I beg leave to submit to your perusal the following observations. If you think them of any importance, I shall be obliged to you for communicating them to the Royal Society. You will find that the chief subject of them is the present state of the city of London, with respect to healthfulness and number of inhabitants, as far as it can be collected from the bills of mortality. This is a subject that has been considered by others; but the proper method of calculating from the bills has not, I think, been sufficiently explained.
No competent judgment can be formed of the following observations, without a clear notion of what the writers on Life Annuities and Reversions have called the Expectation of Life. Perhaps this is not in common properly understood; and Mr. De Moivre’s manner of expressing himself about it is very liable to be mistaken.
The most obvious sense of the expectation of a given life is, “That particular number of years which a life of a given age has an equal chance of enjoying.” This is properly the time that a person may reasonably expect to live; for the chances against his living longer are greater than those for it; and, therefore, he cannot entertain an expectation of living longer, consistently with probability. This period does not coincide with what the writers on Annuities call the expectation of life, except on the supposition of an uniform decrease in the probabilities of life, as Thomas Simpson has observed in his Select Exercises for Young Proficients in the Mathematicks (London, 1752), p. 273. It is necessary to add, that, even on this supposition, it does not coincide with what is called the expectation of life in any case of joint lives. Thus, two joint lives of 40 have an even chance, according to Mr. De Moivre’s hypothesis, of continuing together only 13½ years. But the expectation of two equal joint lives being (according to the same hypothesis) always a third of the common complement, it is in this case 15 years. It is necessary, therefore, to observe, that there is another sense of this phrase which ought to be carefully distinguished from that now mentioned. It may signify “The mean continuance of any given single, joint, or surviving lives, according to any given table of observations:” that is, the number of years which, taking them one with another, they actually enjoy, and may be considered as sure of enjoying, those who live or survive beyond that period, enjoying as much more time in proportion to their number, as those who fall short of it enjoy less. Thus, supposing 46 persons alive, all 40 years of age, and that, according to Mr. De Moivre’s hypothesis, one will die every year till they are all dead in 46 years, half 46 or 23 will be their expectation of life: that is, The number of years enjoyed by them all will be just the same as if every one of them had lived 23 years, and then died; so that, supposing no interest of money, there would be no difference in value between annuities payable for life to every single person in such a set, and equal annuities payable to another equal set of persons of the same common age, supposed to be all sure of living just 23 years and no more.
In like manner, the third of 46 years, or 15 years and 4 months, is the expectation of two joint lives both 40; and this is also the expectation of the survivor. That is, supposing a set of marriages between persons all 40, they will, one with another, last just this time, and the survivors will last the same time; and annuities payable during the continuance of such marriages would, supposing no interest of money, be of exactly the same value with annuities to begin at the extinction of such marriages, and to be paid, during life, to the survivors. In adding together the years which any great number of such marriages and their survivorships have lasted, the sums would be found to be equal.
One is naturally led to understand the expectation of life in the first of the senses now explained, when, by Mr. Simpson and Mr. De Moivre, it is called, the number of years which, upon an equality of chance, a person may expect to enjoy; or, the time which a person of a given age may justly expect to continue in being; and, in the last sense, when it is called, the share of life due to a person. But, as in reality it is always used in the last of these senses, the former language should not be applied to it: and it is in this last sense that it coincides with the sums of the present probabilities that any given single or joint lives shall attain to the end of the 1st, 2d, 3d, &c. moments from this time to the end of their possible existence; or, in the case of survivorships, with the sum of the probabilities that there shall be a survivor at the end of the 1st, 2d, 3d, &c. moments, from this time to the end of the possible existence of survivorship. This coincidence every one conversant in these subjects must see, upon reflecting, that both these senses give the true present value of a life-annuity secured by land, without interest of money.
This period in joint lives, I have observed, is never the same with the period which they have an equal chance of enjoying; and in single lives, I have observed, they are the same only on the supposition of an uniform decrease in the probabilities of life. If this decrease, instead of being always uniform, is accelerated in the last stages of life, the former period, in single lives, will be less than the latter; if retarded, it will be greater.
