DISCUSSION

Newton, The Mathematical Principles of Natural Philosophy
Part 2 (Book I: Definitions III–VII)

MCKEON:1 We started our discussion of Newton last time, and you will recall we found that in the Preface to the first edition something was going on which focused attention on the method. The relation between what we have read in Galileo and what we’re reading in Newton once more emphasizes the importance of dealing carefully with the different aspects of meaning of the terms used. The argument about the relation of mathematics and physics or mechanical art is one that runs throughout history, and it could be any one of them. In other words, at all times, from the Greeks down to today, there’s a perfectly good reason for distinguishing mathematics from physics; and there’s also a perfectly good reason for sticking them together, which would be assimilation, one of our sacred words. Therefore, as soon as Newton begins talking about the two as being the same, the Greeks doing it one way, the moderns doing it differently, you have an indication that something which looks like a method is under discussion.—There is, incidentally, an essay by Einstein in which he deals with exactly the same problem; he comes out somewhat differently, but the continuity of the discussion is quite clear.²—Therefore, you begin asking the question, What method is it? Since Newton is saying in effect that there’s no difference between the method of physics or the mechanical method and the method of mathematics, it could either mean that he was using a universal method, that is, one method for every subject, or a particular method. And the way you tell the difference is, again, perfectly simple. Namely, if he is using a reductive process, that is, one in which you say that mathematics is really a kind of physical motion, then it’s a particular method. If, on the other hand, he is saying that the problems of motion are problems of change in general and, therefore, we measure them, that would be operational, or if you relate them to total motion, that would be dialectical; and then, in either case he’d be using a universal method. And from the first it’s perfectly clear that Newton is going in the direction of the particular method.

Therefore, the first difference from Galileo appears; that is, Galileo used an operational method, Newton a method which we tentatively identified as logistic—we will get more evidence about that today. I think that the way in which you go ahead and test them becomes apparent. Namely, if what you are concerned with in motion is something that can be stated in terms of variables to be measured, this is operational; and obviously, if they are variables being measured, you can talk about the measurement of a body as well as of its motion. You’ll notice, there’s nothing about mass or inertia in itself that makes it one method or the other; you can treat mass as well as velocity and acceleration by the operational method. But if you take a look at the language that Newton uses on page 12 in his first two definitions, where we began to dig into this, it is almost as if he had the distinctions which we’ve been making in mind. Notice, he’s using our word: “The quantity of matter is the measure of the same, arising from its density and bulk conjointly.” Now, if he had said, “We have all sorts of measuring devices, machines and so forth, and, therefore, we can measure matter as well as anything else,” then this would be operational and would bring everything that he says in. But he’s saying the reverse: he is saying that it isn’t the measure of the matter that is there; rather, he’s saying—and the language is not in a form where it’s widely used today, but it makes perfectly good sense—that the quantity is the measure, not that the foot rule is the measure. Therefore, we’ve got to get the peculiarity of the body in from the first. Consequently, when he goes on in the second definition to say, “The quantity of motion is the measure of the same,” it’s not that you have some kind of speedometer that is the measure; it is, rather, the amount which you have—the body, with its mass, and the velocity, with the peculiar characteristics of both—that would give you the quantity of motion. It is the quantity of matter and the velocity joined together.

Now that’s the point which we had reached last time. What I would like to do now is to go rapidly through the remaining definitions and see whether we can learn more about the way in which what Newton is doing is related to what Galileo did. But let me pause first. Are there questions before we go on? Are there any doubts, hesitations, shocks, tensions, inhibitions? . . .

All right, let’s go on, then, to the third definition. Let’s do them one by one from this point. Is Mr. Wilcox here?

WILCOX: Yes.

MCKEON: Mr. Wilcox, do you want to tell us about the vis insita or the innate force of matter, how we get it, how that adds to what we’ve said?

WILCOX: Well, he’s talking about the inertia of the object, the body. He says that the way he thinks . . .

MCKEON: Well, now, remember, don’t tell me what he is saying; it always puts me in a bad mood. One of the devices that would be simple would be to tell me how this differs from impeto or momentum of Galileo. Obviously, in one sense, it’s the same thing. Obviously, he builds on it: this is the beginning of the concept of inertia. Are they the same or different?

WILCOX: Well, the big contrast is that this only works when another force is trying to change the state of the object.

