Philosophy 167: Class 13 - Part 8 - De Motu Corporum in Gyrum: Definitions, Hypotheses, and a Comparison with Horologium Oscillatorium.
Smith, George E. (George Edwin), 1938-
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All right, let me jump into Day Mo two. I'm not gonna go through it text by text. You can read it on your own. I'm gonna try to help you look back at it and read it on your own. So it starts with three definitions. Centripetal Force, I named that by which a body is impelled or attracted towards some point regarded as its center.
Clearly that's a take off on Huygens' position, Centrifugal force and in the Principia he just acknowledges that. He just flat says he's taking it from Huygens' Centrifugal force. Which he did not know, Huygens meant by that that tension in the string. So, his Centripetal force is the pull on the object that goes by the third law of motion, the tension in the string.
Okay, definition two, and the force of that is innate in force of a body that is innate. The word's cincita, there is a perfectly good Latin word innate, but I'm accepting the ((Herrivel)) translation here. A body I call that,incita means, I guess insipid? No, it's not insipid in, but intrinsic to.
A body I call that by which it endeavors to persist in its motion following a straight line. And definition three, while resistance is that which is the property of a regularly impeding medium. Hypothesis one, in the ensuing nine propositions, the resistance is nil, thereafter it's proportional jointly to the speed of the body and the density of the medium.
That was not the original hypothesis one, it's scratched out. The scratched out version reads, in the following propositions, nothing is acting on the body except the centripetal forces. That gives Tom White's side grounds for saying the last two propositions were added as an after thought, because Newton went back and replaced the original hypothesis one with that.
Hypothesis two. Every body by its innate force alone proceeds uniformly into infinity following a straight line unless it is impeded by something from without. Notice no mention of force there? In fact, the first place the word force is used in conjunction with inertia, is publicly as in the Principia.
Hypothesis three, a body is carried in a given time by a combination of forces to the place where it is borrowed by the separate forces acting successively in equal times. All that means is the two forces act at the same time, you end up at the same point as if one force acts followed by the other and he treats that again as impulse.
He goes back and forth throughout this between impulse forces and continuous forces. And finally hypothesis four which you'll see in a moment is on the margin. He had to find rate space to put it in. The space which a body urged by any centripetal force describes at the very beginning of it's motion he's in the doubled ratio at the time.
And the key phrase there is, describes at the very beginning of its motion. It's interesting to compare these to the three hypotheses of Horologium. Two gives you the parallelogram rule, and three says the two are independent but they're talking about motions, not forces or uses. The first is, of course, a version of inertia for purposes for clock.
So, it's even possible that Newton's three, that he started with, were meant to parallel the three from Horologium Oscillatorium. Then he noticed he needed the distance in the very beginning of the motion as proportional to the square of the time, which Huygens seems to think he's proven in the proposition one.
But the proof is a question begging proof, so it could well be that there's a play on Horologion here. There's certainly, this whole thing is written in very much the style of Huygens' Horologion, as I'll comment later, all right. These are the original figures, this is material from a paper of mine that I'll put on supplementary material next week that Curtis Wilson put together for me.
I was an invited lecturer at St. John's and then Curtis put it into a paper cuz I didn't have time, but I used the original figures throughout, because I think they're Newton's own drawn figures are much more interesting than what shows up in the long run. So theorem one we've already seen, all orbiting bodies described by radii drawn to the center area's proportional to the times.
And what I emphasize there that may not be apparent to you because you're not accustomed to doing geometrical mathematics instead of symbolic mathematics. You've got to have ways of representing force and time geometrically, and force is gonna be represented by a displacement on time square, and times gonna be represented here by area, but it's always the problem when you do things geometrically.
Once you learn that, once you learn how to represent quantities geometrically, a lot of other things get very simple. But that's not skill any of you have naturally developed, because you've had symbolic math, and you don't have to think when you have symbolic math. It's so nice, of course, you, nevermind.
