Newton, The Mathematical Principles of Natural Philosophy
Part 4 (Book I: Laws of Motion and Corollaries I–VI)
MCKEON: We concluded our discussion of the definitions last time, and we’re ready now to go on to the axioms, or the laws of motion. Let me recall to you what we’ve accomplished. We found that from the way in which the basic conceptions are laid down, it was important for Newton to differentiate between real motion and apparent motion. On these grounds we came to a definite conclusion: we’d been able to gather that the interpretation was entitative; that the apparatus which he had used in the method thus far was a particular method, not a universal method like Galileo’s, and that it depended on quantities of matter and motion and was, therefore, logistic; and that he had ended with the discovery of a single cause to account for all motions, which gave him a principle that was stated enthusiastically in comprehensive terms. Our final discussion of the ways in which you would find this absolute motion was a bit brief. Let me recall it to you a little more fully than it came out last time.
Obviously, it might be supposed that we could easily get the properties of absolute rest and absolute motion. In point of fact, if you are dealing with a distant object, something like what the fixed stars are, you can’t be sure of absolute rest. And motion itself tends to be observed in relation to adjacent bodies; therefore, the properties of absolute motion would be extremely difficult to get. The causes are easier because we can get that from our definitions. In other words, in the case of absolute motion, the only cause of motion or ceasing to move would have to be the application of a real force; whereas in the case of relative motions, it’s quite apparent that the apparent motion may simply be the result of the movement of the observer and would, therefore, not involve real forces. Consequently, we ended with the observation of effects; and it was in connection with the observation of effects that certain forms of centripetal motion, such as the motion of water in a pail which was spun or the motion of balls whirled about at the ends of a string, gave you, in the case of the latter, a direct relation between the tension observed and the position of the balls circulating around a center. You could, therefore, identify absolute motion in terms of these effects.
With this as a background, we’re ready to take a look at the axioms, or laws of motion, before we go on to the corollaries about the motions of bodies. There are three laws, and here our entitative interpretation is once more apparent. Again, I’d like to do the discussing on the level of a general understanding rather than on that of a specific or technical understanding. Consequently, the question I want to ask is, Why are there three laws? Obviously, from what we know and what we’ve learned about motion, we could have done a variety of things. Mr. Roth, why do you imagine there are three?
ROTH: I think that there are three laws to take into account changes in the forces which generate the causes of motion.
MCKEON: Why three? Couldn’t there be the same start with, say, eight or nine interactions of forces?
ROTH: Well, obviously because . . .
STUDENT: I think it has to do with the progression from the laws based first on considering the body itself, independent of matter and matter’s properties, which is inertia, and then to what would be impressed forces and the effects of the impressed forces on inertia.
MCKEON: Well, can you state that a little bit more in terms of the kind of progress that we have been observing thus far?
STUDENT: Well, I tried to do it in terms of the definitions. In other words, I asked why were the laws different from the definitions, and then I had trouble answering that.
MCKEON: Well, why shouldn’t the laws be based on the definitions?
STUDENT: I’m sure they should be, but in what sense they are different is where I had trouble.
MCKEON: Well, if you have definitions which are, in effect, definitions, what are they definitions of?
STUDENT: Quantities of motion, matter, inertia, forces.
MCKEON: Anyone? It’s the schematism I’m looking for. What are the definitions definitions of?
STUDENT: Measures of quantity.
MCKEON: He doesn’t define a pint or a quart. There are lots of measures that he’s left out.
STUDENT: Well, you still have a uniform motion and accelerated motion.
MCKEON: Really? . . . I would say these are definitions of two kinds of things, or at least they could be grouped under two kinds of things. If they were two kinds of things, what would they be? Yeah?
STUDENT: Are they natural motion and violent motion, the impressed motion?
MCKEON: Yeah?
STUDENT: Causes and effects?
