Trigonometry · a Light

The sine of one degree, to ten sexagesimal places

Every table of sines has to start somewhere, and the hard case is one degree, which cannot be reached by the usual halving and adding of angles already known. At Samarkand in the 1420s, Jamshid al-Kashi turned it into a cubic equation and solved that equation by repeated approximation, each round of arithmetic adding another correct digit, until he had the value to ten places in the base sixty notation astronomers used. No instrument of his day, or for a long time after, could measure an angle anywhere near that finely. He did the same with the circumference of a circle, taking the value of what we call pi far past any practical need. Ulugh Beg's star tables were built on his figures.

Al-Kashi · Samarkand, 1420s