For the purposes of this paper the term “curve-fitting” is not synonymous with “graduation.” It is restricted to graduation by means of an explicit algebraic or trigonometrical formula: graphic and summation methods are excluded. Even so the subject is unlimited, and this paper cannot provide more than an explanation of a few general principles, and a brief exposition of some of the problems of special interest to the actuary.

Our starting point is this: we have decided to fit a curve (vx) to a series of crude figures (ux) and the problems that confront us are:—

(a) What formula are we to ascribe to vx?

(b) How are we to choose the constants in vx in the best possible way?

(c) How are we to test the goodness of fit and to compare the results of different graduations?