54 ON PROBABILITY. 
annual rate of increase 1.0061. We may, therefore, suppose the births to 
have increased in geometrical progression, during the century previous to 
Dr. Haygarth’s observations, at the rate of 1.005, without fear of an error 
which can disturb the accuracy of our results. The deaths at the same time 
were about equal to the deaths forty years previously, a result which is con- 
firmed by direct calculation. These data are sufficient to correct the table 
of mortality, and it is obvious that the error of our hypothesis is altogether . 
of an order to be neglected, for a small inequality in the rate of increase . 
will not affect the result, unless it be of long period. 
91. Table (1), page 56, contains the observations of deaths by Dr. Hay- 
garth at Chester. 
Table (2) has been calculated upon this hypothesis, namely, that the 
births increased during the century previous to Dr, Haygarth’s observation 
in a geometric progression, of which the common ratio was 1.005, 
Table (3) shows the values of annuities on such lives. 
© Table (4) shows the values of single premiums for insuring £1. pay- 
able at death; and 
Table (5) shows the values of annual premiums for insuring £1. pay~ 
able at death. 
Tables (8), (4), and (5), have been calculated from Table (2) by Mr. 
David Jones of the Royal Exchange Assurance Company. .- 
- 92. Mr. Finlaison has recently published very extensive tables of mortality, 
formed from the government tontines and annuitants, which are rendered 
extremely valuable by the accuracy of the materials from which they have z 
been deduced, and by the very great care and attention which have been # 
bestowed on them by the author. . 
Mr. Finlaison (in his valuable report to the Lords of the Treasury) ex- 2 
plains at length the manner in which he made use of the records of the B 
tontines. Mr. Finlaison observes, ‘ that the facts shown in these observa- 
“ tions, bear conclusive testimony that the rate of mortality in England has, 
¢¢ during the last century, diminished in a very important degree, on each sex 
‘“ equally, but not by equal gradations, nor equally at all periods of life ; and 
¢ that while in regard to the males it seems in early and middling life to have 
* remained for a long time as it stood about fifty years ago, in respect of the 
¢ females it has during the same time visibly and progressively diminished 
“to this day by slight but still sensible gradations.” 
Whether life has generally improved or not, it is certain that epidemics 
are of much less frequent occurrence now than they were formerly, which 
circumstance must have a very favourable influence. 
93. Mr. Griffith Davies has published tables of annuities taken from state- 
ments of Mr. Morgan, in his addresses to the general courts of the Equi- 
table Society, and in notes added by him to the latter editions of Dr. Price’s 
¢ Observations on Reversionary Payments.” In Mr. Morgan’s address to 
the general court held on the 24th April, 1800, he stated that the decre- 
ments of life among the members of the Equitable for the preceding thirty 
years, had been to those of the Northampton 
from 10 to 20 as 1 to 2 from 40 to 50 as 3 to b 
20... 30... 1,. 2 B50 ..:60 .. 5." 7 
80 .. 40 ..3.. 5 60... 80. .4£..5 
which statement is confirmed in his subsequent addresses. In a recent 
publication, Mr. Morgan admits that he was not then aware of the great 
number of instances in which there are several policies upon one and the 
same life, but this circumstance cannot very materially affect Mr. Davies's