MORTALITYLAWOF

THE Law of Human Mortality is that which determines the proportion of the number of persons who die in any assigned period of life, or interval of age, out of a given number of persons who enter upon the same interval; and, consequently, the proportion of them who survive that interval.

Tables showing how many out of a great number of children, as 10,000, or 100,000, born alive, die in each year of their age; and, consequently, how many complete each year; exhibit this law through the whole extent of life, and are called Tables of Mortality.

This section is divided into three parts. In the first, we deliver the history of this branch of knowledge, with as much brevity as appears to be consistent with the chief object, which is that of conveying correct and useful information.

In the second part, we demonstrate the whole theory by common arithmetic.

In the third part, a new table of mortality is given, constructed on the principles previously explained; some observations are made on the comparative merits of the different tables that have been published; which were purposely omitted in the historical part, when the tables they relate to were mentioned, to avoid discouraging such readers as might not be previously acquainted with the theory; and the faults are explained, which render most of those tables really of no use, since others, more correct, have been constructed.

The first table of mortality was constructed by Dr. Halley, from the Mortuary Registers of Breslau, for five years

ending with 1691; and was inserted in his paper on the subject in the Philosophical Transactions for the year 1693, with many judicious observations on the useful purposes to which such tables may be applied.

No further information of this kind was communicated to the public, until William Kersseboom of the Hague published there three tracts on the subject (in 4to.) The first, dated March 1, 1738, was entitled, Eerste Verhandeling tot een Proeve om te weten de probable menigte des volks in de provincie van Hollandt en Westvrieslandt. The second, dated May 15, 1742, Tweede Verhandeling bevestigende de Proeve om te weten de probable menigte des volks in de provincie van Hollandt en Westvrieslandt; and the third, dated August 31, 1742, Derde Verhandeling over de probable menigte des volks in de provincie van Hollandt en Westvrieslandt.

A good account of the first of these tracts has been given by Mr. Eames, in the Philosophical Transactions for 1738; and rather a meagre one of the other two, by Mr. Van Rixtel, in the same Transactions for 1743. It is therefore unnecessary to repeat here, any thing contained in those accounts; but as they give no satisfactory information concerning the construction of Mr. Kersseboom's table of mortality (which he called a Table of Vitality), it will be proper to supply so material a defect in this place.

In his first tract, the author informs us that he constructed his table from registers of many thousand life-annuitants, in Holland and West Friesland, which had been kept there from 125 to 130 years previous to the date of his publication; and showed how many of the nominees, or lives the annuities depended upon, were, at the time of their nomination, under one year old, between one and two, between two and three, and so on for all ages.

An exact account was also kept of the age at which each life of every class failed; whence it clearly appeared, what degree of mortality prevailed at every age above one year. But because very few children were nominated at or near their birth, he could not, from these registers, determine the mortality under one year of age. He therefore had recourse to mortuary registers and other observations; from exact accounts of which he found, with sufficient certainty, as he says, that out of 28,000 born alive, 5500 died under one year. He also informs us, that, for this purpose, he made use of the observations of divers learned men in England and elsewhere, especially Major John Graunt's, upon the number of the people and the rate of mortality; and upon taking an average of the whole, he found it to differ but little from that just stated.

And this appears to be the only ground for the assertion made by most writers on this subject (probably copying from each other without having seen the original work), that Kersseboom's Table of Mortality was constructed from observations made upon annuitants in England as well as in Holland; also, that it was formed partly from observations made upon the inhabitants of some Dutch villages.

He first published his Table of Mortality in his second Tract, and in his third, he gave abstracts of the registers from which it was constructed. These were contained in twenty-nine tables, twenty-two of which were for the two sexes separately; in the rest the sexes were not distinguished; and the ages at which the lives failed were generally given to the exactness of half a year.

The numbers of lives, whose current year of age at the time of their nomination was given precisely in these tables, were,

And none of these nominees were above twelve years of age at the time of their nomination.

These, however, are only specimens of M. Kersseboom's labours. He says there were so many lives in the registers, that he had not the courage to undertake extracting the necessary particulars for more than 50,000 of them; and in that, he was greatly assisted by his friend Thomas von Schaak. Of all the lives, not more than 1 of 120 was past 55 years of age at the time of nomination.

Nicholas Struyck, in his Aanhangsel op de Gissingen over den staat van het Menschelyk Geslagt, en de Uitrekening der Luffrenten, published at Amsterdam in 1740, at the end of the quarto volume, commencing with his Inleiding tot de Algemeene Geographie, gave, from registers kept at Amsterdam for about thirty-five years, two tables of observations made upon the duration of the lives of 794 males, and 876 female annuitants separately; and two tables of mortality he had constructed from them for the two sexes; both beginning with five years of age. These two, taken together, differ but little from that of Dr. Halley; they represent the mortality to be considerably greater than Kersseboom's: having been constructed from so few observations, they are not entitled to much confidence, and appear to have been very little known or attended to.

This work of Struyck gave occasion to the publication, in the same year, of a small tract in quarto, by Kersseboom, entitled, Enige Aanmerkingen op de Gissingen over den staat van het Menschelyk Geslagt, &c. wherein he accused Struyck of plagiarism, with but too much appearance of justice.

Neither Kersseboom nor Struyck gave any information as to the manner in which they formed their tables of mor-

tality from the observations on which they were grounded. M. Kersseboom informs us, that he submitted his table to Professor S'Gravesande, some years previous to its publication, and obtained his approbation of it for calculating the values of annuities on lives.

In the year 1742, Mr. Thomas Simpson, in his Doctrine of Annuities, (see the article ANNUITIES) gave a table of mortality for London, being the same that had previously been constructed by Mr. Smart, at twenty-five and all the greater ages, but corrected at all ages under twenty-five years, on account of the greater number of strangers who settle in London under that age, which occasioned, till the commencement of the present century, a constant excess of the burials above the births. This correction Mr. Simpson made by comparing together the numbers of christenings and burials; and observing, by means of Dr. Halley's table, the proportion between the mortality in London and Breslaw above twenty-five years of age.

In 1746, M. Deparcieux published (at Paris in 4to.) his Essai sur les Probabilités de la durée de la Vie Humaine, in which he gave six new and valuable tables of mortality; one of them constructed from the lists of the nominees in the French Tontines, principally those of the years 1689 and 1696, and the rest from the mortuary registers of different religious houses; four of these showing the mortality that prevailed amongst the monks of different orders, and the fifth, that which obtained amongst the nuns in different convents of Paris. Those for the monks and nuns, with the exception of the tables of Struyck, mentioned above, were the first ever constructed for the two sexes separately.

The Essay of M. Deparcieux is written popularly, and with great perspicuity; he has given the most satisfactory accounts both of the data his tables were constructed from, and the manner of their construction.

In his thirteenth table, he included with the five tables of mortality of his own construction; that of Mr. Smart for London, as corrected by Mr. Simpson, Dr. Halley's, and M. Kersseboom's, together with the expectation of life at, or its average duration after each age, both according to his own and M. Kersseboom's table for annuitants, and for every fifth year of age according to each of the other tables; the fractional parts of a year being always expressed in months, and not in decimals.

Dr. Halley first, and Struyck after him, had given the probable duration of life after several ages, according to their respective tables, that is, the term at the expiration of which, the persons now living at any proposed age, will be reduced by death to one-half their present number.

But Deparcieux appears to have first given the average duration of life after any age, and showed how to calculate it correctly from tables of mortality. On account of the scarcity and value of M. Deparcieux's Tables of Mortality, Mr. Milne has reprinted them, with the expectations of life just mentioned, in his Treatise on Annuities, with a short account of their construction; it is therefore unnecessary to pursue the subject further here.

In 1760 M. Deparcieux published (at Paris in 4to.) his Addition à l'Essai sur les Probabilités de la durée de la Vie Humaine, with five tables; three of them relating to life annuities deferred on a peculiar plan, we consider to be of no interest or value at this time: the two others are tables of mortality constructed from statements of the numbers of deaths that took place at different ages, without knowing the numbers of the living at the same periods of life. He obtained the data for the first of them from a clergyman on the frontiers of Normandy and Perche, whose accuracy in all he undertook, he could rely upon; and who gave him the names of the parishes from the registers of which he had extracted the information; but strictly enjoined him not to disclose his name in the event of his making use of

Mortality, the documents. In these the sexes were not distinguished.

