10 - Finite Differences: Interpolation and Quadrature

Summary

Preliminary Remarks

The method of interpolation for the construction of tables of trigonometric functions has been used for over two thousand years. On this method, one may tabulate the values of a function f(x) constructed from first principles (definitions) for x = a and x = a + h, where h is small, and then interpolate the values between a and a + h, without further computation from first principles. For sufficiently small h, one may approximate the function f(x) by a linear function on the interval [a, a + h]. This means that, in order to interpolate the values of the function in this interval, one may use the approximation f(a + λh) ≈ f(a) + λ(f(a + h) - f(a)), 0 ≤ λ ≤ 1. In his Almagest of around 150 AD, Ptolemy applied linear interpolation to construct a table of lengths of chords of a circle as a function of the corresponding arcs. These are the oldest trigonometric tables in existence, though Hipparchus may well have constructed similar tables almost three centuries earlier. In Ptolemy's table, the length of the chord was given as 2R sin θ, where R was the radius and 2θ was the angle subtended by the arc. Later mathematicians in India, on the other hand, tabulated the half chord; when divided by the radius, this gives our sine.