Philosophy 167: Class 12 - Part 9 - Newton on Circular Motion: the Solution for Uniform Circular Motion, and the Inverse-Square Tendency of the Planets to Recede from the Sun.
Smith, George E. (George Edwin), 1938-
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Next document, this is a two page document. I think I included the actual manuscript. It's in Latin. I'm gonna focus on this first page. But the second, and it's Newton's handwriting, the second page, I'll comment in just a moment on the material down here. What's the date on this?
Well, the standard number is 1669 question mark. Okay, we know it occurred after the plague years because there is a forerunner of this manuscript written on the back of a sheet of vellum. On the front of which is a court document setting up a deed, and he would not have had that in Cambridge, he would have had that at home.
It's very typical of him when he wants to write something down, and can't find a sheet of paper. He grabs whatever is convenient, and what's on the reverse side can be almost anything, okay? But he wants to write it down. So there's an early version of this that we are very, very confident happened while he was at home during the plague years.
And then he comes back and does this in Latin, and it's written as if for publication. I'm sorry, I went too far. He's made one significant change in wording, made a second change here, cancelled it, made a slight change here. Now I'm gonna focus on that page, but let me look briefly here.
This clause does the following. It says take his solution which is on the first page for uniform circular motion. Put it together with the three halves power rule of Kepler's. And you get to conclude, in his phrasing here, you get the ratio of the five. He has six, so he includes the earth, yeah, the earth is in there.
Of the six planets, their their tendency to recede from the center. You get their ratios in an inverse square proportion consecutively, Mercury, Venus, Earth, Mars, Jupiter, Saturn. And that's clearly the inverse square. He sees, and way before anybody else did, because nobody else had the solution for circular motion, except Huygens.
And he didn't publish his proof. Newton's the first to publish a proof for uniform circular motion. It's in the Principia. But he had it independently of Huygens, well before, and he has recognized the relationship between inverse square and three-halves power given uniform circular motion. So in effect, he's saying, he's not saying the forces of gravity on the six planets are inversely proportional to their distance from the sun.
What he is saying is their tendencies to recede from the center are inverse square proportional to their distances from the sun. Then he gives an account, just single sentence, on conical pendulum and indicates it can be used to measure surface gravity. Now, you read all this. This is again from the Waste Book.
His first solution for uniform circular motion occurred when he was still in college or just afterwards, in the Waste Book. The clever idea is take an object in a square and have it start bouncing off the center. It's gonna keep that motion going forever if it's perfectly elastic.
Now replace the square by an eight-sided figure, replace it by a 16-sided figure, go to infinity on this, so it's a polygon with more and more sides. What happens in the limit when the polygon reaches the circle? You get the v square r solution for it. He repeats that in the Principia, he's proud of that.
So I include that, just down here. You just notice, side of a circumscribed polygon 6, 8, 12, 100, 1,000 sides. The force of all reflections is the force of the body's motion as the sum of those sides to the radius of the circle. And so if the body were reflected by the sides of an equilateral circumscribed polygon of an infinite number of sides, the force of all the reflections are to the force of the bodies motion as all those sides to the radius, okay.
And doesn't give you the full solution up here though he looks at the conical pendulum and expressly indicates as I recall, notice his reference to Galileo. A body undulate in a circle by all its undulations of any altitude are performed in the same time with the same radius.
He starts describing the conical pendulum. Yes, he made the force of gravity, the motion of things falling where they may not be hindered by air, may very exactly be found from the conical pendulum. All of that is definitely long before Huygens publishes his results on that. Now he may have heard by the grapevine that gravity was being measured by a conical pendulum, because Huygens very freely spoke about it when he was in England.
And it could have filtered, but Newton has that. Here, he gives a solution, uniform circular motion, that's essentially the same as Huygens', except Huygens was looking at what he called the centrifugal force which is the tension in the string holding the object. Newton's not thinking that way at all.
The endeavour from the center, quote, is of such a magnitude that in the time ad, which I said very small, it would carry it away from the circumference to a distance db. So far so good? db is gonna be a measure of the tendency to recede. So is it for Huygens, okay.
Now, since the endeavor provided were to act in a straight line in the manner of gravity, would impel bodies through distances which are the square of the times, okay? So at least at the very beginning, the distance db has to correspond to the square of the times. But now he invokes Euclid 336, that I emphasized to you several weeks ago, bd over ba is equal to ba over be.
Now, back to quoting him. But since the difference between be, that's the full length of that line. And de, the diameter, and also between ba and da is supposed to be small infinita, I substitute one for the other in each case. So that bd, that little increment, divided by da, and that can be a straight line now, equals da divided by de.
What's da a measure of? It's a measure of time, because it's moving uniformly in a circle. Now he says, and we get to the key step, but then in the time in which the body goes through the full circle, the endeavour would carry the body a distance equal to bd times the circumference squared over ad squared.
What's that? The time for the circle versus the time for the small increment ad. bd times that, which of course automatically comes out the circumference squared divided by de once you look up here. You do the substitution from the prior step. What's the circumference squared over de? That's the total distance you're going to travel.
What's the? Well it's going to be, think of it as an acceleration, think of it, the easiest way to think of it is f, do it in terms of a force. The distance you travel is going to be the acceleration times time squared. So now that you know what the distance is for the total time, you can figure out what the acceleration has to be by dividing that distance by the total time squared.
And that's the conclusion. Hence the endeavors from the centers of the diverse circles are as the diameters divided by the squares of the periodic times. We would say, at the radius divided by the square of the periodic times, because we prefer to work on radii. But since he's doing proportions, then the diameter and the radius have the same proportional relation, okay?
r over p square the same thing proportionally to r omega squared, which in turn is the same as v square over r. So he's got the same solution Huygens came up with, but he happens to do it in terms of radius over p squared, period squared. So it is a fully correct solution for uniform motion in a circle using the same fundamental result from Euclid that Huygens had used and announcing it, fair enough?
Independently of Huygens, has to be because this was definitely done before Huygens published a proof which was not in his lifetime. He published the result in Horologium, but this is definitely before Horologium.