Okay with that said, let's take the Apollonian parameters. I'm not gonna run through these. These were all identified before Apollonius, but by the time of Apollonius they had been singled out as the relevant parameters for doing orbital research. And, if you think about it for a moment, none of them are straightforward things, that you just look at and see.

That is they're not parameters that are part of natural language, they're concocted parameters. Concocted in every case in an effort to describe something you're observing in a way that has regularities to it that are absent under other descriptions. So the very act of trying to form a theory means you've gotta discover these parameters.

They don't come for free, and this can be very, very difficult to do. Ptolemy had the great advantage that he inherited all of these from the Apollonian period. And then as I comment here, the italics, each of these was linked to specific observable phenomenon, many of them to the same phenomenon that Ptolemy used.

They were each measurable. They were each assigned a physical significance. But the great thing is the physical significance kept changing through the ages. So when we get to comparing it cuz they have a different physical significance etc., but it's the parameters themselves that are enabling you to do the theory.

Now the issue is, these are the good parameters. Now, I have to find the relations among them. But, look! It can't be that way, because you're simultaneously discovering the parameters and getting robust relations between them. The two have to go together. In fact, that's the hard part, in general, of constructing theories is the two have to go together.

Fair enough? And I'm gonna illustrate that now with Galileo's first of two new sciences, Strength and Materials. So the question is, it's beings, he's only interested in beings and he's hanging weight on them. And the question he wants to answer is, what parameters govern fracture. If he can get the parameters then he can start predicting fracture and do counter factual etc.

Well, it's obviously some combination of the dimensions of the beam and the applied load, the weight, but fine, what combination. His attempts at strengths and materials got very little of anywhere. And when you look at it and look retrospectively, big time retrospectively, he didn't have the right parameters.

The first right parameter is stress. Stress and strain, of course, but stress. That emerges in the 1820s, that we finally get stress as a generalization of pressure. Shear stresses were discovered by Coulomb, but the general notion of stress finally emerges as elasticity theory in continuum mechanics is worked out by a group of people in the 1820s all in France.

Poisson, and being principle names of that. What might interest you, this of course is my engineering field,, stresses a very mediocre predictor of fracture. If you depend on stress for fracture, well I'll do it this way. If I go out and measure the stress at which specimens fracture, the spread in the data will surprise you.

In fact, I'll tell the story about myself. I was a young engineer working on a very, very advanced project and went to metallurgists trying to define what a lower safety margin would be, because we were ready to run a lower safety margin. And I said I wanna see the actual data that this design curve of allowable stress is based on.

And he showed me the data and I remember I was 19 years old. I said you gotta be kidding, the data had huge spread to them, it was very discouraging. Well in the 1920s a man whose name I didn't recall and I didn't have a chance to look it up.

He became the chief designer of jet engines that Rolls Royce Griffith. That's the name. Griffin or Griffith one of those two. Discovered new parameter called stress intensity. And that's the one that tends actually to govern real fracture. He discovered it for brittle fracture, fracture like of glass. But it applies pretty much across the board, the principles, the following.

Almost any real metal etc., have small defects in it. The question now is, what stress level is needed for that defect to become an unstable crack, and propagate at the speed of sound? I hope none of you have ever heard a crack propagate at the speed of sound.

If you have, it's a very scary sound, it's quite loud and quite scary. I have photographs, if you want to see, of an airplane where that happened and part of the fuselage came off. Fortunately only one person died in the process, but the crack was quite audible to all the passengers sitting there.

So, at any rate, stress intensity, to tell you how strange it is, I'm sure nobody here has heard it unless they've been trained as a mechanical engineer. It has the following units, units of stress times the square root of distance. That is pounds per square inch times square root of inches, in English units, and it essentially correlates to energy.

In fact, I'll say what it is. When an infinitesimal increase in the defect releases enough strain energy to keep the defect growing and it takes off, becomes unstable and takes off. And that's what stress intensity measures. So Galileo had no way of getting near that if took another hundred years after we had stress identified and all the theory of elasticity to come up with it.

I'll do this one last way. If you go to any engineering textbook on failure, on a cracking formation, fracture is the right word. You will find that there are five standard criteria for fracture and they don't agree with one another. And the field can not agree on which one is appropriate.

Okay. So Galileo didn't have much of a chance of getting anything