 So let's summarize our results so far. Aribata divided the circumference into 12 equal razes, then divided each razes into two parts, found the half cord of the half arc and the cord of the complimentary arc, found the arrow of the half cord, which allowed him to find the cord of the half arc, which could then be used to find the half cord of the quarter arc. And put together, this gives us a table of half cords where each unit is one-twenty-fourth of the circumference. But it may be a little hard for us to think about this because we're so used to thinking in terms of degrees. So one-twenty-fourth of the circumference corresponds to an angle of 15 degrees and so we can include our degree lengths. So it's important to recognize while these numbers are as exact as we want them to be, the step size is too big for practical use. So we need a way to find even smaller half cords of half angles. Now there's no reason we couldn't continue this process and find the half cord of a half unit of a quarter unit and so on. But for reasons that only become clear when you try to calculate these half cords, Vasgari chose not to find half cords of fractions of a unit arc. Instead, he made the unit smaller. So if each rossus is divided into four unit arcs, then using the new units, we already know the half cords of two, four, six, eight, and ten unit arcs. We can then find the half cords of one, three, and five unit arcs and the half cords of the complementary arcs, eleven, nine, and seven units. So for example, if we divide each rossus into four unit arcs, we'll find other half cords and let's start off with the fact that the half cord of two of these smaller unit arcs is 890. So we'll start with the half cord of two unit arcs, 890, and the half cord of ten unit arcs, 3321. So remember, the half cord of the complementary arc is a part of the radius and the remaining part is going to be the arrow, 117. The arrow and the half cord are two sides of a right triangle whose third side satisfies and so that length will be, this is the cord of two unit arcs and so the half cord is half that amount, or 449. And again, the half cord of one unit arc is one side of a right triangle with hypotenuse equal to the radius and so the other side satisfies. So it will be. And since the quarter circle has been divided into three rossus at four units apiece, 12 unit arcs, then this amount is the half cord of the complementary arc, 12 minus 1, 11 unit arcs. And we can make a further subdivision. If each rossus is divided into eight unit arcs, then the quarter circle is 24 of these unit arcs and we'll have calculated all the half cords of all the even numbers so we can find the half cords of all the odd numbers. When each rossus is divided into eight units, then the full circle is going to be divided into 12 times 896 equal arcs. And so each arc is 196th of a circle, which works out to be three and three-quarter degrees. And at this point, are we about to stopped, although we could obviously continue and what we now have is a table of half cords that correspond to the values of radius 3438 sine theta for theta, a multiple of three and three-quarter degrees. And historically, this is the first time that we get what we would properly call a table of sine values. It's not quite our modern trigonometric table because we do have this factor of the radius incorporated into these values, but it is closer to our tables of sine values than Ptolemy's table of cords. And later mathematicians follow the Indian example and concentrate on the half cord.