The other table of mortality M. Deparcieux constructed from statements sent to him by M. Wargentin of the numbers of deaths of males and females separately, which took place in the different intervals of age in Sweden and Finland, during the three years 1754, 1755 and 1756. Those two tables have the same faults as others constructed from similarly defective data; and we consider them to be of no value.

M. Deparcieux states, (p. 28) that in 1744, he suggested to M. Aubert, the commissary who at that time prepared the Bills of Mortality for Paris, the expediency of distinguishing the sexes in the columns of births and deaths, which had not been done previously, but was in consequence of this commenced with the year 1745, and has been continued ever since, as we have already observed in our account of the Parisian Recherches Statistiques.

M. de Buffon, at the end of the second volume of his Histoire Naturelle, published in 1749, inserted a table of mortality that had been constructed by M. Dupré de Saint Maur, from the registers of twelve country parishes in France, and three parishes of Paris; which M. de Buffon informs his readers that he inserted in his work the more willingly, since these were the only kind of documents, or combinations of them, from which the probabilities of life among mankind in general, could be determined with any certainty. Yet this was a very faulty table, and the numbers of annual deaths were so injudiciously distributed, according to the ages, that it often represented the mortality in one year of age to be three or four times as great, and in some cases, six times as great, as in the next year. Some remarks of M. Kersseboom on this table may be seen in the Philosophical Transactions for 1753. M. de Saint Cyran corrected some of its most obvious errors, and inserted both the original and his corrected copy in his Calcul des Rentes Viagères. (Paris, 1779, in 4to.)

Mr. Simpson, in the Supplement to his Doctrine of Annuities, published in 1752, gave some further explanations of the corrections he had made in Mr. Smart's table of mortality for London; and made some very judicious observations on the difficulties that attend the construction of tables of mortality from the mortuary registers only, of large towns.

In the Nouveaux Mém. de l'Acad. Roy. de Berlin for the year 1760, there is a paper by the celebrated Euler, entitled Recherches générales sur la Mortalité, et la Multiplication du Genre Humain, wherein the subject is treated algebraically. He assumes that the population is not affected by migration, and that the annual births and deaths are always as the contemporaneous population; consequently, that the number of the people increases or decreases in geometrical progression. Then he gives several theorems exhibiting the relations that would obtain between the annual births and deaths and the population, and determines the law of mortality upon these hypotheses, but does not shew how it may be deduced from actual observations independent of hypotheses; neither does he undertake the construction of any table of mortality, but, by way of example, gives that of M. Kersseboom, with the changes of the numbers which become necessary, in consequence of his altering the radix from 1400 annual births to 1000.

Süssmilch took great pains in collecting the numbers of annual deaths in the different intervals of age, which he published in his Göttliche Ordnung; and four tables of mortality formed from these data are to be found in the same work; that in the second volume (§ 461), which has many imperfections, was formed by himself; the three others, being the 21st, 22d, and 23d, at the end of the third volume, were constructed by his commentator Baumann, according to the more correct method of Lambert.

The first edition of Dr. Price's Observations on Rever-

sionary Payments, appeared in 1771, containing his observations on the proper method of constructing tables of mortality from bills which shew the numbers dying annually at all ages, and three new tables of mortality constructed from the London, Norwich, and Northampton bills.

The second edition of the same work was published in 1772, and contained, in the Supplement, much interesting and valuable information which did not appear in the first, together with five new tables, intended to exhibit the law of mortality that obtained, 1st, in the district of Vaud, in Switzerland; 2d, in a country parish in Brandenburg; 3d, in the parish of Holy Cross, near Shrewsbury; 4th, at Vienna; 5th, at Berlin. The first formed from bills of mortality given in the Mémoire of M. Muret; and the 2d, 4th, and 5th, from those given by Süssmilch in his Göttliche Ordnung; the 3d was from the parish register only of Holy Cross. But we consider none of those tables as now of any value, on account of the defects in the data from which they were constructed.

At the end of the first volume of the work of J. H. Lambert, entitled, Beiträge zum Gebrauche der Mathematik und deren Anwendung, published at Berlin in 1765, 8vo, he gave a chapter on the certainty of inferences deduced from observations and experiments; and the example with which he concluded the illustration of his theory, was the deduction of the law of mortality in London from the bills of mortality there; by means of a curve, of which the absciss being proportional to the age, the corresponding ordinate was proportional to the number of survivors of the same age.

In the third volume of the same work, published in 1772, the ingenious author treated the subject at much greater length: $x$ being the age, and $y$ the corresponding ordinate to the curve of mortality for London, proportional to the number of survivors of that age out of a given number (10,000) born alive, he gave this equation to the curve,

($e$ being the number of which unit is the hyperbolic logarithm), which determined the numbers in the table of mortality very near the truth, until 96 years of age, beyond which it was not intended to be used.

M. Lambert also constructed a table by which he intended to exhibit the law of mortality that prevails among mankind in general, from the 23d and 24th tables in the second volume of Süssmilch's Göttliche Ordnung, which gave the numbers of deaths, in the different intervals of age, in seventeen country parishes in the mark of Brandenburg, and from the London bills for thirty years; supposing, with Süssmilch, (Gött. Ord. t. i. § 34), that the country people are double the number of those residing in towns.

By an extract of a letter from M. Lambert to Gaeta and Fontana, given in their Italian translation of Demoisne's Treatise on Life Annuities, (Discorso Preliminare, part iii.) it appears, that all his attempts to find a posteriori an equation which should determine the relation between the age and the number of survivors in this last table, proved fruitless; the formulae he arrived at having been either too long and intricate, or too incorrect. This is the less to be regretted, since there is no doubt that M. Lambert's table did not represent the true law of mortality, as he made no allowance for the effect of the increase of the people by procreation; and it is singular he did not see that that law might be correctly determined from the numbers of the living, and the annual deaths at all ages in Sweden and Finland, given in M. Wargentin's paper in the Stockholm Transactions for 1766, which paper he himself quotes.

Lambert appears to have first demonstrated clearly the principal properties of tables of mortality, in doing which he made use of the differential and integral calculus; but

Mortality, as he could not determine the equation to the curve of mortality, that resource did not avail him much.

Flourencourt treated this subject algebraically in the third chapter of his Political Arithmetic,¹ where he gave a perspicuous view of it, as it had been previously treated by Euler and Lambert; but added nothing himself that was original, except three new tables of mortality; one for males, another for females, and a third for both sexes without distinction; deriving his data in each case from the Göttliche Ordnung of Sissmilch. He also gave a new copy of the table of mortality M. Deparcieux had constructed from the registers of the nominees in the French tontines; assuming 10,000 for the radix, and inserting the numbers under three years of age, nearly according to M. Kersseboom's table; this, however, does not differ materially from the original table of Deparcieux.

The fourth edition of Dr. Price's Observations on Reversionary Payments was published in the year 1783, and contained new tables of mortality for Warrington and Chester, also for all Sweden and Finland, and for Stockholm separately, in which the sexes were distinguished. Those for the whole kingdom were constructed from enumerations of the living, and registers of the annual deaths, in each interval of age, during twenty-one years; those for Stockholm during nine years. The tables for Sweden and Stockholm were the first ever constructed from the data that are requisite to determine the law of mortality among the bulk of the people, and were sufficiently accurate representations of that law, for the times and places in which the observations were made.

In a paper of M. Henrich Nicander, inserted in the Transactions of the Royal Academy of Sciences at Stockholm, for the first quarter of the year 1801, he gave two tables of mortality for all Sweden and Finland, in which the sexes were distinguished, but they were not properly constructed; and the mean duration of life which he gave in them at each age, was very erroneous, especially in early life. In that paper he asserted, without offering any demonstration or proof, that, in what we have called the curve of mortality above, if an ordinate be drawn through the centre of gravity of the portion of the area cut off by the ordinate at any assigned age, on the side of the more advanced ages, the part of the base, or of the axis of the abscissæ, intercepted between these two ordinates, will measure the mean duration of life after such assigned age. And the mean duration of life after each age, which he has given, was determined in this manner.

