Thus it appears that 61.26116, or 611. 5s. 24d. is the premium which ought to be immediately paid to fecure 100%. on the decease of a person aged 45 years, at 3 per cent. per annum, according to the probability of life for London.

300

103)6309.9(61.26116 Answer.

CASE II.

When the Premium is to be paid in fixed annual Payments, during the whole Continuance of Life.

QUESTION 2. What money fhould be given in equal annual payments, during the life of a perfon aged 45 years, to fecure 100%. on the decease of the faid perfon; interest at 3 per cent. per annum?

In this cafe, the value of the affurance in one prefent payment is to be found as in the foregoing cafe, which value divided by the value of the life, quotes the fum to be paid annually during the life of the perfon:-Thus, 61.26116 divided by 12.3, quotes 4.98, or 47. 195. 7d. which is the fum to be paid annually during life, in order to fecure the fum of 100l. at the extinction of the faid life.

If the foregoing questions be repeated, reckoning the interest at 31 per cent. per annum, the premium will be less, viz. in one present payment at 31 per cent. it will be 577. 11. and the annual payments 44. 195. 7id.

Thus it appears upon what very easy terms a large fum of money might be secured at the decease of any person, if the premium be paid by annual payments. Hence the great advantage of inftitutions for the affurance of lives, provided they be properly conducted, and managed by perfons fufficiently skilled in numbers to avoid errors in making their calculations, which are most detrimental to focieties of this nature, and from which there are hardly any of these institutions exempt.

Afsurances of this nature might be extended confiderably more than they are at prefent; and rendered not only subfervient to the parochial poor, but also of infinite advantage to the nation at large, particularly to the revenue in a financial refpect, were the fubject to meet the approbation of the legislature; and perhaps more pecuniary affistance might be derived from establishments of this nature, under proper modifications, than from any other mode of funding, and creating permanent debts, as I fhall prove in another treatise.

When an eftate, or a perpetual annuity, is to be affured for the duration of another life, after the failure of the affured life, inftead of affuring a grofs fum, the value of a fingle payment will be the value of the life fubtracted from the perpetuity, and the remainder multiplied by the annuity, or by the rent of the estate. And the value in annual payments to begin immediately, will be the single payment divided by the value of the life, increased by unity. Therefore, an affurance of an estate or annuity, after any given life or lives, is worth as much more than the affurance of a correfponding fum, as 100l, increafed by its intereft for a year, is

greater

greater than 100l. Thus the prefent values, in fingle and annual payments, of the affurance of an estate of 51. per annum for ever, and of 100%. in money, are to one another as 1051. is to 100l. The reafon of the difference is, that the algebraical calculations, by which thefe values are determined, fuppofe the grofs fum and the first yearly payment of the annuity are to be received at the fame time, after the expiration of the life or lives.

The examples here given will be found fufficient to infruct any perfon in the method of finding the value of annuities, in all cafes of reverfions: as alfo in the principles of affurances upon lives.

CHAP.

SECT. I.

OF THE ORIGIN AND NATURE OF LOGARITHMS,

LOGARITHMS are certain artificial numbers, which are

the ratios of other natural numbers; and are the indices of the ratio of numbers to one another; or, a series of numbers in an arithmetical proportion, anfwering to as many others in a geometrical proportion, and in fuch a manner, that o in the arithmeticals is the index of 1 in the geometricals. Logarithms were invented for the eafe of arithmetical calculations, where the numbers, or operations, are large.

The nature of logarithms depends upon thefe axioms: if a feries of quantities increase, or decrease, according to the fame ratio, it is called a geometrical progreffion, as the numbers 1, 2, 4, 8, 16, 32, which are multiplied by 2: if the series or quantities increase, or decreafe, according to the fame differ ence, it is called an arithmetical progreffion, as the numbers 3, 6, 9, 12, 15, 18, &c. which increase by 3, which is therefore called their common difference. Now, if underneath the numbers proceeding in a geometrical progreffion, be placed as many other numbers, proceeding in an arithmetical progreffion, thefe laft are called the logarithms of the firft; as in the following:

Terms 1. 2. 4. 8. Logarithms o. 1. 2. 3. VOL. II.

16. 32. 64. 128. 256. 512.
4. 5. 6. 7. 8. 9.
L

In

In this progreffion, o is the logarithm of 1, the first term: 1 the logarithm of the second, which is 2; and 2 the loga"ithm of the third term, 4, &c.

These indices, or logarithms, may be adapted to any series in a geometrical progreffion; and, therefore, there may be as many different kinds of indices, or logarithms, as there can be different kinds of geometrical progreffions; as may be seen in the following feries :

Here the fame indices, or logarithms, ferve for any of the fix under-written geometrical feries, from which it appears, that there may be an endless variety of fets of logarithms, adapted to the fame common numbers, by varying the fecond term of, the geometrical feries, as this will change the origi nal series of terms, whofe indices are the numbers 1. 2. 3. &c. And by interpolation the whole fyftem of numbers may be made to enter the geometric feries, and receive their proportionable logarithms, whether they be integers or deci

mals.

The logarithm of any number is the index of fuch a power of fome other number, as is equal to the given one involved to the power denoted by the index of the other number. Thus, if N be equal to r", then the logarithm of N is n, which may be either pofitive or negative, and r any number whatever, according to the different fyftems of logarithms. When N is one, then n is o, whatever the value of r may be ; and, confequently, the logarithm of 1 is always o in every fyftem of logarithms. But in the common logarithms, r is equal to 10; fo that the common logarithm of any number is the index of that power of 10, which is equal to the faid number: thus the common logarithm of N=10", is the

3

index