MCKEON: O.K.

WILCOX: So in this way, it’s going to be different.

MCKEON: All right. Is it resistance to change that occurs when a body is in motion or only when it’s at rest?

WILCOX: Both.

MCKEON: But it’s discovered only when an external force is present. Obviously, you have good ground for your answer. Why do you think Newton is different from what Galileo is saying? Because after all, we said that the impeto was the cause of the motion, cause both of the beginning motion and of the change in motion.

WILCOX: Well, to get back to the resistance to change, it’s basically a lack of cause. I mean, there’s nothing not to happen.

MCKEON: It’s not a lack of a cause. Put it the other way, that is . . .

WILCOX: Well, that it’s always reactive?

MCKEON: Yes. This is a cause that is resistant to change, whereas the other was a cause of change; that is, this is a continuation of motion or even of acceleration. In other words, this was one of the fights that the Cartesians and the Newtonians got into. For instance, take my watch on the table. How much inertia does it have, particularly when nobody is pushing it? At one extreme were the Cartesians, who argued that while at rest, the watch had infinite inertia. The Newtonians, on the contrary, said, It’s a meaningless question unless there’s something pushing it; then, if something is pushing it, you would find out how much inertia it has in terms of that external cause, but it is in terms of a resistance to change. The same thing would go on if my watch were moving. Then, if you asked what the innate force is, you would say, Well, if this goes on moving, it can’t change. But if you ask how much would be necessary to bring it to rest, how much to bring it to a slower state, with the external cause you can measure for each of these. One final question: Why is the focus on “innate,” a funny word?

WILCOX: It’s something that’s within the body itself; it’s not itself caused by any external force.

MCKEON: Well, put that last answer in a different form. It’s in there because it’s not external. That is, there are two kinds of causes, inside and outside, and the inertia is an inside cause.

All right, Mr. Stern, tell us about definition IV in relation to III and in relation to Galileo.

STERN: Well, this is the impressed force, as opposed to the inertia, which is defined in terms of a resisting force; and this is the force that’s doing the pushing. And this force exists only while the action lasts, as opposed to the idea of an resisting force, a force which continues after the impression has been made. So that you measure inertia only when the force is active; the force itself is only a force when it is active.

MCKEON: Well, again you’ve been telling me what he says. Break it down into one or two things you want to say about it. . . . Well, let me start you. It is obvious: if in definition III we have defined an inside force, in definition IV we have to talk about the outside force; therefore, the formal basis is perfectly clear. But on the material side, that is, the content side, it takes on an obvious significance; namely, an impressed force is something outside pushing, whereas the innate force is the condition of the thing, which would depend on what an impressed force could do with it. In other words, to push something which has very little innate force is an easier matter than to push something which has a great deal of mass or that’s coming at you very fast. Is there anything else here?

STERN: Well, this marks a transition where you’re having an impressed force which, coming from the way it’s impressed, could be equal to a percussive force or to a centripetal force.

MCKEON: All right, tell me the difference between those two. . . . This is a good answer. Can you indicate the distinctions?

STERN: Well, a centripetal force is the . . .

MCKEON: Give us the first one . . . .

STERN: Percussion, I guess, is what happens . . . Well, the distinctions are determined in the argument. It’s going to be able to make centripetal clearer because of the argument that the . . .

MCKEON: Clean up the first two. There’s a good reason for the order.

STERN: I would say that the outside of the center of gravity . . .

MCKEON: No, you don’t need any center until you get to the last one; that’s the reason why he doesn’t bring the center in. Suppose I came in with great enthusiasm and said, This table has been rickety for a long time and I bought myself a twopenny nail. I want to fix it. I put the nail in, I’ll press it in. What would the class say? [L!] What would you say?

STERN: I’d say you couldn’t do it.

MCKEON: What?

STERN: I said that it was a poor percussive method.

MCKEON: No, you would say, There is another mode which would be more effective, Mr. McKeon, than pressure. If I asked you what and you said percussion, I would say, Why this method?

STERN: The difference, I guess, would be whether there’s direct application to the object.

MCKEON: Well, when I pressed, I pressed right on the nail. That’s direct contact.

STUDENT: Isn’t percussion two forces coming together as a collision, whereas pressure is one force against a body?

