Now the deep point about this. You're gonna calibrate it how? You're not gonna calibrate it shooting a cannonball absent air resistance. You're gonna calibrate it by shooting it with air resistance. So we're gonna have a theory here of motion without air resistance, and we're gonna calibrate it by taking measurements with air resistance.
Now notice what that does up at the first two points up there. For a given initial velocity, charge cannonball and cannon, measure the range for a 45 degree angle. That's actual range sub 45 for all other angles multiply the value of the amplitude versus the angle table divided by 10,000 by actual, excuse me.
In the amplitude versus angle table divided by 10,000 by actual range. So what are we getting here? We're getting a predicted range, which is equal to the theoretical range over the theoretical range of 45 degrees. Those are apps and air resistance. And we're multiplying by the actual range, which incorporates air resistance's effects, right?
What are the air resistance effects going to be? Well they're going to subtract from the theoretical range right? There's gonna be a loss due to air resistance and of course there's gonna be a loss at every angle. So qualitatively, those losses are gonna tend to cancel out. The theoretical ratios can be pretty good because the losses, which are gonna occur in numerator and denominator, may be roughly the same.
Now let's hope for the best, suppose the amount of distance you lose due to air resistance Is proportional to the total distance. In that case the ratio of the theoretical distances will give you the exact right predicted answer. As if you're lucky enough that the resistance loss is proportional to the theoretical range.
Then when you do this, put the ratio in there, you're getting absolutely exact predictions in error resistance from a conclusion that from a theory that's about nowhere resistance. Everybody see how that works? If you're lucky enough that the air resistance effects is wonderfully proportional to the distances. Then the air resistance effects cancel out in numerator and denominator and you get exactly the right prediction.
This is how engineers, and I say this as an engineer, this is how engineers work. We take physics theories that aren't exact for our case, we calibrate them and hope the calibration sweeps under the rug, all the things that our theory is not taking into account. Calibration can be an incredibly useful, powerful tool in prediction.
But now, ask yourself the question. What does it prove about the theory of motion in the absence of air resistance? I've heard countless physicists say, well, the evidence for physics is it works in engineering practice. Well in engineering practice, we calibrate all over the place. In effect, sweeping under the rug.
It's a favorite expression of mine in this regard. Sweeping under the rug any discrepancies between theory and the actual world. How well we succeed in sweeping them under the rug depends almost entirely on whether we find a way to calibrate it so that when we start doing ratios, the effect cancels out a numerator, denominator.
It's a very, very clever thing to do. And that's essentially what Galileo is counting on when he does this. And as I say, it's inordinately clever. It's the first place I've seen anybody do a classic engineering move. When I saw it, I almost started laughing because I've done it too many times, knowingly cheating.
That is, I know I don't know the physics so let's see if I can get away with the calibration. Everybody see that? Because it's really very neat. The tables are exactly set up to do the calibration I describe and in the process, hopefully, smudge out the consequences of the effects not taken into account.
And that may be what he meant when he said there is no science of air resistance, but if we do it for the case that we can handle mathematically and apply it to the world maybe we can take care of things. Unfortunately resistance lost is not really proportional to the theoretical range.
And so even calibrated, Galileo's tables were pretty much worthless for the artillery people. It led to a series of further books, probably the most important that I know of is by John Collins, a person you'll hear about who had very important connections to Newton. Collins did do a book trying to get better artillery tables taking into account air resistance.
Rupert Hall's dissertation, Rupert Hall is one of the most important figures of the development of 20th century history of science. This is a cleaned up version of his doctorial dissertation. This book was published in 1952. At home I have a hardbound version of it, but Cambridge Press finally reissued it in paperback.
It gives the history of efforts on ballistics in the 17th century. One of the things it says is that people in the field never trusted any calculations of tables. So they did everything trial and error. That doesn't mean they weren't interested. They would look at them, they wouldn't work, and they'd dismiss it.
But that's why it was still a problem at the time of Babbage in the middle of the 19th century. And it's also why it was still a problem at the time of Eckhart in 1943 and 44. For those who don't know, von Neumann got into computers because he was at Aberdeen, Maryland for a different thing connected to the atomic bomb.
And he was standing at a platform waiting for a train and introduced himself to Eckhart and asked what Eckhart was doing. And Eckhart described the computer he was doing and von Neumann immediately recognized he could solve the neutron diffusion problem if he had a computer like that. So he dropped what he was doing in Los Alamos and went to work on computers for the next few years.
It's a nice story, but it is very much, artillery is very, very central to the history of computations, just because it's such hard problem. Fair enough? I hope you see that because it's quite striking. So now we have a questions. The question is what kind of evidence can we get for parabolic projection?
We have the successful prediction of 45 degrees. We have the possibility of symmetry on either side of 45 degrees, that showed up in the tables. And we have the possibility that these tables would actually work. Suppose they do work, what sort of evidence is it for? Parabolic projection?
Now the answer is if it's symmetric on either side, there's pretty good evidence that the actual trajectory is not that different from a parabola, it would seem, because it's the parabola that's producing that effect. But for this to be successful, to successfully predict ranges, it's not at all clear what evidence you have.
And that's why I always balk when people say, here's the evidence for physics, it works in practical application. Because in so many practical applications and good Tom Cleveland is very much here, in so many practical applications, what we do as engineers far exceeds anything science. Now there are sciences that's not true of, electrical engineering is probably the one, at least up to a point it's least true.