Newtons definition of centripetal force precedes his application of the Galilean principle which he reduces to 3 sorts or kinds of vis as a centripetal force.the absolute, the accelerative and the motive.
It is to be noted that in defining centripetal force the first presentations describe it as drawing a body away from its natural or innate rectilinear motion, that being the propensity of its innate force. However this is rather a stronger notion than the original Greek idea of a ” good” or “straight ” line allows. Such a good line if it be true to its nature in the motion may in fact be a curve! In any case the assumption that motion is naturally rectilinear is hardly empirical by any degree!
Supposing now that the innate motion merely attempts to continuously recede from the centre which impels it from its recession hardly spoils the plot, but in fact enhances it in the many and congenial examples he gives of a centripetal force in action, or behaving as he wishes it to be described.
We may also have missed the import of his second example, that such a force is as an example magnetic, and as a third example but of an unknown force, gravitational!. So in proposing this definition it is not to define gravity but all kinds of force that tend the motion of a body continually to ward a centre, and so doing by altering the bodies innate velocity Leo, as well as its general recessive course, we’re it to be viewed from the vantage of the pulling force centre!
There is then a ballistic discussion mediated on the introduction of the idea that the velocity of a body is altered by a centripetal force, in which the relative view is again switched as the explosive power of gunpowder now impels a body to some velocity which gives it increasing range , until at last a velocity should be reached by which the projectile completes one orbit! The argument being so clear and logical that it has been remarked that it alone was sufficient to convince his peers that he knew whereof hew rote. But it should be noted that with increasing peed the missile was said to pass out of orbit to continue into gods universe unhindered. . This vision was clearly advanced so that he might bring the moon into his discussion forthwith and deny ny argument that the moon was drawn out of its way by the same centripetal gravity that governed the ballistics of the projectile.
This having been advanced he leaves it to the mathematicians( that is the qualified astrologers) to determine this exact velocity , and the intensity of this centripetal force so that any body in orbit might just remain so , falling so as to miss the earth continually, but not to come crashing down onto the earth, nor yet to fly away.
Now some have read this and failed to see the complexity of what Newton set out to do. As a consequence they have not grasped the complex nature of the orbital problem in particular, the many forces as impulses or resistive forces or impelling or expelling forces that are herein described and employed to convey the conceptual framework he thn goes on to address. Therefore ome have rejected his later more careful description of circular and orbital motion in which centrifugal force is admitted to the description. It is empirically obvious to all that such a force is needed to account fully for observed behaviours, but some have sought to make velocity the source of this centrifugal force. The changes in in the tension on a string would properly imply both a centripetal and centrifugal force, which when removed by swinging a projectile in an elliptical arc results in the projectile crashing to the ground under no centripetal orce by the string! Thus the string not only must supply centripetal force but also centrifugal force to keep the projectile in that correct attitude to orbit.. The velocity itself if it was solely rectilinear would behave tangentially in breaking loose of the projectile, but instead it behaves as if it has another velocity imparted by a centrifugal force compounded with the supposed rectilineal one ( by those who would argue that rectilinear is a straight tangential velocity instantaneous to the orbital curve).
That being said we must now proceed past the moon to the Jovian system observed by Galileo to follow newtons establishment of his force triumvirate.
Centripetal force is of 3 kinds, and each kind is a measure/ Metron or measured quantity the absolute vis measure is a quantitative appreciation of a central agency. His example is the many kinds of lodestones nd their corresponding intensity. Thus should we specify a particular lodestone, or even Jupiter, that is a particular absolute vid.. Thus absolute means an isolated or specific potency under investigation, and the sole cause of all that is considered in following discussions.. It is defined as acting from the centre.
Later Newton adds that this absolute vis is the cause or central causative agentin the system. I have taken that to mean in the universal system god, but in a local system, clearly it is a specified quantitative object.it has to be quantitative because he is establishing a quantitative fractal system as designed by Galileo.
The accelerative vis is next defined. There are more examples of how the accelerative vis varies around the specified object or centre.. Thus if we isolate a quantity from an extensive magnitude, that is only the beginning. We must now examine that quantity carefully snd specify som sub quantity measure!in other words, taking the object as a whole we must now divide it into multiple parts..in this case taking the intensity of the absolute vis we must now render it into its behavioural parts. In this case , as he later shows, he desires to establish an inverse square proportionality as the parts. This is his taccrlerative vis proportion, which he will later empirically and geometrically and astrologically establish.
He quotes one observation of Galileo directly, that accelerative vis imparts the same velocity regardless of mass at the sae height or distance from the accelerative bis centre.. But the absolute force for each identified system will be different, so the subsystem accelerative force will have a different value but the same law of calculating the measures..
Now I took the accelerative vis to be an instance of a universal vis, an absolute vis, but clearly it is a scale of behavioural action centred on the centre of the absolute vis.. Thus in the universal view accelerative vis describes how the various parts of the universe should behave relative to the centre. Immediately it becomes apparent that this system cannot have a universal application! No one knows where the centre of the universe might be!thus despite its general applicability it is not a universal system but a local empirical one.
In that regard, yhe recent expositions on dark matter seem premature. If we choose a galaxy as an object we can use this system to estimate whether an inverse square law pplirs to the galactic syste, the galaxy is assumed as the absolute vis nd an accelerative vis is designed for it from the galactic centre. When this is done, it apparently does not work! So dark matter was invented o explain this” anomaly”. This ord Anomally means infringement of alas, but it is us who imposed this law on the galaxy in the first place, clearly our law is not a correct description for galactic systems! We do not need to invent dark matter, we just need to get the correct law, which may beone of the other laws Newton tried throughout his astrological principles.
Finally he describes the motive vis measure or Metron. This basically allows the observer to sum up the behaviour of masses or parts of a system which are being impelled by the accelerative vis to move toward or about the centre. In very particular, at the same level one mass may be blanced on a weighing device against not her. Thus the motive vis measures parts against parts under the influence of the same accelerative vis and uses the balance or ahookes spring to establish proportionalmeasures like mass, weight etc.
Again Newton relates this to the astrological Jovian system, in which the moons might now be weighed against each other within the one accelerative force of the absolute Jovian system
There is no as yet direct statement that the accelerative force if the centre itself is in motion imparts this additional motion to the parts it. Accelerates, but this will be advanced later.