## Philosophy 167: Class 13 - Part 12 - A Controversial Scholium in De Motu: Deriving the Ellipse from the Inverse Square.

Smith, George E. (George Edwin), 1938-

2014-12-02

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Problem three. A body orbits an ellipse. There is required to find the law of centrifugal force tending to a focus of the ellipse. And I've given you the answer. The law is its force has to vary as inversely with the distance to the center. Or in other words one over SP square.

Then there's a scholium. The scholium has both then and now caused major controversy. Scholium the major planets orbit therefore an ellipsis having a focus at the center of the sun and with their radii drawn to the sun described areas proportional to the times exactly as kepler supposed. Okay, now that's the question Hook asked him, what's the trajectory?

And that's the question Hauley seems to have asked him, what's the trajectory under inverse square and the answer he's saying is in the lips, but what did he prove? If it's in the lips, it's inverse square, that's the converse at least one person recently out in I'm not gonna dignify his name even says this is an example of how really less intelligent Newton was than we make him out to be.

He made an elementary logic error that only school children make of confusing an if then statement with now, I find that silly. Okay? But there is a question. What did he mean? In the 17th, 18th century, it was Johann Bernoulli who said, he doesn't have a prove of the converse.

He just asserts it. Bernoulli of course had done things symbolically and had a proof. Very neatly symbolic. It falls right out, symbolically. It's an if, and only if, for a closed trajectory. But Newton didn't have it. There are two possibilities. There's, Newton himself in the second edition added a proof.

The proof, I'll just summarize it now, you'll see it next semester. The force determines the curvature at every point. The curvature uniquely determines the curve. So it's an if and only if. That is, any ellipse has a unique schedule of radii of curvature. That's a summary of the proof he offers, which even then, people have rejected.

Maybe he already saw that here. He knew an awful lot about curvature and ellipses. He may even have worked some of this out already with curvature then decided not to present it that way because his knowledge of curvature was too tied to the and he didn't wanna encumber them.

Anybody here with. I have a different proposal. You have to write on it so you have to judge for yourselves. My proposal is he had already shown in the circular case that least to a first approximation you have inverse square forces to give you the three halves power rule.

Now the question I pose then, what’s causing the velocity to vary? And equally, what’s the figure going to be when the velocity varies? Whats the answer here? You don’t need another force to produce the ellipse. The same inverse square force that gets you the three as power rule, will hand you the ellipse as a second approximation.

First approximation is circle. Second approximation with the exact same force you get an ellipse. That's the basis for doing. He's doing it in successive approximations, which is a form of reasoning he uses all over the place in the percipian. There's a virtue to that. Think of every account we've heard so far of why it's an ellipse.

Two separate forces. Kepler has it going around in a circle with a secondary force bringing it out. Bareilly has it back and forth, overcompensating for the Centrifugal conatus, and being pulled into much in the Centrifugal conatus than throwing it out, going back and forth between those. Here for the first time you're saying no no we only need one force here, we got the force originally from just the three has rule, it turns out we don't need another force, and that is a step forward.

It's a big step forward actually. So, that's my explanation for the scholium, but understand, I emphasize when I say it's mine. It means it's conjectural.