765 



MORTALITY, LAW OF. 



MORTALITY, LAW OF. 



766 



practical, of the laws which are found to regulate mortality among 

 mankind in this country. 



Uncertain as is the life of any one individual, it is now very well 

 known that if two different numbers of individuals, at or near the 

 same age, be taken, the number that will be left at the end of a few 

 years will be nearly the same, if they exist during that time under 

 similar circumstances. No tables, however different the station and 

 circumstances of the person from whose lives they are made, differ 

 from one another by anything like the amount which might be sup- 

 posed likely by one who turns his thoughts rather to the existence of 

 one individual than of a large number. A little consideration will 

 make the probability of something like permanence in the distribution 

 of mortality very great d priori. That harvests fluctuate in goodness 

 is very well known ; but it is also obvious that if the fluctuations upon 

 a whole country had been as great as those upon an individual field, the 

 human race must long ere this have been starved off the face of the 

 earth. If, in the same manner, the mortality of races had varied as 

 much as that of families, it is impossible that the population of any 

 country could have gone on in a gradual and regulated state of increase; 

 or supposing that large fluctuations had compensated each other, the con- 

 sequence must have been such a disproportion of the numbers living at 

 different ages as it never has occurred to any one to imagine possible. 



The law of mortality, theoretically speaking, is a mathematical 

 relation between the numbers living at different ages ; so that, having 

 given a large number of persons alive at one age, it can be deduced by 

 the law what number shall survive any given number of years : prac- 

 tically speaking, it is, in the absence of such a mathematical law, the 

 exhibition in a table of the numbers surviving at the end of each year. 

 Thus, DE MOIVRE'S HYPOTHESIS (namely, the suppositiori that out of 

 86 persons born one dies every year till all are extinct) is an asserted 

 theoretical law of mortality ; while the Carlisle table, presently given, 

 is a practical one. 



If // represent the number of persons living at the age of x, out of a 

 certain number a at a certain previous age ( usually the time of birth), 

 then if a line varying with x be made the abscissa of a curve, and 

 another varying with y its ordinate, this curve may be called the curve 

 of mortality. Its form, as deduced from a given set of observations, 

 may lead, by comparison with known curves, to an equation which, 

 more or less accurately, connects y and x. 



Besides De Moivre's hypothesis, others have been given, the principal 

 of which we shall notice in order. 



A curve following a mathematical law may be drawn through any 

 points, however great their number or irregular their distribution ; but 

 the greater the number of points, the more complex will be the equation 

 of the curve. With an equation of a high degree (the tenth, perhaps, 

 or the twelfth), any given table of mortality might be very nearly 

 represented ; but such complexity would be useless, and it has there- 

 fore never been attempted. Similarly, by using arcs of different 

 curves, a near representation might be attained ; but such a method, 

 being practicable in many different ways, would not possess the 

 interest attaching to one simple and uniform law, and would only 

 attract attention by offering facilities for the actual calculation of life- 

 contingencies. 



In 1765 Lambert presented an equation of the following form, as 

 representing very closely the London table (i is the base of Napier's 

 logarithms; : 



y = 10000 /9 



a being = 1 : 13-682, and ft = 1 : 2-43114, and y being the number 

 surviving at the age of x, out of 10,000 born. This form, if it could 

 be made to represent other tables, by an alteration in the constants 

 would be one of great practical utility ; but we are not aware of any 

 attempt having been made to extend it. 



Mr. Benjamin Oompertz, in 1825, presented to the Uoyal Society a 

 memoir ' On the Nature of the Function expressive of the Law o 

 Human Mortality.' As this ingenious paper contains a deduction from 

 a principle of high probability, and terminates in a conclusion whicl 

 accords in a great degree with observed facts, it must always be con 

 nidered 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 

 ary aa an act of justice to Mr. Gompertz, whose ideas have been 

 adopted by a writer on the subject, in a work published in 1832, with 

 out anything approaching to a sufficient acknowledgment. (See on 

 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 

 nishe* from that time forwards ; the evidence of such diminution beinj 

 the increased proportion of deaths in a given time. The proportion i 

 found, in moat 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 pro 

 portions * in equal times ; so that the intensity of mortality, suppose 



* The word In the eighth page of the memoir cited is portions, which is a mis 

 print or an oversight, as the formula Immediately following shows. II x be th 



time, n bz lote equal portions In equal times, and n , I ~ z equal proportions. 



nversely 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 q* d x x b is the decrement of that number in the 

 ime d x, where 6 is another constant ; and this gives dy= abq'ydx, 

 fhich integrated is of the form 



'here q, I, 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 

 f the values of the formula for different ages. The more convenient 

 orm of the above is 



log y = log I no. wh. log. is (log. log y + x log q) 



here log log ,</ is taken without reference to the sign of log rj, and 

 he upper or lower sign is used according as y is greater or less than 

 unity. 



Among other comparisons, Mr. Gompertz has made one with the 



Jarlisle table from the age of 10 to that of 60, and another (deducing 



afferent values of l,rj, and q) from 60 to 100. The two formula! 



btained are, using log -1 for the phrase " no. whose logarithm is," and 



x meaning the age of the parties, 



log y = 3-88631 log'- 1 [2-75526 + "0126 x\ 



log y = 3-79657 - log -'{ 374767 + '02706 x} 



In the first set of ages the discordance between the formula and the 

 able is only in one instance as great as half a year ; that is, there is only 

 me instance in which the number deduced from the formula as alive 

 it 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. Gompertz'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 approximate 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. (See the ' 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 ' Encyclopedia 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 ? represent the number 

 born, and I, the number who survive to the age of x. 



The formation of such a table might require a century of obser- 

 vation. To avoid this, the law of mortality must be 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 be 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 .r, of whom k, + i have 

 survived to the age x+ 1, it follows that the proportion who die in a 

 year is (k, k,+ t ) : k,, which may be represented by m f . A table 

 of the values of m, might be called a table of the yearly rates of 

 mortality. 



A table of yearly rates may be converted into a table of decre- 

 ments, as follows. Assume a number 1 to be born, then from the 

 table of yearly rates, 



Z, = (1 - m ) /, I, = (1 - ,) I,, Ac. 



If the population, say of a town, remained unaffected, or sensibly 

 unaffected by immigration or emigration, that is, if all who were born 

 in the place, and no others, were registered in the burials of the place, 

 the burial registers would form a mortality-table, provided the rate of 

 increase of the population were steady and known. For, firstly, if the 

 population had remained stationary for a long time preceding the com- 

 mencement of registration, the yearly deaths and births being equal, 

 and if the mortality had also remained stationary, the burials of any 

 one year, the parties being distributed according to age, would show 



