e
necessary, while this law was in force, to value life annuities charged
upon a testator's estate. Aemilius Macer (A.D. 230) states that the
method which had been in common use at that time was as follows:--From
the earliest age until 30 take 30 years' purchase, and for each age
after 30 deduct 1 year. It is obvious that no consideration of compound
interest can have entered into this estimate; and it is easy to see that
it is equivalent to assuming that all persons who attain the age of 30
will certainly live to the age of 60, and then certainly die. Compared
with this estimate, that which was propounded by the praetorian prefect
Ulpian was a great improvement. His table is as follows:--

+-------------+-----------+------------+-----------+
| Age. | Years' | Age. | Years' |
| | Purchase. | | Purchase |
+-------------+-----------+------------+-----------+
| Birth to 20 | 30 | 45 to 46 | 14 |
| 20 " 25 | 28 | 46 " 47 | 13 |
| 25 " 30 | 25 | 47 " 48 | 12 |
| 30 " 35 | 22 | 48 " 49 | 11 |
| 35 " 40 | 20 | 49 " 50 | 10 |
| 40 " 41 | 19 | 50 " 55 | 9 |
| 41 " 42 | 18 | 55 " 60 | 7 |
| 42 " 43 | 17 | 60 and | |
| 43 " 44 | 16 | upwards | 5 |
| 44 " 45 | 15 | | |
+-------------+-----------+------------+-----------+

Here also we have no reason to suppose that the element of interest was
taken into consideration; and the assumption, that between the ages of
40 and 50 each addition of a year to the nominee's age diminishes the
value of the annuity by one year's purchase, is equivalent to assuming
that there is no probability of the nominee dying between the ages of 40
and 50. Considered, however, simply as a table of the average duration
of life, the values are fairly accurate. At all events, no more correct
estimate appears to have been arrived at until the close of the 17th
century.

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