MCKEON: Is it a question about these are two forces coming together and the other is a force acting on a body?

STUDENT: As to percussion, you can achieve higher momentum because you have mass which is moving with a certain velocity; whereas with pressure you may have the same mass but much slower velocity.

MCKEON: You’re on the edge of the distinction, but it really is not a distinction. In other words, if I had any piece of foam rubber and the same twopenny nail, I could get the velocity right at the beginning.

STUDENT: Could it have to do with the distinction maybe being the time the force is applied having to be defined?

MCKEON: The nice thing about people who work with simple ideas of this sort is that they often go into simple ideas that they’ve had before and, consequently, time is one; therefore, you’re in the right direction. But what you just said wouldn’t make any difference. What’s the difference between the way in which I put a twopenny nail into wood and the way in which I would put a thumbtack. What would be important?

STUDENT: The distance?

MCKEON: No.

STUDENT: Isn’t it in terms of the inertia of the body itself?

MCKEON: There’s inertia in both cases.

STUDENT: Well, the difference is in the quantity of inertia in each.

MCKEON: With sufficiently strong thumbs and the right scale, you can get all the same quantity.

STUDENT: Well, why is it easier with the thumbtack if sufficiently different pressures may apply?

MCKEON: Because the forces will make a difference, but . . . Yes?

STUDENT: Could it be that in pressure they’re in contact again, whereas in percussion you have only momentary contact?

MCKEON: No, the difference is that in the case of percussion, the body that moves is in motion before contact: the hammer moves through the arc before it hits. Consequently, once the cause is moving the object, all the other characteristics—that is, that they are in contact, have momentum, have force, have the same bulk—all of these could, with proper conditions, be arranged. But in the case of percussion, you have a movement prior to contact causing a movement and, therefore, a different kind of problem. In pressure, you have the bodies in contact from the first. One of the interesting things in the early days—and this is not usually pointed out in histories of physics—is that the early laws of motion were of two kinds. On the continent, they were usually called the laws of shock and they were percussion laws; that is, people were talking about what happens when two bodies came and hit each other. In England, they were laws primarily of inertia, that is to say, how motion continued or ceased to continue. From shock or from percussion you can get around to the problems of inertia; you can do the reverse. But Newton made the initial difference very considerable. Here he’s saying that impressed forces are of different origin. You can hit bodies—that’s percussion—and you can push them—that’s pressure. And now, Mr. Stern, we’re into your favorite kind.

Fig. 24. Analysis of Circular Motion in Newton.

STERN: Yes, the centripetal; that is, you have some force pulling or exerting a new force on the direction of a body in motion or at rest.

MCKEON: That’s true of percussion and pressure, isn’t it?

STERN: Well, it’s pulling it.

MCKEON: Pulling it? Notice, there’s something curious here. One of the things that I would have expected you to notice in this definition is that from this point on centripetal force is going to be very important. He doesn’t mention centrifugal force; he works just with centripetal. Why is this important?

STERN: Well, centripetal force is the force a body exerts on a center, as opposed to . . .

MCKEON: Is the force on the center?

STERN: If you look at the counterforces, that’s the centripetal force.

MCKEON: If we wish now to analyze, in the terms that Newton gave us, the motion of a body which is at the end of a string swung around this way, we say—and we take it as dogma—: There are two forces operating here, and they are perpendicular to each other; that is, the body tends to fly off in a straight line, AB, the string holds it in in a straight line, AC, and we get a curved line as a result, AD (see fig. 24).

STERN: The only one that’s external is the centripetal. The centrifugal force derives from the inertia of the body, though in a sense it’s in your right arm.

MCKEON: That, to an extent, is correct, but that’s on a higher level of speculation than what I was dealing with here. What we are saying here is that we’re going to deal with impressed forces and there are three varieties. What I wanted to do is to find out what is involved in these three and why it is that from this point on the centripetal is going to be the important one.

STUDENT: This may be too simple, but isn’t it his philosophy of nature. And if you quantify the various physical motions of nature . . .

MCKEON: Galileo began by saying that he wouldn’t speculate about a lot of odd figures. He would, however, speculate about the mathematical relations that are exemplified by motion; and he then proceeded to talk about freely accelerated bodies, namely, the fall of bodies in a straight lines. He didn’t get any curves at all. He did eventually when he thought he could get to them, but neither uniform motion nor naturally accelerated motion were involved in the curves.

