might be more readily determined.-This he gives, without any demonstration, in the following words: There is a law of motion, which holds in the cafe, where a body is deflected by two forces, tending conftantly to two fixed points;' which is, that the body, in such a cafe, will defcribe, by lines drawn from the two fixed points, equal folids in equal times, about the line joining the faid fixed points.'

The article before us contains a demonstration of this law; of which it is fufficient to add, that it was communicated by that excellent mathematician the late William Jones, Efq; to Mr. Robertfon, who apprehended it to be highly worth preferving in the Philofophical Tranfactions.

Article 16. Obfervations on the Expectations of Lives, the Increafe of Mankind, the influence of great towns on population, and particularly the fate of London with respect to healthfulness, and number of inhabitants. In a letter from Mr. Richard Price, F. R. S. to Benjamin Franklin, Efq; LL. D. and F. R. S.

The ingenious Author of thefe obfervations, who has few fuperiors, and perhaps not many equals in difquifitions of this nature, proposes chiefly to confider the prefent ftate of the city of London, with respect to healthfulness and number of inha bitants, as far as it can be collected from the bills of mortality. Though this be the main fubject which he undertakes to difcuís, he has thrown out feveral incidental obfervations which are well worthy of notice. We fhall endeavour to give fuch an abstract of his inftructive and entertaining paper, as may lead our Readers to form fome judgment of its defign and execution and, while we would avoid extending this article to an immoderate length, it will be our wifh to do the Author all the juftice which our limits will allow.

The first object of his attention, the meaning of which he proposes accurately to determine, is that which writers on the fubject of annuities have called the expectation of life and this is the more neceflary, becaufe many have either entertained wrong notions, or failed to express themselves with proper precifion on this head. Mr. De Moivre himself has not fufficiently guarded his readers from miftakes. The expectation of life, according to this Author, is that which Mr. Simpson and Mr. De Moivre have called the fhare of life due to a perfon, and fignifies the mean continuance of any given fingle, joint or furviving lives, according to any given table of obfervations: that is, the number of years, which, taking them one with another, they actually enjoy, and may be confidered as fure of enjoying; thofe who live beyond that period, enjoying as much more time in proportion to their number, as thofe who fall fhort of it enjoy lefs. Thus, fuppofing 46 perfons alive, all 40 years of age, and that, according to Mr. De Moivre's hypothefis of an equal

decrement

decrement of human life through all its ftages, one will die every year till they are all dead in 46 years, half 46 or 23 will be their expectation of life; 46 being, by the aforementioned hypothefis, the complement of life, or what it wants of 86 the utmost probable extent of life. In like manner, the 3d of 46, or 15 years and four months, is the expectation of two joint lives, both 40: and the number expreffing this period, multiplied by the number of fingle or joint lives to which the expectation belongs, added annually to a fociety or town, gives the whole number to which such an annual addition would in time grow. Whence it appears, that the particular proportion, which becomes extinct every year, out of the whole number conftantly exifting together of fingle and joint lives, muft, wherever this number undergoes no variation, be exactly the fame with the expectation of those lives, at the time when their existence commenced, e. g. If it was found in any town, where the number of births is equal to that of the burials, that a 20th or a 30th part of the inhabitants die annually, 20 or 30 would be the expectation of a child juft born in fuch a town. These expectations are easily found for all fingle lives, by a table of oblervations, according to a general rule given for that purpose. The expectation of a life of 20 is, by Mr. Simpson's table, formed from the bills of mortality in London equal to 28, 9.

The Author, having premifed thefe general remarks, proceeds to the principal point he has in view. The expectation of an infant juft born, in London, is, by Mr. Simplon's table, 20 years. This number, multiplied by the yearly births when they are equal to the burials, gives the number of inhabitants in London. The medium of yearly births, for the last 10 years, has been 15,710. This number, multiplied by 20, gives 314,200, which is the number of inhabitants in London, on the fuppofition that it fupported itself without any fupply from the country but fince the burials have, at an average for the last 10 years, been 22,756, and therefore exceeded the chriftenings by 7,246, there must be a yearly addition of this number from other parts of the kingdom to recruit the wafte. Suppofe these then to be all of the age of 18 or 20 years, and therefore their expectation equal to 30 years, 30 multiplied by 7,246 gives 217,380, which must be added to the former number, and the fum, or 531,580, fhews the number of inhabitants in London, were the bills perfect. But thefe give too small a number both for the births and burials; allowance muft therefore be made for the deficiencies in the bills, and likewife for those who migrate and die out of the city. The Author allows for thefe 6000 in the births and 6000 in the burials. This multiplied by 20, the expectation as before stated, gives 120,000; which, added

to the foregoing number, gives 651,580 for the number of in habitants complete.

The Author then fhews how to allow for an inequality in the births and burials; and makes several calculations on the fuppofition, that the defect is, in any proportion, either on the one fide or the other: and he concludes, upon the whole, that 651,580, though fhort of the number of inhabitants commonly fuppofed in London, is very probably greater, but cannot be much lefs, than the true number.

