Philosophy 167: Class 7 - Part 8 - Galileo's Tables: Symmetry about 45 Degrees, and the Testability of the Theory.

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

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Synopsis: Description of Galileo's theoretical ballistic tables in the absence of air resistance, and the interesting fact that they are symmetric around a 45 degree angle.

Philosophy and science.
Galilei, Galileo, 1564-1642.
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So this is the comment, and now we see what Sagretto says about it. The force of necessary demonstrations is full of marvel and delight, and such are mathematical demonstrations alone. I already knew, by trusting to the accounts of many bombardiers, that the maximum of all ranges of shots for artillery pieces or mortars, that is that shot which takes the ball farthest, is the one made at elevation at half a right angle.
Which they call at the six point of the, actually I don't know if I put it or Drake put it in the Tartaglia as gunner square. But to understand the reason for this phenomenon infinitely surpasses the simple idea obtained from the statement of others. Or even from experience many times repeated.
Now, immediate question. Do we understand it? We get the successful prediction from motion in the absence of air resistance, and we're predicting something for air resistance. It's not transparent how much we understand here. Okay? Continuing. You say well, the knowledge of one single effect acquired through its causes opens the mind to the understanding and certainty of other effects without need of recourse to experiments.
Now that's a great thought, okay? I have evidence that I've got the cause of this. The same cause then, I can run inferences off without having to do experiments, cuz I now know the cause. Right? So from the fact that I know 45 degrees, I can draw other conclusions.
What other conclusions? That is exactly what happens in the present instance. For having gained by demonstrative reasoning the certainty that the maximum of all ranges of shot is that of elevation of half of right angle. The author demonstrates to us something that has perhaps not been observed through experiment.
And this is that of the other shots that are equal, those are equal in range to one another whose elevation exceed or fall short of half a right angle by equal angles. That is symmetric on either side of 45 degrees, are ranges short of the maximum range, okay?
And that falls right out of his equation. Did they know that? As far as he knows, no they do not. Is it true with air resistance? I don't think so. I'm too lazy to go figure it out. Figuring it out, the easy way to figure it out is simply get ballistic tables.
Until laptop computers became miniaturization of computers became available on board, every navel ship with guns was a big table with calculations table on how to shoot for any distance et cetera. And it called ballistic tables, and that's what you did. You looked it up in the table to see what you wanted to do.
So you could go look at ballistic tables. Just and aside for a moment there, those tables had to be calculated. And I've seen ballistic tables, because my father-in-law was a gunner during World War II and brought home his book of ballistic tables. They're immense. What may intrigue you is the first modern computer was designed, of course, in the 19th century by Charles Babbage.
Who paid for it? Artillery people, to do exactly these tables. The first 20th century computer was designed by Eckert and Mauchly at the University of Pennsylvania with Jon Von Norman discovering about it and jumping into the fray. Who sponsored it? Aberdeen Merrill at the center of artillery in the United States.
That was its application, to compute these things, okay? So the idea of actually being able to compute on the spot by simply looking at the tangent of an angle, it's a wonderful thought. And as I say, I'm too lazy to dig out a ballistics table and see how non-symmetric.
It is about 45 degrees. There's no issue. 45 degrees is the longest range. They weren't crazy. Trial and error actually teaches you things. But I doubt seriously, just from the nature of the loss of equable motion, that it's gonna be symmetric about the two. That is the horizontal motion, that is going to be symmetric, with air resistance present, okay?
But here is a possible source of evidence. Let me ask you a question. Suppose it turns out to be right with air resistance present, what's it telling you about the theory? The theory is about motion in the absence of air resistance. It's produced one prediction with the presence of air resistance that's quite notable.
Suppose it gave you the other one, too? How strong is that evidence for the theory of what's happening in the absence of air resistance? I want you to think about it. You'll get a good chance to think about it on the next paper. Okay, because the next paper is gonna worry about just such questions.
What Galileo proceeds to do is give you tables, calculated tables like artillery tables. That's what they're meant to be. So the independent variable here is the angle of elevation of the cannon. Oh, I left something out by the way. Let's back up, I'm sorry. When he says this about the reverse direction, notice something.
Galileo can't prove that if I shoot the object off at an angle, I get exactly the same parabola as if I drop it off a table. That is, the two work the same way with the appropriate velocity, final velocity being the velocity back up. He can't prove it.
In 1644, Torricelli publishes an absolutely beautiful book, redoing the stuff in Galileo and extending it. Galileo had died two years earlier. Torricelli was his, along with Viviani, his principal proteges. And he actually proves the whole parabola, rigorously proves that result, that Galileo essentially has to assume a semi-parabola can be reversed and form a full parabola.
So that issue disappeared. And when it did, the principle book that gets referred to from then on is Torricelli 's book, not Two New Sciences. Because there are so many more results in it, and so much more is proved in it. In fact at one point uses Belissima the beautiful Torricelli.
The most beautiful, I'm sorry, the most beautiful Torricelli, referring this work and what he did to extend Galileo. All right. Now that you know that, we're gonna look at these tables and see what's happening with them. The table on the right, given amplitudes of semi-parabola described with the same initial speed, okay?
And then the second table is altitudes of semi parabolas described with the same initial speed. And we start from 45 degrees and make it 10,000, length of the semi parabola 10,000, which the range of the full parabola would be 20,000. And we do the heights the same way for 45 degrees.
They start at 10,000 height and go down. Now the obvious question is, whats he doing here? Well, these are really ratios. These are ratios of amplitude versus an amplitude of 10,000. Right? These numbers are all ratios, ratioed relative to an initial value. I'll come back and explain that in a moment, and the same for the altitude.
Over here, given the altitudes and sublimities of parabolas of constant amplitude, namely 10,000, computed giving. Computed for each degree of elevation. So, in this case, we're determining the sublimity and the altitude given the amplitude. But the amplitude here is simply being set to 10,000, so we're being given the relative sublimity and altitude for the amplitude.
If the amplitude is 5,000, we have to multiply in there by one-half. In other words, the tables are being normalized for amplitudes of length 10,000. Non-dimensionalized is another phrase I could use, normalized. But that means to use them, what do you do? You take your cannon, you take your standard cannon ball, you take the amount of powder.
You run the experiment a few times and measure the range at a 45 degree angle. You then multiply that range, you ratio that range to the range of 10,000, and the table now becomes concrete. It's no longer ratios, it's specific values. In other words, the tables are totally functional, but they have to be calibrated.
Calibrated to the artillery gun you're using, or to whatever else you happen to be working with. Does that make sense? Okay, cuz if that makes sense to you, and I use the word calibrated, I'm now gonna show you just how ingenious this really is. So I'll pause a moment.
Everybody see how the tables work? They don't give you the numbers, they give you the numbers relative to something else. It's like Ptolemy's using 60 always as the radius of the different. Everything else is ratioed to that. Here we're ratioing everything to a given amplitude. You can think of it in a different way.
You have a given amplitude, choose a unit of measure such that it comes at 10,000. Now, by that unit of measure, you get all these other things. Okay? Lots of ways of thinking about it.