It is necessary to add, that the number expressing the former period, multiplied by the number of single or joint lives whose expectation it is added annually to a society or town, gives the whole number living together, to which such an annual addition would in time grow. Thus, since 19, or the third of 57, is the expectation of two joint lives whose common age is 29, or common complement 57, twenty marriages every year between persons of this age would, in 57 years, grow to 20 times 19, or 380 marriages always existing together. The number of survivors also arising from these marriages, and always living together, would, in twice 57 years, increase to the same number. And, since the expectation of a single life is always half its complement, in 57 years likewise 20 single persons aged 29, added annually to a town, would increase to 20 times 28.5 or 570; and when arrived at this number, the deaths every year will just equal the accessions, and no further increase be possible.
It appears from hence, that the particular proportion that becomes extinct every year, out of the whole number constantly existing together of single or joint lives, must, wherever this number undergoes no variation, be exactly the same with the expectation of those lives at the time when their existence commenced. Thus, was it found that a 19th part of all the marriages among any body of men, whose numbers do not vary, are dissolved every year by the deaths of either the husband or wife, it would appear that 19 was, at the time they were contracted, the expectation of these marriages. In like manner, was it found in a society, limited to a fixed number of members, that a 28th part dies annually out of the whole number of members, it would appear that 28 was their common expectation of life at the time they entered. So likewise, were it found in any town or district, where the number of births and burials are equal, that a 20th or 30th part of the inhabitants die annually, it would appear that 20 or 30 was the expectation of a child just born in that town or district. These expectations, therefore, for all single lives, are easily found by a Table of Observations, shewing the number that die annually at all ages, out of a given number alive at those ages; and the general rule for this purpose is “to divide the sum of all the living in the Table at the age whose expectation is required, and at all greater ages, by the sum of all that die annually at that age, and above it; or, which is the same, by the number in the Table of the living at that age; and half subtracted from the quotient will be the required expectation.” Thus, in Dr. Halley’s Table, the sum of all the living at 20 and upwards is 20,724. The number living at that age is 598; and the former number divided by the latter, and half unity subtracted from the quotient, gives 34.15 for the expectation of 20. The expectation of the same life by Mr. Simpson’s Table, formed from the bills of mortality of London, is 28.9.
These observations bring me to the principal point which I have had all along in view. They suggest to us an easy method of finding the number of inhabitants in a place from a Table of Observations, or the bills of mortality for that place, supposing the yearly births and burials equal. “Find by the Table, in the way just described, the expectation of an infant just born, and this, multiplied by the number of yearly births, will be the number of inhabitants.” At Breslaw, according to Dr. Halley’s Table, though half die under 16, and therefore an infant just born has an equal chance of living only 16 years, yet his expectations, found by the rule I have given, is near 28 years; and this, multiplied by 1238 the number born annually, gives 34,664, the number of inhabitants. In like manner, it appears from Mr. Simpson’s Table, that, though an infant just born in London has not an equal chance of living 3 years, his expectation is 20 years; and this number, multiplied by the yearly births, would give the number of inhabitants in London, were the births and burials equal. The medium of the yearly births, for the last 10 years, has been 15,710. This number, multiplied by 20, is 314,200, which is the number of inhabitants that there would be in London, according to the bills, were the yearly burials no more than equal to the births: that is, were it to support itself in its number of inhabitants without any supply from the country. But for the last 10 years, the burials have, at an average, been 22,956, and exceeded the christenings 7,246. This is, therefore, at present, the yearly addition of people to London from other parts of the kingdom, by whom it is kept up. Suppose them to be all, one with another, persons who have, when they remove to London, an expectation of life equal to 30 years. That is, suppose them to be all of the age of 18 or 20, a supposition certainly far beyond the truth. From hence will arise, according to what has been before observed, an addition of 30 multiplied by 7,246, that is 217,380 inhabitants. This number, added to the former, makes 531,580; and this, I think, at most, would be the number of inhabitants in London were the bills perfect. But it is certain that they give the number of births and burials too little. There are many burying-places that are never brought into the bills. Many also emigrate to the navy and army and country; and these ought to be added to the number of deaths. What the deficiencies arising from hence are, cannot be determined. Suppose them equivalent to 6000 every year in the births, and 6000 in the burials. This would make an addition of 20 times 6000 or 120,000 to the last number, and the whole number of inhabitants, would be 651,580. If the burials are deficient only two thirds of this number, or 4000, and the births the whole of it, 20 multiplied by 6000, must be added to 314,290 on account of the defects in the births: and, since the excess of the burials above the births will then be only 5,246, 30 multiplied by 5,246 or 157,380, will be the number to be added on this account; and the sum, or number of inhabitants will be 591,580. But if, on the contrary, the burials are deficient 6000, and the births only 4000, 80,000 must be added to 314,290, on account of the deficiencies in the births, and 30 multiplied by 9,246, on account of the excess of the burials above the births, and the whole number of inhabitants will be 671,580.
