 Book 5 of Euclid develops the theory of magnitude and proportion. To Euclid, a magnitude was any geometric quantity. For example, it could be a length, area, or volume. But it could also be an angle, or a circle, polygon, or cube, or any other geometric object. So what can we do with these? Mathematics begins when we have a way to compare two objects. So Euclid defines a ratio. Magnitudes are said to have a ratio to one another when one can be multiplied to exceed the other. And by this Euclid meant that some collection of one magnitude exceeded the other. Let's show that the circle and the square have a ratio. So the idea here is if I have a circle, if I have a square, I want to show that given a circle and a square, a multiple of one can exceed the other. And in this case we might take many circles and cover up the square, and so a multiple of the circle exceeds the square. How about a square and a line? So they can have a ratio of one can be multiplied to exceed the other, but if we take any number of lines, they don't exceed the square. There's always some portion of the square that we are missing. And so the square and the line can't have a ratio. Again, mathematics begins when we can compare two objects. Well, how about comparing two ratios? Euclid gives the following definition from book five, definition five. Magnitudes are said to be in the same ratio, the first to the second and the third to the fourth, when if any equal multiples whatever are taken of the first and third, and any equal multiples whatever of the second and fourth, the former equal multiples alike exceed, are alike equal to, or are alike well short of, the latter equal multiples respectively taken in the corresponding order. Phew, this definition is quite a mouthful, so let's take it apart. So first of all we have four magnitudes, first, second, third, and fourth, so suppose we have four magnitudes A, B, C, and D. We also have two ratios, the first to the second and the third to the fourth, and so this means we have the ratio of A to B and the ratio of C to D. Now I'm going to take any equal multiples whatever of the first and third. So let's take NA and NC where N is a whole number. We also want to take any equal multiples of the second and fourth, so let's take MB and MC where N is a whole number. I note that M and N don't have to be the same number. And we're going to compare these equal multiples, see whether they exceed, equal, or fall short of. So what that means is NA could be greater than MB or equal to MB or less than MB. And the key idea here is that they are alike. So whenever NA is greater than MB, we also have NC greater than MB. If they're equal, we have equality. And if they're less, NC is also less than MD. And if this happens, then the ratio of A to B is equal to the ratio of C to D. Let's dig a little deeper. One way to interpret definition 5 is to view our ratios as fractions. So suppose two ratios are equal. And we'll take the case where NA is greater than MB. Well, if NA is greater than MB, because the ratios are equal, NC must also be greater than MD. I can rearrange this. And remember M and N are some whole numbers. And so this fraction, this ratio of A to B and this ratio of C to D, they're greater than the same rational numbers. Similarly, they're going to be less than the same rational numbers and equal to the same rational numbers. And note that none of these have to be rational or even numbers. What if two ratios are not equal? You could also define the following. When of the equal multiples, the multiple of the first magnitude exceeds the multiple of the second, but the multiple of the third does not exceed the multiple of the fourth, then the first is said to have a greater ratio to the second than the third has to the fourth. So again, if we take this apart, if our ratio A to B is greater than the ratio C to D, then there's some rational number, M, Nths, where A to B is greater than M, Nths. But since A over B is greater than C over D, then C over D has to be less than M, Nths. And if we multiply across, we get that N A is greater than M B, but N C is less than M D. And so that's the situation where the first exceeds the second, but the third does not exceed the fourth. So for example, it shows that the ratio of 9 to 2 is greater than the ratio of the circle to the square on the radius. So we need to take a multiple M of 9 that exceeds a multiple N of 2, but M circles does not exceed N squares on the radius. For example, we might try M equal to 1 and N equal to 4, and we observe that 1 of 9 exceeds 4 of 2. But if I look at the geometry, one circle does not exceed 4 squares on the radius. And so the ratio 9 to 2 exceeds the ratio of the circle to the square on its radius.