On Interpolation, Summation and the Adjustment of Numerical Tables

Interpolation, generally considered, is the process of determining numerical values intermediate to a series of other values that are already known, and it is of extraordinary utility in many calculations, and especially in the formation of extensive tables. If, in the direct calculation of a table, the values to be tabulated depend upon a complicated function, difficult of calculation, considerable labour and corresponding risk of error may usually be avoided by computing the values for a certain succession of equidistant intervals, and afterwards determining by interpolation all other values that may be requisite to complete the table and adapt it to the practical purposes for which it is constructed. In such case the method is made use of as a powerful auxiliary at the option of the computer.