Galileo is gonna leave the world with eight principles. They're all important. I'm gonna run through them. They're all stated in the absence of a resisting media. Direct vertical fall is a uniformly accelerated motion that its distance is proportional to times square. The same speed is acquired for many given height, whether in direct fall or along an inclined plane.

That gets generalized to where it's any trajectory whatsoever. So a rollercoaster works the same way. The same, excuse me, the speed required in descent from any height is exactly sufficient to raise the object back to that height. That's of course an energy principle. These principles are gonna form the foundation for what Leibniz called conservation of vis viva, though Leibniz is at least by Lagrange, is not credited with conservation of vis viva.

Huygens is. For reasons you'll see, Huygens came up with it and Leibniz dubbed it, called it vis viva. But that's a striking result. Same acquired from given height, same for all objects. In the absence of impediments, motion along the horizontal remains uniform, and now my parenthesis, at least over distances small in comparison with the radius of the Earth.

Six, the two components of motion of a body moving uniformly in parallel with the horizontal, and simultaneously falling vertically remain independent of one another. That's a very big deal but the statement at the top, in the absence of a resisting medium, is absolutely crucial to this. Not totally crucial.

If the resistance is proportional to velocity, it still holds. It's only if the resistance is nonlinearly proportional to velocity that you can't separate the two components. That's what Leibniz had terrible trouble understanding, why Newton didn't just do the two components. Newton had the two components solved for resistance as velocity squared, but he knew we couldn't put them together.

And Leibniz's, he's versed in the study of Leibniz more than me. Just wrote Huygens, why doesn't he do the obvious here? And Huygens tries to explain to him, you can't do this. And in the exchange of letters, he just doesn't pick up on it. All right, that's a big deal, and you'll see it reappear in Huygens horologium oscillatorium, it's one of the three fundamental hypotheses that the two remain independent of one another.

Seventh, a body projected horizontally describes to high approximation, notice what I put in there, a semi parabola at least over distances small in comparison with the radius of the Earth. With it's dimensions dictated by the height from which the bodies initial horizontal speed would be acquired naturally and by symmetry a body that is projected at an acute angle upwards describes a corresponding full parabola with it's dimension dictated by the initial velocity and an angle of projection.

And then an eighth the distance of fall from rest from the first second RG over two is the same at every location around the Earth. Those are eight Galilean principles that will carry forward, in some form or another. If you took any physics course whatsoever, from the eighth grade on, you've seen them.

Because that's what the first thing people tend to teach in any physics course, is those Galilean principles. The one that they don't tell you constantly is the path wise independence and I have no idea why they don't tell you that. All right, with that as background we have a bunch of questions we can ask about Galileo's theory parallel to the questions we asked about Kepler's.

For example, do the four fundamental principles of fall in the absence of a resistant medium, and I'm taking four of them to be fundamental, uniform acceleration. Relevance of weight, shape, etc. Path wise independence and return to the height and ascent. Do they hold exactly or only approximately and if the latter or the mean or otherwise.

That was in the absence of the resisting medium its a perfectly legitimate question whether they hold exactly, or essentially exactly, or only approximately. It's a question that we will go on and have people discuss after Galileo. Given that parabolic projection does not hold exactly, I said given that, even in the absence of a resisting medium.

Three questions, what's the true trajectory of a projectile near the surface of the Earth? What would the continuing trajectory be if a body were to continue to the center of the Earth without impediment or resistance? And what is the trajectory in the presence of air resistance? All three of those became major problems.

Newton tries to solve all three of them in the Principia. He fails with the third. The first mathematically reasonable solution for the third is by Johann Bernoulli in 17, 18, 19, as a challenge problem. The challenge problem is to show that the calculus is better than Newton's geometry, and he shows it by actually solving a problem that you would almost never be able to do without calculus, without differential equations.

So it's an important thing historically we will get to next semester. Third, does a body really gain the same increments of velocity in equal increments of time in vertical fall in the absence of resistance everywhere on the surface of the Earth? That is, is the acceleration of gravity uniform around the surface of the Earth?

And I repeated when I first said that he had no way whatsoever of knowing whether that's the case. And fourth how far this is a question he doesn't ask but he calls it locomotion all the time, and he always distinguishes between locomotion and celestial motion and that lead various people to start asking how far up does uniform acceleration go?

Somewhere up there, we get in a celestial range, and it doesn't hold anymore, perhaps. Okay? It's an open question. How far up does uniform acceleration go? And you will see people asking that question, and wondering how to answer it. And of course, Newton gives us an answer as you all realize.