Fig. 17. Conchoid Curve.
Galileo, Dialogues Concerning Two New Sciences
Part 2 (Third Day: Naturally Accelerated Motion)
MCKEON:1 We started our discussion of Galileo last time and pushed through what he has to say about uniform motion, the first section of his discussion, identifying particularly the method, which is important. The method, we said, was operational in that he defined uniform motion in terms of three variables: the velocity, the distance, and the time. Whether or not this is empirical we leave aside, since words like empirical and unempirical are vague. It is an arbitrary definition in the sense of a set of variables, but it is not arbitrary in the sense that any old set of variables will do; therefore, we have to go on to consider other things. But if we limit ourselves—and this is all I shall say with respect to our previous examination of uniform motion—to what he thinks was the originality in this analysis, what is it? He does make a statement which you may have noticed. Remember, the whole point that we went through is that the axioms were deduced from the definition; that the propositions were deduced from the axioms and, therefore, from the definition; and that the method was a very ingenious one. Since you have three variables, what can you say about their relation as you hold one of the variables constant? The result of the entire process was, in terms of our original definition, a way of defining time in terms of distance and velocity, distance in terms of time and velocity, and velocity in terms of the other two. As I say, leave out the value judgments: it is a great achievement to have done this. If someone in economics at the present moment, for example, could discover three variables about which he were to say, “I want to define these, and then let’s take a look at some economic processes in terms of them,” he would be a genius in economics history. There are a great many variables, but none that are in this position exactly. And if this is the case, what is it that he thinks is the originality of his approach? Because this will be relevant to our next step. . . . Any hypotheses? . . .
Well, let me read it to you. We did hit upon it, but I didn’t bang it hard. On page 154, after the definition there’s not a “Note” but a “Caution”: “We must add to the old definition (which defined steady motion simply as one in which equal distances are traversed in equal times)”—in other words, that’s not the originality—“the word ‘any’”—therefore, I would suggest that the word any introduces the originality. Why? . . . I suppose I ought to pounce on one of you. Mr. Dean, do you have any notion why?
DEAN: I believe it is because he’s saying that the earlier computing of velocity by distance and time was done without knowledge of this, and now he’s saying that . . .
MCKEON: That would be related, but . . .
DEAN: Well, in other words, people used to talk this way about translating from one motion to another. I think they would not use equal intervals of time and distance in this way.
MCKEON: Oh, no, no. They kept perfectly good time.
DEAN: Well, I know, but they might . . .2
MCKEON: No, no, no. We will be able to demonstrate when we come to accelerated motions that the variation is in terms of time and not distance, but this has nothing to do with what we do with the uniform motion because that comes first.
STUDENT: Well, it introduces continuity into this. It makes the attempt to postulate the smallest divisions of the . . .
MCKEON: See, let’s do it in terms of the two sets of terms. One, you really wouldn’t have uniform motion—and this is what we said in our discussion—if we merely said that in equal expanses of time, equal distances will be traversed. This is because it would be entirely possible that every five minutes an object would go exactly three blocks but would do it in such a fashion that it would be going very much faster for two-and-a-half minutes and then later not at the same degree of rapidity. Therefore, to get really uniform motion, it must be specified that any unit of time that you take will be a unit in which the distance traversed will be the same. Consequently, this is essential to defining uniform motion. But that means—and remember, we are doing the philosophy and not the physics of Galileo—that the time is continuous; that is, it is divisible as far as you like. But then the distance, likewise, is a continuum, namely, divisible as far as you like; and the velocity is, likewise, divisible. Therefore, there will be no minimum units that would enter into the analysis. There may be minimum units beyond which you can’t be accurate in your observation, but the originality is right here. And it is justified simply in that you would not be defining uniform motion unless you made these specifications when you take this approach. There are other approaches in which you need not get into just this requirement; but when using the operational method, the variables that Galileo has chosen, this is essential.
