Philosophy 167: Class 7 - Part 4 - Projectile Motion: Compounded Horizontal and Vertical Motion, and Some Difficulties of the Theory.

Smith, George E. (George Edwin), 1938-


  • Synopsis: Galileo's model of projectile motion, his assumptions, and problems with his arguments.

    Opening line: "All right. Day four. It starts, this is now the academician."

    Duration: 15:15 minutes.

    Segment: Class 7, Part 4.
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All right. Day four. It starts, this is now the academician. This is not day four with our three characters commenting on it. And it's a fairly remarkable opening statement. I'm going to read it out to you and then comment on it. The first paragraph's not interesting but I'll just read it quickly.
We've considered properties, I didn't look at what that word actually is but in one page from now we can see what word is being translated properties. Existing in equable motion, and those in naturally accelerated motion over incline planes of whatever slope. In the studies on which I now enter, I shall try to present certain leading essentials.
Latin word. Symptomata. Accentia is a perfectly good Latin word that's not the word that's used. Symptomada, and to establish them by firm demonstrations bearing on immovable. Nice noun, mobily, that you can use all over the place, that's used all through the Latin in here. When it's motion in compounded from two movements that is, when it is moved equably and is also naturally accelerated, of this kind appear to be those which we speak of as projections, the origin of which I lay down as follows.
Now, this is the important part I've marked with a red line. I mentally conceive of some movable, projected on a horizontal plane, so it's rolling along the horizontal plane. All impediments being put aside. Now it is evident from what has been said elsewhere at greater length that equitable motion on this plane would be perpetual if the plane were of infinite extent.
But if we assume it to be ended and situated on high, the moveable which I conceive of as being endowed with heaviness, word gravitate we will continue to see that word all the time. Driven to the end of this plane and going on further adds on to its previous equitable and indelible motion, that downward tendency propensianum, which it has from its own heaviness.
That's gravitate again. Thus there emerges a certain motion compounded from equitable horizontal And from naturally accelerated downward motion, which I call projection. We shall demonstrate some of its accidentia, of which the first is this. When a projectile is carried in motion compounded from equitable horizontal and from naturally accelerated downward motions, it describes a semi parabolic line in its movement.
I want to make a couple comments. You notice how Stillman Drake inserts the word motions in there? One of the nice things about Latin, is it has case endings that are masculine, feminine and neuter. And if you've got a prior noun and you have a case ending, you don't have to repeat the noun to the adjective, the adjective carries it.
That's why he has to put moveable in there. Because we can't do it in English. But it's trivially done, motions in the one he put in there, it's trivially done in Latin because there's only one thing the adjective can be tied to. The second thing is tendency. It's a striking word, we're gonna see two, actually there are three words we're gonna see as we go on that are normally translated tendency, but they're used very differently.
Conatus, which is endeavored literally. Tendency, literally the corresponding word in Latin, tendentia. And propensity, the corresponding r2r word propensity. I would have translated this propensity rather than tendency, but I'm not gonna fault Stillman Drake for the way he translated it. Now what I've done on the next two pages, I've given you the Latin.
So it is indeed. Properties is the first in the very first line, it's not the word properties that we associated with it, it's the same one later, accadentia, what we would say accidents, but it says it's not quite a legitimate translation from Latin. So the point I'm making here is watch out for translations.
I understand what the problem is because as you will discover when you get to the Principia, that the new translation of the Principia, new now, 13 years ago, was developed in this class over the years. That is there was a draft and we used it and it kept being revised based on what students had to say about it.
And when I look at it now, well, when I do comment on Newton, I almost always change that translation. And I think of Bernard Cohen rolling over in his grave and Ann Whitman who I knew only through my wife, she was a patient of my wife's, also rolling over in her grave.
But I actually understand exactly what their problem is. A, they have to translate consistently. B, they're trying to convey something to the present reader. When I'm translating it, I'm not especially trying to convey something for the present reader. I'm trying to figure out subtle distinctions that Newton is actually making.
And the most striking one, in the Principia, there are two words translated attraction all through the book. Trahuntor and atrahuntor. They don't mean the same thing. And Newton uses them systematically contrasting with one another. But the subtle distinction between drawing something and attracting something, trahuntor, meaning to draw toward you.
And attracting they thought was to precious to try to stick 20th century readers with. So Anne Whitman, who's probably the best Latinist I've ever been in the same room with, and yeah, that's a fair statement. She was just extraordinary in her sensitivity to Latin pros, and she was a legend and the net effect is I'm not gonna criticize her.
But when we as philosophers look at a passage. And we're gonna put any kind of emphasis on an individual word in translation. There's a very good reason to check what the original Latin was. Time and again. And particularly the term used back there essentials of because it just isn't essentials, okay, it's characteristic features of, but that's not the same word that one would use in Latin.
Okay, as you see now, and so you can compare that whole passage I just translated with interjections. You can go back and compare the Latin. That's what I've given you. And right away, we get through the statement of the proposition, and in jumps segreto and simplicio. I didn't give you any more of the Latin.