STUDENT: Gravity is a force that you have to deal with centripetally.

MCKEON: Let’s go back to the beginning. Gravity is not a centripetal force for the simple reason that, as we have observed in the dogma of this course, there’s nothing about gravity that you would deal with empirically which would make it necessarily a centripetal force. What happens is that in the theory of Newton, it is well to consider it a centripetal force. But it is highly questionable whether in this year, with relativity and quantum mechanics, it’s centripetal any more.

STUDENT: Well, it’s going to be centripetal because he’s talking about a mathematical basis and he says something forms the source of the fall, and to say that it’s gravity or to say it’s something else is not talking about the physical cause but rather . . .

MCKEON: No. We’re not talking about causes. We’ve given a definition of impressed force: we’re talking about it as “an action exerted upon a body in order to change its state, either of rest or of uniform motion.” Then we say the impressed forces are of a different origin, these being the three. I think in some respects maybe I shouldn’t have thrown out your suggestion. Do you want, in terms of what’s gone on, to elaborate a little bit more why we should have these three?

STERN: You’re talking to me?

MCKEON: Yes. I suggested that you went into too high theory because of things having forces. Simplify that; maybe that will be the answer.

STERN: Assuming you attempt to measure an object which has a hypothetical centrifugal force, you cannot measure it. All you can possibly measure is centripetal force.

MCKEON: HOW would you measure it?

STERN: With a spring scale?

MCKEON: Again, you’re dealing with this as if it were operational. The answer is pretty much in the paragraph.

STUDENT: Can I ask a question or at least comment about this? The thing . . .

MCKEON: Sure, if it will help out the class.

STUDENT: The thing that’s probably involved is that this seems different from the other two, which seem to have their origin in the external. As opposed to these, the centripetal is based on, let’s say, that it can either be thrown or impelled or exists in some way inside, so it could be internal as well as external.

MCKEON: Oh, no, no. Centripetal force is always external; in other words, if you have ever played with a sling shot, the moment when you release the external force, the stone flies away.

STUDENT: The only thing I mean is that he does mention it in definition V: he says, “in any way tend.” That is, even if the “tend” is based on its own motion, I realize that it is, in fact, externally moved, but it doesn’t become something other than that to which it tends.

MCKEON: No. The “in any way tend” in definition V is merely to take into account all of the forms in which this circular motion occurs—incidentally, we’re still on definition IV; don’t let this introduce a confusion—and, therefore, he gives you a list which includes gravity, magnetism, and the force by which the planets move. Then he goes on to the stone that I mentioned before in the slingshot.

STUDENT: Anything that can be picked up can be subject to centripetal force?

MCKEON: Yes. Since we began by saying that the innate force, or the vis insita, is present whether the body is in motion or not, it doesn’t make any difference; and in the case of any centripetal force, if you start it, say, in the normal procedure of a sling shot, you take the stone out of your pocket, put it in the sling, and start it. Yes?

STUDENT: But he says centripetal force is what causes a curvilinear motion of the orbits. That’s what centripetal force is.

STUDENT: Isn’t this because both centripetal force and gravity are things that we have definite information about?

MCKEON: No. I think that this last is a little dangerous because it acts as if there were certain facts that would lead us to make those statements. This is, rather, a series of ordinary principles that we’re bringing in that would be relevant to the facts. But facts don’t force us. As I’ve said, the direction that Galileo went in is much closer to the direction which we’re going in today, and there has been talk of reducing the law of gravity to forms which would be explained by the constitution of matter which we’re given by quantum mechanics, in which case all of this talk about circular motion would be irrelevant. Not that Newton was wrong; it was an explanation of what was known then, and it still can be used in many circumstances.