We turn, then, to naturally accelerated motion. As in the case of our first problem, we have an initial statement by Salviati, which runs for two pages; then we have a dialogue between the three of them, out of which you get, third, a series of propositions. I lay out these divisions because I would like, first, to have you tell me what goes on in the initial division, which covers the first two pages. Mr. Davis?
DAVIS: He first tries to lay out his conception of accelerated motion by reference to what happens naturally, and . . .
MCKEON: Well, he says it more precisely than that, doesn’t he? He wants—obviously, anyone would, even a poet—to relate what he says to what happens, in some sense, to natural things. How’s he going to do it? This may all appear pointless if you read it carelessly, in which case it will only fit your preconceived notions. . . .
STUDENT: Well, he wants his definition to exhibit essential features of what has occurred.
MCKEON: Well, you’re skipping the whole . . .
STUDENT: The principles he wants?
MCKEON: No, principles we don’t get to until later. “[F]irst of all it seems desirable to find and explain a definition . . .” [160]. Notice, he could have used another word for “definition”—he could have used a “description,” a “formulation”—but this is a definition, just as he began with a definition of uniform motion. It’s quite clear that in that definition of uniform motion, he begins by telling you what he means: “By steady or uniform motion, I mean . . .” Therefore, what he is saying is, I’m going to tell you what I mean by this definition, and then we’ll see whether it fits the picture. Isn’t this all right? . . . Mr. Davis, if that is what he is saying, how does he go about it?
DAVIS: He starts with an observation, at least he says he does, of a falling body and wonders . . .
MCKEON: No. It seems to me he starts with an observation about helices and conchoids and other figures. . . . Do you have any idea what a helix or a conchoid is?
DAVIS: Yes. It’s an abstract geometric figure.
MCKEON: Well, I don’t want this to trouble you. Have you ever perceived a motion in the form of a helix?
STUDENT: Perceived?
MCKEON: Perceived, yes. Any of you? Did any of you ever wind an induction coil when you were young—or do I take it I am being old-fashioned? Do you know what an induction coil is? If you don’t, all right, skip it. If you took your finger and ran it around the wire after you got it on your cylinder, what would be the motion of the finger?
STUDENT: A helix.
MCKEON: It’s a helix. If you took your finger and ran it down an ordinary wood screw, what would be the motion of the finger?
STUDENT: A conchoid?
MCKEON: What?
STUDENT: It would be a conchoid?
MCKEON: No, no, it’s a helix, too. That is, a helix is a line which is formed by taking a straight line on a plane and wrapping it around a cylinder. You begin with the cylinder, which is circular, but any cylinder will do when you get to higher mathematics. The fact that your wood screw would come to a point would merely mean that you need a little more mathematics, but it’s still a helix. Did you ever move in a conchoid?
STUDENT: A circle?
MCKEON: No. The word comes from shell. If you don’t know what a conchoid is, let me show you. I mean, it’s relatively simple, and it has a beautiful history: Say you take a point—let’s call it A—outside a given line—let’s call it CD—and then draw a set of rays converging on A—that is Freudian: those are the letters that Descartes used when he built conchoids [L!]—merely using the rule that the distance from the end of each ray to its intersection with the line CD will be constant; in other words, the line P1Q1 will be, the line P2Q2 will be, the line P3Q3 will be, and so on, endlessly (see fig. 17). Of course, as the lines get more nearly parallel to CD, the perpendicular distance from the end of the ray to the line CD will be less, and the curve will look something like a good clam shell. A smooth bump if you rode over one is a conchoid. It’s entirely possible there are a whole series of motions when you get to more complex ones that are conchoids; just as, if you had waited for electrodynamics, I think that your helix would have been of some importance.
Well, now, if we are eliminating these, what is it we’re leaving in? . . . Yes?
STUDENT: Motion in a straight line?