MCKEON: No. You see, you would be losing the fact that this is logistic. The logistic method gives you a kind of progression in which you add something each time you go along, but here you’re really landing on two considerations. And those two considerations? . . . They’re definitions of motions and of forces, aren’t they? The first two are definitions of motion because motion requires matter as well as velocity, and the quantity of motion is going to involve the two of them. Next you have two definitions which separate innate forces from impressed forces. Then the rest of them deal with the impressed forces. And with respect to the impressed forces, you really settle down on one, the centripetal, and the three quantities of centripetal force. Well, now, if this is the case, what would be the difference between the laws and the definitions? . . . To begin with, they are laws of motion, and we know what motion is; the definitions tell us what that is. Yes?
STUDENT: Well, in the laws he wants to set up the relationship which would pertain between forces and motions.
MCKEON: Let me emend your statement, and as emended I think it is correct. He wants to set up the relationship between motions in terms of forces. I mean, he could set up the relationship between motions in a variety of ways, but he’s going to ask about the forces. If he wants to set up the relation between the motions in terms of forces, how many laws would he have? Mr. Henderson?
HENDERSON: What’s your question?
MCKEON: We have proceeded by induction thus far and have decided that these three laws are laws in which you relate motions in terms of forces and get axioms. I wanted to know what these axioms would be like and why they are three. . . . If you were doing it, how would you do it? . . . Well, suppose I were doing it and said, Look, we have innate forces and impressed forces; let’s have one law about innate forces and one law about impressed forces and then, to clean it up, have one law about the relation between them. Think that would be sensible? [L!] All right, let’s try this out again inductively. Mr. Roth, what’s the first law about?
ROTH: Well, he talks about motion in terms of impressed forces.
MCKEON: I thought it was a law about the absence of impressed forces. . . . If you’ve got no impressed forces, what’s the law?
ROTH: That it will continue unchanged in motion.
MCKEON: No change occurs. Why does he have to specify that it’s in uniform motion?
STUDENT: If he is to measure accelerative motion, he would need something changed by an impressed force.
MCKEON: Yes. Any accelerative motion would be an impressed force; therefore, he’s saying, if we’re dealing only with forces that are insita, nothing happens. And this practically follows from our definition, doesn’t it? He said that these innate forces are passive; they operate only when an impressed force is present. It’s in the definition, it’s no surprise. All right. Mr. Flanders, what’s the second law about?
FLANDERS: He says what happens to the motion when it’s impressed.
MCKEON: All right, what happens?
FLANDERS: Well, it will change.
MCKEON: It what?
FLANDERS: The change of motion is proportional to the impressed force.
MCKEON: What else?
FLANDERS: Well, it’s headed in the same direction as the force is.
MCKEON: But the law says, “[i]n the right line in which that force is impressed.” Suppose I said, What if I were to impress the force in a curved line instead of a straight line; say I impressed the force by swinging something to hit the object. What then?
FLANDERS: Well, any new kind of direction comes from a straight line.
MCKEON: Well, is that true? If I swing my hammer down, it’s making a circle and, short of an increment in which there is no time, it’s always going circularly.
STUDENT: Well, the way you achieved the circle is from the force impressed directly towards the center at right angles to the point on the circle.
MCKEON: Any curved force you can break up into straight forces. Consequently, the direction in which you impress the force would be the centrifugal force. You’ve been talking about the centripetal force; that’s the force which keeps the thing curving. But the centrifugal force apart from the centripetal would always go in a straight line. Witness the fact that when you let go of the stone in a sling, it goes straight, except for the pull of gravity, which is an instance of the centripetal, too. All right. This, then, is a statement that the changes are all from impressed forces. And you’ll notice, which one of the quantities of force is he talking about?
STUDENT: Both.
MCKEON: Would the accelerative force be proportional to the change in motion and made in the direction of the right line which the force is? Well, how would you have to fix this up to bring that accelerative force in?
STUDENT: Change the velocity.
MCKEON: Change the velocity; otherwise, it’s O.K. All right. Is Mr. Davis here? Tell us what the third law is about.
