We are ready to start changing the subject to evidence problems, confirming that uniformly accelerated motion actually does agree with quote naturalia experimenta. Now there's an obvious problem, let's go back to what it is in orbital astronomy. We had two problems there, neither of which show up at all in the mechanics of locomotion.
The first is we couldn't figure out distances, how far away celestial objects are from the earth. And the second is, we couldn't distinguish between real and apparent motion. That's totally gone. If I drop this object, I'm not worried about real versus apparent. And I'm not worried about how distant it is, because I know how distant it is.
Those problems are gone. They're replaced by three problems. First problem is, when you drop an object even from a reasonable height, it goes rather fast, so fast that the total elapsed time is very, very small. Which means if you can't measure it very accurately, small errors in measurement are on a percentage of the total time larger.
One of the wonderful things about planets is they move so little per night. Actually when you're in a telescope and watching something like the moon, you discover how fast the earth is moving because it goes out of sight within about two or three minutes. You have to be constantly adjusting.
And expensive telescopes have motors to do it automatically, but you've gotta mount it equatorially for them to work. But it's a wonderful thing. You don't have much difficulty getting speeds and times. But time and fall happens so fast, to give you an example, three 100 meter height, you'll see almost that in a few moments.
100 meters, reasonably high, it's like a 30 story building. Five seconds in the absence of air resistance to hit the ground. It's very, very fast. Okay, that's the first problem. The second problem is it's not uniform speed. How do you measure the speeds at any time, what speed is reached?
I can use the mean speed theorem to infer a speed. From the mean speak, but how do I verify that that's right? In fact, that ends up throughout the 18th century being an interesting problem, how they measured speed. I'll tell you about it next week, cuz you'll see it dramatically in said Galileo idea, as a matter of fact.
And then, the third problem, of course, is you're making claims about an idealized circumstance, no resisting medium. We are surrounded by resisting media. So you need some way to control for the effects of air resistance so that your experiments are meaningful. Notice how different, it's a totally different world of experimental evidence problems in this case from what it was in astronomy.
All right, now we turn to his fundamental result, and it will tell you a lot about his style. So proposition two, if a movable descends from rest in uniformly accelerated motion, the spaces run through in any times whatever are to each other as the duplicate ratio of their times.
This is not in the original. This is Drake adding, that is, they are the squares of those times. Duplicate ratio is the way they talked about ratio. It's a ratio multiplied into itself, therefore the square. But again, ratios are comparing one time to another. So we're really comparing two times multiplied into one another or two squares of times.
Corollary, from this it is manifest that if there are any number of equal times taken successively from the instant or beginning of motion, Drake insert. Then these spaces would be to one another, as are the odd numbers from unity, that is one, three, five, seven, etc. You know this, but you may not have ever seen it proved.
I'm not gonna prove it, it's fairly easy to prove, but just think it through. Square of two is four. Square of three is nine. Square of one is one. Square of two is four. What's the difference of between four and one? Three. What's the difference between nine and four?
Five. What's the difference between 16 and 9? Seven. What's the difference between 25 and 16? Nine. I can keep going, folks. It goes forever, okay? The consecutive differences in the squares are in the exact sequence of the odd numbers, one, three, five, seven, etc. I'll come back to why that's such a beautiful result.
I wanna do the second corollary. It is deduced second that if at the beginning of motion there are taken any two spaces whatever, run through in any two times, the times will be to one another as either of these spaces is to the mean proportional space between the two given spaces.
And I've done that for you down there. It's the ratio of the space to the mean proportional of space one and space two. But that's just the ratio of the square roots of the space. Time goes as the square root of the distance covered. You all know that.
You've had it in high school, if you had any physics course at all in high school. That's the one thing they tend to teach you. Some of you may have forgotten it, but it's routine. What's so beautiful about corollary one? Well this is in effect Galileo's style of developing evidence, and we'll see it in other people as well.
Though it's as I say primarily, it just completely characterizes how Galileo works. The problem with experiments is of course there are all sorts of confounding factors. You can't really do measurements of speed. You can't really do measurements of time that accurac,y and you've got air resistance all over.
So what you want to test a theory is some kind of really distinctive phenomenon that A is surprising. B can be checked almost qualitatively, what I want to say is quasi quantitatively. So even if you don't get exactly 1, 3, 5, 7, 9, 11, 13, 15, etc., if it's very close to those, then you can, as it were, brush aside any deviation, saying the basic pattern salient pattern.
Salient's the wrong word here, it's much more strong than that. A really unusual distinctive pattern is being predicted here. Do we observe it in fact? That's his style for experimentation all the time. Don't try to go after exact measurement on routine basis. Find something that's extraordinary and confirm it happens.
And as you look at the rest of day three, next week where I tell you don't bother with the proofs after proposition six, the point I'm making is, he's developing propositions to arrive at something of this sort. A way of testing it that's surprising and can be quasi qualitatively checked.
Does that make sense? Because it's a beautiful idea, and you just go searching for these phenomenon that you wouldn't expect that would stand out very strikingly. And that you don't have to have exact because if it's close, you know you've got something that is very difficult to account for any other way.