Mr. Milne's Treatise on Annuities and Assurances was published in the year 1815; and, in the third chapter of that work, the construction and properties of tables of mortality are fully treated of.

In the second volume of the same work, three new tables of mortality are given; one constructed from very accurate observations made at Carlisle, by Dr. Heysham, who preserved the bills of mortality of the two parishes, which include that city and its environs, and supplied their deficiencies with great care, together with correct accounts of two enumerations of the inhabitants, in which their ages were taken; and a table showing the diseases by which the deaths at all ages were occasioned, is also given.

The fourth and fifth tables in Mr. Milne's work, exhibit the law of mortality which prevailed in all Sweden and Finland, both with and without distinction of the sexes, deduced from the registers kept and the enumerations made there, during the twenty years ended with 1795; which term was subsequent to that wherein the observations were made, from which Dr. Price's tables were constructed.

The seventh table in the same work exhibits the law of mortality at Montpellier for males and females separately, and was constructed from the bills of mortality of that place for twenty-one years, ending with 1792.

The second table at the end of this article, was published in the first edition of it in 1822; the tables of mortality for the lives insured in the Equitable Office, which were constructed by Mr. Babbage and Mr. Davies, were published in 1826; and we have given some account of them as well as of those formed by Mr. Finlaison, from observations on Government Annuities in this country, and published in 1829, in the article Annuities in this work.

MM. Quetelet and Edouard Smits, in their Recherches sur la Reproduction et la Mortalité de l'Homme, in 1832, gave a table of mortality for the towns and the rural districts in Belgium separately, distinguishing males from females, and also for the whole population, without distinguishing the sexes, or the inhabitants of towns from those of the country.

Mr. Morgan, in the above mentioned publication of the Equitable Assurance Society, in 1834, gave a table of mortality for the lives insured in it, (marked C, p. 28), derived from table A of that work; another (D) derived from table B, is not worth a place there.

The second of the tables at the end of Dr. Casper's work on the Probabilities of Human Life, published in 1835, was intended to exhibit the law of mortality in Berlin, with distinction of the sexes; it was constructed from 69,362 deaths at different ages; 36,895 of males, and 32,467 of females, which took place there during the twelve years 1818-1829. And M. Mallet at the end of his valuable Mémoire, published in 1836, has given one for Geneva, in which the sexes are distinguished: it contains both the mean and the probable duration of life after every age, and was formed from the bills of mortality there for the eighteen years 1814-1833. For males, females, and the two sexes without distinction, M. Mallet took so high a radix as 100,000 births; the number of deaths were, of males 5219, females 5688, of both sexes 10,907; and in the column of deaths, on the same line for any age, as the survivors of that age, the author has put the number of deaths in the registers in the next following year, instead of the decrement of life, or excess of the number attaining that above the number attaining the next greater age, which will probably puzzle many readers.

PART II.—On the Construction and Properties of Tables of Mortality.

1. Suppose 10,000 children to have been all born alive at the same instant of time, more than 100 years since; and that the numbers of them who completed and who died in each year of age, were correctly entered in the following table:

¹ Abhandlungen aus der Juristicchen und Politischen Rechenkunst, von Carl Chassot de Florencourt. Altenburg, 1781, 4to.

Mortality, which, then, would evidently be a table of mortality; and this mode of constructing one, were it practicable, would be the simplest possible.

2. But of 10,000 children taken indiscriminately at birth, it is manifest that the number who complete or survive any year of age, will be just the same, whether they be all born at the same time or not; and, therefore, this table might as well have been constructed by noting the times of the births of 10,000 children taken indiscriminately, and registering the time or the age at which each died; for then, after the whole were extinct, it would only be necessary to collect the sum of those who died in each year of their age, and insert it in the third column of the above table (1) against the proper age. The numbers in the second column would then be obtained by beginning with the 10,000 births, and merely subtracting the number in the third from the number in the second column, and placing the remainder in the next line below, in that second column, throughout the table.

3. It is evident that the number against any age in the second column of such a table, is equal to the sum of those in the third column against that, and all the greater ages; that is to say, that the number who complete any year of age is equal to the sum of those who die at all the greater ages.

4. Now let us suppose the population of a place to have remained invariable for one or two hundred years past, during which period 10,000 children have been born alive, at 10,000 equal intervals of time in each year. Also that, there having been no migration, and the law of mortality having been always the same, both the number of the living and that of the annual deaths in each year of age, have remained constant, the whole amount of the annual deaths at all ages, as well as the number of annual births, having been 10,000.

5. Then, if the law of mortality exhibited in the above table (1) be that which obtains in the place just mentioned, that table will represent the stream of life which flows through it, and fills the vacancies left by those who advance in age, or are carried off by death, their successors incessantly following and being followed in the same course.

6. Thus: 10,000 children being born annually at so many equal intervals of time, 7096 will annually complete their fifth year, also at equal intervals; and of these, 112 will die annually in the sixth year of their age.

7. And it is manifest that the number who annually complete any year of their age in such a place, is equal to the sum of the annual deaths at all the greater ages.

8. Let us next suppose, that the constant number of deaths which happen annually in any one year of age, take place at equal intervals of age in that year. For instance, that the four deaths which happen annually, in the ninety-fifth year of age, always takes place at the ages of

and 95 years; or rather, that the last individual dies at the moment before completing the 95th year.

9. Then the number constantly living in any year of age may be determined as follows:

Let us take, for example, the ninety-fourth year, which 14 persons annually enter upon, and 5 die in. Now, if no deaths happened in that year, it is obvious that the 14 persons who annually enter upon it at so many equal intervals (4 and 6), would be all constantly living at 14 equal intervals of age in that year; and if that year of age were divided into five equal intervals, there would be constantly living in each interval $\backslash frac\{14\}\{5\}$ persons; or, in a place

similarly circumstanced, but five times more populous, 14 Monthly Law of persons.

But when five deaths take place at so many equal intervals in the ninety-fourth year of age, (the fifth part of a year being 73 days,) the case is altered. Thus,

Or rather, the oldest life that falls in the 94th year, must be considered to expire the moment before completing that year, as only 9 survive 94.

But the numerators of these fractions being in arithmetical progression, their sum is equal to half the sum of the first and last terms multiplied by the number of terms; or $\backslash frac\{14+10\}\{2\}\; \backslash times\; 5$; which sum being divided by the common denominator, 5, we have the number of the living in the 94th year of age = $\backslash frac\{14+10\}\{2\}$; an arithmetical mean proportional between the numbers who enter upon the first and last of the intervals which that year of age was divided into.

10. Now, the number, 9, who survive their 94th year, is less only by unit than the number 10, who enter upon the last of the intervals that year was divided into; so that if, instead of $\backslash frac\{14+10\}\{2\}$, we take $\backslash frac\{14+9\}\{2\}$, or an arithmetical mean

proportional between the numbers who annually enter upon, and annually survive their 94th year, for the number constantly living in that year, it will only be less by half a life than what has just been demonstrated to be the true number, according to the hypotheses; and the difference would still have been but half a life, although the radix of the table had been 10,000,000 instead of 10,000; the number of the living would, in that case, according to these two methods, have been 1200 and 1199$\backslash frac\{1\}\{2\}$. And the number of the living in any one year of age, even according to the above table, is generally several thousands, so that this difference, which remains always the same, is quite immaterial.

Besides, it is obvious that the above hypotheses can never coincide exactly with the facts. And the above reasoning is evidently applicable to any other year of age.

11. We are therefore authorised to conclude, that in a place, circumstanced as above stated, the number of the living in any year of their age is an arithmetical mean proportional between the numbers who annually enter upon, and who annually complete that year.

12. Thus it appears that

But it is to be observed, that the same numbers occur in

Mortality, the first of these two series as in the second, except the first term of the first, and the last of the second, which are 14 and 0 respectively. Therefore the sum of the second of these two series falls short of the sum of the first by 14, the number who annually complete their 93d year; so that the series of half sums falls short of the sum of the first series by 7, the half of 14. And this reasoning will apply equally to any other age than that of 93 years.

13. Whence it follows, that in a place circumstanced as we have supposed, the number of persons constantly living at any assigned age and upwards, is less than the sum of those who annually complete that and all the greater ages, by half the number who annually complete that year of their age.

