The motion of a disc is of great significance. But it is so curious that we can easily fail to recognise its true nature.
As a child, I was introduced to the disc in the form of a bicycle wheel. At school I was informed of the diameter and the radius but misinformed about the circumference! The perimeter of a circle is not its circumference. The term circumference is in fact Latin for perimeter, but the word is used for the perimeter of any form, including the perimeter of a sector which is part of a disc!
When 2 radii fall on a kilos, they from a sector which has a perimeter which includes the 2 radii. . This perimeter increases with the rotation of the 2 radii, untill it becomes the maximum, which is the perimeter of the kilos without the radii. Thus the word perimeter is flexible in that it can exclude parts of the perimeter not under discussion or observation..
An arc is thus part of the perimeter of a sector, a segment and a circle!, that is an arc is part of the circumference of these forms or figures.
It was clear to me that the distance a wheel tools is the distance around the circumference of the citcle, or the arc of a sector. What I failed to understand, despite Numerous bike rides, car journeys etc, was that the distance the centre moves on a flat surface is the circumference of the circle. It took 50 years to realise that proportion!
Now, working on trochoids and rotations I realise that on a curved surface, the distance the centre moves is proportional not only to the length of the arc the circle moves along, but also the proportion of the radii I of the curved perimeters.
If a circle rolls inside a circular curved surface then the centre moves along a circle concentric to the curved surface and proportional to the radius of that circle to the curvature radius of the surface.
If a circle rolls on the outside, then the centre moves on a concentric circle larger than the supporting surface and consequently further in the proportion to the radius of curvatures of the 2 circles.
But in addition to the proportion of the radii, the centre moves in proportion to the length s of the circumferences of the circlesin contact.
This is an example of a logos Analogos where the logos of the circumferences is dual led with the logos of the radii, not academically but in actual dynamic description of a circular motion in the plane.
The generalisation to space must include at least a third logos representing concentric spheres.
The movement of gears represents this knowledge in its most pragmatic form, and now I can say that the confusion I experienced in studying gears is due ntirely to these incorrect childhood assumptions I relied upon intuitively.
Intuition as to be constantly challenged and updated, which makes it a learned skill and not intuition at all!
It is but a short step to the realisation that logos Analogos is principally and eminently defined by the two magnitudes of concentric spheres, and their many parts.
This is indeed the harmonium men's ram, and the Harmony of the spheres of Pythagoras. Eudoxus lays it all before us in books 5 and 6 and others develop it from there on. In book 3 of the Stoikeia we find some introductory notions of Apollonius which prepare a firm foundation From the later in the course, but more profound notions in books 5 onwards.
We also have the foundation of lineal Strecken as conceived by Prussian geometers, but which Hermann Grassmann developed into a lineal algebra, and the basis for Quaternions which Hamilton developed into a quadratic algebra embedded in a cubic algebra.
Fundamental to these development was the dynamic notion of rotation of a sphere and the rotations of concentric spheres in a harmony.