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 It will be found that the author has been furnished with facts and observations of great value, and that he has endeavoured to present the information they afford, in the forms best calculated for the further prosecution of these inquiries.

In treating of annuities, we think that it may be useful in a work of this kind, to address ourselves as well to those readers who have not, as to those who have, an acquaintance with Algebra; and we shall, accordingly, divide what follows into two Parts, corresponding to these two views of the subject.  PART I.

shall, in this Part, demonstrate all that is most useful and important in the doctrine of Annuities and Assurances on lives, without using Algebra, or introducing the idea of probability; but the reader is, of course, supposed to understand common Arithmetic. In the first 30 numbers of this Part, Compound Interest and Annuities-certain are treated of; from the 31st to the 76th, the doctrine of Annuities on Lives is delivered; and that of Assurances on Lives, from thence to the 108th, where the popular view terminates.

What is demonstrated in this Part, will be sufficient to give the reader clear and scientific views of the subjects treated; and, with the assistance of the necessary tables, will enable him to solve the more common and simple problems respecting the values of Annuities and Assurances. He will also understand clearly the general principles on which problems of greater difficulty are resolved; but these he cannot undertake with propriety, when the object is, to make a fair valuation of any claims or interests, with a view to an equitable distribution of property, unless he has studied the subject carefully, with the assistance of Algebra; for intricate problems of this kind can hardly be solved without it; and those who are not much exercised in such inquiries, often think they have arrived at a complete solution, when they have overlooked some circumstance or event, or some possible combination of events or circumstances, which materially affect the value sought. Eminent Mathematicians have, in this way, fallen into considerable errors, and it can hardly be doubted, that those who are not mathematicians, must (cæteris paribus) be more liable to them.

I. ON ANNUITIES-CERTAIN.

No. 1. When the rate is 5 per cent., L. 1 improved at simple interest during one year, will amount to L. 1·05; which, improved in the same manner during the second year, will be augmented in the same ratio of 1 to 1·05; the amount then, will therefore be $$1{\cdot}05 \times 1{\cdot}05$$, or $$(1{\cdot}05)^2 = 1{\cdot}1025$$.

In the same manner it appears, that this last amount, improved at interest during the third year, will be increased to $$(1{\cdot}05)^3 = 1{\cdot}157625$$; at the end of the fourth year, it will be $$(1{\cdot}05)^2$$; at the end of the fifth $$(1{\cdot}05)^5$$, and so on; the amount at the end of any number of years being always determined, by raising the number which expresses the amount at the end of the first year, to the power of which the exponent is the number of years. So that when the rate of interest is 5 per cent., L. 1 improved at compound interest, will, in seven years, amount to $$(1{\cdot}05)^7$$, and in 21 years, to $$(1{\cdot}05)^{21}$$.

But if the rate of interest were only 3 per cent., these amounts would only be $$(1{\cdot}03)^7$$, and $$(1{\cdot}03)^{21}$$ respectively.

2. The present value of L. 1 to be received certainly at the end of any assigned term, is such a less sum, as, being improved at compound interest during the term, will just amount to one pound. It must therefore be less than L. 1, in the same ratio as L. 1 is less than its amount in that time; but in three years, at 5 per cent., L. 1 will amount to L. $$(1{\cdot}05)^3$$ (1). And $$(1{\cdot}05)^3 : 1 :: 1 : \frac{1}{(1.05)^3}$$, so that $$\frac{1}{(1.05)^3} = \frac{1}{1{\cdot}157625} = 0{\cdot}863838$$ is the present value of L. 1 to be received at the expiration of three years.

In the same manner it appears that, at 4 per cent. interest, the present value of L. 1 to be received at the end of a year, is $$\frac{1}{1{\cdot}04} = 0{\cdot}961538$$; and if it were not to be received until the expiration of 21 years, its present value would be $$\frac{1}{(1{\cdot}04)^{21}} = (0{\cdot}961538)^{21} = 0{\cdot}438834$$.

Hence it appears, that if unity be divided by the amount of L. 1, improved at compound interest during any number of years, the quotient will be the present value of L. 1 to be received at the expiration af the term: which may also be obtained by raising the number which expresses the present value of L. 1 receivable at the expiration of a year, to the power of which the exponent ts the number of years in the term.

3. When a certain sum of money is receivable annually, it is called an, and its quantum is expressed by saying it is an annuity of so much; thus, according as the annual payment is L. 1, L. 10, or L. 100; it is called an annuity of L. 1, of L. 10, or of L. 100.

4. When the annual payment does not depend upon any contingent event, but is to be made certainly, either in perpetuity or during an assigned term, it is called an Annuity-certain.

5. In calculating the value of an annuity, the first payment is always considered to be made at the end of the first year from the time of the valuation, unless the contrary be expressly stated.

6. The whole number, and part or parts of one annual payment of an annuity, which all the future payments are worth in present money, is called the number of years purchase the annuity is worth; and,  Rh