14. From the supposition that the number of persons who die annually in any one and the same year of age, expire at so many equal intervals of age in that year (8), it follows, that for each of these lives which falls before the middle of that year of age, there will be another which will fall just so much after it, and, consequently, that the average quantity of existence during any year of age, for the lives that fail in it, is just half a year.

15. But in taking, for any one year of age, the sum of the numbers in the second column of the table (1) at all the greater ages, each life is counted once for every complete year it survives, after the age first mentioned; and if, to the sum of these, we add half the number in the same second column against that first mentioned age, this half number being the sum of the fractional parts of a year, by which the whole of these lives survive the last year of age they complete (14); the sum total thus obtained will evidently be the whole duration of life after the age first mentioned, enjoyed by all the lives that survive that age in any one year.

16. Therefore, if this last sum total be divided by the number who annually survive that first mentioned age, the quotient will be the mean duration of life after that age which is also called the expectation of life at the same age, being the portion of future existence which an individual at that age may reasonably expect to enjoy.

17. But, by No. 13, it appears, that the last mentioned sum total is also the number constantly living in the place, at and above the age first mentioned (15).

18. Whence, and from No. 16, it follows, that if the number of the living in the place at any age and upwards, be divided by the number who annually complete that age, the quotient will be the mean duration of life after the same age.

19. And, consequently, if the number constantly living at all ages, be divided by the number of annual births, the quotient will be the mean duration of life from birth, or the expectation of life of a child just born.

20. Hence also it appears, that the number of years in the expectation of life at any age, is the same as the number of living persons at that age and upwards, out of which one dies annually.

21. Thus, for example, the expectation of life at 40 years of age being 25.495 years, the proportion of the living in the place aged 40 years and upwards who die annually, is one of 25.495, or, which is the same, 1000 out of 25,495.

22. The numbers represented by a table of mortality to die in any intervals of age, are called the decrements of life in those intervals.

23. And the interval between any age and the utmost extent of life, according to any table of mortality, is called the complement of life at that age, according to the same table.

24. If the decrements of life be supposed to be equal and uniform through its whole extent, and the interval between birth and the utmost extremity of life be divided into as many equal parts as there are annual births, then, one of the individuals born will die at the expiration of each of

these equal intervals of age; and the numbers who survive the several intervals, from birth to the extremity of life, will form an arithmetical progression.

25. Whence it will be found (11), that the number of the living at any assigned age and upwards, will be equal to the number who annually complete that age, multiplied by half the number of years in the complement of life at the same age.

26. And if this last product be divided by the number who annually complete that age, the quotient, that is, half the complement of life, will be the expectation of life at that age (18).

27. The mean numbers of annual deaths at all ages, or, which in this case is the same, the number of deaths in each year of age, that take place during any one year, in a place circumstanced as we have supposed, being given, a table may be constructed as follows, which will answer all the most interesting questions that can be put respecting the population and mortality of the place.

28. Let there be five columns, in the first of which insert the ages 0, 1, 2, 3, 4, ..., 96, 97, 98, 99, and against every age, insert in the fifth column, the given number that died in the year between that and the next greater age; then begin at the greatest age, and proceed towards the least, as follows:—

1st, To the number against any age in the fourth column, add that against the next less age in the fifth, and insert the sum against that next less age in the fourth (7).

2d, To the sum of the numbers in the third and fourth columns against any age, add half the number in the fifth column against the next less age, and insert this last sum against that next less age in the third column (11).

3d, Divide the number against any age in the third column, by the number against the same age in the fourth, the quotient will be the expectation of life at that age, to be inserted in the second column (16).

A complete table of this kind for the two sexes separately, formed from observations made in all Sweden and Finland, during twenty years ending with 1795, will be found in Mr. Milne's Treatise on Annuities and Assurances, being the fourth in that work.

29. Hitherto we have supposed the state of the population to continue invariable for 100 years at the least, on account of the facility with which tables of mortality might be

formed from accurate mortuary registers in such circumstances.

But whether the population be stationary, or increasing, or decreasing, and from whatever causes these changes proceed, provided that they be produced gradually, and not by sudden starts during the time of the observations, the law of mortality may be determined from actual enumerations of the people, and the bills of mortality. Thus,

30. Let the number of persons in each year of their age, that are resident in a place at any one time, be taken, and let an accurate register be kept of the number that die annually in each year of their age, during a term of eight or ten years at the least, whereof the first half may precede, and the second follow the time of the enumeration.

Then, if the number of the inhabitants of every age either increase or decrease uniformly during that term, the mean number of annual deaths in each year of age thus registered, will be the same as if the population of the place had continued throughout that term what it was when the enumeration was made.

31. But if, to the number of the living in any year of age, we add half the number who annually die in the same year, the sum will be the number who annually enter upon that year of their age (11.)

And thus, from the enumerations and registers above mentioned, may be derived the ratio of the number who annually enter upon any year of their age, to the number who annually die in it.

32. But all the observations which have been made with sufficient minuteness, on the mortality during the first year from birth, concur in showing, that many more deaths take place in the first few weeks from birth, than in equal periods of time during the remainder of the first year; and that the nearer to birth, the greater is the mortality among infants. So that the number of the living in successive equal intervals in the first year of age, cannot be correctly assumed to be in arithmetical progression.

33. On this account it is desirable that the annual numbers, both of the children born alive, and the deaths under one year of age, should be correctly registered, as in Sweden.

34. Then, as the number annually born alive, is to the number of annual deaths under one year of age, according to the registers, so is the radix of the table of mortality, to the number dying under one year of age according to that table, which, being subtracted from the radix, the remainder is the number who complete their first and enter on their second year. Whence the numbers, both of survivors and annual deaths, at all the greater ages, may be determined in the order of their succession by No. 31.

35. If, instead of the number of the living in each year of age being taken only once, according to No. 30, that operation be performed several times during the term for which the mean number of annual deaths in each year of age is given; then, the mean number of the living in each year of age throughout that term, must be deduced from the given numbers; and, being substituted for the number at the middle of the term according to No. 30, the law of mortality may be determined with more certainty, than when the people are only numbered once.

36. Both in enumerations of the people, and in bills of mortality, the numbers are, however, almost always given only for intervals of age of several years each. For the manner of interpolating the numbers in each particular year of age, the reader is referred to Mr. Milne's Treatise on Annuities and Assurances, arts. 180 and 181.

37. Hitherto we have only considered the determination of the law of mortality amongst the whole of the inhabitants of a place, of all ages, ranks, and conditions; and until within the last eight years, no statements of facts relating to particular classes of the people had been published, except those of M. Deparcieux, in his valuable Essai, sufficiently numerous and correct to be available for the purpose.

But the inquiries made by order, and with the aid of government in this country, into the mortality amongst various classes of nominees, on whose lives annuities depended, published in Mr. Finlaison's report to the Lords of the treasury in 1829, and already noticed in the article ANNUITIES in this work; with the publication in 1834, of the still more interesting, important, and distinctly detailed observations on the mortality that has taken place among the lives insured in the Equitable Assurance Society, form most valuable accessions to our knowledge of the subject; we therefore proceed to show how the law of mortality among those classes of persons may be deduced from these documents. We shall have compassed that object, when we have determined out of a considerable number of persons who entered upon each year of age, how many died in that year, and, consequently, how many survived it; and the Equitable Assurance documents are so disposed, as to afford great facility in effecting this. The following table is extracted from them, with the addition of the column marked B.

38. According to the common mode of expressing the ages, which is adopted both in the government documents

Mortality, and in those of the Assurance Society, the year of age stated is always that which was last completed; but wherever that is the case, it should be distinctly stated, and kept steadily in view. We confine ourselves in this table to the 31st year of age, and mark it 30+ to avoid ambiguity; for the same reason (+) is put over the ages in column A, to show that each number in it denotes the age last completed.

39. Column B has been added for the sake of illustration; it shows at what earlier ages the different lives entered the society, which afterwards completed the 30th year of their age in it, and entered their 31st: thus, out of 40 insured in the 8th year of age during the period of the observations, only 5 entered their 31st while insured in the society, the rest having passed out of it either by death or otherwise, at earlier ages, as the preceding columns of the original table fully show; and of the 373 persons who were insured in their 22d year, 124 remained insured till they entered their 31.

