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Thus it appears that 61.26116, or 617. 55. 24d. is the premium which ought to be immediately paid to fecure 100l. on the decease of a perfon aged 45 years, at 3 per cent. per annum, according to the probability of life for London.

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 100l. on the decease of the faid perfon; interest at 3 per cent. per annum?

In this cafe, the value of the affurance in one present payment is to be found as in the foregoing case, 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 4/. 195. 7d. which is the fum to be paid annually during life, in order to secure the fum of 100l. at the extinction of the faid life.

If

If the foregoing questions be repeated, reckoning the intereft at 3 per cent. per annum, the premium will be lefs, viz. în one present payment at 3 per cent. it will be 57%. 11. and the annual payments 41. 195. 7d.

Thus it appears upon what very easy terms a large fum of money might be fecured at the decease of any perfon, 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 na ture, and from which there are hardly any of these institutions exempt.

Affurances of this nature might be extended confiderably more than they are at prefent; and rendered not only fubfervient to the parochial poor, but also of infinite advantage to the nation at large, particularly to the revenue in a financial respect, were the fubject to meet the approbation of the legislature; and perhaps more pecuniary affiftance 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 gross 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 fingle payment divided by the value of the life, increased by unity. Therefore, an affurance of an eftate or annuity, after any given life or lives, is worth as much more than the affurance of a correfponding fun, as tool. Increased by its interest for a year, is

greater

greater than 100l. Thus the present values, in fingle and annual payments, of the affurance of an estate of 51. per annum for ever, and of 100l. in money, are to one another as 105l. is to 100l. The reafon of the difference is, that the algebraical calculations, by which these 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 instruct any person in the method of finding the value of annuities, in all cases of reverfions: as alfo in the principles of affurances upon lives.

1

CHAP.

CHAP. XI.

OF LOGARITHMS.

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 such a manner, that o in the arithmeticals is the index of 1 in the geometricals. Logarithms were invented for the ease of arithmetical calculations, where the numbers, or operations, are large.

I

The nature of logarithms depends upon these axioms: if a series 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 decrease, according to the fame difference, 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 first; as in the following:

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

YOL. 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 firft term: 1 the logarithm of the fecond, which is 2; and 2 the logarithm 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:

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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 sets of logarithms, adapted to the fame common numbers, by varying the fecond term of the geometrical feries, as this will change the original series of terms, whofe indices are the numbers 1. 2. 3. &c. And by interpolation the whole system of numbers may be made to enter the geometric feries, and receive their proportionable logarithms, whether they be integers or decimals.

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, ris equal to 10; fo that the cominon logarithm of any number is the index of that power of 10, which is equal to the said number: thus the common logarithm of N=10", is n the

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