The two leading formulæ at present in use for interpolation by central differences are due to Newton‡ (Methodus Differentialis, Prop. III), though often called by the name of Stirling, who republished them about 20 years later.

According to the Darwinian theory of evolution the members of a community less fitted to their environment are removed by death. But this process of natural selection would not permanently modify a race, if the members thus removed were able before death to propagate their species in average numbers. It then becomes an important question to ascertain how far duration of life is related to fertility. In the case of many insects death can interfere only with their single chance of offspring; they live or not for their one breeding season only. A similar statement holds good with regard to annual and biennial plants. In such cases there might still be a correlation between duration of life and fertility, but it would be of the indirect character, which we actually find in the case of men and women living beyond sixty years of age—a long life means better physique and better physique increased fertility. On the other hand, there is a direct correlation of fertility and duration of life in the case of those animals which generally survive a number of breeding seasons, and it is this correlation which we had at first in view when investigating the influence of duration of life on fertility in man. The discovery of the indirect factor in the correlation referred to above was therefore a point of much interest. For it seems to show that the physique fittest to survive is really the physique which is in itself (and independently of the duration of life) most fecund.