 The sign and diameter numbers may be connected with something of a mystery. In his quadrature of a circle, Archimedes had to approximate the square root of 3. He used 265 1.53 is less than the square root of 3, which is less than 1,351 7.80. It's worth noting these are accurate to 4 decimal places, and we have no idea how he got these values. But we can obtain them through a relationship similar to the side and diagonal relationship. So to begin with, we might ask ourselves, self, where do we see square root of 3? And if we think about that, we see that square root of 3 shows up in a rectangle where the diagonal is twice the base. And if we have a rectangle where our diagonal is twice the base, then square root of 3 will be the relationship between the sides of the rectangle. And so this means we want to find a base and side relationship. Suppose we have a rectangle where the diagonal AC is twice the base AB. If we subtract side BC from the diagonal, we leave AD. And as before, if DE is perpendicular to AC, then we have a smaller rectangle with the same relationship. And what we want is the relationship between the base AB and the side BC. And so working backwards, we have base AD and side DE. Our new base AB, well, that'll be AE plus EB. And since our diagonal, AE is twice the base AD, that means AB, the new base, is 2AD. And EB and DE are equal. So I can replace EB with DE. And so our new base is 2AD plus DE. That's 2 times the old base plus the old side. How about the new side BC? So the thing to remember here is that the new diagonal AC had the side BC cut off. So that's the segment here DC. And so BC is AC minus AD. But again, the diagonal AC is twice the base AB. And we have an expression for AB. It's 2AD plus DE. And so 2AB is going to be 4AD plus 2DE, from which we'll subtract AD. And that tells us that the new side is 3 times the base plus 2 times the side. And so this gives us a relationship between the base and side of a rectangle where the diagonal is twice the base. And so maybe we'll begin with a base of 1 and a side of 1. And our relationship says that the new base is twice the old base plus the side, so it will be. And the new side is 3 times the old base plus twice the old side. And so this ratio 5 to 3 gives us a first approximation to the square root of 3. And now Lather rinsed for Pete to find additional base and side numbers. And it doesn't take us too long until we get to 153 and 265. And our comedies used 265, 153 as one of the approximations for square root of 3. But wait, there's more. If you think about what our rectangle looks like, our side is nearly twice the base. And so we might let our base be 1 and our first side will make it 2. And that should actually give us a better approximation. So we'll find our new base, twice the old base plus the side, and our new side, 3 times the old base plus twice the side. And again, we'll find additional base and side numbers until we get the Archimedean approximation, 1351, 788 as the other approximation to square root of 3.