The Kuklos is a disc, as such it is a material object. All material forms hae a boundary, but a Kuklos has a special boundary called a Periphery. That is not all that is special about the Kuklos. It has astonishing symmetry, so astonishing tha it can be defined by it
the notion of symmetry is a tautology, for it is in fact defined by the Kuklos. Summetria on the other hand is a more general notion defined by the social experiences of coming to an agreement on a measure or metron. It is usd to symbolise or signify a binding contract because of the social significance of a common metron. In the Aramaic traditions in the bible equally sharing an animal was signifying the same notion of sharing a common meal or portion. Fairness and justice both derive from this idea of summetria.A summetria is therefore a binding contract which links parties. The notion was taken into the "mathematical" idea of a group, and in its development into group theory.
Group theory however splits the longstanding relationships of Astrology into smaller, defined sub groups of relations inorder to study them specifically. Consequently the development of group theory and ring theory leads to increasing specialization in mathementics, making it harder to see the big picture, which is astrological. Even in Grassmann's time he recognised this development, but in fact it had been going on since Aristotle set up the Peripatetic teachers. Each struggled against the other to establish their curriculum or discipline as being the most imortant thing to study, and so strove to distinguish themselves, and thus their subject. from this competitive set up came the move into specialisms and the renaming and re-branding of subjects and the development of subject boundaries: the mess we are so proud of today!
So the astonishing symmetry of this disc meant that there was a unique measure called the diameter. This measure is used in all cyclic shapes, and is sometimes called the diagonal. The diameter is a measure, but the line that is associated with this measure is part of the boundary of the hemikuklos. Thus we confuse a measure with a line very esily in a kuklos. too disinguish the line we have to consider or "look at" the hemikuklos. Why make this distinction? because a kuklos is a disc and it has no line across it unless we draw it or cut it. to find where to draw this line is problematic! However a hemikuklos has this line as part of its boundary. "Matching" the hemikuklos to the kuklos enables us to find this line easily. Matching then is a fundamental procedure in the Stoikeioon, and it is one of the common notions of duality.
Similarly we can argue that we require the demikuklos to find the ortogonia, which is a rotation metron, but the line associated with it is part of the boundary of the demikuklos.
The kuklos thus provides us with both measures for diameter, a length, and orthos, that is an arc. An arc, rather than an angle is important, because it relates to spherical forms. Angles are a late notion that entered onto the scene sometime in the renaissance, before that all understood gonu to be the rotation of the "knee" joint and the arc it described.
There are 2 ways to describe or draw a circle, which kinematically are different. The first is to drag a stiff v shaped form around one of its ends keeping that end "fixed". This is the rotation method. In it everything rotates, and these things rotate in synchrony and symmetry and relativity, harmony and profound resonance. This is the basis of Pythagoras's measure, often called the harmony of the spheres. As nusical as it sounds, and it is related to music, it is in fact a description of scale measurements. For harnony reade Octave interval and you will see the ratios of this interval written out. Pythagoreans suspected and believed that these balanced relations were universally important in all dynamic systems. Kepler went a long wy to proving them right to hold this insight. Ok so we say they were not completely "right" but who ever is?
The second method is to draw a circle freehand, Freehand means that you the drawer are consantly adjusting the motion of your arm to describe a circle. We can characterise that motion as small corrective motions while attempting to draw an arc. If we do that we find that we actually apply 4 corrective motions: right and down, down and left, left and up. nd up and right. We call these stepwise motions and we can apply them to whatever level of precision we want.
Now, due to Descartes we actually have done a lot of work on the stepwise corrective motions mathematically, but the rotational motion not so much. IN particular we have used transforms from the stepwise to describe the rotational. In so doing we have glossed over the fundamentally different kinematics. We have covered over something that is fundamentally important and inherent in the dynamics and kinematics of the periphery of a kuklos. The tangent is in fact only half the answer!
As descried above, there is so much more in the rotational method, which routinely is ignored in he mathematical treatment of it, or which is pulled in to justify a sep in the stepwise treatment! The tangent has been the tool of choice to unlock the secrets of the periphery of the kuklos, but the tangent has been considered as not part of the kuklos, and this is the difficulty.