MCKEON: Well, let me put it this way. We are leaving in a mathematical equation, that is, d = vt; and we’re going to ask a question with respect to motion, since it is observed in nature that it is sometimes accelerated. Consequently, let me read you his sentence with this in mind: “but we have decided to consider the phenomena of bodies falling with an acceleration such as actually occurs in nature and to make this definition of accelerated motion exhibit the essential features of observed accelerated motions” [160]. In other words, we’re going to make an equation of the accelerated motion and try to get it to fit what we’ve observed so that it’s the same kind of thing. We’re not doing it in a vacuum; we are making an equation and then trying to make it fit what actually happens.
All right, if this is our project—we’re still on page 160—, Mr. Flanders, how does Galileo go about it?
FLANDERS: He said that we must determine the distance before we absolutely determine the time . . .
MCKEON: Well, let me get . . .
FLANDERS: In uniform motion, equal distances are covered in equal portions of time, and . . .
MCKEON: No. The question I am asking is, If what we’re going to do is to set up a definition and then go around and see whether this corresponds to what is observed, how do we get the definition? . . . Let me read you the sentence—and this part is a bad translation—: “And this, at last, after repeated efforts we trust we have succeeded in doing. In this belief we are confirmed mainly by the consideration that experimental results are seen to agree with and exactly correspond with those properties which have been, one after another, demonstrated by us” [160]. I say it’s badly translated because that first sentence—“And this, at last, after repeated efforts . . .”—has a word in the original which would best be translated, “after long contemplation, I’m convinced that I have discovered such a definition.” The word is “contemplation.” He’s convinced because having worked with the definition, deduced the results, and then looked at his experiments, the two correspond.
Well, let’s go on to the next sentence, and this is part of the answer. How did he do this?
FLANDERS: Well, he did it by looking for the simplest explanation . . .
MCKEON: He did it by observing that the design of nature is simplest, that if anything ever swims or flies, the devices used will be the simplest. Therefore, if we’re going to set up a definition of uniformly accelerated motion, it would likewise be the simplest. Isn’t that what he’s saying? In other words, it seems to me that what you try to do when you have a man like Galileo leading the way is to get behind him, find out exactly what he is saying, and see what it means. Obviously, if Aristotle had said it, this would be the subject of great criticism because how do we know that nature is simple? Who would say that a bird flies more simply than a jet plane flies, for example? This, obviously, is irrelevant to our question. What is this exceedingly simple method which leads him to set up his definition? We’re now on page 161. . . . Yes?
STUDENT: Well, he’s watched falling bodies.
MCKEON: He what?
STUDENT: He watches a falling body.
MCKEON: Your edition says that? What mine says is, “when we consider the intimate relationship between time and motion.” We don’t have to watch any falling bodies for that. He wants the equation out of all of that. Let me read it to you, and notice, he’s broadening this question consistently. This is page 161: “If now we examine the matter carefully”—notice, he’s not saying that we examine falling bodies; we’re examining “the matter,” which would be the question—“we find no addition or increment more simple than that which repeats itself always in the same manner.” You’d never find that out by looking at a falling body, would you? Bear in mind, what I’m aiming at in this discussion is that we all read carelessly. You have been putting your ideas on the page; there’s no ground for it here.
Notice, this would follow from what he said about uniform motion. If the great originality of the definition of uniform motion is that one specifies that in equal times, choosing any one, equal distances will be traversed, and we now want to go on to uniformly accelerated motion, then the important thing would be in the same time to make equal increments of motion. That is, we were already involved in the recognition that velocity will have to be a continuum, time will have to be a continuum, and space will have to be a continuum. Therefore, if we’re going on the uniformity, we say, Within an equal time make an equal increment. We have already in our uniform motions compared different velocities: they are always uniform. We will now go on and ask, if we’re going to make velocities that accelerate, let us do it by adding an increment of velocity in equal times. Consequently, what’s the definition that we come up with?
STUDENT: Well, accelerated motion is one where in equal time intervals, equal differences of speed take place.
