 The side and diagonal numbers play an interesting role in the history of mathematics. The Ann of Smyrna in the first century described the side and diagonal numbers as follows. The first side and diagonal are one, then we can generate additional side and diagonal numbers by adding the side and diagonal to get a new side, then adding twice the original side and the original diagonal to get a new diagonal, and then repeat indefinitely. So if we take our first side and diagonal as one, we'll get a new side by adding the side and diagonal, a new diagonal by adding twice the original side plus the original diagonal, and this gives us a new side and diagonal number so we can find the next side by adding side and diagonal, and the next diagonal twice the side plus the diagonal, and again we have a new side and diagonal so we can find the next side and diagonal numbers, and we can continue this as far as we want. So there are many questions we can ask about the side and diagonal numbers. How did they find this relationship? What were they used for? Were they used at all? And the answer to all of these questions is we don't actually know. However they can be derived from our attempt to apply the Euclidean algorithm to the side and diagonal of a square. So remember when we tried to apply the Euclidean algorithm to find the greatest common measure of the side of a square and its diagonal, we took a square A, B, C, D with diagonal AC, we marked off CE equal to CB, and if we draw EF perpendicular, AE and EF are the same, and EF is equal to FB. So the problem that we ran into with the Euclidean algorithm is this reproduced a smaller square where we had the side and the diagonal. But our mathematics failing to solve one problem often gives us insight into another. So if we take the side of our square A, B, it's going to be the diagonal of the smaller square plus FB, but FB is equal to EF, which is the side of the smaller square. And similarly that diagonal AC, well that's AE plus EC, and remember EC and BC are the same, and BC and AB are equal, so BC is really AF plus FB. But wait, there's more. Remember AE, EF, and FB are all equal. So AE plus FB, well that's really two EF, and AF is our diagonal, and so our new diagonal is two sides plus the old diagonal, and this is the side and diagonal relationship described by Theon. So what can we use these numbers for? Since the ratio of the diagonal to the side of a square is square root of two, the side and diagonal numbers provide an approximation for the square root of two. So if we pull in our side and diagonal numbers, the second side and diagonal numbers, 3 and 2, gives us an approximation to the square root of 2, 3 halves, or about 1.5. The next pair of side and diagonal numbers, 7 and 5, give us the approximation 7 fifths, 1.4. The next pair, 17 and 12, give us the approximation 17 twelfths, and we can continue to produce these side and diagonal numbers as far as we want, getting better approximations to square root of 2. Now if these were actually created using this application of the Euclidean Algorithm, it's possible to apply several variations on a theme. So one possibility, suppose we have a rectangle whose height is twice the width. Well let's draw a picture. So let's apply the Euclidean Algorithm and see where we go. Now in the square the two sides were equal, so it didn't matter which one we chose. Since the rectangle has one side twice as long as the other, it'll be convenient to work with the shorter side. To do that, we'll need to subtract our base twice. So subtracting the width twice from the diagonal leaves af. And as before, if we make fg perpendicular to ac, then fg and gb are equal. And we have another rectangle that's similar to the original, and so fg is 2af. Now we can work backwards. So suppose we start with our side af and our diagonal ag. So our new side ab, well that's ag plus gb. But remember gb and fg are the same. And because our height is twice the width, fg is 2af. So that means our new side ab is our old diagonal plus twice the side. Meanwhile our diagonal ac, well that's ac is af plus fc. But remember fc and bc are equal. And bc is twice the base ab. But we have an expression for ab. It's ag plus 2af. And so that means our new diagonal is 5 times the side plus twice the diagonal. And so we might summarize. Add the diagonal and twice the side to get the new side. Add 5 times the side and twice the diagonal to get the new diagonal. And so again we can start off with a side and diagonal of 1 and compute the new side and diagonal numbers. So our next side, we're going to add the diagonal and twice the side, which gives us 3. Our next diagonal, 5 times the side and twice the diagonal, 7. The next side is the diagonal plus twice the side, 7 plus 2 times 3. The next diagonal, 5 times the side plus twice the diagonal. And we can keep going. And we might note that this ratio, 123 to 55, if we square it, we get, which is almost 5. And so 123 to 55 is a good approximation to the square root of 5. Actually, since the diagonal of a rectangle whose height is twice the width is about twice the side, we might begin with a side of 1 and a diagonal of 2. And if we do that, we find our sets of side and diagonal numbers are. And this last one, 161.70 seconds, we find that it's a even better approximation to the square root of 5.