What's the proof? Okay, what's the principle of center of gravity? Well, it's a balance. And it says there's gotta be a certain ratio between the two bodies. The distances of the two bodies to the center of gravity shown here. And if they're moving, they always have to remain in the same ratio relative to that point.
So, I did it the way I think Newton would have done it. It looks at it first in terms of circular motion. And what it has to be is the sun and Jupiter have to be moving in tandem. The sun around in a very small orbit and Jupiter in a much larger orbit.
Both circular in the case I've drawn it around the same position. I'm not worried about that at the present moment. In the Principia, Newton generalizes this to be any orbit whatsoever, and in particular an ellipse. But if it's two bodies, they always have to remain in the same relative distance to the center of gravity for the center of gravity not to be affected.
Fair enough? But we know something about what that has to be. Why? Because we know the magnitude of the force of the sun on Jupiter Therefore we know the centripetal tendency of Jupiter toward the sun. We also know the magnitude of the centripetal force towards Jupiter and therefore assuming it reaches all the way to the sun, we know what, how the sun has to respond.
I've done it in a modern way. This is meant to be changes. In motion, changes in speed actually, which includes a vector concept. So the change in Jupiter in a circular orbit has to at every moment be R over PJ squared. And that has to be proportional to this force because the magnitude of the force is given by a cubed over p square around the sun and the distance has to be the distance from the sun to Jupiter in the denominator squared.
And we can write the same thing down for the sun responding to the force of Jupiter. And we know that these two, the two Rs are the same, the periods are automatically the same, because they have to go together. So when we do a ratio here, we end up with RHR over RJ as being inversely proportional to the strengths of the centripetal tendencies toward the two bodies, both are which we know.
In fact, what Newton knew at this time you'll see shortly he's gonna be asking for more precise values. Newton knew that the centripetal tendency toward Jupiter is approximately one-thousandth of this centripetal tendency toward the sun. Correct number is around 1060, the number he uses is around 1063 but he doesn't have good enough data, but it's one thousandths.
If it's one thousandth, then of course this drawing is greatly exaggerated because the sun is only gonna be, the center of gravity is gonna be one thousandth of the distance between the two shifted from the center of the sun. Fair enough? The striking thing here though is knowing what the two forces are in magnitude, that's this, and knowing what the force has to do, RH over, the radius over period square, we get these distances fixed.
But now, oops, now we can take the worse case. The worse case is with all six of them lined up in the same direction. And all we have to do, and you'll see later actual evidence for this, that he was in a position to be able to say this, is even if the centripetal tendency toward all six planets were as large as the one towards Jupiter, the distance would be 6 over 1,000.
Now it's not a uniform distance. It's gonna be more complicated. But all he says, for if any position of the planets in and if any position of the planets their common center of gravity is computed this either falls in the body of the sun or will always be close to it.
And the close to it is going to be six times the Jupiter. So he needs nothing like the law of gravity at all to reach this conclusion. It's an absolutely straightforward conclusion from looking at the concept of center of gravity. Fair enough? The reason Curtis Wilson asked me to give the talk that became the talk that I, became the paper that he put together off my talk in the St. John's Review called How Did Newton Discover Universal Gravity, this was the key.
That I had shown he, he really didn't have to throw universal gravity out as a mere hypothesis. He's going to be lead to it in a step by step fashion which is what he always claimed. So this is the first step. And it is a kinda proof they compare it to the system though.
The sun is not at rest of it. And of course who says that the centripetal tendency towards Jupiter extends all the way to the sun? He's just allowing that as a possibility. With the obvious thought if the centripetal tendency of the sun goes all the way to Saturn.
What's stopping the ones towards Saturn and Jupiter reaching all the way back? Sure they're a lot weaker, but they're inverse squared. They're inverse square, they're not gonna stop anywhere they're just gonna, okay? So that's what I claim led to a so called Copernican scholium in all its respects.
Any questions on that? Before you're through next semester you will have heard the Copernican scholium, that one part so many times you'll be tired of it. Just imagine him, put yourself in his position. You got this wonderful clean set of 11 propositions with everything exact. All you need is the inverse square.
You still have the inverse square, but now unfortunately there are too many of them, and if there are multiple ones all hell's gonna break loose. So the simple picture collapses and now the obvious question, how can I still use the simple picture? We're not gonna see the answer to that tonight, that's in the Principia.
In fact I'll make a stronger statement. His response to that problem opened the way to answering the question this whole two semesters is about. How did we first come to have high quality evidence in any science? He found a way to turn the very complexity into stronger evidence that anybody had ever seen before.