DAVIS: According to the third law, the two might interact. If the force is impressed on one body, the body which impresses the force will receive an equal force impressed upon it in the opposite direction.
MCKEON: How do we know that?
DAVIS: Well, take your case of, say, a hammer. Not only does the nail move, change relation, and go where you’re driving it, the hammer also stops.
MCKEON: No. What I would like is a little bit more than that. Well, let me be simpleminded again. Here’s a body at rest; I have an impressed force and I drag it along [McKeon drags his pipe across the table]. Obviously, in dragging it along I’ve moved it; the hand pulling has done something which the pipe sitting there can’t resist or equal. Suppose, in much the same fashion, I take these same two fingers and apply them to the leg of the table and I pull. The table doesn’t move—let’s assume that’s so; I mean I could move it with proper conditioning. [L!] Let’s take these as two examples. I could say that I’ve refuted the third law. Remember, I’m simpleminded. Mr. Milstein?
MILSTEIN: Well, he said that it’s not a change in velocity but, rather, a change in the motion, and the motive force includes the mass.
MCKEON: The problem that we’re dealing with is that, given these two cases, in the first case you had a change in motion and a change in velocity, but in the second case you didn’t have a change in either. You’re also going to have the problem of what to do with the relation between motion and acceleration, which is the motive force; but in this case your answer does not fit the difference between them.
MILSTEIN: Well, I think that the explanation of why you could pull one and not pull the other is in terms of the motive force upon each, that the mass of the table is involved with the motive force of the table, as opposed to the force . . .
MCKEON: Yes, but how do I know that? Let’s take it another way. I pull the table gently and it doesn’t move, and I pull as hard as I can and it still doesn’t move. The amount of force that I’ve put into it in the two cases is vastly different, but the result in the motion of the table is the same: no motion. And yet I say the two forces are the same.
MILSTEIN: The two forces? Which two forces?
MCKEON: The action of my hand pulling the table and the reaction of the table not moving. In other words, it would seem to me that the table didn’t move the same amount. . . .
STUDENT: That’s wrong.
MCKEON: The table didn’t move at all.
STUDENT: The table didn’t move in a different sense.
MCKEON: No, the table didn’t move at all; consequently, it would be just sophistry to say that when the table . . .
STUDENT: Well, the inertia exhibited by the table can only be in proportion to the force impressed upon the table.
MCKEON: Do you want to explain what that means?
STUDENT: Is it that the impressed force has nothing to do with velocity or motion but, rather, a certain action?
MCKEON: No, impressed force has to do with both velocity and action.
STUDENT: Well, what says it is an action?
MCKEON: It’s an action which effects a change of motion and rest.
STUDENT: Yeah, but we’re talking about the resistance of the table, which can only be exhibited insofar as far as there’s a . . .
MCKEON: Every action is always opposed by an equal reaction. The actions of the two bodies are mutual; so apparently what’s being pulled acts as much as I’m pulling. It’s a mutual action.
STUDENT: What I was trying to say was that can happen without motion.
MCKEON: Tell me why and how. I mean, what we have said thus far is that to get real motion is kind of hard; what you see is motion relative to things around. We don’t want any of these relative motions. If it doesn’t move relative to anything, it isn’t moving. We can’t play that game here, can we? We’re being scientific; we’re not making things up.
STUDENT: It seems as if we’re going to define resistance.
MCKEON: I need to define resistance?
STUDENT: You need a definition.
MCKEON: How does he define it?
STUDENT: Isn’t it resistance insofar as the body can maintain its present state of motion or rest despite the force impressed?
MCKEON: Well, I know; but then, what is it that the action of the opposed force is? Remember, I began with two examples: in one I dragged the pipe, and in the other I tried to drag the table but the table didn’t move. What’s the difference between the two?
STUDENT: It’s inertia of a body at rest.
MCKEON: But what’s the difference?
