Philosophy 167: Class 9 - Part 6 - Quantifying the Tendency to Recede: the Sling Argument, and the Force Displayed by the Tension in the Sling.
Smith, George E. (George Edwin), 1938-
view transcript only
Next thing I'm gonna do, there's a passage that starts with article 56 and I'm giving it to you. There are five pages in Latin. I'm not expecting you to read Latin. But I think this is the most important passage in all of Descartes' Principia for the future, side by side with the passages I gave you last week on the Laws of Motion.
Cuz it's here that he finally goes in the detail on exactly what the tendency are endeavored to receive from the center is. So, it starts at the top. What is light and as it mounts to what that say. What this connotes to move is among. How to understand.
What this connotes is to move among inanimate things. So why should an inanimate thing be making an endeavor? Connotes is endeavor. The Miller's translated strive. I prefer endeavor because that's the standard, but you'll see me most of the time in the rest of the course. Very, very often just using the word connotes.
It's a nice word and it fits the Latin very much. I mean, it captures what they're trying to say very well. So, how can it be intelligible that something that's inanimate has some sort of endeavor to it. That's the first one of these. In 66, how is a movement, I can't quite see it, so I'll jump to their translation.
I don't like their translations in a couple of spots, why I am including this partly. How there can be such endeavors towards diverse movements In the same body at the same time. Okay, and that involves this diagram where you're tending to move along this line. That's an endeavor you have to stay and rectilinear motion at the tangent.
And there's another endeavor. Being produced by the sling, to instead bring you into a circular motion away from the tangent. And the question is, how can both of these be happening at the same time? And the answer is a complicated answer, and this does halfway decent job in I think at least a half-way decent job in Schuster's article.
The key notion is the determination of where the thing is going to go. So it has tendencies and something now has to determine what happens with those tendencies. And that's what he's in effect showing here in the diagram. How this circular movement gives an endeavor to receive from the center.
That's the next of the apostilles. And now he gives you the story with the ant with the sling, how the ant's gonna be moving out along that thing. By the way, that's reminiscent of Archimedes Spiral. Remember the spiral is a point moving along a ray and the ray moving in a uniform angular motion.
It's comparable to this and now the point I'm leading up to, this is the striking one. How much would this force, how much would the endeavor of this force be and would be, I'm saying, because the word is seat? Which reminds me, most of you are probably seeing our alphabet in the 17th, 18th century for the first time, s has two forms in the alphabet at this time.
Anywhere except the end of the word, it looks like an f without the bar going all the way through. So that word up here is actually sit, not fit. S-I-T. At the end it's our s, and that lasts, in all print, not just Latin. All print. French, English, etc.
Not German of course. Cuz they had a totally different alphabet. But all through the 17th and 18th century it finally changes. It takes a while to get used to reading and seeing the letter s there where it looks like an f. And the striking thing there, that's actually in the subjunctive, and this is of course the story of the object tending, this is really close to the Archimedes spiral, spinning out.
And the last of these, I've given you the Latin. So you can go back and look at the English text carefully. And I'm about to show you why it's so important if you get into detail. Why the English text is not necessarily as interesting or reliable as a literal translation is going to be.
So these are the three diagrams that he's using in here. One for the sling and you can see the rays drawn out to the points where it would be along the tangent. You can see the same thing along here. Where the ant, if it's not resisted, is just gonna keep crawling and being along the tangent.
And finally this object that's, in this case it's going to be determined because it's in a balance in here, he can't move further, because the pressure on the two sides is holding it. And it will, therefore, move along with the ray. Now, the thing I'm leading up to is this passage.
I'm gonna give you the Miller translation, then go back to it. Because it's this one sentence that's gonna be, is hugely important. This culminates. Four articles. It's the last sentence in four articles describing how the centrifugal tendency. Centrifugal is Huygens' word invented for this. How the tendency to recede from the center works.
