 Talwami constructed a table of chord lengths in a circle with a radius of 60. Indian astronomers of the 5th century began considering the half-chord of the half-angle. The earliest documented appearance is in the work of Aryabhata. Aryabhata included the value of 24 of these half-chords in his Aryabhatiya. Aryabhata himself did not describe how he found these values, but the computation of these half-chords is described by Bhaskara in the 12th century. To produce the half-chord length, Bhaskara begins by dividing the circumference of a circle into 12 equal parts, which are known as harases. Horizontal and vertical lines are drawn to form rectangles, and additional lines are drawn to form triangles. Since the length of a chord depends on the diameter of the circle, Bhaskara chose a circle with a radius of 3438. This number might seem to be a little bit arbitrary, so let's take it apart. This number actually comes from the following process. If we let one minute of arc be equal to one linear unit, then the circumference of the circle, well that's 360 degrees times 60 minutes of arc per degree, 21,600. And if our circumference is 21,600, the diameter will be approximately 6,876. So the radius will be half the diameter, 3438. And Bhaskara's general procedure begins with the whole chord of some number of rasas. Half of this gives the half-chord of the half arc. Now it's important to remember that the half-chord of half the arc is actually half the chord of the full arc, and importantly not a chord. Also typically the half is implied, so while we could say the half-chord of half of four rasas, we know that half of four is two. So we would just call this the half-chord of two rasas. So for example let's try to draw that half-chord of two rasas. So remember two rasas are two of the 12 equal arcs. But since we're talking about the half-chord, this amount is actually the half-arc, so the full arc is four rasas, and the half-chord is half the chord of the full arc. Now this half-chord forms one side of a right triangle whose hypotenuse is the radius, so we can find the other side using the right triangle theorem. And note that this actually gives us another half-chord, and if you look at the geometry this is actually the half-chord of the complementary arc. There's one other important quantity here. The arrow of the half-arc is the remaining portion of the radius. But again this is one side of a right triangle whose other side is the half-chord of the half-arc. So we can find the hypotenuse again using the right triangle theorem, which gives us the actual chord of the half-arc, and half of that is the half-chord of the quarter-arc. What this means is that starting with any half-chord we can repeatedly apply the right triangle theorem to get additional half-chords. So Bascara begins with a division of each rasas into two units. Now as our starting point we might recall that the chord of one-sixth of the circumference is equal to the radius. Since the circumference itself is 12 rasas, then one-sixth of the circumference is two rasas, and since each rasas is two units then two rasas is four unit arcs. And so that means the chord of four unit arcs will equal the radius 3438. So we'll mark out that chord of four unit arcs in the central angle. Now from here we can find the half-chord of two unit arcs is 1719. And this gives us a right triangle with hypotenuse 3438, the radius, and one side 1719. And so the other side satisfies x squared plus 1719 squared equals 3438 squared. And we can solve this for x. And the value of Bascara used is 2978, which is actually the value we'd find rounding up. And remember this is actually the half-chord of the complementary arc. So the quarter circle is three rasas divided into two arcs. It's six unit arcs, and so this amount is the half-chord of six minus two four unit arcs. But wait, there's more. The remaining part of the radius is, and this is what we would call the arrow of two unit arcs. And we now have a right triangle with sides 1719 and 460. And so our hypotenuse can be found. And again, if we solve this and round up, we get the value that Bascara used 1780. And the important thing to understand here is this is the full-chord of two unit arcs. So half of this is the half-chord of one unit arc, 890. And again, this is one side of a right triangle with hypotenuse equal to the radius. And so we can find the other side. And this is the half-chord of the complementary arc. And again, since the quarter circle is six unit arcs, the complementary arc is going to be five units. And so now we know the half-chord of five unit arcs. And so we now have the half-chord of one, two, four, and five unit arcs. But what about the half-chord of three unit arcs? To find the half-chord of three unit arcs, we'll need to find the full-chord of six unit arcs. So the chord of six unit arcs, remember, it's two units per rosses. This is three rosses. Well, that's the hypotenuse of a right triangle with two sides equal to the radius. And so the chord length satisfies the right triangle theorem. And we solve. And so using the same division of the circle, the Ascara finds the chord of six unit arcs, is 4,862. And that gives us our half-chord of three unit arcs, 2,431.