It is not to be expected, at this period of time, that anything new remains to be discovered in the Doctrine of Annuities, especially by one not conversant with analysis; yet a useful application may occasionally be found of the results of former investigations, the more so should such have remained in comparative neglect.

The announcement of the relation which subsists between an annuity payable yearly, and when it is payable half-yearly, quarterly, monthly, &c., the rate of interest continuing the same, is due to the illustrious De Moivre, and was made by him in the second edition of his treatise of Annuities on Lives, published in 1743. It was reproduced in the subsequent editions of the same work, and is to be found, with some errors of the press corrected, in that appended to the third edition of his Doctrine of Chances, 1756.

Although I have not strictly followed De Moivre, having computed my factors before I found they were described in his work, I here transcribe his rules with the demonstration of them, remarking that, the notation he employs being at variance with that to which we are now accustomed, and likely, therefore, to cause embarrassment to the student, I have substituted the modern notation throughout.