MCKEON: He says, “[W]e may picture to our mind”—notice, the method of development is here; that is, we are constructing in our minds—“a motion as uniformly and continuously accelerated when, during any”—any, the word again—“equal intervals of time whatever, equal increments of speed are given to it” [161]. So that this is the way in which we will proceed to our definition, and the final summation before the dialogue begins is that “it in no way disagrees with right reason if we accept the statement that the intention of velocity takes place in proportion to the extension of time.”3 The definition is repeated, then, on the top of page 162: “A motion is said”—again, notice the arbitrary language—“is said to be uniformly accelerated, when starting from rest, it acquires, during equal time-intervals, equal increments of speed.”
Are there any questions, then, concerning this first step of what we have done? . . . We have made use of our beginning in the analysis of uniform motion. With the variables we have constructed a definition of uniformly accelerated motion. Let’s now see if we can reconstruct what the dialogue is about. Sagredo begins with his interventions. Mr. Roth?
ROTH: He wants to . . .
MCKEON: Well, does Sagredo object to this?
STUDENT: He doesn’t object to the definition, but, as he says, the definitions are arbitrary.
MCKEON: In other words, he agrees with what we’ve been saying.
ROTH: He wants to know if the definition says anything about nature, about the . . .
MCKEON: No. He has a scruple, he says. The scruple is an objection which Salviati welcomes because the Author also raised and answered it in discussion with himself. And we will now have a series of questions. What’s the first one? It’s Sagredo’s question, and Simplicio comes in on part of this. What’s his first difficulty?
ROTH: I’m not sure. I think that . . .
MCKEON: You’re not sure?
ROTH: He sees a problem in that in the definition, as the time approaches zero, the speed increment also approaches zero, the zero meaning that it’s at rest; yet he sees from observation that there seems to be a big jump between rest to motion, and this jump is not really as small a measurement as the difference should be according to the definition.
MCKEON: Is this related in any way to what we’ve been bringing up in the discussion?
ROTH: Well, initially he says that we must consider it this way because it does apply to nature and . . .
MCKEON: Yes?
STUDENT: Then it oughtn’t to make any difference.
MCKEON: If time were infinitely subdivisible, then within a finite time you never would get an actual motion. So even though Sagredo is coming right along, he is arguing that if you are starting from a state of rest and time is infinitely subdivisible, then there would be no initial speed unless there were a sudden change. Sagredo’s being a Heisenberg before his time; this is a quantum of speed from which you get your initial impeto.
STUDENT: But I thought Heisenberg argued that . . .
MCKEON: Well, let’s leave out the historical analogies. If this is the difficulty, what is the reply? . . . Miss Frankl?
FRANKL: Well, it’s answered by explaining an experiment which can be performed. If you take an object and put it on a building surface, it will press down a certain degree; and if you lift it just a hair’s breadth off, the difference in the degree that it settles down is so slight as to be unnoticeable.
MCKEON: What’s the difficulty with this experiment? . . . Page 164: “See now the power of truth; the same experiment which at first glance seemed to show one thing, when more carefully examined, assures us of the contrary.” In other words, if you take the experiment alone, unless you analyze it and do your reasoning, it will prove both cases equally. What did he do, therefore, with respect to this?
FRANKL: The next thing he says he needs to establish such a fact is that one can think of a stone thrown up.
MCKEON: Look, all the experiment did was to put a heavy body on something soft; then you lift it and drop it, and it goes further down. You can observe these differences. Then he takes a second example. You drop a block—a pile driver is what he has in mind—on a stake; and if you drop it from higher up, you drive the stake in further. Consequently, it’s not merely the weight but the velocity that enters into the driving force.
STUDENT: Isn’t this just a different equation for what force equals?
MCKEON: Unfortunately, Galileo never got force straight; consequently, don’t go around handing this to him.
STUDENT: Well, somewhere he talks about both the velocity and the weight of the bodies.
MCKEON: We now talk about their impact. If he is talking about weight, he hasn’t a good idea of mass as something separate; if he hasn’t any idea of mass, he doesn’t know what force is. Now, maybe he has an idea of mass, but force he doesn’t have; consequently, the nicety of the notion of force which Newton and Leibniz brought out does not belong here. . . . But that’s irrelevant; that is, in the form in which he is presenting it, you don’t need the idea of force.
