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 S3. In the same way as we represent Absolute Time A by the real numbers R, we represent Absolute Space by three-dimensional Euclidean space, E3: this is the set of points in the Cartesian product R3 = R x R x R plus the Euclidean distance rule between any two points in R3.
The Euclidean distance rule states that given two points A, B in S3, then the distance d(A, B) is given by the rule
where (xA, yA, zA) are the coordinates of A in R3 and similarly for B. Such a rule has all the properties of a metric (q.v. Appendix).