Instead, I'm gonna give you what they actually said. These are two remarkable statements. They give you four things wrong with the idea of the parabola. We haven't even started looking seriously at the parabola yet. We have yet to look at the proof of Proposition 1. But, Segretto and Simplicio are gonna descend on poor Salviati with four complaints.
First, I'll read it out. It cannot be denied that the reasoning is novel, ingenious, and conclusive being argued ex-supposition. That should be suppositionea. The n doesn't belong there, that's a mistake by me. That is, by assuming that the transverse motion is kept always equable and that the natural downward motion likewise maintains its tenor of always accelerating according to the squared ratio of the time.
So he's saying here's the assumption, the equable motion continues completely independently when the ball rolls off the table and the downward motion develops as it rolls off the table, no crosstalk between the two at all. Also, that such motions or their speeds in mixing together do not alter, disturb or impede one another.
That's a strong claim that they remain independent of one another. In actual air, they definitely do not. But that's not obvious to tell you how un-obvious it is. When Lydents read Newton's Principia, he could not understand why when Newton does projectiles for air resistance, and other motions with air resistance.
He doesn't just treat the components of motion and put them together vectorally. You can't. You can if resistance is proportional to velocity. It's not proportional to velocity. It's proportional to velocity squared and that creates crosstalk. So it's a perfectly legitimate complaint to raise there. What's the grounds for saying the two components remain independent of one another?
You'll see that assumed again. It's not the last time you're gonna see that. In this way the line of the projectile will not degenerate into some other kind of curve. But this seems to me impossible, for the axis of our parabola is vertical just as we assume the natural motion of heavy bodies to be.
And it goes to the end of the center of the Earth. Yet the parabolic line goes ever widening from its axis, so that no projectile would ever end at the center of the Earth. Or if it did as it seems it must, then the path of the projectile would become transformed into some other line quite different from the parabolic.
So everybody should understand if it's a parabola, it's gonna go further and further away from the center of the earth, and that makes no bloody sense. Okay. So there are two challenges here. One challenge is what's the trajectory before it hits the Earth? And question, can it really be a parabola given that it's not gonna continue as a parabola.
Then the second question became a very, very famous question. What is the trajectory if it continued below the surface of the Earth? That became a quite celebrated problem. Mersenne had already raised it as a problem, and various people, Descartes included, had proposed solutions that Mersenne showed did not work, so it was an open problem at the time.
And Galileo does not try to solve it. In fact, to my knowledge, the first solution is by Newton, well it is in the Principia, but it actually occurs in the letter to Robert Hooke in 1679. You'll see it in this semester, it will be the next to the last week of this semester.
Okay, so two objections so far. Namely crosstalk, why no crosstalk in the second? What's the trajectory given that it can't be a parabola all the way to the center of the Earth. Now, Siglitio pops in. To these difficulties, I add some more. One is that we assume the initial plane to be horizontal which would be neither rising nor falling, and to be a straight line.
As if every part of such a line could be at the same distance from the center, which is not true. For as we move away from it's midpoint towards it's extremities, this line departs ever farther from the center of the Earth, and hence it is always rising, everybody understands?
If I take a flat plane at the surface of the earth and extend it, I'm gonna get a deceleration from gravity cause it's getting further away from the center of the Earth. That's the problem with saying motion along the horizontal is equable. If it's the horizontal at the surface of the earth it can't be and that's the point being raised here.
For as we move. I've already said that part and hence it's always rising. One consequence of this is that it is impossible that the motion is perpetuated or even remains equable through any distance. Rather it would be always growing weaker. Besides, in my opinion it is impossible to remove the impediment of the medium.
So that this will not destroy the equability of the transverse motion, and the rule of acceleration for falling heavy things. All these difficults make it highly improbable that anything demonstrated from such fickle assumptions can ever be verified in actual experiments. All right. Two different claims. One is he undercuts Galileo's claim to the principle of inertia because the horizontal plane is not horizontal.
Maybe what Galileo meant is around the circumference of the Earth, which is true. Remove impediments, let a ball roll down and have a groove around the surface of the Earth. It will keep rolling around and around and around, totally free of impediments. That's true. But of course, that's not motion in a straight line.
It's perpetual circular motion. And the other complaint is no way with air resistance are you going to have equable motion along the transverse. Now, fact of the matter is, they already had artillery. Okay, and that’s one of the things that in play here is artillery. And they knew something about artillery, they knew the shape of the projection.
Anybody who’s been to a baseball game and seen a long fly ball knows the shape of the projection. It is not symmetric about the highest point, it rather goes off at an angle and then drops to a much sharper angle. And to just show you a contemporary of Galileo, this is Thomas Harriot, had developed a mathematical theory trying to get curves like that for projection.
Ones that, as you can see, round off trying to get the effect of almost vertical final descent. So others were working on this problem but they considered the parabola as irrelevant because they knew. They knew that the actual artillery had much more of this shape, and the problem of course now is to actually get that shape, and it's not that Harriot got it, but he certainly explored it in interesting ways.
You'll notice the title of this book is the The English Galileo. Because he did so many things in common with Galileo. But of course, he didn't publish, which makes him very different. All right, you got the point? We got four objections to the whole project of day four right at the outset.