Dr. Brakenridge makes it 751,800: but our Author apprebends that in both his methods of eftimating, the Doctor proceeds on wrong principles. He determines, fays the Author, the number of houfes by a method too precarious to be depended on; and then allows 6 perfons to a house, which is undoubtedly too large an allowance; 5 being, at an average, an allowance large enough for London, and too large for Eng land in general. By the fame reasoning it appears, as he fuggefts in a note, that the number of people in England, which the Doctor ftates at 5,340,000, ought to be reduced to 4,450,000. The other method by which Dr. Brakenridge proposed to determine the number of inhabitants in London, was from the annual number of burials, adding 2000 to the value for omiffions, and fuppofing a 30th part to die every year. This, which the Doctor apprehends a moderate fuppofition, our Author plainly fhews to be much too large, and at the fame time points out the ground of his mistake,

Dr. Brakenridge obferved, that, at the time of his calculation, the number of inhabitants in London was 127,000 less than it had been. The bills have lately advanced, but ftill they are much below what they were from 1717 to 1743. And our Author finds by calculation, that, taking the medium of annual birth and burials for 20 years, viz. from 1716 to 1736, the number of inhabitants in London was 84,260 greater than it is at prefent: fo that London has, for the laft 30 years, been decreasing; and though now it is increasing, he very juftly apprehends, that the additions lately made to the number of buildings round it are owing, in a great meafure, to the increafe of luxury, and the inhabitants requiring more room to live upon. And he fhews, by taking the medium of annual burials for feveral years, from 1655 to 1768, in the 97 parishes within the walls, that, though fince that period London has doubled its inhabitants, yet, within the walls, they have decreased; and fo rapidly for the last 30 years, as now to be reduced to one half. The prefent increase of London, were it long to continue, is an event more to be dreaded than defired. The more London increases, unless general population should

increase

increase likewife, the more the reft of the kingdom must be deferted; the fewer hands must be left for agriculture; and, confequently, the lefs must be the plenty and the higher the price of all the means of fubfiftence, Moderate towns, being feats 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 compenfated by any advantages.

The diminution of the number of the present inhabitants of London, compared with those of 40 years ago, is ftated on the fuppofition that the proportion of the omiffions in the births to thofe in the burials was the fame then as it is now. But this appears not to be fact.-These omiffions were greater formerly, which makes the difference in the number of inhabitants lefs confiderable than it seems to be from the face of the bills. The reason of this difference the Author suggests in two or three particulars. He obferves, that it is probable that London is now become less fatal to children than it was. But notwithstanding the great improvements in London, fince the year 1752, and other circumstances which very much contribute to the prefervation of children, and the health of its inhabitants, he proceeds to fhew the amazing difference, efpecially to chil dren, between living in great towns and in the country. In London, according to the most moderate computation, half the number born, die under three years of age; in country villages and parishes, the major part live to mature age, and even to marry. It appears from the account given by Dr. Heberden, and published in the Philofophical Tranfactions, vol. lvii. that in the ifland of Madeira, about two thirds of all who are born live to be married; fo that the expectation of a child just born, is about 39 years, nearly double that of a child just born in London. In Madeira, a 50th part only of the inhabitants die annually. In London, above twice this proportion die annually. In the four provinces of New England, there is a very rapid increase of the inhabitants; notwithstanding which, at Bofton, the capital, the inhabitants would decrease, were there no fupply from the country; for, from 1731 to 1762, the burials have all along exceeded the births. So remarkably, the Author obferves, do towns, in confequence of their unfayourableness to health, and the luxury which generally prevails in them, check the increa of countries. Healthfulness and prolificness are, probably, causes of increase that are seldom feparated.

In conformity to this obfervation it appears, from conparing the births and weddings in countries and towns, where registers

of

of them have been kept, that in the former, marriages, one with another, feldom produce lefs than four children each; generally between four and five, and fometimes about five. But in towns feldom above four, generally between three and four, and fometimes under three.

Dr. Heberden obferves, that in Madeira the inhabitants double their own number in 84 years: but that is a flow increase compared with that which takes place amongst our colonies in America. In the back fettlements they double their number in 15 years; and through the northern colonies in 25 years. In New Jerfey the inhabitants double their own number in 22 years. In New England the original number of settlers, in 1643, was 21,200: they have had no extra-acceffion fince; yet, in the year 1760, they were increased to half a million. They have therefore all along doubled their number in 25 years; and, if they continue to increafe at the fame rate, they will, 70 years hence, in New England alone, be four millions; and, in all North America, above twice the number of the inhabitants in Great Britain.

All these are observations of very confiderable importance: were they properly attended to they would, in all likelihood, be the means of producing a new fyftem of political arithmetic. They would, at leaft, recommend pacific and healing measures: as prudence and fafety forbid our infulting and provoking a growing power.

The Author concludes with wifhing that the London bills were more perfect and extenfive than they are. (In their prefent imperfect state they can admit of no very accurate and fatisfactory conclufions.) Great advantages would arife from including more parishes, and from diffuting registers throughall the towns and parishes in the kingdom. We should hence derive the neceffary data for computing accurately the value of all life annuities and reverfions. It would enable us to judge of the different degrees of healthfulness of different fituations; to mark the progrefs of population from year to year; to keep always in view the number of people in the kingdom; and, in many other refpects, furnish inftruction of the greatest importance to the ftate.

Mr. De Moivre recommended a regulation of this kind, and expreffed his defire that an account were taken, at proper intervals, of all the living in the kingdom, with their ages and occupations. This would, in fome degree, anfwer the purposes which our Author has mentioned.

We cannot help expreffing our concern, that the principles on which the Author's reafoning is founded, are not as folid and fatisfactory as the reafoning itself. We are fully perfuaded that the method here propofed of eftimating the number of inhabi

tants