Well you know I'm sympathetic with Newton. I regret not having had, I had a year of really good Geometry and I'd wish I'd had more. Theorem two, where bodies orbit uniformly in the circumferences of circles. The centripetal forces are the squares, the arcs simultaneously described, divided by the radii of their circles.
Squares of the arc are the same thing as the velocities of course. Corollary five. The squares of the periodic times are as the cubes of the radii. The centripetal forces are reciprocally as the squares of the radii and conversely In other words. The three halves power rule holds if and only if the force on concentric circles varies as inverse square and that you saw Halley announcing he had discovered, Renard had obviously discovered it.
Huygens did not discover it as he told us in the 1690s in a letter because the vertices had blinded him to there being anything interesting to do in calculational Astronomy. And he said that very clearly bitter that he had not gotten all the results Newton got in both this and book one.
Theorem three is the one I was featuring, it becomes proposition six in the Principia so you saw it on the prior side as listed as proposition six because that's from a lecture I gave at the American Mathematical Society. Theorem three, if a body P in orbiting around the center S shall describe any curve line APQ.
And if the straight line PR touches the curve at any point and to this tangent from any other point Q of the curve there be drawn QR parallel to the distance SP now just QR is supposed to be parallel to SP. Exactly what angle it's supposed to be is an interesting question but I'm not gonna worry about that now.
Huygens, if you recall, had the following, take the arc and take the straight line the same distances and draw the line between those two. Newton's not doing it that way. And if QT be let fall perpendicular to this distance SP, I assert that this centripetal force is reciprocally as the solid SP squared times QT squared divided by QR provided that the ultimate quantity of that solid when the points P and Q come to coincide as always [UNKNOWN.
Now, quick comment about that did it with limit just because i thought it'll be easier for you to follow. Newton is very very comfortable with the fact that QT is approaching zero as QR approaches zero. He's comfortable with it precisely because the ratio doesn't have to be zero, in fact the ratio isn't gonna be zero and we're gonna see it provably not zero in a whole bunch of cases.
That's Newton's picture of how to do limits geometrically. Namely, take ratios of two geometric quantities and now have both of them move towards zero. Is the ratio itself defined with a very definite limit? It could be various possibilities, infinity, zero or anything in between, but if it's got a very definite limit, it's perfectly okay to talk that way.
And he'll defend this in section one of the Principia, where he lays this kind of math out with some care. All right, there's one funny thing here in this corollary five. So here's what the Scholium theorem two says. The case of the fifth corollary holds true in the heavenly bodies.
The squares are the periodic times, whereas the cubes are their distances from the common center round which they resolve. That it does obtain in the major planets revolving around the Sun, and also in the minor ones. Orbiting around Jupiter or Saturn, astronomer's agree, he does not know that's true of Saturn at this point as you will see next week.
How sure he is, it's true of Jupiter? You'll see him again next week querying is it really true? So he's asserting this with some confidence that maybe now, there's a difficulty here, I'm not gonna read this out. You can read it for yourself, I'll just make the point.
As I've stressed from the very beginning of this semester this year. The planets are very very nearly circular. Mercury's the most off, where the minor axis is 2% shorter than the major axis. So draw them on a board, they look circular but that doesn't mean the motion is uniform.
In Mercury it s a 50, I give the amount, 50% greater at the point of maximum speed to minimum speed and the difference you can see, it's one plus E over one minus E and even for Mars it's 20% greater. So what's the legitimacy of using a theorem about uniform circular motion in concentric circles?
To draw the conclusion this is true of the orbits, the only answer can be take those orbits to a first approximation to be uniform, take the mean of them so to speak. The three has power rule but that totally leaves open, what are we to say about the variation in speed?
There's obviously a loose end in this, and I can't believe Newton was not aware of it. I can't believe everybody wasn't aware of it. Indeed, I'm fairly confident that's why Hook is asking his questions. Because sure the three has power rule gives you inverse square for circles but it's clearly not circles.
So, what are you gonna get with an actual inverse square of a rule?