INTEREST

, is a sum reckoned for the loan or forbearance of another sum, or principal, lent for, or due at, a certain time, according to some certain rate or proportion; being estimated usually at so much per cent. or by the 100. This forms a particular rule in Arithmetic. The highest legal Interest now allowed in England, is after the rate of 5 per cent. per annum, or the 20th part of the principal for the space of a year, and so in proportion for other times, either greater or less. Except in the case of pawn-brokers, to whom it has lately been made legal to take a higher interest, for one of the worst and most destructive purposes that can be suffered in any state.|

Interest is either Simple or Compound.

Simple Interest, is that which is counted and allowed upon the principal only, for the whole time of forbearance.

The sum of the Principal and Interest is called the Amount.

As the Interest of any sum, for any time, is directly proportional to the principal sum and time; therefore the Interest of 1 pound for one year being multiplied by any proposed principal sum, and by the time of its forbearance, in years and parts, will be its Interest for that time. That is, if r = the rate of Interest of 1l. per annum, p = any principal sum lent, t = the time it is lent for, and <*> = the amount, or sum of principal and Interest; then is prt = the Interest of the sum p, for the time t, at the rate r; and consequently , the amount of the same for that time. And from this general theorem, other theorems can easily be deduced for sinding any of the quantities above mentioned; which collected all together, will be as follow: 1st, the amount, 2d, the principal, 3d, the rate, 4th, the time.

For example, let it be required to find in what time any principal sum will double itself, at any rate of Simple Interest. In this case we must use the 1st theorem , in which the amount a must be = 2p or double the principal, i. e. ; and hence ; where r being the interest of 1l. for one year, it follows that the time of doubling at Simple Interest, is equal to the quotient of any sum divided by its Interest for one year. So that, if the rate of Interest be 5 per cent. then is the time of doubling.

Or the 4th theorem immediately gives .

For more readily computing the Interest on money, various Tables of numbers are calculated and formed; such as a Table of Interest of 1l. for any number of years, and for any number of months, or weeks, or days, &c, and at various rates of Interest.

Another Table is the following, by which may be readily found the Interest of any sum of money, from 1 to a million of pounds, for any number of days, at any rate of Interest.

Multiply the principal by the rate, both in pounds; multiply the product by the number of days, and divide this last product by 100; then take from the Table the several sums which stand opposite the several parts of the quotient, and adding them together will give the interest required.

Ex. What is the interest of 225l. 10s. for 23 days, at 4 1/2 per cent. per annum?

Another ingenious and general method of com-| puting Interest, is by the following small but comprehensive Table.

N. B. This Table contains the interest of 100l. for all the several days in the 1st column, and at the several rates of 3, 3 1/2, 4, 4 1/2, and 5 per cent. in the other 5 columns.

To find the Interest of 100l. for any other time, as 1 year and 278 days, at 4 1/2 per cent. Take the sums for the several days as here below.

For any other Sum than 100l. First find for 100l. as above, and take it so many times or parts as the sum is of 100l. Thus, to find for 355l. at 4 1/2, for 1 year and 278 days.

First, 3 times the above sum,

When the interest is required for any other rate than thosein the Table, it may be easily made out from them. So 1/2 of 5 is 2 1/2, 1/2 of 4 is 2, 1/2 of 3 is 1 1/2, 1/3 of 3 is 1, 1-6th of 3 is 1/2, and 1-12th of 3 is 1/4. And so, by parts, or by adding or subtracting, any rate may be made out.

Compound Interest, called also Interest-upon-Interest, is that which is counted not only upon the principal sum lent, but also for its Interest, as it becomes due, at the end of each stated time of payment.

Although it be not lawful to lend money at Compound Interest, yet in purchasing annuities, pensions, &c, and taking leases in reversion, it is usual to allow Compound Interest to the purchaser for his ready money; and therefore it is very necessary to understand this subject.

Besides the quantities concerned in Simple Interest, viz, the principal p, the rate or Interest of 1l. for 1 year r, the amount a, and the time t, there is another quantity employed in Compound Interest, viz, the ratio of the rate of Interest, which is the amount of 1l. for 1 time of payment, and which here let be denoted by R, viz, . Then, the particular amounts for the several times may be thus computed, viz, As 1 pound is to its amount for any time, so is any proposed principal sum to its amount for the same time; i. e. the 1st year's amount, the 2d year's amount, the 3d year's amount, and so on. Therefore in general, is the amount for the t year, or t time of payment. From whence the following general theorems are deduced: 1st, the amount, 2d, the principal, 3d, the ratio, 4th, the time. From which any one of the quantities may be found, when the rest are given.

For example, suppose it were required to find in how many years any principal sum will double itself, at any rate of Interest. In this case we must employ the 4th theorem, where a will be = 2p, and then it is . So, if the rate of Interest be 5 per cent. per annum; then , and hence nearly; that is, any sum doubles in 14 1/5 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 facilitate the calculation of Compound Interest for any sum, and any number of years, at various rates of Interest.

The use of this Table, which contains all the powers R^t, to the 20th power, or the amounts of 1l. is chiefly to calculate the Interest, or the amount, of any principal sum, for any time, not more than 20 years. For example, required to find to how much 523l. will amount <*> 15 years, at the rate of 5l. per cent. per annum Compound Interest.

In the Table, on the line 15 and column 5 per cent,

See Annuities; Discount; Reversion; Smart's Tables of Interest; the Philos. Trans. vol. 6, p. 508; and most books on Arithmetic.

INTERIOR Figure, Angle of. See Angle.

Interior Polygon. See Polygon.

Interior Talus. See Talus.

Internal Angles, are all angles made within any figure, by the sides of it. In a triangle ABC, the two angles A and C are peculiarly called Internal and oppo- site, in respect of the external angle CBD, which i<*> equal to them both together.

Internal Angle is also applied to the two angles formed between two parallels, by a line intersecting those parallels, on each side of the intersecting line. Such are the angles a, b, c, d, formed between the parallels EF and GH, on each side of the intersecting line.—The two adjacent Internal angles a and b, or c and d, are together equal to two right angles.

Internal and Opposite Angles, is also applied to the two angles a and b, which are respectively equal to the two n and m, called the external and opposite angles.

Also the alternate Internal angles are equal to one another; viz, a = d, and b = c.