STUDENT: Well, this means he has the idea of distance related to velocity, which is the way everything changes here, but it is the distance rather than the time that’s involved.
MCKEON: Distance is related to velocity; in fact, it’s even related to the . . .
STUDENT: But it appears at this point that velocity increases as a function of the distance rather than of the time.
MCKEON: The velocity does increase as a function of the distance, but not directly.
STUDENT: Yes.
MCKEON: Well, you’re pushing ahead now. What I want to do is to find out what he means when, on page 164, he says, “But without depending on the above experiment, which is doubtless very conclusive,”—notice—“it seems to me that it ought not to be difficult to establish such a fact by reasoning alone.” How does he do that by reasoning alone, Miss Frankl?
FRANKL: Well, he thinks of calling it a proposition in his mind; he’s imagining it, and I . . .
MCKEON: He’s been doing that all along; that is, he didn’t say, Look, I’ll do it with a pile driver and watch it drive a stake in. . . .
FRANKL: And then he says that . . .
MCKEON: No, no; I mean, you need to realize you haven’t answered it. I mean the fact that imagination is involved is irrelevant; otherwise, we’d need to get into a long psychological discussion. What is it that reason alone would give us?
FRANKL: Well . . .
MCKEON: Let’s look at the example. The example we’ve had thus far is that something heavy on something soft, like leather, is lifted and then dropped, and something hard is pushed into something else. What are we now going into?
FRANKL: Something having to stop has to go through every degree of slowness, so . . .
MCKEON: What is it our example has shown? Our example thus far has been either to take something that’s at rest, lift it up and drop it, or take something at rest and then drop it.
STUDENT: We’re going the other way: we’re going up.
MCKEON: We’re going to throw it up. Why is this reason alone rather than what you mentioned before? . . . Yes?
STUDENT: He explains that if you throw the stone up, the stone will ascend and will go through every stage of slowness as it’s going up.
MCKEON: Well, this may be true, but finding out exactly what this is might not help us. What is it that the imagined experiment of throwing a stone up adds which is not present in dropping the stone?
STUDENT: It’s thrown up and then comes down.
MCKEON: It has come to rest and then has begun to go in the opposite direction. Otherwise, if you’re talking merely about how something at rest in terms infinitely divisible ever gets an initial velocity, you can’t work it out. But this is reason alone in that if the object going up begins to come down, it must have come to rest and, therefore, have gone through the process of losing all of its speed and from rest gaining its speed. You notice, as the argument is presented, this is an essential step that gets out of the difficulty of the mere observational examples and, therefore, has a unique importance. It is by reason itself that we are proceeding.
Well, that was the first question. We now have a second one. What is it, then, that now is raised as a question? We’re still on page 164. . . . It’s Simplicio who brings it in now.
STUDENT: Well, something thrown up slows down in infinite steps, yet one can see them stop, and we have to get . . .
MCKEON: Yes. Simplicio’s problem is exactly the opposite of Sagredo’s. That is, Sagredo wondered how a body at rest would ever get into motion if time was infinitely subdivisible. The rational proof that Salviati gives removes his difficulty but runs into exactly the opposite one, namely, that, as Simplicio says—notice, the language again is the same—, “if the number of degrees of greater and greater slowness is limitless,”—in other words, we’re dealing now with the process of losing velocity rather than one of acceleration—“they will never be all exhausted, therefore such an ascending heavy body will never reach rest.” What’s the answer to that?
STUDENT: Well, isn’t it the same one of infinite divisibility, where this body doesn’t remain at a velocity over a period of time?
MCKEON: Yes.
STUDENT: That’s after we . . .
MCKEON: Yes. That’s what needs emphasis because, remember, this is what we said the any in our definition removed. In other words, the only way Simplicio would be correct would be if you thought of the parts of time as being such that through some period of time, however small, the body retained the same velocity; but if it is losing speed continuously, then the impossibility of a moving body coming to rest is removed. Well, now, these may seem to be dialectical and scholastic difficulties, but quite obviously our Author, as recorded by Salviati, didn’t think this was the case.
