Scalar motion is another term that is often used with very little understanding of its meaning. Scalar has already been defined, so let us examine the term motion and its connection with the concept of a scalar:

Motion

"changing place or position."

Motion is a simple enough concept to understand, but when you consider it in the context of "scalar motion", it becomes like "military intelligence"—a contradiction in terms. How is it possible for quantity possessing "magnitude only" to change place or position, when both concepts are totally foreign to the idea of a "magnitude only" scalar? It can't, and there lies the problem with the term "scalar motion."

Exactly what is meant by the term, "motion," when associated with the concept of "magnitude?" The answer is found in how we express the concept of motion as speed—an inverse relation between some "quantity of spatial distance", s, and some "quantity of time," t, as s/t. In other words, speed is just a ratio of space to time and therefore motion, and in a more generic sense is simply a ratio of quantities.

It is important to understand that the concept of motion is a subset of ratio, because ratios deal with magnitudes and motion deals with quantity (magnitudes of something, namely space and time). In essence, we have two similar concepts: that of scalar ratio (generic) and that of scalar motion (specific to space and time).

Scalar Motion is therefore the projectively invariant cross-ratio, with specific aspects of space and time.