Philosophy 167: Class 7 - Part 3 - Striking Results in Two New Sciences: Proposition XXX, Proposition XXXVI, and Galileo's Achievement in Day 3.
Smith, George E. (George Edwin), 1938-
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Proposition 30 says something very different. And notice, I didn't even list it as problem, but it's gotta be a problem. Drop the vertical bd from a point b in the horizontal line ac. In which take, any point c. And in the vertical, take a distance be equal to bc drawing ce.
I say that of all inclined planes from point c to the vertical bd. Ce is that along which descent will be made to the vertical in the shortest time of all. Okay, now I'm translating all that. Let alpha be the angle of the plain makes with the horizon, eg, angle bcf.
Let d be the distance along the horizontal from the apex of the incline plane to the vertical. And s the length of the incline plane. What we want is the one for the shortest time. Algebraically, we would write it as the distance is one half the distance along the incline plane.
One half times the acceleration of gravity, times the sign of the angle, times T-square. And the height d, excuse me, the horizontal distance that the inclined plane covers that you're trying to cover in the shortest possible time is s times the cosine of alpha. That's just simple trigonometry.
Substituting the one and the other and you get d equals one-half a sine of alpha times cosine of alpha times t square. And indeed, you may not be able to see this immediately depending on how much calculus you've had. But the quantity t square minimizes given d fixed if alpha's 45 degrees.
The way to think of that is, one half sine of alpha, cosine of alpha. For those who don't remember, this is the sine of 2 alpha. Sine of 2 alpha reaches its maximum at 90 degrees. And therefore, alpha gives you the maximum at 45 degrees. And whatever that maximum is, it's the minimum time.
That's another striking result. Fastest time. And he now elaborates that in a rather tricky thing, Proposition 36 of the Scholium following. The last one was the Proposition 30, but he continues with this. So first he says, let the circumference cbd be no more than one quadrant of the vertical circle with it's lowest point at c.
To which is raised the plane cd and let to planes be deflected from the ends d and c to some point b taken on the circumference. I say, that the time of descent through both the planes is briefer than the time of descent through dc alone. That's a striking result.
Let's say, you want to go faster. Don't go in a straight line. Go in a diverted, not at all apparent. And then, he adds in the Scholium to it. From the things demonstrated, it appears one can deduce, okay? That's not a proof, folks. That one can deduce that the swiftest movement of all from one terminus to the other is not through the shortest line of all which is the straight line ac.
But through the circular arc. Okay, that turns out not to be true. It turns out that it's true as far as he's concerned, with the sequence of planes. But if you ask actually, what's the shortest, what's the trajectory of fastest shortest time descent. It became a famous problem first solved in the early 1690s by Jacob Bernoulli who then, put it out as a challenge problem.
We're told by Newton's niece that he stayed up most of the night solving it and submitting his solution anonymously. That Yacka Breunulli famously said in a letter, I can tell that this is from Newton by, what is it? The footprint of the lion.
From the paw of the lion, that's right, very good. So we'll see that problem come up again. The answer's the cycloid and the cycloid's going to get very important in two or three weeks. But it's not the circle, but it's still, he's got a striking result. And this one, in principle, you might consider testable except how do you maintain rolling through the arc of a circle?
Even if you can maintain it pretty well through a 45 degree incline plane. It's just not clear what you can do there. All right. Let me end with day three and field any questions on it. It's the quote at the end that's rather nice by Segreto. Remember, Segreto is the learned outsider commenting on all this.
It appears to me that when we grant that our academician with. Appears to me that we may grant that our academician was not boasting, when at the beginning of this treatise, he credited himself with bringing to us a new science concerning a most ancient subject. When I see with what ease and clarity, from a single simple postulate, he deduces the demonstrations of so many propositions.
I marvel not a little that this kind of material was left untouched by Archimedes, Apollonius and Euclid, and so many other illustrious mathematicians and philosophers. Especially seeing that many and thick volumes have been written on motion. Now, of course this is Galileo praising himself, I trust you all realize that.
But it is very striking. He has a definition and what Stillman Drake listed as a postulate, the Pathwise Independence Principle. And everything else is being proved from that. And it's remarkable how much you could get out of it. Others who had messed around with uniformly accelerated motion like the Oresme and the Mertonians, got nothing like this development.
They were happy to get the mean speed theorem. And try to get uniformly accelerated motion, but they didn't really succeed with the latter. The most they got was mean speed theorem. So, it is a very striking thing, and it gives you the impression it's very striking. But now, the question is exactly what is the so special about the achievement.
Noam Chomsky, I am going to appeal to him for a moment. He, for years, and by years, I now mean 60 years almost. It is 60 years in his case, has emphasized the contrast between predictive power and explanatory power. And the he's talking about grammar. And he in effect says, a grammar that has a very, very large number of independent principles may be very good at predicting things.
But it doesn't explain much how a child acquires it, because there's just so many principles. By contrast, a grammar that has very few fundamental principles. That's going to be very powerful for explaining, but typically, they don't predict. So the striking thing is when you get a very large range of predictions from very, very little material and the two generally don't go together.
This is a case in which they do. That's one way of looking at it. Another way of looking at it is just the sheer question answering power of the theory. How many questions you can answer thanks to the theory. But I leave to you what's so powerful about this theory.
It, of course, is gonna become a model for other people. As I say, Newton never read this, as far as we can tell. But he certainly knew what Galileo did and he had read Huygens’ Horologium Oscellatorium which you’ll see in three weeks from now. And that is very much a Galilean type development except it goes way past Galileo.
It does all sorts of problems and all sorts of motions Galileo can’t do. So it's in effect an extension of Galileo. But it's very much in the Galilean style in almost all respects. I'll say it this way. Newton's Principia's model on Huygens’ Horologium Oscellatorium and had Newton not published his Principia or Horologium Oscellatorium would unquestionably be the most important book published in the 17th century, and wouldn't even be a close question.