40. This table shows that of 4692 persons who entered on their 31st year of age in the society, 265 went out of it alive in the same year of their age during the observations, 51 were still living and insured at the termination of the observations, and 762 came into the society in that year of their age; the individuals of those three classes amount together to 1078 persons. But of the 265 who went out of the society alive during that year of age, it may reasonably be assumed, that whatever number of them at the time of their exit exceeded 30½ years of age by any interval of time, as many would fall short of that middle age by the same interval, when they went out; and, therefore, that the whole of them on an average may properly be considered to have been exposed to the action of the Law of Mortality during one half of that year of age; so that the mortality among them must have been the same as it would have been among half the number of exactly similar lives, in passing through the whole of that year.

In the same manner it appears, that of the 762 persons who were insured at various periods of the 31st year of age, and of the 51 persons who remained alive and insured in the society at various periods of that year of age when the observations terminated; may, each of them on an average, be properly assumed to have been exposed to the action of the law of mortality in the society during one half of the 31st year of age; and we are therefore entitled to conclude that the mortality among them was the same as it would have been among half the number of exactly similar lives in passing through the whole of the 31st year of age.

But 4692, the total of the numbers in column C includes all those 1078 lives, the same as if they had all entered upon their 31st year of age in the society, and had all passed through it except those which were carried off by death. It is, therefore, manifest, that 539, the half of their number, must be subtracted from 4692, and the remainder, 4153 must be taken as the number of persons entering on the 31st year of their age, and continuing exposed to the law of mortality in the society during the whole of that year, among whom thirty-two deaths take place in the same year of age.

This, when the principle is clearly understood, is certainly a very simple operation and easily performed; and its application to every other age in the original table of data is exactly the same as to the 31st year.

41. In constructing the table of mortality, supposing that in proceeding from the earliest age we have arrived at the completion of the 30th year, or the entrance on the 31st, and have determined the number of survivors at that limit to be 4305; since

we find 33 to be the number of deaths which will take place in the 31st year of age out of 4305 persons who enter on it; and, consequently, that 4272 enter on their 32d year,

according to the table we are constructing; the method of Mortality, completing it is obviously the same throughout, and can be Law of. attended with no difficulty after the valuable documents requisite for the purpose have been obtained.

42. What has been shown here respecting the determination of the law of mortality amongst insured lives, applies also, and with rather more facility to the nominees on whose lives annuities depend. The life annuities sold by government in this country not being redeemable, are always continued during the whole of the lives they depend upon; therefore, with regard to them, column E in the above table is left blank; and in the case of the old English Tontine, which commenced in 1693 and ended in 1783 with the life of the last survivor, column D will also be quite blank.

PART III.—On the Law of Mortality as deduced by the preceding methods from actual observations; and on the comparative merits of the different Tables of Mortality that have been published.

43. When the uniformity of anatomical structure in different individuals of the human species is considered, and the great power possessed by the human body, of so adapting itself to the circumstances it is placed in, as to avoid injury from changes in those circumstances, it appears natural to expect a priori, that, where the circumstances of the people are not greatly different, the law of mortality will be nearly the same. And, from a comparison of the best tables of mortality yet constructed, we are induced to believe that this expectation will be realized, whenever a sufficient number of good observations shall have been made, under circumstances sufficiently varied.

44. We know of no observations that have hitherto been made and published, from which the law of mortality may be correctly determined, except the following:

1. 1. Those of M. Deparcieux in France.
2. 2. The Swedish.
3. 3. Dr. Heysham's at Carlisle.
4. 4. Dr. Cleland's at Glasgow.
5. 5. Mr. Finlaison's on the nominees of life annuities granted by government in this country.
6. 6. Mr. Morgan's on the lives insured in the Equitable Assurance Society.

Those of Deparcieux, Finlaison and Morgan, were made only on select classes of the people; the Swedish are incomparably the most numerous and extensive; and whilst Dr. Cleland's exhibit the mortality in a large manufacturing town, Dr. Heysham's will, we believe, be found to be best authenticated, and most correct.

45. The climate of Sweden is so unfavourable to the products of agriculture, and the number of the people is so great in proportion to the quantity of food produced, that unfavourable seasons there, are generally followed by distressing dearth, and the destructive epidemical diseases constantly attendant upon famine, which raise the mortality, when they occur, much above what it would otherwise be; and both in that way, and by weakening the constitutions of those who survive them, they materially increase the average mortality deduced from observations made during any considerable number of years. Of this the reader will find ample proofs drawn from authentic sources, in the 10th, 12th, and 13th chapters of Mr. Milne's Treatise on Annuities.

46. For these reasons, the mortality in Sweden deduced from many years' observations, will be found to be higher than in the more temperate and fruitful parts of Europe. And we shall probably make the nearest approach to the general law of human mortality in the temperate climates, that can be made from the Swedish observations, by selecting a period in which no remarkable epidemics prevailed. Such a period was that of five years, 1801-1805; during

Mortality, which, according to a statement of M. Nicander, in the Law of. Transactions of the Royal Academy of Sciences at Stockholm for the year 1809, the population and mortality were as stated in Table I. at the end of this article.

47. From these data, the second table at the end of this article has been formed. The numbers in the columns for males and females separately, having been determined according to Nos. 36 and 31-35; assuming that, of 20,000 children born alive, 10,219 are males, and 9,781 females, in the ratio of 275,599 to 263,812.

The numbers against each age in the columns for the whole population without distinction of sex, are arithmetical mean proportionals between the corresponding numbers in the columns for males and females separately, against the same age.

48. From the table last mentioned, Table III. has been deduced by No. 16, exhibiting the expectation of life at every fifth year of age; or its mean duration after that age.

49. Vaccination commenced throughout Sweden and Finland in 1804, during which year, the number vaccinated was 38,255; and, in the year following, 42,839.

The number of deaths by small-pox there, during the year

Annual average number, 4,680

Whilst the annual average of the ten years ending with 1803, was 6962. (Vet. Ac. Handl. 1809, and Mr. Milne's Treatise on Annuities, art. 698.)

50. Therefore if we assume that, had vaccination not been practised in the years 1804 and 1805, the annual average number of deaths during the five years ending with that last mentioned, would have been greater by 2,282 than it actually was, and that these 2,282 additional deaths would have all taken place under five years of age, both assumptions will be near the truth; and it will follow that the annual mortality under five years of age, which actually was but one of 13.534, would have been one of 12.629, had vaccination not been introduced. Its introduction cannot have affected the first three tables above five years of age; and under that age, not quite so much as has just been stated.

51. Of all ages, and both sexes, there actually died annually, during these five years, one of 40.901; had vaccination not been practised at all, the annual average mortality would not have been so great as one of 39.759.

52. Table IV. exhibits the mean duration of life after every fifth year of age, according to twelve different tables of mortality; the first six, A, B, C, D, E, F, having been

constructed from the requisite data (30 and 38,) the last from six, M, N, O, P, Q, R, from mortuary registers only.

53. The numbers in the first column A have been taken from table III. in Mr. Milne's Treatise on Annuities, and those in B from table E (p. 28) in the Equitable Assurance Society's publication.

54. Deparcieux's table C constructed from great numbers of accurate observations on the nominees in the French Tontines, resident principally in Paris and its environs, represents the duration of life too small after 60 or 65 years of age. (See Mr. Milne's Treatise on Annuities, articles 867 and 868.)

55. Column D has been taken from the 45th table in Dr. Price's Observations, E from the 5th in Mr. Milne's Annuities, and F from the 3d table in this article. All these tables represent the duration of life in Sweden and Finland, after 45 or 50 years of age, to be less than according to the others; and it might reasonably be expected, a priori, that the excessive cold in Sweden would be unfavourable to the prolongation of life in old age.

56. Of the less correct columns, M has been deduced from the 7th table in Mr. Milne's Annuities, and N from the 42d in Dr. Price's Observations; but, as the Montpellier and Chester tables, just referred to, give the expectations of life only for males and females separately, the numbers in columns M and N against each age, are arithmetical mean proportionals between the expectations for males and females against the same ages in those tables; which, though not quite correct, is fully sufficient for our present purpose.

