Given any scatter of points we define 2 kinds of sequencing.
The first is sequencing by observer orientation;
The second is sequencing by observer translation.
Given points a,b,c,d…. The observer either positions itself on dot a or establishes itself as an extra dot O. Then by Turns (!) the observer includes each of the dots in a sequence. It is this " by turns " that establishes the sequence of orientations that maps the sequence created from the points.
If a and b are any 2 points then a+b is the orientation of an observer at a looking directly at b . It is clear that a +b is not the same as b + a, but they can be defined as contras of each other
Thus a + b + c +… Represents a sequence of orientation changes which are defined as a rotation. This is thus a rotation sequence.
Now let the observer orient himself a + b, then define ab to be the observer looking directly at b from a precisely when the observer is a + b.
Then ab means a projection of a onto b, a direction drawn from a to b by the observer translating in a straight line onto b. it could also be thr system of points that define a straight line at each point between a and b that are precisely on a straight line.
They could equally be defined as the straight line H generated by ab.
If ab is a line then abc cannot be a line. It has to be a projection of a point onto a line or a line onto a point.
To translate around a sequence of points I have to do ab + bc +' cd…………..in which the observer must be oriented correctly at each pont for the form ab ,bc… etc. to make sense..
Although rotation seems to be a magnitude, it is a special kind of magnitude: a self referetial one. We could define rotation as the fundamentl ratio Logos in which any 2 magnitudes of the same kind have to be copared. it is a matter of fact that unless this minimal condition is fulfilled we cannot subjectively determine rotation, let alone objectively. The 2 magnitudes used are the radius and the arc on a sphere with one being fixed either objectively or subjectively or both. Any change is compared against the fixed magnitude, usually the radius or diameter of the sphere.
If the arc is fixed, then the centre of the circle moves away or closer but the arc curls or straightens as it does so. This behaviour is used to define curvature.
This behaviour of the circle is perhaps the clearest example of quadratic proportionality or perhaps mor generally hyperbolic proportionality. It intimates that some magnitudes we experience do not vary in a simple logos Analogos relationship like projective geometrical ones with fixed constraints. Our experience of variation in sequence is more complex than that.