Every supposition in these calculations seems to me too high. Emigrants from London are, in particular, allowed the same expectation of continuance in London with those who are born in it, or who come to it in the firmest part of life, and never afterwards leave it; whereas it is not credible that the former expectation should be so much as half the latter. But I have a further reason for thinking that this calculation gives too high numbers, which has with me irresistible weight. It has been seen that the number of inhabitants comes out less on the supposition, that the defects in the christenings are greater than those in the burials. Now it seems evident that this is really the case; and, as it is a fact not attended to, I will here endeavour to explain distinctly the reason which proves it.
The proportion of the number of births in London, to the number who live to be 10 years of age, is, by the bills, 16 to 5. Any one may find this to be true, by subtracting the annual medium of those who have died under 10, for some years past, from the annual medium of births for the same number of years. Now, tho’, without doubt, London is very fatal to children, yet it is incredible that it should be so fatal as this implies. The bills, therefore, very probably, give the number of those who die under 10 too great in proportion to the number of births; and there can be no other cause of this, than a greater deficiency in the births than in the burials. Were the deficiences in both equal, that is, were the burials, in proportion to their number, just as deficient as the births are in proportion to their number, the proportion of those who reach 10 years of age to the number born would be right in the bills, let the deficiencies themselves be ever so considerable. On the contrary, were the deficiencies in the burials greater than in the births, this proportion would be given too great; and it is only when the former are least that this proportion can be given too little. Thus, let the number of annual burials be 23,000; of births 15,700; and the number dying annually under 10, 10,800. Then 4,900 will reach 10 of 15,700 born annually; that is, 5 out of 16. Were there no deficiencies in the burials, and were it fact that only half die under 10, it would follow, that there was an annual deficiency equal to 4,900 subtracted from 10,800, or 5,900 in the births. Were the births a third part too little, and the burials also a third part too little, the true number of births, burials, and of children dying under 10, would be 20,933, 30,666 and 14,400; and, therefore, the number that would live to 10 years of age would be 6,533 out of 20,933, or 5 of 16 as before. Were the births a third part, and the burials so much as two-fifths wrong, the number of births, burials, and children dying under 10 would be 20,933, 32,200 and 15,120; and, therefore, the number that would live to 10 would be 5,813 out of 20,933, or 5 out of 18. Were the births a 3d part wrong, and the burials but a 6th, the foregoing numbers would be 20,933— 26,833—12,600; and, therefore, the number that would live to 10 would be 8,333 out of 20,933, or 5 out of 12.56: and this proportion seems as low as is consistent with any degree of probability. It is somewhat less than the proportion in Mr. Simpson’s Table of London Observations, and near one half less than the proportion in the Table of Observations for Breslaw, where it appears that above 9 of 16 live to be 10, and that one half live to be 16. The deficiencies, therefore, in the births cannot be much less than double those in the burials; and the least numbers I have given must, probably, be nearest to the true number of inhabitants. However, should any one, after all, think that it is not improbable that only 5 of 16 should live in London to be 10 years of age, or that above two thirds die under this age, the consequence of admitting this will still be, that the foregoing calculation has been carried too high. For it will from hence follow, that the expectation of a child just born in London cannot be so much as I have taken it. This expectation is 20, on the supposition that half die under 3 years of age, and that 5 of 16 live to be 29 years of age, agreeably to Mr. Simpson’s Table. But if it is indeed true, that half die under 2 years of age, and 5 of 16 under 10, agreeably to the bills, this expectation must be less than 20, and all the numbers before given will be considerably reduced.
