On 11/9/2020 at 7:39 AM, Boydstun said:
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I note also, concerning the Greek ascription of circular motion as fundamental natural state and explanatory base for celestial bodies that the Greeks did not have the elementary equation for circular motion, which was not derived until Huygens and Newton found it.
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Here is that discovery, as I conveyed in a work from 1995:
V2N2 pp. 27-28 “Space, Rotation, Relativity”
It was Huygens who named the tendency of a body in circular motion to recede from the center the centrifugal (“flees from center”) tendency. He succeeded in quantifying the strength of this tendency; he gave us the formula of what we, after Newton, take to be centrifugal force. Imagine sitting on the edge of a merry-go-round holding a plumb bob. The bob hangs vertically initially while the merry is still, but moves outward from the hand as the speed of the merry increases. Let the speed of the merry be made constant. Huygens computed from pure kinematics that were the bob released, rather than held by a wire, it would travel outward a distance proportional to the rotation speed squared divided by the merry’s diameter and directly proportional to the time interval since release squared. This is so provided that time interval is small, and small is all we require for a formula of the instantaneous parameters of circular motion.
Huygens knew that distance in one of Galileo’s expressions for free fall is also proportional to the square of the time interval. Huygens still thought of the tendency of bodies to fall and their centrifugal tendencies as an inherent force, or power, they exhibit in those situations. He did not yet have clearly our Newtonian concept of force as external cause acting on the body. A whirling body has a centrifugal tendency, and like the falling tendency of bodies, it yields not a uniform motion but one proportional to the duration squared, at least for short durations. So for Huygens, as for everyone after him, a body rotating at uniform speed is [classified as] undergoing an acceleration. Huygens initiated what we should now call the dynamics of circular motion, as he quantified the centrifugal tendency, by determining what rotational speed the merry would need to for the bob to have a centrifugal tendency equal to its gravity.
V2N3 p.53 “Space, Rotation, Relativity”
Newton imagined a square inside of which is circumscribed a circle. Let the sides of the square be banks of a square billiards table (without pockets). Launch a ball so as to strike and rebound at a point, on each of the four banks in consecution, midway between corners; the very point at which the imaginary circle touches the square. The angles of incidence and reflection will be 45°. In making one complete circuit, Newton figured, the force that the ball exerts on the banks in reflections is to the force of the ball’s linear motion (the ball’s linear momentum) as the path length of the ball’s circuit is to the length of the circle’s radius. Newton then showed that whatever regular polygon is fitted about the circle (replacing the square about the circle), the ratio of the ball’s reflecting forces exerted in a complete circuit to the force of the ball’s linear motion is always as the circuit’s length to the circle’s radius. Then if we allow the sides of the polygon to become infinitely short and numerous, the polygon becomes the circle, and we have that the ball’s force exerted on its circular container in one revolution about that circle is 2πr/v, where r is the circle’s radius. Then the instantaneous force the ball exerts outward when in circular motion, the endeavor of the ball to flee the center, is m(v·v)/r, as Huygens had earlier found, unbeknownst to Newton at the time of his own discovery.