So whats he do now. Now I am gonna turn to the publication you've seen. The publication I assigned. It really consists of two parts. Number one. Number two, it is dialectic in the classic philosophic sense. He's taking a series of hypotheses that he thinks every Cartesian will accept and showing the Descartes claims in his rules of impact are false.
Okay, so he's going to show in effect Descartes is inconsistent with his own positions because taking, working with Cartesian hypotheses you can show that the very claims Descartes makes are false. So what are the hypotheses. Number one, anybody once moved continues to move if nothing prevents it at the same constant speed along a straight line.
Okay, that's of course our principal of inertia, etc. Hypotheses two. Whatever the cause of the rebound of hard bodies for mutual contact when they collide with one another we pause it, and when two equal bodies with equal speed collide directly with one another from opposite directions, each rebounds with the same speed with which it approached.
That's a Cartesian principle, I think it's rule number one. If they approach one another this way, and they're hard bodies and they're equal in size, they bounce off the same way. Hypothesis three. When two bodies collide with one another, even if both together are further subject to another uniform motion, they will move each other with respect to a body that is carried by the same common motion, no differently than if this motion extraneous to all were absent.
That's the principle of relativity. Okay, so in effect saying, the same thing happens whether you refer to one frame or another. Now what he does is use number 1 and number 2 to take other problems, change the frame of reference so that it meets the conditions of one and two, solves the problem right off of those, transfers back to the original problem.
So, we have to equal bodies approaching one another at different velocities. Adjust one in the boat, so that, relative to the boat, they're moving in exactly the same speed. Invoke hypothesis two to get the solution relative to the boat. Now transform back relative to the ground. Now I say this generally, we've got two frames of reference, they're moving with respect to one another.
We've got a prototypical solution for one frame of reference. Convert any other problem into the prototypical frame problem for that frame of reference, by simply adjusting the philosophy of the other frame of reference and that's how we get solutions. And the first two propositions reject two principles of Descartes.
I can't remember which ones they actually are at the moment, so I have to look at notes. Proposition one, it says, "Exchange at rest." So what happens, according to Descartes, on exchange of rest is, as I recall, if this ball is at rest and this one strikes it it bounces back perfectly and the one at rest doesn't change, if they're equal size.
The contest is won by the one at rest. To the contrary, they come back at opposite speeds. And then the other rule that's false, exchange at unequal speeds, that's rule three. At any rate, the point that's made, propositions one and two show that Descartes' rules six and three are wrong.
And he gets that only out of the relativity principle and two Cartesian principles. Now what you can't guarantee is Cartesians will accept the relativity principle. That's the only fly in the ointment in this part at all, but he's now gonna do some more. Hypothesis Four. If a larger body meets a smaller one at rest, it will give it some of its motion, and hence lose something of its own.
Descartes said, if a smaller, if a larger body meets a smaller body they go off together. Oh no, I'm sorry, I have, I'm misleading you. That hypothesis is Descartes. He says that very thing. The larger body meets a smaller. Proposition three. A body, however large, is removed by impact, is moved by impact by a body, however small, and moving at any speed.
Descartes had expressly denied that. If one body's larger than another, the smaller one can't make it move. Now you could see how he's gonna solve that. He's gonna take this case, change the frame of reference and then change it back. And sure enough, when you do that, the smaller body's causing the larger one to move.
Hypothesis five. When two bodies meet each other, if after impulse, one of them happens to conserve all the motion that it had, then likewise nothing will be taken from or added to the motion of the other. That's a Cartesian principle because the total motion has to stay the same.
Consequence. Whenever two bodies collide with one another, the speed of separation is the same with respect to each other as that of approach. That violates almost every one of the Cartesian rules. Okay. Two more consequences before we get fancy. This of course is the one that you already saw.
If two bodies each collide again at the speed at which they rebounded. You didn't see this was the next one. Which they rebounded from impulse, after the second impulse each will acquire the same speed at which it was moved toward the first collision. That is if you could somehow or other collide and then collide a second time you restore the original position.
And then proposition six is the one. When two bodies collide with one another, the same quantity of motion in both taken together does not always remain after impulse what it was before, but can be either increased or decreased denying Descartes conservation of motion. He does not, in the paper you read, state the principle of conservation of momentum.
