Mr. Benjamin Gompertz, in 1825, presented to the Royal Society ; memoir On the Nature of the Function expressive of the Law o Human Mortality.' As this ingenious paper contains a deduction fron a principle of high probability, and terminates in a conclusion whiel accords in a great degree with observed facts, it must always be con sidered as a very remarkable page in the history of the inquiry befor us. We enter into some detail of it the more readily, that it is neces nary as an act of justice to Mr. Gompertz, whose ideas have beei adopted by a writer on the subject, in a work published in 1832, with out anything approaching to a sufficient acknowledgment. (See of this point the Journal of the Institute of Actuaries,' vol. ix., part 2 July, 1860.) There is in the human constitution a power of resisting the effect of disease, which increases from birth up to a certain age, and dimi Milnee from that time forwards; the evidence of such diminution bein, the increased proportion of deaths iu a given time. The proportion i found, in most tables, not to be altered by equal quantities in equa times, but to diminish in a greater ratio as life goes on. Mr. Gompert assumes that the " power to oppose destruction " loses equal pr.( portions in equal times ; so that the intensity of mortality, suppose * The ward in the eighth page of the memoir cited is portions, obi& is a mil print or an oversight, as tho Annetta immediately following shows. If s be It time, a b x loses eqnal portions in win times, and a, b 2 equal proportions.
aversely proportional to this power, must be represented by the ormula a q', where a is its value at the commencing age from which years are reckoned, and q a constant depending on the rate of ncrease of the intensity. If, therefore, y be the number living at the nd of x years, y . a dxx b is the decrement of that number in the ime d x, where b is another constant ; and this gives d y = abq"ydx, vhich integrated is of the form y = vhere q,l, and g are to be determined. This can be done by three values f y out of the given table ; and the result, hitherto purely hypothetical, an then be compared with the other parts of the table, by calculation If the values of the formula for different ages. The more convenient orm of the above is log y = log 1± no. wh. log. is (log. log g + x log q) where log log g is taken without reference to the sign of log g, and he upper or lower sign is used according as g is greater or less than Among other comparisons, Mr. Gompertz has made one with the iarlisle table from the age of 10 to that of 60, and another (deducing Efferent values of 1, g, and q) from 60 to 100. The two formula] )lotained are, using log 1 for the phrase "no. whose logarithm is," and r meaning the age of the parties, log y = 3'88631 + '0126 x log y = log I { + '02706 x} In the first set of ages the discordance between the formula and the table is only In one instance as great as half a year ; that is, there is only one instance in which the number deduced from the formula as alive at a given age represents the number living in the table at an age so distant from the given age as half a year. Several other comparisons,
with other tables and different constants, give equally satisfactory results. Few who know the best tables of mortality will be inclined to think that their probable error is within half a year ; so that, as we now stand, Mr. Oompertz's principle, namely, "that equal proportions of the power to oppose destruction' are lost in successive small equal times," is as well established for a large portion of life as any of the tables. It is remarkable that the approxiinate accuracy of the common method of finding an annuity on three joint lives, by substituting for two of the lives one life of the same value, is a consequence of the approximate truth of Gompertz's law. (Seethe Assurance Magazine,' July, 1859, vol. viii. p. 4.) We now come to the practical exhibition of the law of mortality in tables. A very good account of the history of the subject, by Mr. Milne, appears in the Encyclopredia Britannica ' article 'Mortality,' the references in which may be consulted by those who desire informa tion on the state of the question in foreign countries. We shall in this article confine ourselves principally to English tables.
The obvious and simple mode of forming a table of mortality would be to take a large number of infants born alive, all of the same sex and in the same station of life. If the numbers left alive at the end of every year were noted, until all had become extinct, a column of ages, accompanied by an opposite column noting the number of survivors, would be a table of mortality in the most usual form. Such a table might be called a table of decrements. Let 1 represent the number born, and 1, the number who survive to the age of .r.
The formation of such a table might require a century of obser vation. To avoid this, the law of mortality must he assumed stationary ; that is, it must be presumed that, out of those who reach, say age 70, the proportion who die in a year is now what it will be when an infant new-born reaches that age. This being assumed, let the members of a community be counted, and their ages registered ; at the end of a year it will appear what proportion of each age has died. If the pro cess be repeated in succeeding years, other sets of events are obtained, which may all he put together into one table, when the number has become large enough to secure the observed events representing the average, and to destroy the effects of accidental fluctuations. If then altogether k, persons have attained the age x, of whom I have survived to the age x+ 1, it follows that the proportion who die in a year is (k, : which may be represented by sa,. A table of the values of m, might be called a table of the yearly rates of mortality.