And, in the fame manner, we shall find, that the Value of an Annuity on a life, for the limited time comprehended between the ages of 42 and 49, is worth 5.3492 years purchase.

Now, if it were required to determine the Value of an Annuity on life, to continue from the age of 42 to 70, we must proceed

thus:

It has been proved, that an Annuity on life, reaching from the age of 54 to 70, is worth 8.3365 years purchafe; but this Value, being estimated from the age of 49, ought to be diminished on two accounts: First, because of the Probability of the life's reaching from 49 to 54, which Probability is to be deduced from the Table of Obfervations, and is proportional to the number of people living at the end and beginning of that interval, which, in this cafe, will be found 302 and 357: The second diminution proceeds from a difcount that ought to be made, because the Annuity, which reaches from 54 to 70, is estimated 5 years fooner, viz. from the age of 49, and therefore that diminution ought to be expreffed by; fo that the total diminution of the Annuity of 16 years will be expreffed by the fraction 302 which will reduce it from 8.3365 years purchafe to 5.5259; this being added to the Value of the Annuity to continue from 49 to 54, viz. 4.0374, will give 9.5633, the Value of an Annuity to continue from the age of 49 to 70. For the fame reason, the Value 9.5633, eftimated from the age of 42, ought to be reduced, both upon account of the Probability of living from 42 to 49, and of the discount of money for 7 years, at 5 per cent. per annum, amounting together to 3.8554, which will bring it down to 5.7079; to this adding the Value of an Annuity on a life to continue from the age of 42 to 49, found before to be 5.3492, the fum will be 11.0571 years purchase, the Value of an Annuity to continue from the age of 42 to 70.

3575

In the fame manner, for the last 16 years of life, reaching from 70 to 86, when properly discounted, and alfo diminished upon the account of the Probability of living from 42 to 70, the Value of those last 16 years will be reduced to 0.8; this being added to 11.0571 (the Value of an Annuity to continue from the age of 42 to 70, found before), the fum will be 11.8571 years purchase, the Value of an Annuity to continue from the age of 42 to 86; that is, the Value of an Annuity on a life of 42; which, in my Tables, is but 11.57, upon the fuppofition of an uniform decrement of life, from an age given to the extremity of old-age, fuppofed at 86..

It

It is to be obferved, that the two diminutions, above-mentioned, are conformable to what I have faid in the Corollary to the second Problem of the first edition, printed in the year 1724.

Those who have fufficient leifure and skill to calculate the Value of joint Lives, whether taken two and two, or three and three, in the fame manner as I have done the first Problem of this tract, will be greatly affifted by means of the two following Theorems:

If the ordinate of a curve be; its area will be

I

If the ordinate of a curve be; its area will be

22

ar

N°. VII.

N°. VII.

The Probabilities of human Life, according to different Authors.

Table I, by Dr. Halley.

Age Living. Age Living. Age Living. Age Living. Age Living.Age Living.

O1400 Age Living. Age Living. Age Living. Age Living. Ag Living. Age Living.

Remarks on the foregoing Tables.

The first Table is that of Dr. Halley, compofed from the Bills of Mortality of the City of Breslaw; the best, perhaps, as well as the first of its kind; and which will always do honour to the judgment and fagacity of its excellent Author.

Next follows a Table of the ingenious Mr. KerfJeboom, founded chiefly upon Registers of the Dutch Annuitants, carefully examined and compared, for more than a century backward. And Monfieur de Parcieux by a like ufe of the Lifts of the French Tontines, or long Annuities, has furnished us Table III; whofe numbers were likewife verified upon the Necrologies or mortuary Registers of feveral religious houses of both Sexes.

To these is added the Table of Meffieurs Smart and Simpson, adapted particularly to the City of London; whofe inhabitants, for reasons too well known, are fhorter lived than the reft of mankind.

Each of these Tables may have its particular ufe: The Second or Third in valuing the better fort of Lives, upon which one would chuse to hold an Annuity; the Fourth may ferve for Londen, or for Lives fuch as those of its Inhabitants are fuppofed to be: while Dr. Halley's numbers, falling between the two Extremes, feem to approach nearer to the general courfe of nature. And in Cafes of combined Lives, two or more of the Tables may perhaps be usefully employed..

Befides thefe, the celebrated Monfieur de Buffon has lately given us a new Table, from the actual Obfervations of Monfieur du Pré de S. Maur of the French Academy. This Gentleman, in order to strike a just mean, takes three populous parishes in the City of Paris, and fo many country Villages as furnish him nearly an equal number of Lives and his care and accuracy in that performance have been fuch as to merit the high approbation of the learned Editor. It was therefore proposed to add this Table to the reft; after having purged its numbers of the inequalities that neceffarily happen in fortuitous things, as well as of those afifing from the careless manner in which Ages are given in to the parish Clerks; by which the years that are multiples of 10 are generally overloaded.

But this having been done with all due care, and the whole reduced to Dr. Halley's Denomination of 1000 Infants of a year old; there refulted only a mutual confirmation of the two Tables; Mr. du Pre's Table making the Lives fomewhat better as far as 39 years, and thence a small matter worfe than they are by Dr. Halley's.

We may therefore retain this laft as no bad ftandard for mankind in general; till a better Police, in this and other nations, fhall furnish + Hiftoire Naturelle, Tome II.

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