Well, suppose we go on. Let’s merely leave this as a question. The point that I would have gone on to make is that from this point on in definition V we will deal, first, with centripetal force in some detail—notice, all of our definitions have been of the quantity of something: the quantity of matter, the quantity of motion—and then, at the end of this definition, we will turn to a statement of three kinds of quantity. The final sentence of the definition reads, “The quantity of any centripetal force may be considered as of three kinds: absolute, accelerative, and motive” [15]. Let me indicate that there is a reason for the order of the terms there. You’ll notice, it would be the case that for any instance that you pick, for example, a slingshot, you could talk about the absolute, the accelerative, or the motive force of that same centripetal force. Therefore, he is going about getting a device by which to do the presentation of his facts. Now, the reason that I’ve done this in advance is to simplify in preparation for my question. What I want to ask first is, What does he present as the explanation of centripetal force in definition V? Then I’m going to go on to ask, What is the relation between the three quantities of centripetal force that he sets up? With that warning, Mr. Milstein, will you tell us about definition V and what the centripetal force is that will be critiqued in these further dimensions?

MILSTEIN: Well, what’s going on is that you have a center of force and a body which is going to be attracted, I suppose, to that center; and this centripetal force of attraction of bodies is going to draw the body . . .

MCKEON: No, no. You’re being a metaphysician with this business of forces of attraction. All of these are things that Newton himself denied had any meaning; they didn’t mean action at a distance or anything of this sort. Do it in descriptive terms. What is it that he is undertaking to talk about?

MILSTEIN: Well, a force which acts perpendicular to the body, to a kind of . . .

MCKEON: But why? Why does he want to talk about this?

MILSTEIN: Because this is the force which keeps a body moving in a circle.

MCKEON: No. Now you’re being a metaphysician again. We don’t know anything about that really, do we, about what a force is? What we are looking for, rather, is a way to describe and write the equation for something.

STUDENT: Bodies that move in straight lines or not in straight lines. Because he has in the earlier definitions described the behavior of bodies, or given various equations for, moving in straight lines, he’s now trying to see the behavior of a body when it is not in a straight line.

MCKEON: I thought you were coming along very well, but you left out the crucial part of your response. What happens to the straight line in the process? Remember, what we have distinguished in III and IV are innate forces and impressed forces; and an innate force is the kind that won’t appear unless something is pushing a body, and impressed forces are the kind that push. What’s the line characteristic of all impressed forces?

STUDENT: Well, you have an equilibrium.

MCKEON: No. Let me put it this way: the answer is that it’s a straight line, too. This is the remarkable point here; this is the reason why I introduced figure 24. We got rid of our curves and got two straight lines out of this, and of the straight lines the only one which is an impressed force is the one holding it in. That’s why I thought that with a little push we could have gotten here. In other words, where there is in nature a curvilinear motion, the impressed force is a straight line to the center, so that all of our impressed forces are now going to be straight lines. We’re also on the edge of the other difficult question: the reason why, of the impressed forces, the kind that push or the kind that hit are relatively unimportant, but the ones that operate in such fashion as to explain complicated apparent motions, which is what in figure 24 the force represented by AC is going to do now, they will be important. So that the crucial part of your sentence was that in explaining all these complicated motions, he wants to get an impressed force which is also in a straight line, isn’t it?

STUDENT: That was one of the main things troubling me before, but it’s not part of the distinction here. The thing that is peculiar with the centripetal force is that it is in a direction. The others didn’t do that, and that seemed to make a difference.

MCKEON: Did you ever drive a nail into a piece of wood? Give me the direction of the varying forces involved.

STUDENT: Yes. But I mean to say, the others are origins in some external role. They’re nondirectional origins.

MCKEON: If you draw a picture of percussion, you get another curve. There is the curve of a hammer coming down but, again, it causes a straight line. It’s the same kind of problem if you want to be subtle about it. But let’s get on. Yes?—I don’t understand: the class obviously is more interested in metaphysical questions than in physical questions!

STUDENT: I was thinking about the question of why centripetal force and the other two forms, being some force with different actions only when actually acting on a body, yet may go on in the body when the action is over. Since he’s using the logistic and not the operational method, would it not be to his advantage to talk about force which could be uniformly constant and always going on in the orbit so he wouldn’t have to measure it?

MCKEON: He’s going eventually to spend a good part of his time talking about the orbits, but it is still the case that the orbits continue in their path only as long as the pressure is there. All sorts of odd things happen when the pressure is removed: you have novas, you have a body disappearing. All of this happens as a result of the logistic method.

STUDENT: But if you talk about percussion and pressure, then you have to make it clear in the examples what that measurement is when you’re talking about pressure.

