 If we know the chords of angles alpha and beta, Telami's results allow us to find the chords of angles alpha minus beta and one half alpha. We can then use the new chord length to find other chord lengths. Now as you can probably guess, actually calculating the chord lengths, especially if you're working in sexic ensembles, is quite a task and only a sadistic monster would require you to find them. So I won't ask you to do more than 0, 10, 12, 15 of them. But the computational process is not actually as important as the question of how you could find them. So let's explain how you could find the length of the chord, subtending the arc of 12 degrees. So remember, Telami already found the chord of 60, 72, 90, and 120 degrees. So we might consider a few differences. So taking the chords whose lengths we know, 120 minus 90 is 30, 120 minus 72 is 48, and notice that if we have 48 twice, we get 12. And so if we want to find the chord of 12 degrees, we'd use the known values of the chords of 120 and 72 degrees to find the chord of 48 degrees, have the now known chord of 48 degrees to find the chord of 24 degrees, and again, now we know the chord of 24 degrees, so we have the now known chord of 24 degrees to find the chord of 12 degrees. Now, since we've divided the third conference into 360 parts, it would be nice if we could find the chord of 1 degree. We can't. In fact, the smallest whole-degree angle we can find using Telami's theorem is the chord of 3 degrees. Now, while there's no reason why we have to have the chord of 1 degree, because we are measuring our angles in degrees, it would be nice if we had that chord. And so to get the chord of 1 degree, Telami proves an important approximation theorem. If two unequal chords are inscribed in the same circle, the ratio of the greater to the less is less than the ratio of the arc of the greater to the arc of the less. And what this allows us to do is it allows us to find bounds on the length of a chord as long as we know the length of a chord subtending a greater and a lesser central angle. For example, let's approximate the chord of 80 degrees. So we'll want to find two angles, one less than 80 degrees and one greater than 80 degrees, where we know the length of the chord. So let's start off with the chord of 60 degrees. And so the chord of the greater angle is to the chord of the lesser angle. That's the chord of 80 to the chord of 60 in a ratio less than the ratio of the angular measurements, 80 to 60. Now, we know the actual length of the chord of 60 degrees. That's 60 parts. And this ratio of 80 to 60 is the ratio of 4 to 3. And that tells us the chord of 80 degrees is less than 4 thirds of 60. And that means the chord of 80 degrees is less than 80 parts. Now, we also want an angle that's greater than 80 degrees. So let's take the chord of 90 degrees. So again, the chord of the greater is to the chord of the lesser in a ratio less than the ratio of the angles. And again, we know the chord of 90 degrees. That's 84, 51, 10. And we're arranging our proportionality. We know that the chord of 80 degrees is greater than 8 9ths the chord of 90 or 75, 25, 29. And so we have a lower bound on the length of the chord of 80 degrees. Now, since our angles of 60 degrees and 90 degrees are not very close to 80 degrees, our bounds are very loose. We could get better bounds if we could find the chords of angles closer to 80 degrees. So now, Ptolemy finds the chord of one degree. Now, remember the smallest whole degree chord that we can find is the chord of three degrees. But we can also take half of that. So we can find the chord of one and a half degrees. At least we could find it in principle. Ptolemy actually found it 13415. And also the chord of three quarters of a degree. That one isn't so bad if you're starting with the chord of one and a half degrees. So we want the chord of one degree, which is somewhere in between these two. So our theorem says that the chord of the greater is to the chord of the less as the less than the angles. So that's the chord of one to the chord of three fourths is less than one is to three fourths. And that says the chord of one degree is less than one and a third of the chord of three fourths. But we know that chord of three fourths. And so this gives us an upper bound on the chord of one degree. We can also use that chord of one and a half degrees. So again, the chord of the greater is to the chord of the less as the angles are to each other. And we're arranging our proportionality that says the chord of one degree is greater than two thirds the chord of one and a half degrees. We know the value of that chord. And that gives us a lower bound on the chord of one degree. And what's important here is that to two sex or decimal places, the upper and lower bounds are the same. And so Ptolemy concludes that the chord of one degree is one to fifty.