Upon the whole, I am forced to conclude from these observations, that the second number I have given, or 651,580, though short of the number of inhabitants commonly supposed in London, is, very probably, greater, but cannot be much less, than the true number. Indeed, it is in general evident, that in cases of this kind numbers are very much overrated. The ingenious Dr. Brakenridge, 14 years ago, when the bills were lower than they are now, from the number of houses, and allowing six to a house, made the number of inhabitants 751,800. But his method of determining the number of houses is too precarious; and, besides, six to a house is, probably, too large an allowance. Many families now have two houses to live in. The magistrates of Norwich, in 1752, took an exact account of both the number of houses and individuals in that city. The number of houses was 7,139, and of individuals 36,169, which gives nearly 5 to a house. Another method which Dr. Brakenridge took to determine the number of inhabitants in London was from the annual number of burials, adding 2000 to the bills for omissions, and supposing a 30th part to die every year. In order to prove this to be a moderate supposition he observes that, according to Dr. Halley’s Observations, a 34th part die every year at Breslaw. But this observation was made too inadvertently. The number of annual burials there, according to Dr. Halley’s account, was 1174, and the number of inhabitants, as deduced by him from his Table, was 34,000, and therefore a 29th part died every year. Besides, any one may find, that in reality the Table is constructed on the supposition, that the whole number born, or 1238, die every year; from whence it will follow that a 28th part died every year. Dr. Brakenridge, therefore, had he attended to this, would have stated a 24th part as the proportion that dies in London every year, and this would have taken off 150,000 from the number he has given. But even this must be less than the just proportion. For let three fourths of all who either die in London or migrate from it, be such as have been born in London; and let the rest be persons who have removed to London from the country or from foreign nations. The expectation of the former, it has been shewn, cannot exceed 20 years, and 30 years have been allowed to the latter. One with another, then, they will have an expectation of 22½ years. That is, one of 22½ will die every year. And, consequently, supposing the annual recruit from the country to be 7000, the number of births 3 times 7000 or 21,000, and the burials and migrations 28,000 (which seem to be all high suppositions), the number of inhabitants will be 22½ multiplied by 28,000, or 630,000.
I will just mention here one other instance of exaggeration on the present subject.
Mr. Corbyn Morris, in his Observations on the Past Growth and Present State of the City of London… (London, 1751), supposes that no more than a 60th part of the inhabitants of London, who are above 20, die every year, and from hence he determines that the number of inhabitants was near a million. In this supposition there was an error of at least one half. According to Dr. Halley’s Table, it has been shewn, that a 34th part of all at 20 and upwards, die every year at Breslaw. In London, a 29th part, according to Mr. Simpson’s Table, and also according to all other Tables of London Observations. And in Scotland it has been found for many years, that of 974 ministers and professors whose ages are 27 and upwards, a 33d part have died every year. Had, therefore, Mr. Morris stated a 30th part of all above 20 as dying annually in London, he would have gone beyond the truth, and his conclusion would have been 400,000 less than it is.
Dr. Brakenridge observed, that the number of inhabitants, at the time he calculated, was 127,000 less than it had been. The bills have lately advanced, but still they are much below what they were from 1717 to 1743. The medium of the annual births, for 20 years, from 1716 to 1736, was 18,000, and of burials 26,529; and by calculating from hence on all the same suppositions with those which made 651,580 to be the present number of inhabitants in London, it will be found that the number then was 735,840, or 84,260 greater than the number at present. London, therefore, for the last 30 years, has been decreasing; and though now it is increasing again, yet there is reason to think that the additions lately made to the number of buildings round it, are owing, in a great measure, to the increase of luxury, and the inhabitants requiring more room to live upon.