So this is their translation. We see too that the stone which is in a sling makes the rope more taut. As the speed at which it is rotated increases. That's well known, right? The faster you spin something, the more taut the sling becomes. And, since what makes the rope taut is nothing other than the force by which the stone strives, that's connotes, to recede from the center of its movement, we can judge the quantity of this force by the tension.
Judge the quantity of this force by the tension. Here's the Latin of the same thing. I'm not gonna go through it. You can read that sentence for yourself. Here's a literal translation of that sentence. And we experience the same thing with the sling. By means of the greater speed, to be sure at which the stone in it rotates, the rope is stretched all the more.
And, indeed, this tension, given rise to by the force alone by which the stone endeavors to recede from the center of it's motion, displays to us the quantity of force of this kind displays to us the quantity of the force. The tension in the string displays the quantity of the force.
Fair enough? Two people took the idea seriously of all right let's figure out exactly what that quantity is. They did so independently. Huygens and Newton, and their doing so will be central to the next class and the one class jumped over after that, because that's the beginning of both of them having a full account of circular motion.
And it's in Huygens' account that he introduces the term centrifugal force. What is centrifugal force? The tension in the sling. It's literally identified as the tension in the sling. So, this is a tremendous idea, we can quantify this. We can quantify it by the tension. And that's picked up on very dramatically.
I'm even more struck by the very idea that tension displays the quantity of the force. Why would he say that? Well, look at a history here. You see in the top diagram, it's in Galileo, but it's lifted out of Steven and Work on Statics. What's the idea? How much does the weight H have to be to balance weight G when G is on an inclined plane?
The answer of course is less than the weight G. Right? How much less? By the sine of the angle. And that's how Steven works out relationship of balance with the sine of the angle. What's the common feature here? The tension in the string that's connecting g and h.
What's the amount of that tension? It has to be the weight of h hanging down. But they knew all about that. How did you think they tuned stringed instruments? They hung a weight on the end to get the tension they wanted. So the whole idea of the tension is one that we're displaying the quantity of the force, producing the tension in the tension, by simply seeing what's producing the tension.
The middle diagram is Huygens' in his work on centrifugal force, it's right at the beginning, the same idea, the tendency to fall in this case down an incline plane is to be measured by the tension in the string that's holding it. And then of course the bottom diagram is Descartes.
So what we've got here is a notion already around in statics that forces produced statically tension in strings. And the tension in the string can be used as a measure of the force. The force required to maintain a body on an incline plane. Which is much less than on a pulley maintaining it vertically.
That's the great advantage practically of an incline plane. Is precisely we can use so much less force to carry something up. And in all the cases, the idea is that we ought to be able to quantify this. Well, in these cases, it's static. We're quantifying it just by the amount of weight, or the amount of weight needed to offset it.
Question, how do we quantify it in the case where it's moving uniformly in a circle? That's now posed as a problem proposed by Descartes. And the only thing that's odd to me about this is, and I may be jumping the gun here. When I first noticed this I knew of no other Descartes, no other, I'm not a Descartes scholar, I knew of no other Descartes scholar who had singled out this sentence.
And I found it extraordinary because both, both Huygens and Newton not only read this sentence, it's absolutely clear that's what they were doing in developing their theories. They're using the same words from this sentence. And I couldn't be more transparent, if I go back, that they were using the very words from this sentence.
The only thing that's not there is the word excuse me, the word or which is to draw or pull. That's the word that Huygens starts using is a form of Any rate, is that at least clear to people? Cuz I consider this the one part of all of part three that had the greatest influence on the subsequent development of our science.
Didn't have the greatest influence at the time, it had a great influence on two people. They happened to be the most important two people in the second half of this century, and the two people from whom Modern science derives more than anybody else in the 17th century. But you'll come to appreciate Huygens next week, you'll have to take my word for this at the present moment.
But it's an extraordinary idea and it's opening the way to quantify this tendency to recede and what's needed to resist it.