Here, r being the interest of 11. for 1 year, it follows, that the doubling at simple interest, is equal to the quotient of any sum divided by its interest for 1 year. So, if the rate of interest be 5 per cent. then 100 5 20, is the time of doubling at that rate. Or the 4th theorem gives at once =3 the principal sum, r the rate or interest of 1. for 1 year, a = the whole amount of the principal and interest, there is another quantity employed in Compound Interest, viz. the ratio of the rate of interest, which is the amount of 1. for 1 time of payment, and which here let be denoted by R, viz. R = 1 + r, the amount of 17. for 1 time. Then the particular amounts for the several times may be thus computed, viz. As il. is to its amount for any time, so is any proposed principal sum, to its amount for the same time; that is, as 14: R:: : :ɲR, the 1st year's amount, R2, the 2d year's amount, R3, the 3d year's amount, Therefore, in general, Rt a is the amount for the t year, ort time of payment. Whence the following general theorems are deduced: 1st, a = · Rt, the amount, From which, any one of the quantities may be found, when the rest are given. As to the whole interest, it is found by barely subtracting the principal from the amount ɑ. Example. Suppose it be required to find, in how many years any principal sum will double itself, at any proposed rate of compound interest. In this case the 4th theorem must be employed, making a = 2; and then it is, log.a-log. log. R log. 2p-log.f log. R log. 2 log R So, if the rate of interest be 5 per cent. per annum ; then R=105 = 105; and hence log. 2 ⚫301030 log. 1 05 021189 14 2067 nearly; that is, any sum doubles itself in 14 years nearly, at the rate of 5 per cent. per annum compound interest. Hence, and from the like question in Simple Interest, above given, are deduced the times in which any sum doubles itself, at several rates of interest, both simple and compound; viz. The following Table will very much facilitate calculations of compound interest on any sum, for any number of years, at various rates of interest. The Amounts of 17. in any Number of Years. The use of this Table, which contains all the powers, R2, to the 20th power, or the amounts of 17. is chiefly to calculate the interest, or the amount of any principal sum, for any time, not more than 20 years. For example, let it be required to find, to how much 5231. will amount in 15 years, at the rate of 5 per cent. per annum compound interest. In the table, on the line 15, and in the column 5 per cent. is the amount of 17. viz. this multiplied by the principal gives the amount or and therefore the interest is Note 1. 2.0789 523 When the rate of interest is to be determined to 1087-2647 10871. 58. 31d. 564l. 58. 31d. a year, or express a any other time than a year; as suppose to express that time, and R must be taken the amount for that time also. Note 2. When the compound interest, or amount, of any sum, is required for the parts of a year; it may be determined in the following manner: 1st, For any time which is some aliquot part of a year :Find the amount of 11. for 1 year, as before; then that root of it which is denoted by the aliquot part, will be the amount of 11. This amount being multiplied by the principal sum, will produce the amount of the given sum as required. 2d, When the time is not an aliquot part of a year Reduce the time into days, and take the 365th root of the amount of 17. for 1 year, which will give the amount of the same for day. Then raise this amount to that power whose index is equal to the number of days, and it will be the amount for that time. Which amount being multiplied by the principal sum, will produce the amount of that sum as before. And in these calculations, the operation by logarithms will be very useful. OF ANNUITIES. ANNUITY is a term used for any periodical income, arising from money lent, or from houses, lands, salaries, pensions, &c. payable from time to time, but mostly by annual payments. Annuities are divided into those that are in Possession, and those in Reversion: the former meaning such as have commenced; and the latter such as will not begin till some particular event has happened, or till after some certain time has elapsed. When an annuity is forborn for some years, or the pay. ments not made for that time, the annuity is said to be in Arrears. An annuity may also be for a certain number of years; or it may be without any limit, and then it is called a Perpetuity. The Amount of an annuity, forborn for any number of years, is the sum arising from the addition of all the annuities for that number of years, together with the interest due upon each after it becomes due. The The Present Worth or Value of an annuity, is the price or sum which ought to be given for it, supposing it to be bought off, or paid all at once. Let a the annuity, pension, or yearly rent; = n = the number of years forborn, or lent for; m the amount of the annuity; vits value, or its present worth. Now, I being the present value of the sum R, by proportion the present value of any other sum a, is thus found: a as R : 1 :: a: the present value of a due 1 year hence. In like manner R is the present value of a due 2 years, be the present values of a, due at the end years respectively. Consequently the sum R3 R a 1 + &c. = (− +· + R4 - R R2 of 3, 4, 5, &c. of all these, or continued to n terms, will be the present value of all the n years' annuities. And the value of the perpetuity, is the sum of the series to infinity. But this series, it is evident, is a geometrical progression, 1 having both for its first term and common ratio, and the R number of its terms n; therefore the sum v of all the terms, or the present value of all the annual payments, will be When the annuity is a perpetuity; n being infinite, R is |