It should be remembered, that the number of inhabitants in London is now so much less as I have made it, than it was 40 years ago, on the supposition that the proportion of the omissions in the births to those in the burials was the same then that it is now. But it appears that this is not the fact. From 1728, the years when the ages of the dead was first given in the bills, to 1742, near five-sixths of those who were born died under 10, according to the bills. From 1742 to 1752 three quarters; and ever since 1752 this proportion has stood nearly as it is now, or at somewhat more than two-thirds. The omissions in the births, therefore, compared with those in the burials were greater formerly; and this must render the difference between the number of inhabitants now and formerly less considerable than it may seem to be from the face of the bills. One reason why the proportion of the amounts of the births and burials in the bills comes now nearer than it did to the true proportion, may, perhaps, be that the number of Dissenters in considerably lessened. The Foundling Hospital also may have contributed a little to this event, by lessening the number given in the bills as having died under 10, without taking off any from the births; for all that die in this hospital are buried at Pancrass church, which is not within the bills. See the preface to A Collection of the Yearly Bills of Mortality from 1657 to 1758 Inclusive… (Thomas Birch, ed.; London, 1759), p. 15.
I will add, that it is probable that London is now become less fatal to children than it was; and that this is a further circumstance which must reduce the difference I have mentioned; and which is likewise necessary to be joined to the greater deficiencies in the births, in order to account for the very small proportion of children who survived 10 years of age, during the two first of the periods I have specified. Since 1752, London has been thrown more open. The custom of keeping country-houses, and of sending children to be nursed in the country, has prevailed more. But, particularly, the destructive use of spirituous liquors among the poor has been checked.
I have shewn that in London, even in its present state, and according to the most moderate computation, half the number born die under three years of age; and I have observed that at Breslaw half live to 16. At Edinburgh, if I may judge from such of its bills as I have seen, almost as great a proportion of children die as even in London. But it appears from Graunt’s accurate account of the births, weddings, and burials in three country parishes for 90 years; and also, with abundant evidence, from Dr. Short’s collection of observations in his Comparative History, and his treatise entitled, New Observations…on the City, Town, and Country Bills of Mortality; that in country villages and parishes, the major part live to mature age, and even to marry. So great is the difference, especially to children, between living in great towns and in the country. But nothing can place this observation in a more striking light than the curious account given by Dr. Thomas Heberden, and published in the Philosophical Transactions (LVII [1767], 461-3) of the increase and mortality of the inhabitants of the island of Madeira. In this island, it seems, the weddings have been to the births, for 8 years, from 1759 to 1766, as 10 to 46.8; and to the burials as 10 to 27.5. Double these proportions, therefore, or the proportion of 20 to 46.8, and of 20 to 27.5 are the proportions of the number marrying annually, to the number born and the number dying. Let 1 marriage in 10 be a 2d or 3d marriage on the side of either the man or the woman, and 10 marriages will imply 19 individuals who have grown up to maturity, and lived to marry once or oftener; and the proportion of the number marrying annually the first time, to the number dying annually, will be 19 to 27.5, or near 3 to 4. It may seem to follow from hence, that in this island near three-fourths of those who die have been married, and, consequently, that not many more than a quarter of the inhabitants die in childhood and celibacy; and this would be a just conclusion were there no increase, or had the births and burials been equal. But it must be remembered, that the general effect of an increase, while it is going on in a country, is to render the proportion of persons marrying annually to the annual deaths greater, and to the annual births less than the true proportion marrying out of any given number born. This proportion generally lies between the other two proportions, but always nearest to the first; and, in the present case, it is sufficiently evident that it cannot be much less than two-thirds.
In London, then, half die under three years of age, and in Madeira about two-thirds of all who are born live to be married. Agreeably to this, it appears also from the account I have referred to, that the expectation of a child just born in Madeira is about 39 years, or double the expectation of a child just born in London. For the number of inhabitants was found, by a survey made in the beginning of the year 1767, to be 64,614. The annual medium of burials had been, for eight years, 1293; of births 2201. The number of inhabitants, divided by the annual medium of burials, gives 49.89, or the expectation nearly of a child just born, supposing the births had been 1293, and constantly equal to the burials, the number of inhabitants remaining the same. And the same number, divided by the annual medium of births, gives 29.35, or the expectation of a child just born, supposing the burials 2201, the number of births and of inhabitants remaining the same; and the true expectation of life must be somewhere near the mean between 49.89 and 29.35.