If we take a demikuklos we see no tangent, we see just the arc an the right angle. However, if one now flips the right angle along one of its legs we immediately see the tangent. Tis is nothing new. We know that a radius meets a tangent in a right angle. What is new is that the gnomon, or the right angle is a fixed relationship involved in the dynamics of the describing of the periphery of the kuklos. We only need the demikuklos to see this. The demikuklos thus embodies the tangential relationship, and we may use it to describe this relationship. However, when we do that we justify a circular rotational motion independent of tangential initiators,but rather inherent within the demikuklos of the form Kuklos. Tus we can backtrack to this rotational motion being inherent in the kuklos form.
This is kinematically distinct and requires careful analysis.
For example, simply switching to radians to describe tis motion is a misleading step in the analogy between stepwise and rotational descriptions because the relationship is actually based on the sine and cosine ratios.
Also there is a great tendency o use he term constant in a loose fashion, when kinematically the rotation method is far from constant. there is also a tendency to use zero as an evaluaton rather than as a condition. One other thing:from misapplying calculus we tend to assoiate an inward acceleration with a second derivaive of the angle change, but in fact the inward acceleration must vary as the rate of change of the angle, that is the first derivative. If you use the demikuklos you can see why.
The circular discs used in a pulley did not solve the issue of circular motion for Newton, but they went a long way toward the solution .Pulley systems were the ideal model for dynamic and static equilibria. They explained why objects stayed at rest or in uniform motion in space and why hey had to be pressed out of a straight line by another force. That is Newtons first law, modeled in a pulley system.
Although circular motion was essential to their action it could not be fully explained by their action and interactions. Nevertheless, his third law was also demonstrated by the pulley systems.
Now we can understand the subtle confusion that arises from Newtons definitions, because the Tangent and the chord can never be equal in nature to the curvr! Newton says so in Lemma 7, however the quantitative measure of the same can bear the same numeral. All Newton et al claim is that this being the case , one may start ones analysis at the moment when this is the case. In using the measure for this purpose,much confusion has resulted over the nature of these curves, which remain unaltered, and the nature of the method of reasoning he is using. the things studied do not change their nature, rather we as analysts change our perspectives to enable a sound basis to our analysis. Thus starting from this nascent position, enables Newton to equate like for like, and hopefully safely conduct us to the correct starting point for the various analyses.
It is clear, almost from the outset that Mr Newton desires to use the Euclidean rectilinear forms, with parakkek properties as his method of compounding the various forces and accelerations and velocities, because they strikingly and convincingly present themselves readily in all his experiments using pulleys. but it is also clear that they must be applied in situations of static or dynamic equilibrium. It is to the evocation of these states that Newton bends his method of fluents and fluxions . In safely determining when these forms may be applied, and how their dynamic development may safely proceed over finite time, Mr. Newton did not shirk his responsibility to be scrupulous/ So it is rather unworthy of some less scrupulous to impugn him, not so much of error, as underhand methodology..
Being thus thoroughly confused they make the profoundest of errors themselves, which Mr Newton hoped to avoid of himself.
The Kuklos is a pinnacle form worthy of much study as it leads on to all general curves. However ideal it stands as a real entity in our kinematic understanding of motion, and indeed founds it rather than confounds it. It as torque is possibly the most nascent of the forces we have occasion to meet in experience, yet nevertheless its influence in all curved forms is tangible, if weak. Without this ideal form and its weak mechanical counterpart, there would be no mechanics.
The tangential principle. or as it is better known the principle of the right line in kinematics obscures the principle of the kuklos or the spiral principle both of which are evident in mechanics everywhere. it is kinematically dominant because we make it so. the western Europeans made it so as a standard form, but it is not at all naturally dominant. Why do we push the curve into the realm of compound forms instead of fundamental forms?
Of the three the right line, the circle and the spiral, the circle is the cornerstone form.All other curves or lines derive from it.
Parallel curves depend on the kuklos . Thus any curves with the same centre of curvature and and concentric kissing circles are parallelfor all those parts for which these conditions are met.
Also curves are parallel if they represent s translation of the curve between parallel lines,and have the same curvature.
When a disc roll alung its periphery the centre of the kuklos is translated horizontally the distance of the amount of periphery. But should the kuklos slope and describe a closed circle then the distance is in the proportion of the cone it describes, and the cone it forms with the centre it rotates around.
In a plane circle, any circle rotating around its periphery, move its centre further if it is outside the plane circle than if in. Thus the motion of the centres do not measure the periphery of the plane circle.
If rotating a circles centre around a larger circle by rotating the circle against another, the rotation of the cirle to keep pace with the rotation of the radius is found using parallel line and radian measures
To find how far the radius in a rotating circle must move, transform to lengths and the transform back to the dimensions of throtatin circle.