In most branches of mathematics, the actual use of fundamental processes in the form which the first definitions suggest, is often supplanted, either by processes of greater skill, or by the use of pure reasoning. In the subject of which this paper treats, there has not been much attempt to connect formulæ by reasoning. The actual exhibition of successive annual results has been the only method extensively employed; and it throws the required total result into a series of terms: this series is either algebraically summed, or calculated term by term for insertion in a table.

The present paper is intended to show that this summation of algebraical series may be dispensed with, at least in questions of annuities certain : and also that common points of principle, which the ordinary methods leave altogether out of sight, will reduce many questions of life annuities to an absolute coincidence of form with the corresponding questions of ordinary annuities.