252 



the subject in a controversial point of view, and by expressing a hope 

 that the difficulty will in a very few years be satisfactorily cleared up. 



On the Nature of the Function expressive of the Law of Human Mor- 

 tality, and on a new Mode of determining the Value of Life Contin- 

 gencies. In a Letter to Francis Baily, Esq. F.R.S. &;c. By Ben- 

 jamin Gompertz, Esq. F.R.S. Read June 16, 1825. [Phil. Trans. 

 1825, ;>. 513.] 



This paper, which professes to be a continuation of former re- 

 searches on the same subject printed in the Transactions of the Royal 

 Society, is divided into two chapters. In the first the author consi- 

 ders the nature of the law of those numbers in tables of mortality, 

 which express the amount of persons living at the end of ages in 

 regular arithmetical progression. He remarks that for short intervals 

 the law approaches nearly to a decreasing geometrical progression, 

 and that this must be the case whatever be the strict expression for 

 the law of mortality, provided the intervals do not exceed certain 

 limits. But he further remarks, that this property will be found to 

 belong to very extensive portions of tables of mortality, and instances 

 Deparcieux's tables, where from the age of 25 to that of 45, the 

 numbers living at the end of each year decrease very nearly in geo- 

 metrical progression. 



Considering however the whole extent of such a. table, it will be 

 found that the ratio of this geometrical progression is not the same 

 in all parts of the table. But before he enters on this consideration, 

 the author draws some consequences from the hypothesis of a geo- 

 metrical progression being the strict law of nature after a certain age. 

 One of these is the equality of value of all life annuities commencing 

 after that age. Another is, that the want of instances in history of 

 persons living to very enormous ages (waving those of the patriarchs,) 

 is no proof that such may not be the law of nature, as he shows by 

 calculation, that out of 3,000,000 persons of 92, not more than one 

 should on this supposition reach 192. This leads him to some gene- 

 ral considerations on the causes of death, after which he resumes the 

 consideration of the general law of the tables. 



To find this a priori, he supposes that a person's resistance to 

 death decreases as his years increase, in such a manner, that at 

 the end of equal infinitely small intervals of time, he loses equal in- 

 finitely small portions of his vital powers. He further supposes, that 

 among any given number of persons of equal vital powers the pro- 

 bability of death is the same, but that among all, it is inversely pro- 

 portional to the vitality. These postulata being assumed, he enters 

 into an analytical investigation, the result of which is a represen- 

 tation of the law of life by such a function as is sometimes called a 

 double exponential, that is, a geometric progression in which the 

 ratio is itself variable in geometric progression. 



He then proceeds to examine the coincidence of this law, with se- 

 veral tables of the best authority, such as those of Dcparcicux, 