57. The number in column O against each age has been derived from that given by M. Mallet in his table of mortality as the mean duration of life in Geneva from and after that age, by subtracting one half (0.5) from each of them; which will be found to be a necessary correction.

58. Column P has been derived from Lambert's table for mankind in general, already mentioned in the historical part of this article, in which he gives a column headed mean age. Thus, against the age of 20 in that column, stands 54.3, by which he means that persons who survive 20 years of age, do, on an average, attain the age of 54.3 years; so that their expectation of life at 20, will be 34.3. But his numbers in that column are all too great by $\backslash frac\{1\}\{2\}$, or 0.5, as he has himself demonstrated; the last, therefore, should be 33.8; and

1 The author of this article in reading the note on the 4th and 5th pages of Mr. Arthur Morgan's introduction to the valuable work above mentioned, when that gentleman was so good as to send it to him on its first publication, had noted with a pencil in the margin, that the practice there stated was not quite correct; but the circumstance had long passed from his recollection, when he showed in No. 40 of the present article, how the law of Mortality in the Society might be determined from the large and valuable table marked A; therefore, by the words "age on admission," at the head of the extreme left hand column over the age in the same horizontal line with the number of lives admitted in the first division of the large column with that age at its head, he was naturally led to conclude that that was, as it ought to be, the number of lives which really were insured during the observations, when that was the age they had last completed. But by the note referred to, it appears not to be so; on the contrary, none of the lives insured during any year commencing with the first of January, are entered in these statements as having come into the Society until the first day of January next following the day of their actual admission, and then each is stated to be one year older than it was when insured; although whatever deaths may happen amongst them before the first of January next following the commencement of the insurances, are entered in the statements as having taken place amongst the lives previously insured. The manuscript being wanted immediately, was in the hands of the printer, 400 miles from the author, before these circumstances were called to his recollection, by his accidentally seeing his original note on the subject; he is therefore anxious to give this explanation. The principle of the method of determining the law of mortality in such cases, remains just the same. But it now appears that the number stated in Table A to have been admitted at any age, should have been stated to be so at the next younger age; and any calculator who may choose to employ himself on the subject, had better make that correction before proceeding farther. The probable error arising from the fault will not be great; but it is incumbent upon us to state the right method of proceeding.

Morty. 59. The reason assigned by Lambert for voluntarily ad-
Law. mitting this error at each age, as well as the corresponding
one in the number of the living at and above the same age,
into his table, was, that he did not consider the data in his
possession enabled him to determine the duration of life
within half a year of the truth.

60. In both these errors M. Lambert has been followed
by J. C. Baumann, in constructing the 21st, 22d, and 23d
tables inserted at the end of the third volume of Süssmilch's
Göttliche Ordnung which were intended to represent respec-
tively, the law of mortality among the country people in the
churmark of Brandenburg, amongst the whole population
of the churmark, and amongst the inhabitants of London.

61. The numbers in column Q were calculated by De-
parcieux from Dr. Halley's table; and those in column R
have been extracted from the 18th table in Dr. Price's Ob-
servations.

62. Upon comparing the numbers in the first six of these
columns, which are more correct, with those in the last six,
which are less so; it will be found, that at the early periods
of life, its future mean duration according to the tables
formed from mortuary registers alone, is less than according
to those formed from the requisite data; also that the dif-
ference is greater the younger the lives are, and diminishes
while the age increases, so as at 60 or 65 years of age to be
little or nothing, and to continue small, and variable both
in kind and magnitude, through the rest of life.

63. This appears to have arisen from the number of the
people having varied but little during the first 35 or 40
years of the century that ended at or about the middle of
the term in which the observations were made; and having
increased considerably by procreation, during the remainder
of that century; such increase having been slow at first,
but gradually accelerated afterwards.

64. Table V. is calculated to illustrate this part of the
subject. The columns A and B represent the law of mor-
tality among the whole population of Sweden and Finland
without distinction of sex, having been merely copied from
Table II.

Column C shows the proportion of 10,000 annual deaths
in Sweden and Finland that took place in each year of
age, on an average of five years ending with 1805. And
the number in column D against any age, being the sum of
of those in column C against that and all the greater ages,
would be the number who annually attain to that age, if the
number of the people of every age had remained stationary
from the year 1700 till 1805 (7).

65. The table of mortality formed by the columns C and
D therefore, is that which Dupré de Saint Maur, Süssmilch,
Lambert, Baumann, Florencourt, Muret and others, for want
of the mortuary registers of a whole country, endeavoured
to form by combining the registers of different town and
country parishes.

Those tables also have the same faults, which have been
formed in a similar way for particular towns or other com-
paratively small districts, from the bills of mortality alone,
where the population had been increasing during the cen-
tury preceding the commencement of the observations or
statements in the bills of mortality which each table was
formed from, and also during the period of the observa-
tions.

66. But it has been ascertained by repeated enumera-
tions of the people in Sweden and Finland, that the hypo-
thesis of their number having remained stationary for the
last 100 years or more, is far from the truth. And by
comparing columns A and B with C and D, it will be seen
in what manner, and to what degree, the falsity of the hypo-
thesis in this case, has vitiated the table derived from it.

67. To facilitate this comparison, columns E and F have
been added. Taking the age of five years for an example;
the numbers against that age in columns C and D show,

that, according to the hypothesis, out of 5988 children who
annually enter upon their 6th year, 144 die in it; while it
appears by columns A and B, that out of 7096 children
who enter upon that year of their age, only 112 die in it:
$112\; :\; 7096\; ::\; 144\; :\; 9123$, so that 9123, inserted against
the age of five years in column E, is the number of children
annually entering upon their 6th year, out of whom 144
really die in the same year of their age; and the mortality
as represented by the hypothetical table in this case, is to
the true mortality, as 9123 to 5988, or as 3 to 2 nearly.

Then the number in column F against any age, is always
the excess of that in E above that in column D against the
same age.

68. Columns B and C both containing 10,000 deaths, it
will be seen that in column C, they are greatly accumulated
at the early ages, in comparison with those in column
B; and that in old age, the deaths in column C are much
less numerous than in B; which are necessary consequen-
ces of the people increasing by procreation; the numbers
of the people in a progressive population, in comparison
with a stationary one, being greater in early life, and less
in old age. And, while the law of mortality remains the
same, the numbers of deaths at the different ages, must nec-
cessarily be distributed in a similar manner.

69. This enables us to see clearly how the principal dif-
ferences have arisen between the correct and incorrect
tables of mortality AB and CD; whilst the number of annu-
al deaths at all ages (10,000) is the same in both, the pro-
portions of that number are necessarily in column C of the
increasing population greater, and the deaths are more
densely distributed at the early ages, and less so at the ad-
vanced ones than in column B of the population which has
consisted of the same number of persons of every age, and
has produced the same number of annual births and of
annual deaths at every age for 100 years past. The in-
creasing population is necessarily attended with a cor-
responding increase of the annual births and of young persons;
while the elderly persons are only those left by the law of
mortality out of the corresponding small numbers that were
born annually, 50, 60, 70, or 100 years back. And as col-
umns A and B represent the true law of mortality that pre-
vails among the people, columns C and D cannot do so,
although constructed from observations on the same peo-
ple; for in an increasing population it is not true that the
sum of the deaths happening annually above a given age
will be the same as the number of persons annually arriv-
ing at or completing that age; it always falls short of it,
and the more so as the given age is younger, there being
deficiencies at all ages above that.

Columns E and F show what the errors of the hypothet-
ical table are at the different ages, and they are of a sim-
ilar kind in all tables similarly constructed from records of
the number of deaths at the different periods of life in an
increasing population; and such are probably 99 out of 100
of those hitherto published; many of them for people in-
creasing their numbers much faster than the Swedes have
done, and the errors of such tables must be greater than
those of the table CD.

70. Table VI., which needs no further explanation than
is placed at the head of it, will also illustrate the difference
between tables of mortality formed from the requisite data,
and those constructed from mortuary registers only.

It is better fitted for this purpose than Table IV., with
which, however, it will be found to correspond very well.
But the 4th table has other uses which this has not.

71. From what has already been advanced, it would ap-
pear probable, that the number of annual births in Sweden
and Finland had been nearly stationary, and rather decreasing
than increasing, upon an average, from about 1700 till 1735.

