# Absolute space

In *The Principia,* Newton also commented upon Absolute Space, the physical space in which we move about:

**Absolute space, in its own nature, without relation to anything external, remains always similar and immovable. Relative space is some movable dimension or measure of the absolute spaces . . . [Newton, 1687]**

It was appreciated in Antiquity that Absolute Space has what appears to be intrinsic properties: it has three dimensions, it appears unbounded, it seems directly linked to the surface of the Earth, and there is a distance between any two points in it. Because it has three dimensions, we shall denote the abstract ‘thing’ described by Newton by *S*^{3}. In the same way as we represent Absolute Time *A* by the real numbers R, we represent Absolute Space by three-dimensional Euclidean space, E^{3}: this is the set of points in the Cartesian product R^{3} = R x R x R plus the Euclidean distance rule between any two points in R^{3}.

The Euclidean distance rule states that given two points *A, B* in *S*^{3}, then the distance d(A, B) is given by the rule

where *(x _{A}, y_{A}, z_{A})* are the coordinates of

*A*in R

^{3}and similarly for B. Such a rule has all the properties of a metric (q.v. Appendix).