What I do wanna do is spend a moment on Archimedes. I've added slides for the first time on Archimedes, principally with the thought that Galileo modeled his work on Archimedes work, two works in particular on floating bodies which is the work that Archimedes is truly legendary for. The fundamental result any body displaces in water equal to its own weight, etc.

But the one I'm gonna be showing you is no less important. When I say modeled on, I'm actually consciously using the concept Kuhn introduced originally when he started using the word paradigm, a word he abandoned later in life. Because it had to use, he had used it in so many ways he thought the only thing he could do was quit using it.

But his original idea, and I'm almost quoting now, is an achievement from the past that you model your work on. And Archimedes is very much an achievement from the past that Galileo modeled his work on. So this is from the Equilibrium of Planes. I've just given you a couple of excerpts.

One's very important, the others are not necessarily so important. The two postulates, three postulates I wanna emphasize, equal weights at equal distances are in equilibrium. And equal weights at unequal distances are not in equilibrium, but incline towards the weight, which is at the greater distance. So we don't actually get specifics of motion.

We simply say when you're in equilibrium and what happens when you lose equilibrium, you incline in the direction. But note, no detailed description of motion. Similarly, the second one, if when weights at certain distances are in equilibrium, something be added to one of the weights. They are not an equilibrium but inclined towards that weight to which the addition was made.

And correlatively, if anything be taken away from one of the weights, they are not an equilibrium but inclined towards the weight from which nothing was taken. I'll let you look at the others, but you'll notice what happens here, you start proving propositions one by one. And you can look at them, same way I want you to be looking at the ones in Galileo, particularly next week.

Very often you prove a proposition in order to enable, put you a position, it's enabling, put you in a position to prove a more important one later. So when you look at these and look at them in order, proposition one, two and three. If you ask why are one, two, and three there in the sequence they're in, the best way answer it is to see how they're appealed to later because that's almost always what's going on.

You're building something up. That's true in Euclid and he's simply mimicking Euclid to a very great extent. The next slide gives, it's the classic reference for what's called the Principle of the Lever. You won't quite see it as the Principle of the Lever, but let's read. I'm not gonna go through the proofs, I'm simply gonna do the proposition itself.

Two magnitudes, whether commensurable or incommensurable, I'll come right back to that, but notice two different propositions, balance at distances reciprocally proportional to the magnitudes. So if you know the bodies you have, and you wanna lift one of them. You first set this up reciprocally by the distance and then you add weight to the one you don't wanna lift in order to lift the other one.

Okay, it's in effect what you do here, and that's the Principle of the Lever. It's important for many different reasons. One of the most important in this course is in the, I guess it's the 11th week next semester, we're gonna see Johann Bernoulli announcing in print to three years after Newton died, Newton doesn't understand the principal of the lever.

Which is correct. He certainly understood the principal of the lever, but he didn't, anywhere we've got rotational motion he gets confused between forces and torques. He doesn't understand what we now call a torque is, and it's fairly striking. Bernoulli is not wrong in saying that. The conclusion Bernoulli draws, the Vortex Theory A planetary motion can therefore be completely defended and his paper won a prize for that, that's wrong.

But he certainly gonna appeal to the fact that Newton doesn't appreciate rotational motion in the way that's needed here. The other comment to make here, commensurable versus incommensurable. This is important, it's gonna come up again. So I need to make sure you know exactly what it means for two magnitudes, and we're talking here geometric magnitudes.

The term magnitude is being used quite consciously. They're not numbers. Two magnitudes are commensurable if there is a third magnitude that goes into each of them an integral number of times. Any third magnitude. And to be incommensurable is then to deny that there is any magnitude, that goes into both of the other two, an integral, integer number of times.

And of course they had discovered that, for example, the hypotenuse of an isosceles right triangle, in which it's length is by our standards, the square root of two if the sides are one. The hypotenuse of an isosceles right triangle is incommensurate with the sides. There is no length that will go an integral number of times into both.

And to go an integral number of times gets translated in Euclid as measure. To measure something is to in effect, take off some length an integral number of times. Fair enough? And the oddity here is he proves it both ways. He feels the needs to prove it both for commensurable and incommensurable.

Why? Because for the commensurable case, it's relatively straightforward to set up the ratios and do the proof. For the other case, the extra actually has to add some weight to reduce it to a commensurable case. That's the way he ends up proving it. So you can read those proofs for yourselves, but what you're looking at here is something Galileo knew very well.

This was published, memory serves me right, it was published in Latin off the printing press in 1567, but it's around that time. It's when Galileo was very young, or just before he was young. It may have been 1547. I'm flipping between the two dates. At any rate, you have the two proofs, and then you go right into center of gravities, starting with the center of gravity of rectangles.

But this is, again, one of these places where he's building up to more and more complicated shapes by starting with simple shapes, and having them enable proofs of later shapes. So that's the model on which Galileo was working and he knew very well and pretty much assumed everybody else knew it well.

And certainly the Principle of the Lever, which comes into play very big time in 1691 by Johann Bernoulli's brother, older brother, Jacob Bernoulli. In solving a particular problem that you'll get to know and when we get to Hogan. It becomes a central development in modern mechanics. The turning of the Principle of the Lever into a means of handling rotational motion and conjunction with translational motion is a very big deal.

That's by the way, why Bernoulli says Newton doesn't understand it, because Newton doesn't seem to understand rotational motion. The two places he should understand it, he says very silly things.