All right, striking predictions. Here's another one coming out of proposition six. I'm not gonna discuss propositions three through six much. We do with algebra just by writing down velocity is acceleration times time. And distance is one half of the acceleration times time square. We do everything that's done in proposition three through six of getting the inter relations between the different parameters.

Algebra does have some virtues. You can also write the things down and not understand them at all, which is the history of me doing electromagnetism, I just do the equations blindly and pay no attention to what the phrases mean. But this is a striking result. If from the highest or lowest point of an vertical circle any inclined planes whatever are drawn to its circumference the times of descent along those planes will be equal.

And that's an if and only if when you go down to the corollary two it's also deduced that if. From the same points there descend a vertical and an incline plane over which descents are made in equal times. They are inscribable in a semicircle of which the diameter is vertical.

So, in effect what we've got here is another striking thing. Why should it be along the circle that these two things hold. Now of course here we get a rolling falling problem because you're not gonna fall along an incline plane unless its very steep and if you're gonna go down vertically, its hard to make it roll but not recognizing the distinction between the two.

This is something you could use to test the theory where you don't have to have them exactly equal if they're near enough equal. And as you depart from the circle they violate that, than you get the same kind of thing. I'll give you more examples. So I'm not going to talk a lot about the rest of day three.

I've got to single out three or four important things that I want to move today for. I had, I told you in effect, to skim day three. What I told you to do is after proposition six, just read the propositions and corollaries. And skip the proofs. One reason to skip the proofs is we can do everything in them algebraically so much faster but the other reason is they don't need proofs and the results don't matter that much to the subsequent history largely because most of the experiments can't be done but the structure of it, we have those fundamental results.

Propositions one through three. Then four through nine, we enable ourselves to compare different inclined planes and times and speeds, etc. Then propositions two through 26 are problems where we have an initial speed and we divert it from one plane to another. So we're coming down an inclined plane.

We have a nice smooth transition to the horizontal, then we can have a nice smooth transition back up another incline plane. That doesn't have to be symmetric to the first. And of course one of the key results, is no matter what the second incline plane looks like in compared to the first, the ball will roll down, go flat, go back up to the same height.

And that's independent of inclination. Now of course, again, it's gonna happen only with relatively low inclinations because of rolling versus falling. But fair enough. And that's gonna culminate in the scholium to proposition 23, which I'll put some time in, in just a minute. Then propositions 27 through 31 look at minimum time trajectories.

And the final propositions that culminate in the scholium to proposition 36 are comparisons along different paths. Two remarks there on this slide. Scholium is a term from Euclid. It's a commentary on propositions that are proved and corollaries to them. So it's not required in a scholium you give formal proof.

It is expected to give formal proofs in the body. And in this work, where we've got an academician's treatise, we're gonna have propositions. We're gonna have an introduction, then we're gonna have propositions as they're approved, corollaries. And every once in a while, Scholia commenting on what's going on.

But then the dialogue serves to do all of the rest of the commentary of course, and that's why it's written in the dialogue form. And the other thing in there, 21 theorems, 16 problems, and it's explained here, I'll just make the point. 16 problem, what makes something a problem in Geometry is you have an unknown quantity, geometric quantity, and you have to find it.

But that's not enough. You have to find it by constructing it with ruler and compass. That's what the Euclidean requirement is. So when the statement was made, I'll come right back, Dave, on this. When the statement was made Kepler wanted a solution to his problem of getting where the location of a planet is, given the time.

What he wanted was a ruler compass construction, and when people said it can't be done, what they meant at the time was it can't be done geometrically at all. Now it turns out there's a systematic relation between doing it geometrically and doing it within certain constraints algebraically. So the two are related but it's quite a demand to have to construct the magnitude using ruler and compass.

Let's look at a couple of the striking results that are historically significant. This is a Scholium, proposition 23. I'm not going to read the first part but the second part is quite important, historically. It may also be noted that whatever degree of speed is found in the movable.

This is by its nature indelibly impressed on it when external causes of acceleration or retardation are removed. Which occurs only on the horizontal plane. For on declining planes, there is a cause of more acceleration, and on rising planes, of retardation. From this, it likewise follows that motion in the horizontal is also eternal since it is indeed equitable, it is not weakened or remitted, much less removed.

That passage is the principle basis for saying Galileo put forward the principle of inertia. What we now call the principle of inertia. He's gonna take it back in a slide I have in just a few minutes. But there's a much deeper problem with it. Namely, he clearly and in many places says circular motion is self perpetuated.

Uniform circular motion is self perpetuating, and it's a straightforward violation of what we call the principle of inertia. So I for one always dispute that he deserves to get much credit for the principle of inertia. But the statement is here. And the statement does talk about eternal motion on the horizontal.

And you will next week see Gassendi generalizing this specific statement. And that generalization is the first occurrence of what we now call the principle of inertia. In the public literature. So, we're not, you know? In all fairness, this statement was influential, up to a point. And the idea, it's gonna be followed up.

But you can sort of picture. Okay, an object rolls down an incline plane. Goes a short distance horizontally then goes back up another inclined plane. Lower the second inclined plane and it's gonna have to be longer and longer until it gets back to the same height. What happens if it's flat?

It goes forever cause it can't get to the same height. That's essentially the idea here. Okay. We'll come back to what's wrong with this, but I thought it very important that you see the principle basis for giving credit to Galileo. Almost all of the physics literature gives credit solely to Galileo.

There's no mention of Descartes or Gassendi. I'll give you arguments next week that Descartes deserves the vast majority of the credit. For the principle of, what we now call the principle of inertia. Continuing with this, though, the point I just made, from this we may therefore reasonably assert that if descent is made through some incline plane, after which there follows a reflection.

Through some rising plane the moveable ascends by the impotence received all the way to the same altitude or height from the horizontal. So both of these are laying out that principle of the speed you're required is exactly what you need to take it back to the same height.

And if it's the horizontal it goes on forever.