When two scalars are brought into relationship with each other, a ratio is the result.

Ratio

"the relation between two similar magnitudes with respect to the number of times the first contains the second."

As can be seen in the definition of the ratio, the ratio adds the concept of proportion to the concept of magnitude. This gives rise to three possibilities for the proportions of the ratio: one magnitude is either equal, greater or less than the other. This introduces the concept of a scalar orientation.

Orientation

"to adjust with relation to, or bring into relation to surroundings, circumstances, facts, etc."

The three possible scalar orientations for a ratio of scalar magnitudes A and B, as A:B, are: A=B, A>B and A<B. These relations shall be referred to as scalar orientations, since the definition of "motion" requires a place or location and that concept does not yet exist.

Recall that the minimum scalar magnitude is unity, so the ratios of A/B or B/A will never become undefined since neither A nor B can be zero.

Note that in the low and high orientations, the possible combinations of scalar ratios are unlimited. But, where A=B, only one ratio is possible: unity. The scalar orientation structure thus shows a natural separation across a common "scalar boundary" of unity, which can be used as a reference point, a natural datum. This gives a "place" or "location" in which we can begin to define scalar motion, but it has a problem: given any ratio, there is no way to determine if you are observing A/B or B/A… another reference point is needed to determine the orientation of the ratio, itself, with respect to an observer or environment. This can be found in the invariant property of the cross-ratio.

Invariant

"a quantity or expression that is constant throughout a certain range of conditions."

A cross-ratio is literally a "ratio of ratios", and is the only projective invariant in all strata of geometry. In a scalar sense, it relates two scalar orientations through a ratio, and that ratio remains constant—giving a secondary orientation to the ratios and producing scalar motion. One can also think of it as the ratio of slopes between two lines on a graph.

Note: A "projective invariant" simply means that you can't alter it, regardless of what perspective transformation you apply to it. A perspective transformation is the process of introducing assumptions to coordinates, to produce a reference system (such as a plane at infinity, and the "eye cone"). More on this later.

There are now two things to consider as the basis of scalar motion: the cross-ratio and the ratio orientation.

The cross-ratio introduces the concept of association. In a geometric sense, it is like two points joining to form a line, except here you join two ratios to form the cross-ratio. The result is a concept we call "dimension."

Dimension

"a magnitude that, independently or in conjunction with other such magnitudes, serves to define the location of an element within a given set."

There are only two possible ratio orientations, since there are only two scalar magnitudes involved in a specific relationship: A:B or B:A.