We now go on to a further question. Again, this is Sagredo who brings it up, on page 165. Notice, all we’ve done thus far has to do with the possibility of going from rest to motion, motion to rest, with a uniform change in either direction. What’s our next problem? Miss Marovski.
MAROVSKI: Well, in what—Sagrahdo?
MCKEON: Sagredo.
MAROVSKI: He says that in this acceleration you can find a solution to the question of what the acceleration of natural motion involves. He explains it in terms of impetus and force and . . .
MCKEON: Remember, this is the early Italian form that inertia had, impeto; it’s a good idea. This isn’t one of the bad ideas that Galileo is said to have brought up.
MAROVSKI: He’s talking about the impetus given to the stone when it’s thrown upward.
MCKEON: Yes. You remember in our earlier discussions we were constantly running into the problem of whether the principle was internal or external. This impeto or impetus would be a motion imparted to the body and opposed to the force of gravity; consequently, in the case of coming to rest, the external force would be getting less and less, and the impetus would be what would remain. The answer that Simplicio gives is that this explanation occurs only in cases of natural motion which is preceded by violent motion, as in the case of something thrown; whereas, in the case of a motion starting from rest, it would be different. Sagredo removes the difficulty by talking about degrees of force ranging from a small force to a large force, with the force impelling upward being reduced to a limit. Then Salviati ends this phase of the discussion by saying that investigation of the cause of the acceleration of natural motion should be postponed. There are a variety of reasons that have been proposed; however, what we want to do is to investigate some of the properties of accelerated motion, whatever the cause. Remember, I said earlier that this is a commitment to a kind of method in which the principle is not essential for the definition of the motion: examine what the motion is, including the naturally accelerated motion; then go back to the principle as an additional problem. If the properties demonstrated are verified in the movement of the bodies falling naturally, then we can say that the assumed definition comprises such motion.
Let me sum up what the arguments have been thus far. The arguments have been with respect to the relation of motion and rest, how you get from one to the other in view of the continuity, the infinite divisibility, of time; and from that we got into the question of what would be the cause of the continuity or the brake, which we excluded. We now come to the second main question, which is on 167, the question of whether the variable is time or space. I had hoped to finish both questions today. Well, let’s at least start on the second.
It’s Sagredo who starts the discussion, and it’s very simple. What he does is to substitute space for time in the original definition. That is, where the original definition says that in equal increments of time, the increment of acceleration is the same, he now makes the statement that the incremental velocity you add in acceleration is the same as the increment in space. This, incidentally, is the discussion which has been going on in western Europe since the fourteenth century, and there are writers who take either of the two variables. The distinction of Galileo is that the demonstration which he gives for the choice is one that ends the argument; but, on the other hand, this is the passage where Mach4 says that Galileo made a mistake. Let’s ignore the physical side of it. As I say, I think that Mach is wrong in this argument, but you don’t need to go into it. What I am interested in is how the argument is presented. That is, Salviati begins by saying that this is a common error even the Author—of course, the Author is Galileo; Salviati is speaking for him—committed in the beginning, yet there’s a simple demonstration of the falsity and even the impossibility of this position. Can we, in the time that remains, state how the issue is formulated? What is the problem? Why is it that Salviati says that there’s no reason for choosing the other position?
STUDENT: It seems like before because when an object is falling uniformly, it would have a certain velocity; and he states that the criteria for the acceleration would be uniform in the same time, so the distance covered would be the same, when an object falls, as the time would be in which the object is falling.5
MCKEON: Are you sure? Does that persuade you? I don’t think you’ve stated it quite rightly. You’ll notice, we’re doing the same thing again. That is, we’re going to take units of space and we’re going to do it, he says, in terms of four and eight (see fig. 18). Space I is four feet and space II is eight feet. The velocity of the second would be, let us say, twice the first; so the first is velocity one and the second is velocity two. The first velocity would be one in which four feet were covered in the unit of time; the double velocity would be one in which eight feet were covered in the same unit of time. Now, Galileo’s argument is that if the first unit of time was such that the velocity covered only four feet, there wouldn’t be any time left for the increment since the same unit of time is the one which it will have to cover eight feet. So that no matter what it is, you could not even get started on this second increment because there isn’t any more time left. You notice, this is very much like what we did before. In fact, if we were doing this mathematically, much of it could be derived from the three equations that we got for our uniform motion.