STUDENT: Isn’t body simply the mass of inertia? And so you would then . . .
MCKEON: I know, but then take my third example: the one body, the table, exhibited different amounts of inertia in the two instances, so you can’t say that one body has two different amounts of inertia each time . . .
STUDENT: Well, it exhibits the amount of inertia needed to counteract the force . . .
MCKEON: How do you exhibit inertia?
STUDENT: By not moving.
MCKEON: Really?
STUDENT: By resisting force.
MCKEON: O.K. This body is moving [McKeon rolls a ball on table]; from this time on it goes on moving because of inertia. It’s in motion; it continues in motion.
STUDENT: But without changing state.
MCKEON: Well, then, you don’t exhibit inertia by being at rest.
STUDENT: You have different states of inertia.
MCKEON: What’s that?
STUDENT: If you stay at rest, you exhibit inertia.
MCKEON: You have still not told me how I know we have inertia, particularly when we talk about quantities of inertia.
MILSTEIN: Well, we know that that’s what is wanted, because then it moves with an equal force as soon as it relates to the same inertia.
MCKEON: That’s why I started that way. That is . . .
MILSTEIN: If there’s an equal force opposing it, you’re pulling on the chair’s armrest and the arm . . .
MCKEON: . . . and the object stays still. And it’s . . .
MILSTEIN: And the velocity . . .
MCKEON: . . . a tug of war. There are equal forces; neither wins, unless that gets pulled apart.
MILSTEIN: What about this? The forces are the same, but you have the difference of the mass and the acceleration; and when the mass of the two bodies differs, the velocities of each one would differ inversely.
MCKEON: But we would have to take into account the motive quantity of the force, in which case you would have an equality of the motive quantities and still have to compare them in moving.
MILSTEIN: Well, the movement would differ. The difference between pulling on the table and your hand moving is because the movement is inversely proportionate to the weight, to the mass. The motive forces are equal, but the motive force is a measure of mass and acceleration.
MCKEON: That is, you think that when my hand pulls at the table, the motive force is due merely to what my hand would weigh if I chopped it and took it off?
MILSTEIN: No, it’s both a factor of the mass of your hand and the acceleration of your hand.
MCKEON: Is that what the motive force of my hand would be?
MILSTEIN: Yes.
MCKEON: Is that what you think? . . . I have an electric motor at home, one horsepower. It weighs about fifty pounds. The motive force would be fifty pounds times the velocity of the flywheel? You’d better say, No! [L!]
MILSTEIN: Well . . .
MCKEON: No, no. This is part of what I’m driving at: notice that we have been talking about forces in different senses. When we have talked about our slingshot and so forth, we have noticed that a force was applied to make it rotate; we’ve even said we could increase the force. We’ve talked about the constant force on the string. We have not talked about the changes of force making the stone fly around, and those changes of force could be applied in a number of different ways, not merely by gravitation. That is, in gravity we would have a force which comes directly from the mass and the velocity; but there are others, such as mechanical forces and electrical forces. All these are sources of force which we don’t have to take into account here because what we’re concerned with is what’s holding it together. We will eventually bring in the other forces in these terms. Ergo, with respect to the impressed forces, we don’t know where they’re from. If it is dropping a ball and hitting something with it, yes, an impressed force. If the hammer that we’ve been talking about was merely dropped, that’s an impressed force where the mass and the velocity will do it. But if I use my strength on the hammer, I’ve got another source of force, and you can’t get that by weighing my arm.
