So we've got a series of results. I'm gonna state them fairly carefully now. A sufficient condition not a necessary condition. Well let me back up. What we've got is a series of if then propositions. Every one of the propositions, except the last two, in the if clause is centripetal force.
If it's in centripetal force then this. They're all if then force proposition. No empirical thing in here. Just if thens. Given those if then, what do we have? A sufficient condition for Kepler's area we were to hold. A necessary and sufficient condition for Kepler's three as power area to hold for multiple bodies moving uniformly in concentric circles.
A necessary condition for bodies to be moving exactly in ellipsis In which all departures for uniform motion in a straight line are directed toward a focus on the ellipse. A sufficient condition. For Kepler's three halves power rule to hold for multiple bodies moving in confocal ellipses. A solution for the motion of a projectile along the conic section trajectory.
Under inverse square of forces, that in principle can be applied even to comets. A solution for vertical fall is under inverse square of centripetal forces that allows the difference between this world and uniform acceleration and free fall to be determined. And a solution for Galilean motion under resistance forces that vary linearly with velocity.
Which, in principle, allows the differences between Galileo's solution for free fall and projectile motion and the corresponding motions in resisting media to be calculated. I say, in principle because it's pre-supposing. Resistance varies, as velocity. And my question is the question that you have to figure how and why he answered it.
So what? What conclusions about the actual world do a bunch of if then statements give you? Why are we so much better off with these 11 propositions than we were before we had any of them? And all but the second one on uniform circular motion, no one had ever been near before.
Okay. That's your paper topic in a way. It's also the exciting thing because of course Newton is sitting there asking himself that question, and in asking himself that question he ended up generating a 500 page book, all devoted to the so what question. What can we get out of this?