, using respectively top-down and bottom-up. Top-down is simple at a low level, is primarily vector oriented, involves little or no multiplication, and works fine with single-precision floating point or even fixed point arithmetic. Bottom-up is simple at a high level (if you don't try to follow the individual arithmetic steps when debugging), uses an extra dimension to express projective geometry (needed to represent the part at infinity dropped when passing to affine space), is primarily matrix oriented, involves lots of multiplication, and benefits from double precision. All high-end video cards today use bottom-up exclusively, understandable for rendering which needs projective rather than Euclidean geometry, but its benefits for rigid solid modeling over top-down are less clear to me, though for articulated solid modeling bottom-up seems preferable. What other approaches to defining Euclidean space have been proposed? Vaughan