All right, we’re back where we were. . . . No, it’s a very simple point but a very important one. Notice what we have said. We have said all along that the inertia, the innate force, is passive; it just doesn’t exist unless an active force is operating. Therefore, it is perfectly absurd to ask what is the inertia of a table standing on the floor unless you consider it in respect to what an impressed force is. That’s the reason why you test it. If you want to know what the absolute force of the table is in moving—we have talked about that—then you would do such things as weighing it, and you then get the absolute force. But anything where we are dealing with a relation between an impressed force and an inertia, or the innate force, the equality would be at the moment of exercising the impressed force. If the impressed force is such that all of the gravitational considerations which we’ve been talking about exceed the motive force we are using, then we are justified in saying that the table standing still means an equality of the two. That is, the impressed force and the innate force that comes into actuality in resisting this pressure would be equal, and you would have an infinite gradation from the slightest tug up to the moment when you use enough force to move the table. When you move the table, they are equal still, but the table is moving. Is this clear? You see, it’s in this sense that all of my examples would fit in. The question of whether the object on which the impressed force is exercised moves or does not move doesn’t make any difference. What we are concerned with is the measure of the force used; and whether or not the object moves on which the impressed force acts, the impressed force and the inertia are the same. If it isn’t moved, it will continue to stay still by inertia; if it is moved, it will continue to move by inertia the amount it is moving. Yes?
MILSTEIN: One final follow-up. It will continue moving the amount that it’s moving. I don’t follow.
MCKEON: Once an impressed force—we said this when we defined it—has been impressed, it stops being impressed and the object that has acquired the motion from the impressed force continues to move by inertia at the same rate.
MILSTEIN: I mean it stops at another impressed force.
MCKEON: That’s right. Once it’s in motion, the only way it could be stopped would be by an impressed force . . .
MILSTEIN: O.K.
MCKEON: . . . and when the impressed force stops it, it stays stopped until it’s moved by another impressed force.
STUDENT: What is it the impressed force becomes greater than at the moment the table moves?
MCKEON: It becomes greater than the force which is now actualized. You see, a passive force can become active when an external force operates on the thing; therefore, when I pull on the table, the table pulls back on me.
STUDENT: With equal force.
MCKEON: What’s that?
STUDENT: With an equal force.
MCKEON: With an equal force. Notice that if I am moving it by impulsion rather than by pressure, if, say, I swing my hand around, hit the table and the table doesn’t move, my hand stops and it’s stopped because the table is standing still. Therefore, the table stopped my hand, didn’t it?
STUDENT: Yes.
MCKEON: And it wasn’t moving. That’s the force.
STUDENT: But if your hand had moved the table, you’re saying that the table would still have exerted an equal force with your hand?
MCKEON: Against it.
STUDENT: But it moves.
MCKEON: What’s that?
STUDENT: Then why does it move?
MCKEON: It would move in terms of the total forces that are involved. That is, you would now take the force calculated as impressed—and it need not be merely the mass times the velocity, although in the case of hitting it with a heavy object it could be—then take the mass and the velocity acquired by the table, and they’ll be the same. That’s why they’re equal.
STUDENT: Let’s say that you exerted an impressed force such that it equals the inertia force of the table so that the table does not move, that is, you’re exerting the maximum impressed force where the table cannot move.
MCKEON: You mean another slight amount and it would move.
STUDENT: Right. Then you could have, say, a gradation of impressed force less than this amount and the table will not move. But the table will be exerting equal amounts of force against this impressed force for every point in that gradation.
MCKEON: In other words, you will have a gradation of inertias equaling the gradation of impressed forces.
STUDENT: How do you know?
MCKEON: That is, if you don’t . . .
STUDENT: You can’t measure it.
MCKEON: What’s that?
STUDENT: Can you measure it?
MCKEON: There’s nothing that you need to measure. What we are talking about is what happens in this interaction. It’s entirely possible that the table has this large reserve that it could drag up if I increased the impressed force; but it hasn’t done that yet, so you don’t take it into account. It’s the actual situation which you’re dealing with. This is one of the things that the logistic method does. That is, in any situation you take into account what it is that is being exerted. You have laid down that the only time there is any exertion is in an impressed force, an external force. Consequently, the force which returns would be related to that external force. I don’t know why this should worry us. Miss Frankl?
FRANKL: When the motive force starts this object moving, would it make any difference when you measured the resulting force?
