How's he test this theory? Well, there are three results. I'm gonna do those and then take a break. The first one I've already told you about, the periods of conical pendulum strings of different lengths vary as the square root of their heights, AB and AE. So in other words, keep them the same height and you have the same time.
If if the lengths vary and they're not the same height, we can get the periods of the conical pendulums. So we can test, we can do two conical pendulums at different lengths look at their heights, compare the two. The second result you read, I chose not to emphasize it completely til after break.
He discovers that the following is the case. Take a parabola, rotate it about its central axis, that's a paraboloid. Make a chalice of that shape. Take a ball, like a roulette ball, which of course he knew about, put it in there, and ask at what speed does it stay in the place.
The peculiar thing about the paraboloidal conical shape is whatever speed keeps it in one place, at the same place, at one height in the thing, the same speed does it in every place. It's a universal. Okay, so that's testable, make yourself a paraboloid cone like this. Rotate it on an axis, and simply make sure to put the little ball in there in different spots, and see if it stays fixed at the same thing.
Everybody, I think, can picture what's happening here. Gravity's pushing down, but you're pushing against this surface, and since this surface is on an angle, it's gonna give you, at every place, just the amount you need to compensate gravity with the component of centrifugal force given the slope. That's what he discovered about parabolas.
Then the last result, this is the only one directly about tension in a string. Take a 90 degree circular arc pendulum. Question, where do you have to intercept it so that the string will stay barely taut all the way around? And, his solution is two-fifths of the length of the pendulum, and again, that's a lovely, Galilean result in two respects.
You remember the intercepted pendulum in Galileo? It's actually an idea of Galileo's, where the diagram comes right out of Galileo. But now we know, or at least we can test, whether or not this gives you the right amount. Now you don't have to do the same thing I did with Galileo.
Does it have to be exactly two fifths? Well of course not. You'll say resistance causes a little trouble, but it was very close to two fifths, gives you the taut, you're gonna say this theory's right. Anything that's problematic with it is simply due to resistance. That's what I meant before by a quasi-quantitative test of a very striking result.