Now, remember, he says this is a simple form of the position. Why is it that we couldn’t make the statement, Well, instead of what we’ve done here, if we took time to time and velocity to velocity, we’d still have the same trouble? Why is it that this statement would not be true, whereas our original definition, at least on the level at which we’ve stated it, is correct? . . . Are you all clear on what it is that we’ve said on our simpleminded level? That is, if we’re dealing with accelerated velocity such that at the end of the second space, which is the way we’d have to put it, the velocity is double what it was at the end of the first space, we would never get into traversing the second space because we’d have used up all of our time in the first space.
STUDENT: Well, before you did that, don’t you really want half the time rather than twice the time?
MCKEON: What’s that?
STUDENT: Isn’t it half the time rather than twice the time? The reason is that the space covered is not exhausted in the first half for the fall . . .
MCKEON: It’s the time . . .
STUDENT: Yeah.
MCKEON: . . . that we’re worrying about.
STUDENT: Unless you have something falling twice as fast . . .
MCKEON: No. You see, in the original definition the space is not traversed in the same way; consequently, space through all its divisibility would not be used up in this fashion. You would have a unit of time for the additional space because we are dealing with the successive moments of time; whereas in figure 18 we are dealing with the successive units of space with no time. If the successive units of time are provided, though, we can go on to the next unit of space. The manner of the sequence is important. As a matter of fact, the equation figure 18 is based on, that is, Space = Velocity X Time, contains the answer; that is, if you now write the equation for velocity and time, the relation between them is a direct one instead of the inverse one that we’ll run into when we deal with space as we get further along. But I think that we can postpone this, namely, the question of whether the first formulation is correct, until we get further into the formulation, because in the propositions it will become apparent exactly what this means in more precise terms. Still, the initial form is this simple argumentation which, since it is simple, is one that in the mathematical interpretation can give it an appearance which makes Galileo seem wrong.
Well, we will go on from this point—we’re on page 168. What I would like to do is to skip rapidly over the question of the space and the time and particularly the interpretation of the pendulum, which appears on page 170, to get into the theorems. And in the theorems, here, again, on one level they’re fairly difficult; you get into subtleties. But on the level we’re talking about, I think that you can reduce them to simple terms. We are looking for something like the way in which these three variables will be manipulated; therefore, instead of merely the three definitions that we end with, we shall be looking for the forms of comparison as they come in. Watch the way in which those forms occur because here, again, the simple observations are interesting. Let me give you a question that you can contemplate.
In the very beginning, on page 153, he remarks that in the older treatments very few things have been observed, but he has been able to discover “properties . . . which are worth knowing and which have not hitherto been either observed or demonstrated.” On page 177, you have one of these properties. In fact, he indicates quite clearly that this is the one that he has in mind, namely, that the computation of the moving body from rest will give you, in terms of the distances, the sequence of the odd numbers: 1, 3, 5, 7. This is the observation that he makes; it’s a very curious one. That is, in other words, if you take a look at the sequence of numbers that he’s playing with and then think back to the sequence of numbers that Plato played with when he was talking about time in the motion of the soul, they are constructed in exactly the same fashion. They’re a little different; that is, Plato came out a little differently than Galileo, enough differently to make all the difference in their dynamics. But the basic mode of consideration, as one might expect from the opposition of an operational method to a dialectical method, is quite similar. So that, although I won’t ask you this, have in the back of your mind this question: Given these proportions which we are working out, what is the device by which Galileo nails them down to what is observed? How is it made to operate?
All right. We’ll leave it there and raise our question next time.