Again: A 50th part of the inhabitants of Madeira, it appears, die annually. In London, I have shewn, that above twice this proportion dies annually. In smaller towns a smaller proportion dies, and the births also come nearer to the burials. At Breslaw, I have observed, that, by Dr. Halley’s Table, a 28th part dies annually; and the annual medium of births, for a complete century, from 1633 to 1734, has been 1089; of burials 1256. At Norwich, the annual medium of births, dissenters included for four years, from 1751 to 1754, was 1150; of burials 1214. And as the number of inhabitants was at that time 36,169 (see p. 103 [p. 91 above]), a 30th part of the inhabitants died annually. In general, there seems reason to think that in towns (allowing for particular advantages of situation, trade, police, cleanliness, and openness, which some towns may have), the excess of the burials above the births and the annual deaths are more or less as the towns are greater or smaller. In London itself, about 160 years ago, when it was scarcely a fourth part of its present bulk, the births were nearly equal to the burials. But in country parishes and villages the births almost always exceed the burials; and I believe it seldom happens that so many as a 30th, or much more than a 40th part of the inhabitants die annually. In the four provinces of New England there is a very rapid increase of the inhabitants: but, notwithstanding this, at Boston, the capital, the inhabitants would decrease were there no supply from the country: for, if the account I have seen is just, from 1731 to 1762, the burials have all along exceeded the births. So remarkably do towns, in consequence of their unfavourableness to health, and the luxury which generally prevails in them, check the increase of countries.
Healthfulness and Prolifickness are, probably, causes of increase seldom separated. In conformity to this observation, it appears from comparing the births and weddings, in countries and towns where registers of them have been kept, that in the former, marriages, one with another, seldom produce less than four children each, generally between four and five, and sometimes above five. But in towns seldom above four, generally between three and four, and sometimes under three.
I have sometimes heard the great number of old people in London mentioned to prove its favourableness to health and long life. But no observation can be much more erroneous. There ought, in reality, to be more old people in London, in proportion to the number of inhabitants, than in any smaller towns, because at least one quarter of its inhabitants are persons who come into it, from the country, in the most robust part of life, and with a much greater probability of attaining old age, than if they had come into it in the weakness of infancy. But, notwithstanding this advantage, there are much fewer persons who attain to great ages in London than in any other place where observations have been made. At Vienna, of 22,704 who died in the four years 1717, 1718, 1724, 1725, 109 reached 90 years, that is, 48 in 10,000. But in London, for the last 30 years, only 35 of the same number have reached this age. At Breslaw it appears, by Dr. Halley’s Table, that 41 of 1238 born, or a 30th part, live to be 80 years of age. In the parish of All-saints in Northampton, an account has been kept for many years of the ages at which all die; and, I find, that of 1377, who died there in 13 years, 59 have lived to be 80, or a 23d part. According to Mr. Kersseboom’s Table of Observations, published at the end of Abraham De Moivre, The Doctrine of Chances… (3rd ed., London, 1756), a 14th part of all that are born live to be 80; and, had we any observations in country parishes, this, probably, would not appear to be too high a proportion. But in London, for the last 30 years, only 25 of every 1000 who have died, have lived to be 80, or a 40th part, which may be easily discovered by dividing the sum of all who have died during these years at all ages, by the sum of all who have died above 80.
Among the peculiar evils to which great towns are subject, I might further mention the Plague. Before the year 1666 this dreadful calamity laid London almost waste once in every 15 or 20 years; and there is no reason to think that it was not generally bred within itself. A most happy alteration has taken place, which, perhaps, in part, is owing to the greater advantages of cleanliness and openness, which London has enjoyed since it was rebuilt, and which lately have been very wisely improved.
