I do not pretend that much which is original will be found in this paper. Previous writers, notably the late Sir J. W. Lubbock, and Messrs. Woolhouse, Sprague, and G. F. Hardy, have, so far as actuarial science is concerned, nearly exhausted the subject of Formulas of Approximate Summation; and from time to time examples have been given of the practical use of the formulas, which have indicated the great facilities they afford to computers. But something in the way of numerical illustration remains to be done in order to bring the formulas into more general use, and to supply that want is my present object.