The numbers both of the annual births and deaths, from
the year 1749 till 1803, will be found in Milne's Treatise

Mortality, on Annuities, art. 698: these kind of returns to government were not made before 1749, neither have we any satisfactory account of the population before that period.

72. But the statements in our 7th table corroborate the inferences just drawn from the 5th and 6th, as they shew that during the 43 years ended with 1800, the total population increased, while the proportion above 90 years of age diminished through the whole term, and increased very little during the next 10 years.

The numbers in that table include both sexes, and the long continued diminution of them past 90, cannot be explained by supposing the males to have fallen in battle; for the females were reduced in the same proportion, their number throughout, having been to that of the males above 90 years of age, as nine to five nearly.

From the 7th table, therefore, it appears probable, that the annual births in the years

1698, 1705, 1710, and 1715, were respectively proportional to the numbers..... 907, 637, 837, and 786.

The last number, 786, has been calculated upon the supposition that the proportion of the population in Sweden and Finland to those in Sweden alone, was the same in 1810 as in 1805.

73. It should also be observed here, that the disastrous career of Charles the Twelfth commenced with the eighteenth century, and terminated in 1718, when the country was in such a state of exhaustion as it could not have recovered from for many years; whence there appears reason to believe, that the annual births during the succeeding fifteen or twenty years, did not increase fast.

Cantzaer informs us, that between the 10th of August 1710, and the month of February 1711, near 30,000 persons were carried off by the plague in Stockholm alone. (Mém. du Royaume de Suède, t. i. p. 29.)

74. It will be seen that the numbers in col. F of Table V., in proceeding back from four years of age to birth, continually decrease, contrary to what generally obtains; and as we ascribe the general increase of these numbers, when taken in the retrograde order of the ages, to the annually increasing number of births, so will this anomalous appearance be found to arise partly from the average number of annual births having actually decreased for a few years; for

But it appears to have arisen principally from the practice of vaccination during the years 1804 and 1805, by which the mortality among children, or the numbers in col. C, in a few of the first years from birth, were reduced below what they otherwise would have been (50) while those in col. D remained nearly the same (64); consequently, the numbers in col. E were reduced in nearly the same ratio as those in C (67), and the reduction in col. F was in each case nearly the same as in E (61).

75. The numbers relating to Sweden and Finland in the 7th table, have been derived from the Stockholm Transactions for the years 1766, 1801, 1809, and 1813.

Those relating to Spain and the Spanish possessions in Europe and Africa, including the Canary Islands, from the Censo de la Poblacion de Espana en el año de 1797, mentioned in the first section of this article. These last have been included in this Table, to show the difference in the proportion of aged persons between Spain and Sweden, and still more between the Canary Islands and both.

76. If the population of Spain had remained invariable from 1697 to 1797, the law of mortality there, might have been easily derived from the statements above mentioned of the enumeration in 1797; but in the actual state of things, that cannot be determined without comparing these with exact accounts of the numbers that died annually in each interval of age. And, as was observed in the first section of this article, the author has obtained satisfactory information that no such returns from the parish registers throughout Spain, as are there mentioned, ever were published, nor is it probable they were ever made.

77. When what we have advanced respecting the 5th and 6th tables is clearly understood, it will not be difficult to account for the greater part of the difference between the more and less correct columns in Table IV.

Most of the observations which the German tables were constructed from, were made between the years 1720 and 1750; and those who died then between 60 and 100 years of age, must have been born between 1620 and 1690; in which period nearly the whole of the thirty years' war, ended in 1648, was included, during which, and for several years after, it is probable that the annual births increased little or nothing, if they did not decrease.

78. Amongst the less correct columns of Table IV., those for Montpellier, Chester, and Geneva, agree much better than the rest with the more correct ones, which has probably arisen in each case, partly from the mortality in these three places having really been less throughout life than in most large towns; and partly from the annual births in them, having increased less than in the other places, during the fifty or sixty years preceding the period in which the observations the tables were constructed from were made.

79. The Northampton table was constructed by Dr. Price, from the bills of mortality (from the year 1735 to 1780) of the single parish of All Saints, containing a little more than half the inhabitants of the town; and as the deaths exceeded the births in number, the Doctor applied a correction to the table under twenty years of age, which, if it had answered the intended purpose under that age, as we are satisfied it did not, could have no effect on any of the numbers above the same age; and almost all of the useful applications of such tables, are to ages above twenty.

80. The table so formed could only be correct, provided that the numbers, both of the living and the annual deaths at every age above twenty years, had continued invariable during the 146 years that intervened between 1634 and 1780; provided also, that no migration from or to the town took place, except at twenty years of age, and that the annual increase the population received by migration at that age, was just equal to the excess of the annual deaths above the annual births.

81. But we consider it to be much more probable, that during these 146 years, Northampton partook of the prosperity and adversity that prevailed in the rest of the kingdom; and, consequently, that its population was generally progressive, though sometimes stationary, and sometimes retrograde.

82. We have not room here to support this opinion by numerical statements and calculations, but from the population abstracts, and an enumeration of the inhabitants of Northampton, given in Dr. Price's Observations on Reversionary Payments, (vol. ii. p. 94,) it will be found, that both the annual births, and annual settlers in that town, have been increasing ever since about the year 1715 or 1720; also, that although the burials exceeded the baptisms till the year 1802, the supply by migration was much greater than that excess; and, consequently, that the numbers of the living have been accumulated more at the early ages, and less at the advanced ones, than they would have been had the population remained stationary.

83. Thus it appears, that the faults in the Northampton table are of the same kind as those of the others constructed from mortuary registers only. And the civil war in the time of Charles the First, with the unsettled state of the kingdom for some years before and after it, would probably have prevented or greatly retarded, the increase of the annual births, during the time in which those persons were born, who died past sixty years of age between the years 1734 and 1781, and may account for the table after that age being near the truth; while the comparatively rapid increase of the people during the sixty years ending with 1780, appears to explain the great excess of mortality in that table at the early periods of life.

84. As it was only from the Carlisle and Northampton tables of mortality, that tables of the values of annuities on single and joint lives had been calculated, sufficiently copious to admit of the values of interests dependent upon the continuance or the failure of human life being accurately derived from them. When the first edition of this article was published in the year 1820, the author gave the following comparison between the mortality represented by each of these tables to take place at the different periods of life, with that which had been observed to obtain among the members of the Equitable Assurance Society.

85. From an address delivered at a general court of that Society, by Mr. Morgan the actuary, on the 24th of April 1800, it appears, that according to the result of an annual experience of thirty years, the decrements of life (22) among the members of the society, were to those in the Northampton table,

The same information may also be found in two notes in Dr. Price's Observations on Reversionary Payments, (vol. i. p. 183, and vol. ii. p. 443.)

86. From the preceding statement, the Carlisle table of mortality, (No. II. in Mr. Milne's Annuities, or No. V. at the end of the article ANNUITIES in this work, and the Northampton table, (No. XVII. in Dr. Price's Observations), we have derived the following:—

87. This table shows that the law of mortality exhibited in the Carlisle table is almost exactly the same as that which has prevailed among the members of the Equitable Assurance Society. And although the members of such a society, when they first enter, are select lives, they are not, even then, so much better than the common average, as many persons suppose; for the more precarious a life is, the stronger is the inducement for parties interested in its continuance, to get it insured, so that bad risks are frequently offered to such companies. And many proposals for insurance are accepted by the directors, that are not thought very eligible at the time, in cases where they are not aware of any specific objection to the life proposed.

88. Besides, it is to be considered, that of the number in a society at any one time, but a small proportion can have been recently admitted, and in a few years from the time of admission, the members will generally have come down to the common average of persons of the same ages.

89. It ought also to be observed, that most of the tables of mortality that have been published, have been constructed from observations made upon the whole population of very large towns, such as London, Paris, Vienna, and Stockholm; in each of which there are particular quarters inhabited only by the very lowest of the people, who, unfortunately, are also very numerous, badly clothed and fed, therefore exposed to serious injury from the inclemencies of the weather; extremely ignorant and vicious, indulging in the abuse of spirituous liquors, and inattentive to cleanliness both in their persons and habitations, which last are crowded, badly ventilated, and surrounded with mud and the putrid remains of animals and vegetables. These are the nests of contagious diseases, in which they are generated and kept alive, where they at all times occasion great mortality, though not so much within the last forty-five or fifty-five years as previously, and from which, when circumstances favour them, they spread amongst the rest of the people.

