Al-Biruni's law of interpolation (istikmal)

Interpolation (istikmal) is a method for finding a function's value between two values given in a table; al-Biruni devised his form of it in Book 3 of al-Qanun al-Mas'udi to improve the use of sine tables. ^{1 2}

The editor explains that the common method assumed that within one interval the increase in the sine is proportional to the increase in the angle; in fact equal intervals of angle do not give equal differences of sine, so the common method is inexact. ²

Al-Biruni took two paths. First, he narrowed the intervals: he computed sine tables for every quarter degree instead of every whole degree, wishing he could have done so for every minute but for the length of the work; the editor describes his tables as correct to the seventh or eighth decimal place. ² Second, he improved the method of use: noting that a change in the angle brings gradual change in the differences of the sines rather than in the sines themselves, he set the ratio of the part of the interval to the interval equal to the ratio of the appropriate difference to the difference of the differences, which yields a more exact value. ²

The editor considers that al-Biruni thereby reached the interpolation laws in the simple form later ascribed to Newton and Gregory six hundred years after him, and that his way of measuring differences of angles was the cornerstone of the generating method they used. ^{3 1} He adds that al-Biruni was the first to take unity as the radius in sine tables, so that the tabulated lengths became ratios. ⁴