MCKEON: I don’t think that there is any need to bring that in. Somewhat later we’ll begin talking about instances of this. We will have the finger pressing the stone, the stone dragged by a horse, and the stone whirled around by the sling. In one of these, no motion occurs: when I press down, there’s an equal force pressing back and nothing moves, neither my finger nor the stone. The next time you have the horse dragging the stone and they’re both moving. The third time you have the stone moving and it’s not moving in a straight line. The point of these examples is that for the purpose of comparing motions by means of forces, they are all the same. Consequently, if you take the middle example, the horse comes up to the stone and is hitched onto the stone, the stone having been at rest; when he begins walking, the stone moves. It doesn’t make any difference whether or not you begin during the instant when the stone first moves after being still. You would put into your postulate that you’re not talking about the stone being stuck, that it’s on a smooth surface and there’s the same resistance of air all the way through. Therefore, it’ll move the same the first minute, the second minute, the first second, the first hour, unless the horse gets tired. . . . I thought that these laws of motion were sucked with your mother’s milk, that is, if you did such a thing. Let’s go on. We now go into a series of the motions of bodies—yes?
STUDENT: But what’s moving, isn’t it exerting a force equal to the force which is moving it and isn’t it in the body? And if we’re comparing this force with the force that’s pulling it, isn’t this a minus velocity relative to the equal and opposite one?
MCKEON: It is not a minus velocity. What it’s pulling is moving with the same velocity as all the teams of horses that I’ve ever seen.
STUDENT: That’s true, but if we’re going to talk about the force being equal, won’t the inertia acting as the opposite force be equal? Is it fair to use velocity in the plus direction and also in the opposite?
MCKEON: No. You have one of two situations. Either what you are pulling continues to be a drag, in which case it is exerting an action which is equal and opposite, that is, pulling in the opposite direction. The other possibility is that, having started the stone, it will acquire by inertia the motion that the impressed force had, in which case it will go on and doesn’t need any more force. There is no more impressed force. But as long as there is an impressed force, the other is pulling in the opposite direction and is equal. O.K.?
All right. Well, now, I want to ask again the same simple, nonmetaphysical question about the series of corollaries. We’re going to deal with these laws of motion in connection with corollaries. You have them curiously arranged in your edition: you may have noticed that corollary III seems to be missing. What I would like to know is, with these peculiarities, What is the sequence, roughly, that we go through in these corollaries? Take I, II, IV, and V and tell me about it. Miss Marovski?
MAROVSKI: In the first one, he talks about a single body and two forces. . . . I don’t really see the difference between the second corollary and the first one.
MCKEON: Anyone see a difference between them?
STUDENT: Well, the second one just says that you can resolve any single force into two oblique forces.
MCKEON: Yes. That is, take your parallelogram of forces. If you have two forces operating on the body, you form your parallelogram, and the resulting force is the diagonal AD (see fig. 26).1 If you have any single force, you can get that by any of an infinite number of forces and angles which can exert on the same object a force which would make it go down the line AD. So that the two are related as the inverse of each other. You begin with two forces applied to the same body and you get a single force; or you begin with a single force and you end up with pairs of forces that would give you the same motion. Having taken the two of them, can you give us any moral reflection on them?
MAROVSKI: Moral reflection on the first two?
MCKEON: Yes.
MAROVSKI: Well, from the third law we had . . .
MCKEON: Focus on the first two corollaries. Tell me something about the first two. . . . Such as, where did we get them from?
MAROVSKI: Well, we’re talking about two forces acting on a single body, and the third law is included where we’re talking about relationships between forces that are opposed.
MCKEON: Well, I’m not sure. Did you tell me where we got them from?
MAROVSKI: Where did we get them from?
MCKEON: Yes.
MAROVSKI: From the laws.
MCKEON: All three?
MAROVSKI: Laws I and II.
MCKEON: Is that true? . . . He practically tells you. Yes?
STUDENT: Well, it comes from the second one.
