which is ones experience of ones own internal motions;
which is ones experience of ones own external motions and interactions;
which is ones positing that others have internal motions and experiences as one does and indeed that others exist internally independent of one;
which is positing that one shares a common external experience and interaction with others.
This image illustrates the proof by Huygens, and in fact is one of many demonstrations Newton explored to establish the invariance of the solution. Without knowing that either light or sound had a finite speed, Newton was not aware of the need to explore our apparent experience of acceleration.
All the "proofs" of this formula rely on an intimate knowledge of the behaviour of the sine ratio. Sincee it was the great mathematical enterprise of the centuries after the Arabs gained control of the worlds empires, (and traded with those it did not control), it is to be expected that much was known about the sine values for small rotations. Thus the approximating of the sine ratio by a ratio of equality is the basis for Newton's reasonable lemmae about quantities and ratios becoming dual as the quantities converge.
The value of Newton's exposition is the clarity with which the Euclidean propositions and constructions are laid out before the reader. Thus to anyone familiar with Euclid, first hand, Mr Newton provides compelling demonstration of the applicability of these euclidean notions and the "rightness" of the conclusions. Nevertheless, he did not cease to test the formal derivations against the facts of measurement and observation, and new thinking.
Thus when Cotes revealed to him the arc measure called the radian, and indeed the use of √-1 with the logarithms to show that the sin and cosine could be related in this way ln(ix) = cosx +isinx, Newton did not fail to see yhe application to his derivation of the acceleration quotient, and how it may provide a refinement above the clear and present approximations he justifiably made without concealment.
In the end, De Moivre, Cotes and Newton had a de facto understanding of what w now call vector algebra as it applied to the Euclidean plane, and Kepler understood the stereoscopic application of Euclid to the spherical space around the planets, which we now consider a 3d vector algebra.
What i am plainly saying, and have said, is that Euclids Stoikeioon is a Vector Algebra rhetorically exposited.
There is a constant relationship also between the tangent the chord from the point of tangency and the rotation of the radius . The angle of rotation (or the arc) of the radius leads to the the chord rotating half the arc relative to the tangent, while the new tanget rotate the full arc relative to the tangent. This is true even of a full rotation. But if w then proceed to go again the measuremen must begin again. However, if one moves around a kissing circle with the same tangent one may se the full rotation of the chord back to its original position.
These topological considerations were not new to Euclid and Ptolemy, and in fact they well illustrate the inherent topological nature of the Stoikeioon. Despite the insistence on no unreasonable neusis , the Euclidean school were well versed in topological morphing of figures by para;;e; ;ines and corcles. The notion of duality was and is clear. It is we who later invemted area to attempt to understand what remains the same amongst all these topologica; shifts.
In fact area does nothelp. If anything it confuses. What is clearly presented in Euclid, is the mystery of space and time in our perception of it.