Newton now does three problems, the first problem looks really weird and I trust most of you, if you read with any care, thought why in the world is he doing this? Here's the first problem, suppose the force center, suppose the motion is on a circular arc and the force center is on the circumference of the circle.
How does the force have to vary to keep the object in the circle if the force center is on the circle? Now obviously it can't get to the force center, that's a singularity it's going to go in, so we're doing part of a circular arc toward that and his answer is one over r to the fifth.
That is, it's an inverse fifth instead of inverse fourth squared variation, and the obvious question is why in the world does he have this? And, in a paper of mine that I'll put up later in the course, it's way too early to put this up because it's really about the Principia, I make a point namely by early 1690s, we know he had done the full solution for what you can think of as a Ptolemaic ecocentric circle.
Have the fore center not be at the center of the circle, but have the orbit be a circle. Now remember, Jupiter, Venus and Saturn were undetectably different from just what I described, a circle with a center sweeping out equal areas about a point off-center. Okay, you couldn't tell the difference between that and an ellipse.
So, it's a natural question to ask, what about the eccentric circle? And in turns out, you can read here, I'm not going to read out loud my own prose, it's one thing to read other people's prose, but not mine. I'll explain next semester how I did this, Newton himself gave me the method.
It turns out that the correct answer is given here, the variation of the force goes inversely as the product of sp squared, that's my little r, and pv cubed. Look at pv, it's varying all over the place as you go around. Now that's not a simple rule because it's not in terms of r, what you want is r to an exponent, instead you've got r plus this pv cubed term, okay?
Now you'll notice right away if the point s goes all the way to v, so it's on the circumference, and it's inversely it's pv to the fifth, or r to the fifth, etc. So, the solution pops right out of this. I did it a way Newton showed, where I expressed it all in terms of sp, and so the force to do this kind of circle has to vary inversely, and it's a nice, complicated expression.
sp over the radius of the circle to the fifth, plus 3 times 1 minus the eccentricity times sp over a cubed, plus 3 times 1 minus the eccentricity squared, squared, times sp over a, and finally, 1 minus the eccentricity squared, cubed, sp inversely sp over a. So it's four different terms all varying as R, how do you get it to one term?
Well the simplest way, there are two ways to do it, have sp over a become one so that the center of forces in the middle of the circle, and that all becomes a single constant then. I think of the number's eight is what it becomes at that point.
Alternatively, have the eccentricity go to one, so it's all the way to the radius, and then you only have the term on the far left. So my picture is, he had actually worked out the Ptolemaic circle, back at the time, found it was not a good illustration of how to get a rule in distance alone, therefore, took the singular case out of it.
And there he had very, very good reason to look at the Ptolemaic circle because it's the other candidate for what the orbits are, but that's my guess in a paper that is saying a lot besides that. Okay, now turn to problem two, the central ellipse. I'm going to do some of this math to try to show you how to do it, but I want to make sure I get through the evening.
The problem you most likely had trying to follow these proofs is you didn't have the background from Apollonius, which I understand, you didn't have the background from Apollonius. I'm supplying it now to make it easier to follow the math, but I want to make a quick remark now, here and throughout the Principia, as far as I'm concerned you do not have to pay attention to the details of the math unless I point out this case is worth looking at the details of the math.
We want to look at the physics of the Principia, we want to understand why each proposition is there and what it contributes to the whole. The proofs of them, anywhere the proof is not legitimate, it's only one or two places in the whole book, I'll show you. But the proofs are fine, the proofs are in a form of math that, if you think you have trouble with it, the person we were honoring last night, Steve Weinberg who has real claim now to having contributed to physics than anybody else alive tells me every time we talk about Principia, he just can't follow the math.
Okay, so if he can't follow the math don't feel bad that you're uncomfortable, I get very angry at him, saying it's pure geometry. Why doesn't anybody want to work in geometry, et cetera. And I can't convince him, I finally convinced him to look at a little of it, but not much.
So don't feel bad, it's really not very hard math, it has one major shortcoming, each step you've got to have the insight to see why it's true before you go to the next. When you're doing things symbolically, you don't have to think, but without the insight and the steps, it's just words in this case, right?
So we start from a bunch of things, pv times vg, you can see it up here, is to qv squared as pc squared is to cd squared, pc and cd are both half conjugate axis. The way to think of an ellipse is it's a circle projected on a plane that's at an angle to the original plane of the circle.
Diameters, right angles then become congregant diameters. Okay, if you can picture that, that's the way Apollonius of course thought of it, out of his cone, out of things slicing through it at an angle. That principle at the top is actually the generalization of Euclid, 36.3 for the circular case.
Okay, that's what's neat about it, and that's why Newton would have been sensitive to it. Then, the second thing there, qv squared is to qt squared as pc squared is to pf squared, that's from two similar triangles, if you look at it. QVt and PFC. Why are those similar?
Because the various sides are perpendicular. Two of the sides are perpendicular. Well, one side in common, two sides perpendicular to one another. They have to be similar triangles. Once you have similar triangles, you get the ratios all over the place. It's that you're looking for all the time.
Similar triangles give you ratios. And finally, all circumscribed parallelograms are equal in area where the obvious area is ad times pf. If it's a parallelogram or it's bc times ac times four if it's the rectangle circumscribing it. But those are principles from Apollonius that Newton basically just assumed people knew.
I don't know how many people at the time knew Apollonius. Surely, John Locke did not know Apollonius. That's why he needed help getting through the initial. These are the initial proposition in the Principi cell with the same proofs. Locke could not follow it, so again, don't feel bad if you're struggling.
All right. Here's how the proof goes, now. Remember what we want. We want that little distance, qr, divided by the square of qv squared times pc squared. Okay, because the center of force is c. So we write down pv which is the same as qr because those are parallel lines.
So qr times vg is the qt square. Well that's just the thing we want. To start with and we get right off the top combining the first two lines we get the stuff on the right hand side. But qr just is pv, so we can substitute that in.
And we can collapse this all together using this area here that is this rule for equal areas regardless, that gives us line two. Now we let vg approach two pc. Okay, that is as point q approaches p, v moves toward p and as a result vg that whole length moves toward two pc.
Therefore given the limit this expression qr over pc squared times qt squared, that's what the theorem says we have to get. On the right hand side is the twice of the square of the area, and on the numerator is pc. So therefore to keep an object moving in an elliptical trajectory on a plane, to satisfy Marius, in which all displacements from uniform motion in a straight line are directed to the center of the ellipse, the force has to vary linearly with the distance to the center.
The further away you are, the greater the centripetal force. I've given you now, it's a lovely book. I'm just afraid to pass it around. This is the original edition, I'm proud to say, and needless to say, Bernard found it somewhere and bequeathed it to me. It's from 1738.
It's by William Yule, major philosopher. It's a textbook to help people read the Principia. And it has sections on, conic sections. The first three sections of Newton and the differential calculus. A nice, little thin book, when he was pushing, he was Master at Trinity College. He was trying to reform all education at Cambridge and one of the things, it was very big, is read the Principia.
So, he was helping people by giving them texts. And I've just given you his own account to get the Apollonian requirements. So. You can just go over those and see for yourselves how to get these things. You of course learn conic sections symbolically so you've got parabolas, ellipses and hyperbolas symbolically, not as sections of a cone.