MCKEON: It’s derived from the second one. That is, if you had the first law alone, you wouldn’t have anything you could do; so you take the second law and you take two impressed forces. Then you say, As a result of this demonstration, the body would now be moving according to the law of one impressed force. Consequently, we deduce this from law II in order to get back to law I. The end of the first corollary says, “Therefore it will be found in the point D, where both lines meet. But it will move in a right line from A to D, by Law I” [27].
All right. Mr. Henderson? What is corollary IV going to do?
HENDERSON: This corollary seems to tie in with relative or absolute motion by some kind of connection which is difficult to verbalize the exact relation to.
MCKEON: O.K.
HENDERSON: I think it is more or less an example of relative motion.
MCKEON: Well, I said O.K. because I assumed you’d shake that a bit and let it go. Why is it relative motion?
HENDERSON: Because it’s the center of gravity.
MCKEON: It’s the common center of gravity, which sounds to me as close as we get to a state of absolute of motion.
HENDERSON: Then regardless of . . .
MCKEON: That is, for any given system, if we know the common center of gravity, we’re talking about the absolute, not the relative, motion. Look, put the world on the blackboard again (see fig. 25). He’s not talking necessarily about the world, but O is a common center of gravity of the world. If something odd is happening out here at X, a motion, and if I observe it from my home, which is on this planet out here at Z, and plot it with respect to myself, that’s relative motion. If I then take into account the way in which my planet is moving and this object X is moving, I can talk about the absolute motion of both of them—I don’t think you need absolute and relative because, as I say, you have a limited system. But whenever you have the common center of gravity, with respect to that system it is absolute motion. What is it, in more simple terms, then, that we would be wanting to do here?
STUDENT: Well, say you take an artillery shell and explode it and then you talk about both forces, the lines of force: it can be resolved into what would be a continuation of the shell in space. The shell is moving along in a certain direction when it explodes, and all the little fragments that were moving in their directions can be . . .
MCKEON: Well, this is a very exciting example, but I’m not sure what it does to answer my question.
STUDENT: Well, I think that’s what corollary IV says.
MCKEON: Well, you see, I don’t think you’ve said anything with respect to corollary IV. You’re still at the same point. Look, the first two corollaries took laws I and II, put together two impressed forces, and got a simple motion out of them. Remember, we want to get the relation of motions to forces. What would be the next step in complexity you would want to deal with? Yes?
STUDENT: The issue of more bodies?
MCKEON: You would want to deal with a system of bodies. Suppose you had a whole system of bodies moving relative to each other and you moved this system. Let figure 25, instead of being the universe, be the system. What he is saying is that if you move the center of gravity and if everything else continues the same with respect to that center, then what you have done with respect to a single body will hold for a system of interrelated bodies in exactly the same way. That’s what he says at the conclusion of the discussion: “And therefore the same law takes place, in a system consisting of many bodies as in one single body, with regard to their persevering in their state of motion or of rest. For the progressive motion, whether of one single body or of a whole system of bodies, is always to be estimated from the motion of the center of gravity” [29]. So that we’ve rendered it more complex again: we now have a way of moving a whole system. What’s corollary V do? I hope we have time for one more. I see Mr. Brannan isn’t here. Is Mr. Knox? Mr. Dean? Mr. Dean, you can go on.
DEAN: Would this be carrying it forward for the parts as long as a system doesn’t move circularly? Isn’t he saying that the center of gravity would move but the motions of bodies themselves would continue on unaffected.
MCKEON: Well, clean that up just a little bit more.
DEAN: Well . . .
MCKEON: Let’s get corollary IV. What we have said about corollary IV is that we can take a whole system and move it as we would one body. You can see that. I mean, obviously, if I move my battered briefcase here, I move the brown end and the bluish end and all the black parts at the same time in the same motion. There’d be an infinite number of specifications that you could make out about this briefcase, but they would remain at rest relative to each other; therefore, any motion that the case has as a whole, each of the parts would have. This would be true for a body, and it would be true if I had a system. That is to say, suppose I had a magnetic field with a number of objects held in place and I moved my magnet in such a way that all of the objects were free to follow it and I didn’t disturb the magnetic field. That would mean that the force exercised by the magnet, still or in motion, would move the same; in other words, the interrelations of motion would remain the same whatever happened to the system. Now we’re going on to V, and what do we want to say about it?
