The mathematics of annuities has been very fully treated in Demoivre's _Treatise on Annuities_ (1725); Simpson's _Doctrine of Annuities and Reversions_ (1742); P. Gray, _Tables and Formulae_; Baily's _Doctrine of Life Annuities_; there are also innumerable compilations of _Valuation Tables_ and _Interest Tables_, by means of which the value of an annuity at any age and any rate of interest may be found. See also the article INTEREST, and especially that on INSURANCE.
_Commutation tables_, aptly so named in 1840 by Augustus De Morgan (see his paper "On the Calculation of Single Life Contingencies," _Assurance Magazine_, xii. 328), show the proportion in which a benefit due at one age ought to be changed, so as to retain the same value and be due at another age. The earliest known specimen of a commutation table is contained in William Dale's _Introduction to the Study of the Doctrine of Annuities_, published in 1772. A full account of this work is given by F. Hendriks in the second number of the _Assurance Magazine_, pp. 15-17. William Morgan's _Treatise on Assurances_, 1779, also contains a commutation table. Morgan gives the table as furnishing a convenient means of checking the correctness of the values of annuities found by the ordinary process. It may be assumed that he was aware that the table might be used for the direct calculation of annuities; but he appears to have been ignorant of its other uses.
The first author who fully developed the powers of the table was John Nicholas Tetens, a native of Schleswig, who in 1785, while professor of philosophy and mathematics at Kiel, published in the German language an _Introduction to the Calculation of Life Annuities and Assurances_. This work appears to have been quite unknown in England until F. Hendriks gave, in the first number of the _Assurance Magazine_, pp. 1-20 (Sept. 1850), an account of it, with a translation of the passages describing the construction and use of the commutation table, and a sketch of the author's life and writings, to which we refer the reader who desires fuller information. It may be mentioned here that Tetens also gave only a specimen table, apparently not imagining that persons using his work would find it extremely useful to have a series of commutation tables, calculated and printed ready for use.