4. Call the square of the third number the fourth number,

5. Divide the product of the second payment, and time between the payments, by the product of the first payment. and the rate, and call the quotient the fifth number.

must be a negative quantity; and consequently will be greater than t, that is, the equated time will fall beyond the second payment, which is absurd. The value of x therefore cannot

From this it appears, that the double sign, made use of by Mr. MALCOLM, and every author since, who has given his method, cannot obtain, and that there is no ambiguity in the prob. lem.

In like manner it might be shown, that the directions, usually given for finding the equated time, when there are more than two payments, will not agree with the hypothesis; but this may be easily seen by working an example at large, and examining the truth of the conclusion.

The equated time for any number of payments may be readily found when the question is proposed in numbers, but it would not be easy to give algebraic theorems for those cases, on account of the variation of the debts and times, and the difficulty of finding between which of the payments the equated time would happen.

Supposing r to be the amount of 11. for one year, and the othlog.ar +b

rem for the equated time of any two payments, reckoning compound interest, and is found in the same manner as the former.

6. From the fourth number take the fifth, and call the square root of the difference the sixth number.

7. Then the difference of the third and sixth numbers is the equated time, after the first payment is due.

EXAMPLES.

1. There is 1001. payable one year hence, and 1051. payable 3 years hence; what is the equated time, allowing simple interest at 5 per cent. per annum ?

2. Suppose 4001. are to be paid at the end of 2 years, and 21001. at the end of 8 years; what is the equated time for one payment, reckoning 5 per cent. simple interest?

Ans. 7 years,

3. Suppose 3001. are to be paid at one year's end, and 3001. more at the end of 1 year; it is required to find the time to pay it at one payment, 4 per cent. simple interest being allowed? Ans. 1.248437 year,

COMPOUND INTEREST.

COMPOUND INTEREST is that, which arises from the principal and interest taken together, as it becomes due, at the end of each stated time of payment.

RULE.*

1. Find the amount of the given principal for the time of the first payment by Simple Interest.

2. Consider this amount as the principal for the second

* The reason of this rule is evident from the definition, and the principles of Simple Interest.

payment, whose amount calculate as before, and so on through all the payments to the last, still accounting the last amount as the principal for the next payment.

EXAMPLES.

1. What is the amount of 3201. 10s. for 4 years, at 5 per cent. per annum, compound interest?

2. What is the compound interest of 7601. 10s. forborn 4 years, at 4 per cent. ? Ans. 1291. 3s. 61d.

3. What is the compound interest of 4101. forborn for 21 years, at 4 per cent. per annum; interest payable halfyearly? Ans. 481. 4s. 112d.

4. Find the several amounts of 501. payable yearly, halfyearly, and quarterly, being forborn 5 years, at 5 per cent. per annum, compound interest.

Ans. 631. 16s. 31d. 641. and 641. 1s. 91d.

COMPOUND INTEREST BY DECIMALS.

RULE.*

1. Find the amount of 11. for one year at the given rate per cent.

* DEMONSTRÁTION.

Let r = amount of 11. for one year, and ɲ = principal or given sum; then since r is the amount of

=

3

11. for one year, r2 will be its amount for 2 years, r3 for 3 years, and so on; for when the rate and time are the same, all principal sums are necessarily as their amounts; and consequently as r is the principal for the second year, it will be as 1:r::r amount for the second year, or principal for the third; and again, as 1 : r :: r2 : r 3 = amount for the third year, or principal for the fourth, and so on to any number of years. And if the number of years be denoted by t, the amount of 11. for t years will be r*. Hence it will appear, that the amount of any other principal sum p for t years is pr; for as 1: rt: :p : pr2, the same as in the rule.

If the rate of interest be determined to any other time than a year, as, 1, &c. the rule is the same, and then t will represent that stated time.

r amount of 11. for one year at the given rate per

Then the following theorems will exbibit the solutions of all the cases in compound interest.