 One of the classic geometric construction problems is squaring the circle. But what do we mean by that? Squaring a figure meant to construct a square equal to that of a given figure. But what does this mean? Do the Greeks, this meant you had to cut apart and reassemble one figure to form a square. And this can be done if the figure is bound by straight lines. But it's impossible if the area has a curved boundary. You can't dare rid of the curved pieces and straighten them out. So in that sense, squaring the circle is really impossible. No problem, we'll just create some mathematics that will allow us to do this. And so to deal with problems like this, Eudoxus, who lived around 350 BC, created the theory of ratio and proportion. Now when we say created the theory of ratio and proportion, what we mean is this. Other geometers had been using these ideas implicitly, but Eudoxus gave it a sound theoretical basis. And by invoking ratio and proportion, you could avoid the problem of trying to compare otherwise incomparable objects. To see how this works, we might consider the following. While we might not be able to say whether a given circle and a given square are equal, we can generally say when one is greater than the other. And in fact, Eudoxus defined the following. A ratio exists between two quantities if either can be multiplied to exceed the other. And what Eudoxus meant by this is that if you take either of these two quantities, if you make enough copies of one of them, all those copies together will be greater than the other quantity. And this is true no matter how large or small the quantities are, as long as they're the same type of quantity. So you could compare a small square with a very large circle, but you couldn't compare a line and a square. Because while we could use the square to exceed the line, we have to also be able to use the line to exceed the square. And no matter how many copies of the line we set down, the lines have zero thickness, so they'll never cover the square. Well, if we have two ratios, the next thing we want to know is when they are equal. So suppose we have two ratios, A is to B and C is to D. How can we tell if they're equal? And here, Eudoxus has a definition for equal ratios. Fair warning, it's a bit of a mouthful. So here it goes. The ratio of A to B is equal to the ratio of C to D if, given any equal multiples of the A and C and any equal multiples of B and D, the former equal multiples alike exceed, equal, or fall short of the latter. As I warned you, that's quite a mouthful, but it's actually not so bad if we take it one step at a time. Suppose we take whole numbers M and N. These are going to be our multipliers. So these equal multiples of A and C. I'm going to take M, A, and M, C. Again, a bunch of copies of A and an equal number of copies of C. I'm also going to find N, B, and N, D. Again, equal multiples of B and D. Now, I might not be able to compare the copies of A to the copies of C, but since A and B do have a ratio, I know I can compare the copies of A to the copies of B. And one of three things will happen. M, A will either be greater than, exceed, equal to, or fall short of N, B. And so if the ratios are equal, whenever M, A is greater than N, B, it must also be true that M, C is greater than N, D. Likewise, if M, A is equal to N, B, so must M, C be equal to N, D. And if M, A falls short of N, B, so must M, C fall short of N, D. And this gives us a way of comparing a circle to a square. And if we've squared the circle, we want the ratio of the circle to a square to be the ratio of equality. Now, there's all sorts of neat stuff here in this definition of the equality of ratios, but here's an important caveat. While this does allow us to compare a circle to a square, we can't compare the circumference to a straight line. And that's because we run into a similar problem with equating a circle in a square. We can't take multiples of the circumference and compare them to see which one is greater to a straight line. And in fact, a satisfactory answer to this question would have to wait until the 20th century. On the other hand, at least the theory of ratio and proportion allows us to talk about the problem of squaring the circle. So let's take a look at that next.