The three heads, into which it has been found expedient to divide the general subject of the present paper, all relate to the consideration of a series of numerical quantities which are supposed to represent, either accurately or approximately, some known or implied law which necessarily has an influence in regulating their progression. Assuming that we only know the values of certain terms of the series of quantities, distributed at stated equidistant intervals, it has been shown that all the intermediate or absent terms are most accurately and expeditiously determined by the process of interpolation, already discussed and practically developed.

The Friendly Societies, which are sometimes called Benefit Clubs, have for many hundred years been highly popular in England; not merely because they supplied one of the principal wants of the working classes—namely, relief in sickness or accidents which disable them from obtaining their usual wages—but because, by mutual association, they afforded some scope for the talents and sagacity of some of the members to be recognised and appreciated by others. But by the very nature of their composition they were deficient in the element of stability ; and notwithstanding the best intentions of the members, the Societies have been constantly broken up and reformed.