MCKEON: No. As a matter of fact, in the Newtonian system, the third book is about the système du monde, the system of the world. What he does is to take his general laws and apply them to each of the planets, each of the circumstances. He goes right down into detail, and it’s much more complicated than the question of figuring out how much the head of a hammer weighs and how far it will drive the nail. In other words, for both situations the logistic is in. Yes?

STUDENT: Well, he says that centripetal acceleration is a specific fact resulting from the two forces rather than just one pulling because that would give a normal acceleration as the result of the two forces working together. He gives the illustration that an object could circle the earth without . . .

MCKEON: No. YOU can generalize. In the case of any impressed force, you’ll have an acceleration.

STUDENT: But he’s talking about the orbital acceleration, whereas later he talks about the relationship between the three.

MCKEON: Any curved motion is an acceleration, by definition.

STUDENT: Yes. Well, clearly it’s an acceleration, but a uniform acceleration, and in order to . . .

MCKEON: I don’t think this comes in. We’ve not raised the question of the acceleration yet. See, a change of velocity or a change of direction is an acceleration; therefore, by definition, anything moving in a curved line is an acceleration. Similarly, a held nail that begins to move, that goes from rest to motion, shows an acceleration.

STUDENT: Well, what I was describing, I would say, is the relationship between the two impressed forces that give an orbit around a point.

MCKEON: No. There aren’t two impressed forces.

STUDENT: There are two forces, one impressed and one centripetal.

MCKEON: There’s only one impressed force. All you need is the string.

STUDENT: But you need a force to react against the centripetal force. Without that there’d be no . . .

MCKEON: That would be the innate force.

STUDENT: Yes, that was what I was suggesting. That is, without that, it would be drawn into the center.

MCKEON: No. The point you’re making would hold for percussion and pressure just as much. That is to say, you have, however you move an object, the innate force of the object and the impressed force which starts it moving; therefore, in the analysis of the motion you would need to deal with the interrelation of the two.

STUDENT: Yes.

MCKEON: Consequently, there’s nothing peculiar about the centripetal here. But let’s go on; otherwise we will lose the advantage of all the enlightenment that we’ve gotten thus far. I will not go into detail, but there’s some nice points about the place the projectile comes in. You may remember that the projectile is one of Galileo’s favorite instances. We didn’t get around to reading about the projectile; that came third after uniform motion and accelerated motion. Here there are three questions that Newton asks. What I would like to get in the next ten minutes is some reason—well, leave the reason out. Explain to me what the differences between absolute, accelerative, and motive forces are, what they are as quantities of a centripetal force, a force which is the same. Mr. Kahners? . . . Mr. Brannan? . . . Mr. Dean? . . . Mr. Knox? . . . You don’t have to take them all; you can take one and explain it.

KNOX: Well, it relates to the cause. It depends on what you think the cause is.

MCKEON: Well, you see, I’ve been objecting to what I call the metaphysical form of answer. The word cause does appear in the discussion of the eighth definition, with respect to the absolute force; but he explains it in terms of forces. I think that we can get along without bringing in the cause as something separate. What I want to know really is why we want three different quantities, in other words, three different ways of talking about the quantity of the same force, and how we will know whether we have one or the other. . . . What’s the absolute form? . . . That’s the first. . . .

KNOX: It’s the measure of the same in proportion to the cause.

MCKEON: Well, forget that the word cause is in there. Tell me what “efficacy” means.

KNOX: Well, it means that the effect of the . . .

MCKEON: Take VI and VII together. What’s the difference between the absolute quantity and the accelerative quantity? And you can use the lodestone and the bob.

KNOX: Well, in the accelerative it’s the variable of velocity in time, and . . .

MCKEON: Those are my two variables here, proportional to the velocity and to the time. It changes with the time; that is, velocity is the variable and the time is the given. On the basis of this you can get . . .

KNOX: Well, he introduces the idea of place . . .

MCKEON: No, no. The idea of place is in all of them. For example, in definition VI, it’s even in the definition; space and place are closely related terms that he will explain in later definitions. Time, space, and body will be in all three of these, but in different ways. What I’ve given you is that by holding one constant, you can ignore something about another. What I want to know, therefore, is what is it we’re ignoring as we go from one to another?

STUDENT: Well, an accelerative force is a function of the distance and it varies inversely. To take the illustration of a lodestone, in a lodestone you take . . .

