Philosophy 167: Class 6 - Part 8 - Uniform Motion- the Exactness of Geometry.
Smith, George E. (George Edwin), 1938-
2014-10-7
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All right now we go to uniform motion and I get to start showing you the way he conceptualized things. So staying within the framework of Eudoxian ratios where you are supposed to compare similar items. We get ratios. And I'm putting them up here properly. In a given time, two different objects moving, the ratio between their speeds is the ratio of the distances covered.
In a given time. Now over a given distance, the ratio of the two speeds is inversely as the ratio of the two times. So it's subscript two in the first and subscript one in the second. We would of course replace these by fractions. And think of it as division.
I could do that. And I'll frequently do it in this course, because it's so much easier for us to think of it that way. That's not the way they thought of it. They thought of it as ratios. So, therefore, speed and uniform motion varies directly with distance, and inversely with elapsed time.
The compound of two ratios. The way we would write that is two fractions and multiply them into one another, and of course, you would get the ratio of two feet per second relationships to speeds. They thought of it as compound ratio, it's the equivalent of multiplying the two.
But what proposition six then announces. And notice that six, you had to get there, through the earlier steps. The earlier steps being the ones at the top that I didn't give proposition numbers to. If two movables are carried in equable motion, the ratio of their speeds will be compounded from the ratio of spaces run though, and from the inverse ratio of the times.
And he actually gives a proof where he let's, I can't quite see it, but you can look directly at the note. V and t represent the spaces, s and r represent the times. And ratio of the speeds is represented by the ratio of c to g compounded from the ratio of c to e.
And a ratio of e to g. So we end up over here on the right of lengths representing the speeds being obtained from lengths representing the times and lengths representing the distances. And that's the right word, represent. We have geometric magnitudes representing three different magnitudes. So what I have now done and we'll stop when we get to the end of this segment, I think.
I've given you three pages out of a wonderful website. I had the pleasure, at Dibner Institute, of being asked to be a referee for this guy's tenure at Clark University and I already knew the website. boy did I pour it on, and he's still there, he got tenure.
But you know they were complaining he didn't do original things in mathematics. What he did was take all of Euclid's elements, put it into a website, you can't see it here, you actually, in this case I can't manipulate things. I'll show you in the next curve, but in all the figures you can change the figure by just moving a spot around and start appreciating how robust the theorem actually is because it's not tied to any one figure.
I'll show you that in just a moment. This happens to be what's called Euclid's algorithm for finding the greatest common divisor of two magnitudes. And there happened to be numbers. Okay? That's what you're doing. The numbers, even for Euclid, are represented by lengths and he then tells you subtract the one length down until you have a remainder that's too small to subtract any more.
Then take that one and subtract it from the other one, and when you're through with that, you have the greatest common divisor. And you notice he's using the word measure up here. He's presupposing that these two are commensurable with one another, the two legs are commensurable, and he's also presupposing they're not prime.
I showed you that just because it's the algorithm named after Euclid. The ones that are interesting are the next two. The mean proportion. The mean proportion between a and c is the magnitude b such that a is to b as b is to c. If you look at the bottom here, it ends up being x square corresponds to ab.
Now what did they know about this? They knew the very high likelihood of incommensurability among those lengths because they're dealing with right triangles, okay? So if they put numbers to them, it may be impossible to specify the exact lengths. Because one of the numbers is going to be irrational with no pattern and the best you can do is approximate it.
There's no approximation here. You want the mean proportion between a and b and b and c. You take ab, you extend it to bc. You take the midpoint of that, you draw a circle. And at point b the intersection of the two lines, you raise the perpendicular, that's the mean proportion.
And it's an exact mean proportional between any two lengths whatsoever, whether they're commensurate or not. How do you want to get a square root of something geometrically? Square root of five, I take five, I add one to it to get a length of six. I draw a circle from the midpoint of that, and at the 0.51, I raise a perpendicular to the circle, that's the mean proportional.
And the thing that makes this mathematic special, it's one of the reasons Newton uses this very mathematics in the Principia Is all questions of incommensurability and exactness are gone. Geometric magnitudes are exact. That doesn't mean when you draw a circle it's exact. We have all of the classic Plato objections to the drawn figure not being perfect.
But we're treating these objects as perfect figures. And in this case, this is the way you get mean proportionals. It's a very, very well defined quantity geometrically regardless of what magnitudes you're doing a mean proportional between. Now the nice thing about this is point d can, I'm trying to remember which one can be moved here.
It's probably b, point b is I think the point that can be moved. You can go on the website and start playing. It's all three volumes, all what is it, ten or 11, 13. I can't remember the number of books in Euclid. Does anybody remember? 13, yeah it's 13.
On and on and really learn Euclidean geometry the way it should be learned. Now available on that website. It's great fun. The other one I gave you is the third proportional because it's more tricky. A is to b as b is to an unknown. The unknown is the third proportional, in this case you start off, let's see what he's specifically doing.
The third proportional between ab and ac. So what you do is extend the triangle so that it ends up ab is to ac, as ac is to ce. And the construction, again, gives you the exact magnitude that's the third proportional, the missing unknown. And we can keep doing this.
You know, place after place we can construct things totally exactly without numbers. That doesn't mean we can't put numbers on them, the point is we have complete, exact rigor when we work geometrically. It is not clear it's even possible to have that when we work with numbers. Because of incommensurability.
Again, it's very foreign to you, but how many of you actually had a full year course in Euclidian geometry? Yeah, these people who grew up outside of the Untied States, went to school outside the United States. I'm old enough that I had it. You had it? I'm impressed, in Beverly Hills?
I'm impressed. Good teaching. At Lionel school, Beverly Hills High School?
I was surprised at how hard it was.
Okay. That private school. That makes a difference. Fair enough. They don't have to go by state rules. At any rate, that's the math and I'm giving it to you, starting you with the reading for tonight because I've gotta get you accustomed to looking at rudimentary Euclidean geometry because that's what's in the Principio.
Okay, he will switch to numbers some. He will switch to algebra. He will switch to symbols. But his primary mathematics is geometry and I gave you the first reason. It's exact numbers or not. The other reason is he thought you can do limits completely rigorously, geometrically. And he didn't think you could do limits completely rigorously.
Working symbolically and nobody did figure out how to do it until the 19th century. So, we'll get to that later.
