On Interpolation by Iteration of Proportional Parts, without the Use of Differences

Abstract

Linear interpolation between two values of a function ua and ub can be performed, as is well known, in either of two ways. If the divided difference (ub−ua)/(b−a), which is usually denoted by u (a, b) or u (b, a), is provided, or its equivalent in tables at unit interval (the ordinary difference), we should generally prefer to use the formulawhich is the linear case of Newton's fundamental formula for interpolation by divided differences. If differences are not given, but a machine is available, then the use of proportional parts in the form of the weighted averagethe linear case of Lagrange's formula, is actually more convenient, since it involves no clearing of the product dials until the final result is read.