I am undone! Woe is me !
However it is not as bad as that. Reading the Astological principles further in order to understand the introduction of centrifugal force, I find that Newton ascribes its description to the excellent mr Halleys work on hoological oscillation! Of course he mentions Huygens and Wren.
The first reveal was newtons scholium on the astrological definitions of various quantities and measures and the general relativity of these terms snd ideas. In so doing he describes certain absolutes as concepts the reader is expected to accept without question, either as astrological or commonly accepted among philosophers. Thus without being immersed in his times it is easy to miss certain subtexts implied by certain ideas.
The second reveal is the absolute geometrical nature of the theorems and corollaries by which he derives the centripetal force . In so doing the reader is drawn away from the heavens to geometrical figures. Having demonstrated in corollaries certain ” mathematical” relations, he asks why they may not now be applied to the motions of orbits in space!
Well the reason is, in my opinion that the construction of the curves is so evidently geometrical that one must also suppose that this is how god does it in the heavens! . This is evidently not a sustainable assumption, except if one be a believer in god. Yet I caution that the god who this might be attributed to states plainly that he will not allow this assumption!
The more reasonable circumstance is to say that this method is a geometrical model which must prove itself by empirical observation to be useful, and nothing more. This utilitarian view avoids the difficulties arising from the models short comings.
Now we may see how Newton models a centripetal force , drawing a body from a rectilinear course ( potentially) continually into a curve..mthat the rectilineal motion is but a device is evident by Newtons process of eliminating it to the curve, and stating certain laws in terms of the arcs, not the tangents or chords of them.. And yet there are sod who insist on the rectilineal as the true motion, denying the curve of the orbit.
Now as a second but contrary proof Newton refers to Halleys centrifugal force. This is the force imparted to a curved or circular boundary by an object constrained within it! Thus the object by impulses on the circular boundary transmits a centrifugal force, o which the boundary responds so that the object is turned into such sn orbit as might be considered circumnavigating. Newton observes that the force that expands the circle outwards is equal and opposite to his centripetal force, by which the centre attracts an object into an orbit..
Thus the curved orbit is constrained within 2 encasing methods which both agree in the evaluation of the force required to turn a moving obje with certain velocity into a curved orbit.
Precisely how this is done is not here discussed, but how it may be modelled by infinitely reducing triangular areas containing the curve is. Once again this is a model process we are asked to apply to the behaviours in space, whose mysterious actions remain mysterious, unless you believe this model process to be actually how Nature does it!