In the mathematical subfield of numerical analysis, interpolation is a procedure that assists in "reading between the lines" in a set of tables by constructing new data points from existing points. This rigorous presentation employs only formulas for which it is possible to calculate error limits. Subjects include displacement symbols and differences, divided differences, formulas of interpolation, factorial coefficients, numerical differentiation, and construction of tables. Additional topics include inverse interpolation, elementary methods of summation, repeated summation, mechanical quadrature, numerical integration of differential equations, the calculus of symbols, interpolation with several variables, and mechanical cubature. 1950 edition.