The ultimate object of the following paper is to elicit a practical and efficient method of resolving, numerically, problems of every description involving an unknown quantity. In order to accomplish this end with the utmost generality, the most important step is to determine the inversion of a given function in a suitable and convenient series, involving the differential coefficients which appertain to an approximate value of the variable.

To say that the subject of the rate of mortality among assured lives and annuitants is at present very imperfectly understood, and that the process usually adopted to deduce the rate of mortality from observations upon these classes, is not a proper one for the purpose, and leads to erroneous and misleading results, may seem at first sight to be very rash assertions; but I believe that upon consideration they will be found strictly correct.

our predecessors adopted a ready and (all things considered) a sufficiently effectual method of regulating the charges to be made for granting assurances on lives. They satisfied themselves on the one hand that the mortality table adopted allowed for a much higher mortality than would probably be experienced, and on the other they assumed in their calculations a rate of interest below that which they knew would be realized on their investments. With these precautions they could rely pretty safely upon realizing a considerable profit, and this consideration not unnaturally proved sufficient for the scientific requirements of the time.