Now the, oh, the laws. What is new here is, and this is again Heravel's translation, because law four has the relative motion of bodies enclosed as a given space. With the addition of the thing about the ships. That once again is not relative motion, that's motion among themselves.

I should really changed these slides. I was so busy putting new slides in for today that I've never taught before that I didn't go back and change some of this. Law three, of course, this is the first place in this stuff where law three shows up. As much as any body acts on another, so much does it experience in reaction a bunch of examples.

Then an interesting comment. In fact, this law follows from definitions 12 and 14 in so far as the force exerted by a body to conserve its state, is the same as the impressed force on the other body to change the state of the first. And the change in the state of the first is proportional to the first force, and the second to the second force.

So in effect, the reason he may have cancelled that definition about the internal force being impulse is he's got it here, built into the third law, which came later. So, that may be why he canceled that definition. I don't know. These two are, in effect, the same as in the augmented version of De Motu.

Except for this comment about the motions of a ship from Galileo's dialogue or from Gizendi. I don't know which one he's drawing that from. Now this is where we get important. This is an inserted page that I showed you before, all in Isaac's hand. This page is entirely in Isaac's hand.

He's renumbered the definitions. That's why it's so clear, it belongs here. And he re-does quantity of motion because of this change. Now let's look at them because it's fairly remarkable. The density of a body is the quantity or amount of matter compared with the quantity of space occupied.

By the heaviness, pondus stopped for a moment. Pondus is the word he's using here. It's used all through Lucretius' De Rerum Natura, so it's a commonplace word in this kind of discourse. It's ambiguous in English. You open a dictionary and it will give you two choices, weight and heaviness, okay?

But it's thoroughly ambiguous. I'm following Haravel and translating it heaviness, but you notice I decided to be on the safe side on everywhere the word occurs, I'm gonna show the Latin beside it. Because in a couple, in a few minutes, I'm going to give you a reason to think he really meant, weight all the way through.

But let's go with it. By the heaviness of the body I understand the quantity or amount, that's put in of matter moved. Matter moved, apart from considerations of gravity as often as it is not said of gravitating bodies. As often, maybe I think that's a typo by me, as often as it is not said of gravitating bodies.

The key thing here is, we're gonna use the word for weight independently of whether gravity is involved. That's what's being said. To be sure, the heaviness of a gravitating body is proportional to its quantity of matter. And the agreement, analogia is the word he uses, I translate that agreement.

It's essentially what we would now translate a correlation. But agreement seems to me to be the best. Legitimates setting forth and designating each by the other. Quantity of matter, designated by pondus. Why? Because they're proportional to one another. The agreement is actually to be gathered as follows. The oscillations of two equal pendulums of the same heaviness are counted, and the amount of matter in each will be reciprocally as the number of oscillations made in the same time.

Moreover, experiments carefully made on gold, silver, lead, glass, sand, common saltwater, wood, and wheat, always led to the same number of oscillations. On account of this agreement, and lacking a more convenient word, I sit forth and designate quantity of matter by pondus even when gravitation is not being considered.

All right several comments about this. Comment number one he's now done the experiment that he proposed in the earlier definition. He said you can do this and described the experiment, but now he's done it. When you see in a little bit what's involved in doing the experiment, this is not trivial, and notice how many different types of matter he used in it.

What's this experiment showing? Well technically, it's showing the following. What's the force of gravity by which we hold an object up, two different objects, they weigh differently. What's the tension in the string gonna be? It's gonna be heavier in the one that has more weight than in the other.

At the same time, Galileo and Huygens tell us, you let them go they fall at the same rate. Okay, how can that be? Unless the force that's causing them to fall is proportional to the quantity of matter in the body itself? And that's what he's saying. So this is the same old thing as you all wrote about with Galileo in the second paper, all bodies fall in the same way.

It's the same statement but now being made in terms of forces. The force of gravity acting on a body always yields, at any point, the same acceleration. The change in motion is the product of the mass times the change in speed. And the force is proportional to the change in motion.

So the only way they can all have the same acceleration at any point is the force vary in it's magnitude depending on which body it's acting on. In what rule of proportion? In proportion to the quantity of matter in the body. That's the first thing being said. The second is he's here distinguished what came to be known as mass from weight.

This is where it's drawn. He hasn't got the word mass yet. He's got the word pondus in place of it. And he's then saying that word can be used to designate the quantity of matter. Why? Because the quantity of matter is proportional to weight. Okay, but now, to translate it in English, we'll distinguish between heaviness and weight.

You'll see in a moment what's gonna happen with this. Now this is a big deal because we know something. Are resistance forces proportional to the quantity of the matter on the body in which they act? No. Heavy bodies slow down much less under resistance than light bodies. When you push something, and you try to push a train or you try to push a toy train, they don't resist in the same amount.

You can put the same force in each and the toy train you will move and the real train you won't budge. Why? Because, once again, the force you exert when you push on something is not proportional to the quantity of matter in the body on which it acts.

But gravity is. That makes gravity a very, very different force. And he's not only noting this, he's taking the trouble in the spring of 1685, to do a very elaborate experiment to test it. Fair enough? I'm done with this, but I'm gonna show you more about it in just a moment.

For those who know a lot, what we're looking at here is the principle that Einstein called the equivalence principle that is at the heart of the formation of general relativity. It's usually said gravitational mass is the same as inertial mass, but the experiments to do this that confirm that this was the first such experiment.

We'll come back to the other experiments a little later, because a little later you're gonna see a better description of this experiment, not very long from now.