 The length of a chord or a half chord depends on both the angle and the radius of the circle. Ptolemy used a radius of 60 and worked base 60. Islamic geometries continued this practice and produced tables giving what we would call the values of 60 trigonometric function theta for the different trigonometric functions. We'll use the notation r sine theta equals 60 sine theta. Now simple geometry can be used to give us the following exact values, r sine 60, r sine 45, r sine 30, and r sine 18. The half angle and angle difference formulas can be used to find r sine for other angles, but not r sine of one degree. Since we'd like a table of chords in one degree increments, this is inconvenient. In the 15th century, Jamshid Al-Khashi used an approximation technique to find r sine one degree. Although Al-Khashi worked in base 60, and we should as well, we'll present his work using more familiar base 10. Though for the sufficiently adventurous, I'll post something that shows Al-Khashi's work in base 60. So the angle summation formulas and a little algebra gives us the relationship between the sine of 3 theta and the sine of theta. Now since r sine theta equals 60 sine theta, this gives us sine of theta is 1 sixtieth r sine theta, and so our triple angle formula gives us. And while this is true generally, if theta is equal to one degree, we have the relationship. And this is useful because we know the exact value of r sine of three degrees. So if x is r sine of one degree, then we have r sine of three degrees equals 3x minus 1900th x cubed. Now again, we know the exact value of r sine three degrees, and so to illustrate Al-Khashi's approach, we'll use r sine three degrees, which is approximately 3.14016. Al-Khashi actually used a much more accurate value, and if we substitute this in, we can rewrite our equation as... Now as a side note, this is the type of equation that could be solved using the Chinese method of approximating roots of polynomials. But Al-Khashi used a very different method to approximate a solution. And this is circumstantial evidence that Al-Khashi's method of finding fifth root was original within the Islamic world. To solve this equation, Al-Khashi did what a bad algebra student would do and solve for x this way. The important difference here is Al-Khashi didn't stop and say that this was a solution. He then used a method of successive approximations, and that works as follows. First, we see that x is approximately equal to 1, and so we can write x in expanded form as 1 plus a tenths plus b one hundredths plus c one thousandths and so on, where a, b, and c and so on are the digits in the successive decimal places of x. Now our expansion has an x cubed term, and notice that if we expand, we get 1 plus 3 a tenths plus a whole bunch of other terms where all the remaining terms have a factor of one one hundredths or smaller. And this suggests they won't affect the value in the first decimal place. Consequently, if we let x equal one plus a tenths and substitute into our equation, we get, where on the left hand side we'll use the full value one plus a tenths, and on the right hand side we'll use the approximation x is approximately one. You can think about that as our previous approximation on the right and our new approximation on the left. So we can simplify and solve for a, which gives us, and remember a is supposed to be the next digit of the root, and so that next digit is going to be a equal to zero. Now we know x is approximately one point zero, so adding that next decimal place, we let x equal one point zero plus b hundreds, which gives us, and again on the left we have that new approximation, on the right we have the current approximation. And we solve, and this is going to give us b equal to four where we round down since this is supposed to be the next digit in the expansion. And repeating, we know that x is one point zero four something, so we'll let x equal one point zero four plus c thousandth, which gives us, again new value on the left, old value on the right, and solve. So we know that x is one point zero four seven plus d ten thousandths, and so we get, and so we approximate x to be one point zero four seven one, and we could continue to find more digits, but our accuracy is going to be limited by this approximation that we used for r sine three degrees. Now a few other useful things we found here, we found that r sine one degree is about one point zero four seven one, and keep in mind this is half of one side of a one hundred eighty gone inscribed in a circle of radius sixty, and so the perimeter of this one hundred eighty gone will be one hundred eighty times two times our approximation, and we use this as an approximation to the circumference of the circle with radius sixty, we get an approximation for pi, and Alcache actually used a more accurate approximation for r sine three degrees, and found the first nine six decimal places of r sine one degree, and this could be used to approximate pi to sixteen decimal places, a world record at the time.