90. It is, therefore, obvious, that in such places, the average mortality at every age, must be considerably greater than that which prevails only among the middling and higher classes of society, even in such towns.

91. But the lives upon which leases, annuities, reversions, and assurances depend, are very seldom exposed to the influence of the causes of mortality mentioned in number 89. Whence it follows, that a table of mortality on which those causes have had no great influence, is best adapted to the valuation of such interests.

And these kind of valuations are the most important purposes to which tables of mortality can be applied.

92. The number of years in the mean duration of life from birth according to a table of mortality properly constructed from the necessary data, will, when the population has remained stationary for a century or more, be the same as the number of persons in the whole population, out of which one dies annually (20). When the population has been increasing, the mean duration of life, according to the table, will be less than the number out of which one dies annually in that population; but the difference will be small, except under particular circumstances, as appears by the following statement.

93. The above mentioned table of mortality for Glasgow, which the author has had by him several years, he expects to publish soon.

The high number in the column of differences for females in that place, arises from the small mortality amongst them, which was occasioned principally by the great influx of healthy females between the ages of 15 and 30, at which period of life the rate of mortality amongst them was small; its minimum for them being in their 18th year.

Between the ages of 10 and 15 the numbers of the two sexes were just about equal, whilst between the ages of 15 and 30 the females were more numerous than the males in the ratio of 3 to 2; and this accounts for the great number of females (42.32), out of which one died annually there, which necessarily raises the number out of which one died annually in the whole population of both sexes, and there-

fore the number in the column of differences (in the line numbered 10) for the whole population of both sexes in Glasgow.

94. Similar causes probably produced similar effects, although in a much less degree in Stockholm.

95. When tables of mortality are constructed from the numbers of deaths only in the different intervals of age, without comparing them with the numbers of living persons in the same intervals; such as that formed by columns C and D of Table V. at the end of this article, and the population is increasing. The number of years in the mean duration of life from birth, according to that table, will fall short of the number of the people, out of which one dies annually, by a much greater number than in the case we have just been considering, of the table of mortality having been properly constructed from the necessary data: as the following statement will show.

96. In both of the above statements, where the sexes are not mentioned, the table is for both without distinction.

97. In places where the increase of the population was slow, the numbers set against them in the column of differences are smaller than where the population increased more rapidly.

98. In the tables of mortality for Belgium and Montpellier, the sexes were distinguished, but not in the given number of the whole population, as well as the whole number of deaths in each of those places, and for that reason the sexes could not be distinguished for those two places in this last table.

99. When what we have shewn here is clearly understood,

and the proportion of the people dying annually is known, it will not be difficult to judge whether a table of mortality for that people has been constructed properly from the necessary data; or, what is much more common, and more easily effected, merely by summation of the deaths at all ages. For in the case of the correct table of mortality, the difference obtained in the manner stated above, will probably not exceed 2; nor in the case of the incorrect table, will it be likely to be less than 7, upon the supposition of the population having increased generally for a considerable number of years preceding the termination of the observations; if for a whole century, so much the better.

¹ But omitting the year 1764, in which no observations were made.

100. The probable life at birth, or the age to which half the number born attain, will also be a good criterion for enabling us to judge, by comparison, whether a table of mortality has been correctly constructed from the necessary data or not; provided that, in the places the compared tables have been constructed from, the children have been similarly circumstanced, or nearly so.

In Belgium vaccination was practised during the whole period of the observations, in Sweden only during about $1\backslash frac\{1\}\{2\}$ out of the 5 years' observations.

101. From all that has now been stated, we are entitled to conclude, that the Belgian table of mortality has been constructed either from the registers of burials alone, or only from the statements of the numbers living in the different intervals of age at the time of the enumeration; most probably from the burials alone. M. Quetelet appears now to be aware that such tables are very incorrect. In his late work sur l'Homme et le Développement de ses Facultés, ou Essai de Physique Sociale, 2 tom. Paris 1835, in 8vo, after giving the above mentioned table of mortality for Belgium, in which only the numbers attaining the different ages are given, without the decrements of life, or its mean or probable duration at the different ages, he states the mean duration of life from birth according to that table, and then proceeds thus, (tome i. p. 166):—"D'après le dernier ouvrage de M. Rickman, la vie moyenne serait en Angleterre de 33 ans (32 pour les hommes, 34 pour les femmes).¹ On l'estime en France de 32-2 ans d'après le chiffre des naissances.² Du reste, ces calculs supposent une population stationnaire, et nous aurons occasion de voir qu'ils peuvent conduire à des erreurs assez graves."

102. Reasoning as in Nos. 68 and 69, it will be seen that when the number of the people has been decreasing for a series of years, the deaths will be more densely distributed among the advanced ages, and more rarely in the early periods of life, than if the number of the people at every age, and the law of mortality had remained always the same. Consequently, in the table of mortality constructed from such data, merely by taking the successive sums, in retrograde order, from extreme old age to birth; those successive sums, which are the numbers of the living at the different ages in the table so constructed, will be greater at advanced ages and less in early life than if the population had remained stationary, as stated above. So that the errors of the incorrect table would, in this case, be of the opposite kind to those of the common tables, constructed in a similar way from the deaths only, in an increasing population. From the general increase of arts, manufactures, commerce, and civilization, ever since tables of mortality were first formed, the population has been increasing more or less rapidly in almost every place, for which a table of mortality has been constructed. But in Amsterdam we have an instance of the population having continued to decrease for half a century or more. We here present a view of it since the year 1622.

103. The information on this subject is scanty and difficult to procure. Kersseboom took much pains in endeavouring to make just estimates of the population from the annual births and deaths; and taking it to be 35 times the number of annual births, he estimated the population of Amsterdam in 1742 at 241,000 persons.

Three only of the six numbers stated above, viz. those for the years 1622, 1826, and 1830, were determined by actual enumerations at the times stated. Although M. Smits, the secretary of the Statistical Commission, relative to the population of the Netherlands, is not mentioned in the above table, we are greatly indebted to him. In his Statistique Nationale he gave a table, (No. 10, p. 82), shewing the number of deaths that took place in Amsterdam in each year of the 18th century. M. Lobatto has given them in the place above referred to for each of the ten years 1816-1825; and M. Quetelet, in the part of his work referred to above, has given them for each of the 17 years, 1816-1832, and states that in the year 1777 the mortality was 1 in 27; the number of deaths in that year was 8939, whence the population inserted in the table is derived. M. Lobatto states, on the authority of Professor Van Swinden and of M. Nieuwenhuys, that from 1774 to 1813 there died annually 1 of 26; the deaths in 27 years, 1774-1800, were 233,510, the annual average number, therefore, was 8648.5, and multiplying this by 26, we obtain 224,862, the mean number of the people during these 27 years, which is stated in the above table to have been the population in 1787, the middle year of the 27.

104. The enumeration in 1622 was made in levying a capitation tax, which was exacted with great strictness, even in every receptacle for paupers, the master or owner of it was obliged to pay the tax for each inmate. This enumeration appears not to have been known of, either by Kersseboom or by Struyck at the time of his first publication on the subject; but an original paper, with many particulars which he gives, was afterwards communicated to him. Struyck, after taking great pains with the subject, stated his opinion, that in 1753, Amsterdam did not contain 200,000 inhabitants.

105. M. Lobatto, at the end of his Beschouwing, &c. gives two tables of mortality for Amsterdam, one for males, the other for females, constructed from the deaths that took place during the ten years 1816-1825, with the expectation of life at every age, and we here give the same particulars for it as are given for some others above, but marking the difference (—) as it is of the opposite kind to those which resulted from an increasing population.

¹ Preface to the Population Abstract. 1831.

² Annuaire du Bureau des Longitudes, pour l'année 1834, p. 102.

³ Beschouwing van den aard, de voordelen, en de inrigting der Mantchappijen van Levensverzekering, &c., door R. Lobatto; Amsterdam, in 8vo, 1830.