 Euclid proved two important results on circles. Well, actually he proved a lot more important results, but we'll focus on two of them. First, in Book 3, Proposition 16, Euclid proved the following. The perpendicular to the diameter of a circle at its end point falls outside the circle. No line may be interposed between the perpendicular and the circle, and the angle of the semicircle is greater and the remaining angle less than any acute rectilinear angle. Now this proposition is quite a mouthful, so let's take it apart. First, we say tangent, but tangent is hard to define. Falls outside is more quantifiable. Now Euclid also claims that we can't wedge another line between the tangent and the circle. In other words, the tangent is unique. And the angle of a semicircle? Well, let's see what that is. The proof is a classic proof by contradiction. Let the circle have centered D and diameter AB, and let AE be perpendicular. Now if this line doesn't fall entirely outside, it has to cross the circle at some other point C. We'll draw a CD, and we note that triangle ADC is isosceles, so angle CAD must equal angle ACD. But remember, angle CAD is a right angle. But this produces a triangle with two right angles, which is impossible. Which means our assumption must be false. By a very similar line of reasoning, we can prove that it's impossible to interpose another line. So what about this angle of a semicircle? Well, let's consider a semicircle. And let's zoom in a little bit. We can talk about this region between the semicircle and the tangent line, and we could call that an angle. But it has some strange properties. First, it's going to contain any acute angle CAD. And since the hole is greater than the part, that means the angle of a semicircle is greater. Similarly, if we look at this remaining angle, later called the horn angle, we see that it's contained by the acute angle EAD, and so it's less. And what makes us noteworthy is that EAD can be any acute angle. Euclid's result on the area of a circle is based on the principle of exhaustion. Given two different magnitudes A less than B, it's always possible to find a magnitude C between them, where A is smaller than C is smaller than B. This idea goes back to Bryson and Antiphon, who used it to find the area of a circle. A polygon could be inscribed in a circle. The number of sides could be doubled. And lather, rinse, repeat. So the inscribed polygon always has an area less than the circle, but doubling the number of sides produces a polygon with a larger area that's still less. Of course this doesn't always work. Let's go back to that tangent. Given any acute angle, the horn angle is less. But we can always produce larger and larger horn angles, but the horn angles will always be less. So the principle of exhaustion clearly has limits, which means it needs a proof. And Euclid provides this in Book 10, Proposition 1. Given two unequal magnitudes, if the greater is continually reduced by a quantity greater than its half, there will be left some magnitude less than the lesser magnitude. Euclid's proof relies on the existence of a ratio between the two magnitudes. So remember that a ratio exists if one magnitude can be multiplied to exceed the other. So suppose AB is greater than C. Then some multiple K of C is greater than AB. And so now let's subtract from AB an amount greater than its half and repeat this process K times. At each step, the remainder from AB is less than the corresponding multiple of C. So at the end of this process, the remainder is going to be less than C itself. Book 10, Proposition 1 provides the theoretical justification for Bryson's approach. Given any circle, inscribe any polygon and let S be any area less than the area of the circle. By repeatedly doubling the number of sides of the inscribed polygon, you will eventually obtain a polygon with area greater than S. Now before we can launch into Euclid's discussion of the area of the circle, we'll need one more proposition, Book 12, Proposition 1. Similar inscribed polygons are to each other as the squares on the diameters of their respective circles. This can be proven without the principle of exhaustion and is just a straight area proof. So for the area of the circle, Euclid proves Book 12, Proposition 2 circles are to each other as the squares on their diameters. Euclid proves this by contradiction. First, suppose the square on BD is to the square on FH as the circle is to some area less than the circle. We can inscribe in circle FH a polygon with area greater than S, so inscribe a similar polygon in our other circle. Now by our assumption the square on BD is to the square on FH as the circle is to some area smaller than the circle. Now since the polygons are similar, the squares on their diameters are to each other as the polygons are to each other. But since these are the same ratios, the circle is to S as the polygons are to each other. And we can rearrange our proportionality so the circle is to its inscribed polygon as S is to the other inscribed polygon. But the circle is larger than its inscribed polygon while S was smaller than the inscribed polygon. So this is impossible. Now the second half of the contradiction proof is rather clever because it reduces the problem to the first half. Suppose our square is to the square as the circle is to some area T greater than the circle. So our square is to the square as the circle is to T. Now remember T is greater than our circle FH, which means we can shrink the components of this ratio to S and the circle FH itself, where S is smaller than our circle ABCD. But that's our first case. The square is to the square as some smaller area is to the circle. Now because this is a statement about a proportionality, we can read this as the area of a circle is some constant of proportionality times the square on the diameter. And here's the important thing. Euclid never gave the value of this constant of proportionality. So what is that constant of proportionality? We'll take a look at that later.