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.

The use of the commutation table was independently developed in
England-apparently between the years 1788 and 1811--by George Barrett,
of Petworth, Sussex, who was the son of a yeoman farmer, and was himself
a village schoolmaster, and afterwards farm steward or bailiff. It has
been usual to consider Barrett as the originator in England of the
method of calculating the values of annuities by means of a commutation
table, and this method is accordingly sometimes called Bar