What I do want to show you, though, is what we, I think almost all of us who've worried about this, feel that Hooke contributed with this. This is more complicated than it first appears to be. What we almost all think is that Newton had already been lucky at curvilinear motion, but he had been looking at it in terms of his solution for circles.
Now I'm gonna stop a second, switch to last week's for one moment, and show you something that I failed to stress last week in my haste. It's at the bottom here. After he gets this bouncy motion for it to get a circle with increasing the number of sides on the polygon, notice at the bottom of the page.
If the body B moved in an ellipsis, that its force in each point, if its motion in that point be given, will be found by a tangent circle of equal crookedness with that point of the ellipsis. Now let me translate that. Take a circle that gives you the radius of curvature and the circle of curvature at the ellipse.
You know what the force perpendicular to the trajectory, the force into the circle, has to be, V squared over R. So if you know the velocity on the periphery, you know the size of that circle and you know a component of force perpendicular to the trajectory. Problem, now everybody see, this is from the Weiss book.
This is far earlier he's thinking in terms of curves are made up of series of circles. Now let me drop that and go to my one for tonight, which is my figure. That's not what I wanted to do. Let's make it easier, just close that. So the picture here, and it's actually a picture you've seen but not drawn this way.
I've drawn it the way it's taught in text books today. It's called an osculating circle. The picture here is suppose suddenly you've got this object moving along a curvilinear path and there are two forces on it. One of them's perpendicular to the path that determines the curvature at that point.
The other is tangent to the path and it either accelerates or decelerates. Suppose you suddenly ceased the force that's accelerating or decelerating. Then it would go into that circle permanently. Now think of the curve being made up as a series of such circles, one right after another. They're called osculating circles.
You saw it with evolutes because that's what the evolute was doing, was giving you a series of osculating circles by giving you the radius of curvature, one after another. Both Huygens and Newton looked at the problem of general curvilinear motion from that point of view. But you see immediately the difficulty.
You know one thing if you know the velocity. Huygens had published for an ellipse all the radii for curvature. He had given you the solution for radii of curvature all around an ellipse in part three of Horologium. So you know the radii of curvature, you know the velocities, you know one of the forces.
What's controlling the other force? You've got two unknowns. You've got more unknowns than you have any way to solve. Okay, what did Hooke contribute? He said don't think of this as two independent forces. Think of it as one force directed to a single point in space. Then since you know one component of it, and you know the angle that that component forms with the line to the center, you know the other force.
You've reduced two degrees of freedom to one degree of freedom simply by stipulating the force is always aimed to the same point in space. Especially if you say it's aimed to a focus, but it doesn't matter which point. So what Hooke contributed to Newton was a, the picture, what's happening here are two motions combining.
And b, they're not two independent motions if they're being governed, for example, by an inverse square force directed toward a single point. They are instead mutually determined simultaneously by the force. Given the condition at one point on the circumference and the variation of the force, you should have it all.
Fair enough? So it's a reduction of a more complicated problem to a tractable problem is what, and I say we all. First of all, the vast majority of the people who worked on scholarship on the Principia, I really am sort of the last living one of us. So when I talk about the rest I'm talking about a bunch of people who aren't in a position to contradict me anymore, but I think there's agreement.
We've had more than one occasion we were all together at the Royal Society back in the late 90s for a week, roughly ten of us. And I think there's just total unanimity on this, that, at the very least, what Hooke gave Newton was a way of conceptualizing this problem that made it tractable.
Now notice, you're not doing general curvilinear motion anymore? You're instead doing a particular generalization of circular motion. What generalization? One in which the forces are always aimed toward a center, but they're allowed to vary with the distance to the center, okay? Newton never gave Hooke any credit for anything.
So bad was it that Halley had to intercede in book one to get Hooke's name into it, which he did. He talked Newton into putting the name in. It goes in a string, Wren, Halley and Hooke. So it's even then begrudging. But you'll see the significance of that in a moment.
Whatever, though, Newton never gave any credit whatsoever for getting anything from Hooke, but it seems very clear he had no way of doing this kind of problem until after 1679. Because when we looked at the earlier stuff he's trying to do it with osculating circles, as Huygens tried to do, and it just doesn't work.
And I said why. If you don't constrain the general curvilinear problem, you've got thousands of solutions, thousands of things that could be happening. Constrain it where there's only one force acting at any moment with two components and it's a different problem. Fair enough? I don't want to dwell on that but I want to drive the point home.