DEAN: Well, this would be the opposite. In other words, whatever happens to the parts, the system remains the same.
MCKEON: Do any of you see what’s going on in corollary V? Yes?
FRANKL: I thought that as long as the system was not by its motion exerting any particular force on the bodies, that it was just carrying them along in a right line or staying at rest, the bodies were free to move and interact as if they . . .
MCKEON: Bodies free to . . . ? Corollary V reads, “The motions of bodies included in a given space are the same among themselves, whether that space is at rest or moves uniformly forward in a right line without any circular motion” [29]. It’s not a question of being free to move. He uses in his examples, although it isn’t right here . . .
FRANKL: The ship.
MCKEON: The ship, yes.
FRANKL: The example of the motions on a ship fits in here.
MCKEON: All right. Yes?
STUDENT: He’s saying the relationships are the same whether the whole system is moving or whether it’s at rest.
MCKEON: O.K. Then how is this related to the corollary before this one?
STUDENT: The one before said that it would move, that the common center of gravity would be unifying it, and that the action would be the same with respect to the common center of gravity. He said it makes no difference in this case as long as it’s in uniform motion: the relationships among the members, the bodies within the system, would not change.
MCKEON: Well, why is it that he specifies that it’s moving uniformly on a right line?
STUDENT: Because it doesn’t change the center of gravity.
MCKEON: Well, it amounts very much to that. But notice, what we are saying. . . . Suppose we had a system such as the one in figure 25 with odd motions, and let’s say that X and Y are going around the center of gravity, O. We move the system, and we have specified that X and Y will remain unchanged in their motions. We now say that if we are taking such a system, the motions of the parts will be the same whether it is at rest or in motion, provided it’s going in a straight line. Notice that if I begin swinging it, I’ll affect these motions; consequently, it has to be in a straight line. Therefore, with respect to the system, we are saying that we can take it as absolute: we don’t have to worry whether it’s in motion or at rest. Whether it’s in motion or at rest, the interrelations of the motions of the parts will be exactly the same. From this we will go on to corollary VI, where we will say the same thing. That is, suppose instead of taking the whole system and moving it, we have X and Y and they have all the peculiarities we had before, and what we did was simply apply two parallel forces to the two of them which were equal. What they did relative to each other would be the same. These are our six corollaries.
I brought our Maxwell book in because I thought we would finish Newton this time. We won’t. We will go on next time and talk about the scholium to corollary VI, then make our jump over to page 105 and talk about gravity. Let me give you, then, an assignment with a second jump, which will take you to page 155 in the Optics, where he picks up gravity again. Read about ten pages from there. Then, other things being equal, we will start reading Clerk Maxwell, Matter and Motion. You might begin—no, stay off Clerk Maxwell till we’re finished with Newton because you might otherwise work along on the assumption that, since Clerk Maxwell is very pious with respect to Newton, he’s doing exactly the same thing. He isn’t. So let’s get Newton all cleared up first before I ask what Maxwell does.
STUDENT: How far should we read?
MCKEON: Read the section on gravity, section 5, down through page 112. That’s the end of the section on gravity. Then you have a letter on the ether and gravity, which you might read without forcing yourself—well, you may as well read that; it only goes to page 116. So read pages 105 to 116.
STUDENT: How far should we go in the Optics?
MCKEON: Page 155 is in the questions from the Optics, shortly after Query 28. You can read all of Query 28 if you like, but what you need is a running start before he begins Query number 29, which is on page 156. Query 28 begins on page 150. If you want to be neat, you can begin on page 150.