The facts I have now taken notice of are so important that, I think, they deserve more attention than has been hitherto bestowed upon them. Every one knows that the strength of a state consists in the number of people. The encouragement of population, therefore, ought to be one of the first objects of policy in every state; and some of the worst enemies of population are the luxury, the licentiousness, and debility produced and propagated by great towns.
I have observed that London is now increasing. But it appears that, in truth, this is an event more to be dreaded than desired. The more London increases, the more the rest of the kingdom must be deserted; the fewer hands must be left for agriculture; and, consequently, the less must be the plenty and the higher the price of all the means of subsistence. Moderate towns, being seats of refinement, emulation, and arts, may be public advantages. But great towns, long before they grow to half the bulk of London, become checks on population of too hurtful a nature, nurseries of debauchery and voluptuousness; and, in many respects, greater evils than can be compensated by any advantages.
Dr. Heberden observes that, in Madeira, the inhabitants double their own number in 84 years. But this (as you, Sir, well know) is a very slow increase compared with that which takes place among our colonies in America. In the settlements, where the inhabitants apply themselves entirely to agriculture, and luxury is not known, they double their own number in 15 years; and all through the northern colonies in 25 years. This is an instance of increase so rapid as to have scarcely any parallel. The births in these countries must exceed the burials much more than in Madeira, and a greater proportion of the born must reach maturity. In 1738, the number of inhabitants in New Jersey was taken by order of the government, and found to be 47,369. Seven years afterwards the number of inhabitants was again taken, and found to be increased, by procreation only, above 14,000, and very near one half of the inhabitants were found to be under 16 years of age. In 22 years, therefore, they must have doubled their own number, and the births must have exceeded the burials 2000 annually. As the increase here is much quicker than in Madeira, we may be sure that a smaller proportion of the inhabitants must die annually. Let us, however, suppose it the same, or a 50th part. This will make the annual burials to have been, during these seven years, 1000, and the annual births 3000, or an 18th part of the inhabitants. Similar observations may be made on the much quicker increase in Rhode Island, as related in the preface to Dr. Birch’s Collection of the Bills of Mortality, and also in the valuable pamphlet, last quoted, on The Interest of Great Britain with Regard to Her Colonies, p. 36 [above, IX, 87-88]. What a prodigious difference must there be between the vigour and the happiness of human life in such situations, and in such a place as London? The original number of persons who, in 1643, had settled in New England, was 21,200. Ever since it is reckoned, that more have left them than have gone to them. In the year 1760 they were increased to half a million. They have, therefore, all along doubled their own number in 25 years; and, if they continue to increase at the same rate, they will, 70 years hence, in New England alone, be four millions; and in all North America above twice the number of inhabitants in Great-Britain. But I am wandering from my purpose in this letter. The point I had chiefly in view was, the present state of London as to healthfulness, number of inhabitants, and its influence on population. The observations I have made may, perhaps, help to shew how the most is to be made of the lights afforded by the London bills, and serve as a specimen of the proper method of calculating from them. It is indeed extremely to be wished that they were less imperfect than they are, and extended further. More parishes round London might be taken into them; and, by an easy improvement in the parish registers now kept, they might be extended through all the parishes and towns in the kingdom. The advantages arising from hence would be very considerable. It would give the precise law according to which human life wastes in its different stages, and thus supply the necessary data for computing accurately the values of all life-annuities and reversions. It would, likewise, shew the different degrees of healthfulness of different situations, mark the progress of population from year to year, keep always in view the number of people in the kingdom, and, in many other respects, furnish instruction of the greatest importance to the state. Mr. De Moivre, at the end of his book on the doctrine of chances, has recommended a general regulation of this kind; and observed, particularly, that at least it is to be wished, that an account was taken, at proper intervals, of all the living in the kingdom, with their ages and occupations; which would, in some degree, answer most of the purposes I have mentioned. But, dear Sir, I am sensible it is high time to finish these remarks. I have been carried in them far beyond the limits I at first intended. I always think with pleasure and gratitude of your friendship. The world owes to you many important discoveries; and your name must live as long as there is any knowledge of philosophy among mankind. That your happiness in this, and every other respect, may continually increase, is the sincere wish of, Sir, Your much obliged, and very humble servant,