MCKEON: I know, but what does that mean? In the case of the lodestone, the distance will weigh; and in the case of gravity, the valleys and the tops of the mountains come in.

STUDENT: Well, what he wants to get at is this notion that the weight of a body varies with the accelerative force rather than with . . .

MCKEON: The weight of a body?

STUDENT: Well, the motive force he defines as the accelerative force times the quantity of matter. The weight of the body is equal to the motive force, and that at a greater distance from the center of gravity the weight of the body or the motive force . . .

MCKEON: No, I think you’re confusing weight and mass. They are proportional to each other but they’re quite different. I don’t think that . . .

STUDENT: No, I didn’t say weight was the same as mass; weight was equal to the motive force, which was equal to the acceleration times the quantity of matter, but this was equal to the mass.

MCKEON: No. I’m afraid this isn’t . . . Miss Frankl?

FRANKL: I thought that the absolute force was the force of the lodestone which was at the center and it was the mass times the pressure that it would be exerting. And the accelerative force was measured on the body that it was affecting, and it would be velocity over time.

MCKEON: No, no, no. All of them are the body in motion. Let me, since the time is almost up, call your attention simply to this. Take the middle one; this is the accelerative quantity. Remember, it is the same force that we are talking about; therefore, part of the difficulty I’ve had with many of your answers is that you’ve been talking about different forces. It is one accelerative force that we are talking about. It is sometimes hard to get an example that will link to all three to illustrate it, but the lodestone is one that does run through them as we go along.

Let’s take accelerative force. What we are talking about is the force that would give you a certain velocity in a given time; therefore, the variable that we’re interested in is the velocity: How much velocity will you get? When you go back to the absolute force from this, the absolute force would be the force—this is what the “efficacy” is—that would be measurable in terms of the body and the velocity, that is, you’d need both the velocity and the mass, so that there’s no consideration of time. In other words, a stronger lodestone will have more absolute force than another. With respect to the accelerative force, namely, how does a lodestone of a given absolute quantity move an object, it depends on how close the lodestone is to the object it’s attracting. Therefore, with the same absolute quantity, the velocity will be different as you move your lodestone away from the iron that you’re moving. Are these first two clear before I go on to the third?

STUDENT: But how do you get the absolute force?

MCKEON: You could ask about the absolute force of a lodestone in itself. You can stamp it on a magnet, and this is the absolute force it has. You can sell it, and the Bureau of Standards will say that this is quite all right. This same magnet will, in different situations, have different accelerative forces because if you’ve ever used a magnet and gotten it nearer or closer . . .

STUDENT: Yes, but I wondered, How do you know what the absolute force is?

MCKEON: The absolute force would be measured by taking different-sized bodies and seeing what velocity they have. Therefore, the absolute force is measured in terms of mass times velocity. And if you took a series of magnets at the same distance from a body of the same sort, the velocity would give you the difference of their absolute force.

STUDENT: But I thought the movement would be a relative one rather than an absolute one.

MCKEON: As Newton will say later, the only way in which we ever get to the absolute forces is by means of relative ones. If you make the experiment properly, if you whirl water in a bucket, you will get an absolute force. Similarly, if you take your magnet with bodies that you know the mass of and measure their velocity, you will get the absolute quantity of their forces.

STUDENT: But each one is measured relative to others.

MCKEON: Let’s postpone the absolute/relative issue till later because you can make the distinction between them but the only way in which you will ever make an observation in any laboratory would be with respect to the relative. As he says, the absolute and the relative spaces of a body are the same but not the same in number—we’ll come to that later. Consequently, if you want to know what the space of a body is, you measure it, and that’s the absolute space of the body.

Well, we’re already beyond our time. We’ve distinguished two of the centripetal forces. Suppose we postpone till the next discussion this onward process, bearing in mind that the motive quantity is going to be the important one, merely on the supposition that if you have three, you can state them in the order in which the climax will come in on the last. I’m debating with myself on whether there’s a point in giving you additional reading. Well, let me tell you what the plot is. I had intended you to read through corollary VI, on page 30, and the subsequent scholium. The scholium is a long one and we probably will not go all the way through that. Then, since there’s a good deal in the Principia that is not directly related to our problem of motion, we would jump to gravity, which starts on page 105. Read the section headed “On Gravity,” which goes to the middle of page 112.