Philosophy 167: Class 12 - Part 12 - Motion Along a Cycloid: Newton on the Isochronism, and the Law, of Cycloidal Pendulums.
Smith, George E. (George Edwin), 1938-
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This is a letter from Newton to Oldenburg after he received the complimentary copy of Horologium Oscillatorium. Huygens sent Oldenburg to send to Newton. And he says, the part I'm gonna single out here that I've got the line on, in the demonstration of the eighth proposition on the descent of gravity.
The eighth proposition is this one, and notice what he does. He has a series of inclined planes changing at sharp, single points changing slope. Okay. What does Newton say? There seems to be an illegitimate supposition, namely that the flexures at B and C, that's the points of contact I just showed you, do not hinder the motion of the descending body.
What he's saying is if you go down and do a sudden sharp change, you're gonna get an impact that slows you down. For in reality, they will hinder it, so that a body which descends from A shall not acquire so great philosophy when arrived at to D, as one which distends from E.
If this supposition be made, because a body descending by a curved line meets with no such opposition, this proposition is laid down in order to the contemplation of motion and curved lines. Then it should have been shown that the rectilinear flexures do hinder yet infinitely little flexures which are in curves though infinite in number, do not at all hinder the motion.
In other words, he's got a lousy proof, and I got a better one. They're rectifying curve lines by which Mr. Huygens calls evolution. I have been sometimes considering also, notice the date, 1673, and have met with a way of resolving it, which seems more ready and free from the trouble of calculation than that of Mr. Huygens.
If he please, I will send it to him. The problem is also capable of being improved by being propounded thus more generally. What's he referring to? The stuff on curvature I showed you earlier which is, of course, why I showed it to you. He's referring it to him.
He has a totally general method of curvature. Huygens has one only with evolutis. That's the proposal. Alright so what is we have from Newton, and I'm gonna go fast now, and I'm sorry about this, but it's spelled out relatively well. Newton, we don't know the dating of this manuscript.
It may be after he got horologia. It may be before he got horologium, but the head of the Royal Society had published the paper trying to anticipate Huygens' solution for the cycloid. People knew Huygens had shown the cycloid as isochronous and was using it to measure surface gravity, but they hadn't seen the proof.
So Brauencker publishes a proof. It's possible Newton did it before Braunecker, after Braunecker after he had read Huygens. Once again we can't date it. What's stunning about it, and the non bold I'm gonna let you study on your own just because I'm forced because of time. The accelerations at D and any point P, along the arc DC of the cycloid are everywhere proportional to the remaining length of the arc to C, that is to the arc's DC and PC.
What's he saying? The acceleration at any point is proportional to the distance of that point from the neutral point. Okay, and to do that, he has to use distances of a cycloid. The rectification problem for the cycloid had been solved by Christopher Wren. Quite famously, Wren didn't publish it, interestingly enough.
Pascal published it for him and announced that it was Wren's solution. It was probably Wren's greatest contribution to mathematics, the rectification of the cycloid. Claim, if the acceleration everywhere along the arc DC is as the arc length remaining to C, then the times of descent from any starting point along DC are the same.
What's he saying? He's saying here is a sufficient condition for isochronism. Have the accelerations always be proportional to the distance to the neutral point. That's general. It's true for vibration on a string. It's true along the cycloid. It is a universal truth. It's actually a necessary condition, too.
It's an if and only if. How has Newton approached this problem? Totally different from Huygens. He's figured out, what's the necessary and sufficient condition for isochronism? And shows immediately, it's true for our cycloid. Huygens discovered the same necessary and sufficient condition in the 1680s when he was working on spring balanced clocks and realized the relation between the two.
But Newton had it very early on. This is an example of what I mean when Newton zeros in on the core question. What's the necessary and sufficient condition for isochronism. Generally, he sees it. It's explained here. It's infinitesimal reasoning. And this is a result from Rennes. Because of lack of time I am going to run fast.
He then goes on to obtain the two things. First of all he gets, this step is how do you represent time. You have to represent it geometrically. That's gonna be important next week. For now, I just call your attention, he has to find a way in this diagram to represent time, and he ends up that the arc lengths along the generating circle, the circle that generates the cycloid can represent time.
Once he has that, he derives the law of the cycloidal pendulum which is the same as Huygens law, and shows it can be used to measure surface gravity. He does this in about four pages. Huygens does the same thing in about 20 pages. He does it without presupposing you know the velocities everywhere from the distance of fall, he rather gets that result.
But the core of the whole of this, first of all, it's always the arc, not steps and second, he's got the necessary and sufficient condition, and thanks to Rennes rectification of the cycloid, he can then do the whole thing. I may run about three minutes over. We're at 11 after on my watch which is slightly fast.
So, this I give you as an example. You'll see a generalization of this showing up in the Principia. But I give you this an example of how cleanly he can approach a problem that other people approach much less cleanly. Huygens didn't do anything wrong, but he had to struggle to get there because he didn't have the, he, it wasn't, Newton's use of the calculus here is not central to it.
Rather, what's central to it is going after the necessary insufficient condition for isocronism which, I repeat, is absolutely universal. Any motion that's regular and isochronous, you know in vibration, no matter, in a musical instrument, no matter how large the amplitude is, the frequency stays the same on a string.
That's how violins, etc., have to work. That's this principle. That the frequency is entirely dependent on the fact that the acceleration at any point is proportional to the distance. True in all vibration, etc.