Euclidean space began as a model of the real world and became pure algebra
Ancient Greek geometers wanted a model of the space we live in. Euclid's great step was to prove every property of it from a handful of starting assumptions. Over two thousand years later mathematicians rebuilt the same idea from vectors and algebra, and discovered that there is essentially only one Euclidean space for each number of dimensions.
Euclid's Elements gathered earlier Greek work and derived everything from postulates. Some seemed obvious, such as there being exactly one straight line through two points. One, the parallel postulate, simply seemed impossible to prove. That approach, building geometry from axioms, survives today as synthetic geometry.
In 1637 René Descartes introduced coordinates, turning geometric problems into calculations with numbers. That was a real reversal: until then, numbers had been defined through lengths and distances rather than the other way round. Still, the basic definition of Euclidean space did not change until the late 19th century. Meanwhile Ludwig Schläfli pushed geometry beyond three dimensions and found every regular polytope, the higher-dimensional cousins of the Platonic solids, in spaces of any dimension.
Once non-Euclidean geometries appeared, mathematicians had to say precisely what made Euclidean space special. The modern answer is a set of points acted on by a space of translations, shifts that move every point the same distance in the same direction. An inner product on those translations supplies distances and angles, and the result was proved equivalent to the old axioms.
This abstraction has practical logic. Physical space comes with no preferred origin and no built-in set of axes, so working without choosing them is often cleaner. Distances in the mathematical version are pure numbers, not inches or metres. And since all Euclidean spaces with the same number of dimensions are equivalent, picking an origin and perpendicular axes lets anyone use ordinary lists of coordinates instead.
Source: Euclidean space