That's only in the 1669 paper. But he does state some of the, the kinetic energy principle, the masses times velocity squared. Now, he can't solve the general problem with just Cartesian principles. So he's got to add something more to solve the general problem. What he adds is his version of Torricelli's principle, which we're gonna hear again and again as the night goes on.
For in mechanics, it is a most certain axiom that the common center of gravity of bodies cannot be raised by a motion that arises from their weight. So if we get a motion from a falling, and you go back up, it can't go higher. Okay. And with that he can do a reductio proof for the following case.
If two bodies, the speeds of which correspond inversely to the magnitudes. So the larger body is moving slower, the smaller body is moving faster, in proportion to their magnitudes. Collide with each other from opposite directions, each will rebound at the same speed at which it approached. And he proves that by proving if they don't do that, you will violate this version of Torricelli's principle.
Get a reductio on each part leaving you with, this is the case. Now you can see how he can do any problem. Because, given any two balls approaching one another at any speed of any magnitude. He just adjusts the speed so that the conditions of proposition eight are satisfied.
They're approaching at a speed inversely proportional to their masses. Mass is not his word, that's Newton's word, later. Inversely proportional to their bulks, you solve it in that frame of reference. You take the original problem, transform it into a frame of reference that meets proposition eight, solve the problem there, take it back into the other frame of reference, you got the general solution, and that's how you got that formula I showed before.
So what have we done here? We've taken a Cartesian problem: Impact. Use Cartesian assumptions plus one Galilean assumption if you'll let Torricelli's principle be Galilean and we've given a solution to, a solution to impact for hard bodies, generally of all kinds, hit on impact. And I say it's Galilean, because what you actually do with the Torricelli Principal is you let objects fall from height, that's the way they're going to acquire their velocity.
So if they have acquired their velocity and they're now in impact and they don't rebound in exactly the right way, they won't have enough velocity to get back up. Fair enough? Now, I'll make the point I really wanna make. We've got a Galilean solution to a Cartesian problem.
And to cap it all off, we've got a Galilean way of testing it. And what he does, of course, in the paper you read, he didn't just do one body. If 100 bodies in double proportion are given in order. That's consecutive squares. And the motion begins from the greatest.
One finds by tearing out the calculation according to the precept of the rule set forth in proposition nine. But abbreviated in the compendium that the speed of the smallest body is to the speed at which the greatest is moved approximately as 14 billion, 760,000 to 1. A hundred bodies in a row going from the largest to the smallest in geometric progression gets you a 14 million amplification of the effect of the largest one on the smallest one.
That's a nice striking thing to test. And of course, that's the point of this kids' toy, but of course we need to do it with different sized spheres and to test it that way. Okay, but again, perfect Galilean form of testing, a really stunning result. Now, one is not gonna do it with 100, but two or three would be very compelling.
And by the way, that's how they've started testing this, with ballistic pendiums at the royal society, etc. We'll talk about that later. The royal of society results are disappointing. They had a lot of trouble setting this experiment up. Huygens says he's done the experiment and Christopher Wren said he had done the experiment at Oxford and I believe it's possible to do.
I just don't think the royal society carried it out as well as it might. Last thing about collision. Then we go on to another topic. This phrase at the bottom here, "The result is not alien to reason and agrees above all with experiments.". I want you to hear that phrase 'cause it's gonna come up repeatedly in all of Huygens' work.
Not alien to reason. As we sit back and adopt reasonable principles consistent with the world around us as we normally observe it, we develop mathematics, we get striking results out of the theory. Once we have those, those are testable, when it agrees with those two you're covered. Fair enough?
That's the style. It's not really different from Galileo in style. I wanna go a little bit longer. That's all I'm gonna do about percussion. This is the nearest we get to a general solution till Newton's Principia. And Newton's Principia then gives us the general solution that we use today where we no longer require hard bodies.
And, you'll see it first week of next semester. I hope you're impressed. Now, Wren. I repeat, Christopher Wren got the same general solution. He didn't draw the consequences. Okay. In particular didn't draw the consequence. Decartes' conservation of motion is wrong, but there is a replacement for it, conservation of linear momentum in any one direction.