I do not know what proportion of the users of mathematical tables go beyond the use of proportional parts in interpolation work. But there are undoubtedly, on the one hand, some tables whose construction would be unnecessarily laborious and expensive were they to be at such an argument interval as to render linear interpolation feasible, and, on the other hand, there are those who find it necessary to use such tables. Given such a table, there are various methods of performing the interpolation. There is the use of the Taylor series, f (a + h) = f(a) + hf′(a) + ½h2f″ (a) + ..., which requires a knowledge of the derivatives of the function as well as the function itself. Then there is the use of the Lagrange formula of interpolation, fx = a0f0 + a1f1 + a2f2 + ..., involving a number of tabular values on both sides of the required value.