interest, added to the principal sum, and accumulating every year. As, for instance, if the interest on the above-named sum of money for one year be £234 15s. 34d, compound interest means, that this interest, not being paid to the lender, becomes a new, or additional loan, to be added to the former principal sum; which principal, being thus increased for the second year, the interest is to be calculated thereon, by an entirely new statement and working: the third term being the original loan augmented by the year's interest. The result of this working, that is, the interest for the second year, will then, again, be to be added, and a new statement, and new working take place, for the third year; and, of course, for each of the succeeding years, until you come to the half year, when you will state if £100 produce 45s. that is, half a year's interest, what will the augmented principal sum produce? And thus do you find the compound interest on any sum, and at any rate per cent. per annum, for any number of years. (27) What is the compound interest on £13620 for three years and nine months, 34 per cent, per annum? To find the interest for the three quarters of a year, you will perceive, that you may adopt one of these two methods; that is, you may go on with the statement and working, to a fourth year; and, having ascertained that, by deducting from it one quarter, you have the interest for the three quarters of the year: the other method is, that which is pursued in the former question; namely; take the interest on £100 for the three quarters of a year, as the middle term, and state the question thus, if £100 £2 16s. 3d., what will the augmented principal sum produce? : (28) What is the compound interest on £5217 for 24 years; interest at 4 per cent. per annum. payable half yearly? Here is the same principal sum, and same rate of interest, as in questions 25 and 26, but for just half the time, and the interest made payable every half year; which, is, in other words, a stipulation that the interest shall be calculated, and added to the principal, at the end of each half year; and not at the end of the year, as in the former case; a mode of reckoning which, as you will find, makes a difference in favour of the lender of the money, of no less than £2 12s. 9 d. in the first year; a trifle which people who put out money to interest, are by no means in the habit of disregarding; and the consequence is, that the interest on money so hired, is generally reckoned and added to the principal at the end of each half year, and so it goes, rolling on, enriching the already wealthy lender, and impoverishing the indiscreet, or unfortunate borrower. The statements will, of course, be to be made in this manner, as £100 is to half a year's interest, that is, to £2 5s. Od., so is the principal sum lent, to its interest. And so you go on for each half year, increasing the principal each time by the addition of the interest. (29) What is the compound interest on £13620, for two years, at 33 per cent, per annum, payable half yearly? This question, as to principal and rate of interest, is the same as that numbered 27. And thus I use the same sums, nearly, in order that the difference between calculating interest yearly, and half yearly, may be clearly seen. This difference will be manifest on a comparison of the answer to this question, with the result of the second operation on question 27. (30) What is the interest on £4217 12s. 6d., for 60 days, at 5 per cent, per annum? Five per cent per. ann. That is, £5 for £100, for the term of 365 days. But we would know, what is the interest on this money, at this rate, for 60 days? Let us find the interest on £100 for this time, and then we shall find the rest. If 365 days have £5, what will 60 days have? Suppose the answer be, that 16s 54d is the interest of £100, at this rate, for 60 days. We may then say, If £100 : 16s 5d :: £4217 12s 6d, and this brings the answer to the question. In like manner, may you find the interest on any sum, at any rate, for any number of days; and this is the Rule-of-Three method, the basis of all the other methods of computing such things. There are shorter methods; but, besides that it would take a considerable space in the book to describe them, my pupils would, I hope, forget them, in the pursuit of more valuable knowledge, and in the practice of more useful occupations than that of calculating interest of money; for the doing of which, there are Tables ready constructed, for those who have frequent occasion for such things. Another question,, or two may be useful.. (31) What is the interest on £3715, for 54 days, and for 73 days, at 4 per cent, per annum ? These questions on interest are useful, not only because there is no settling affairs of our own, nor those of others, without an occasional recourse to them; but they are useful, too, in this place, because they afford very fine practice for the learner, in this most valuable rule. I must not, however, close this series of instructions on the reckoning of interest, without giving, on the space I have here to spare, some intimation of my serious disapprobation of the practice of lending out money on such terms. The practice is so general, that we think not of the consequences, and have all forgotten that it has ever been interdicted.. Were accumulation, indeed, the sole end of our existence; did virtue consist in "the heaping up of riches," and true piety in the worshipping of them and their possessors, we should all now be right, in this country; the wisest, the most virtuous, most pious people that the sun ever shone upon; and the "grinding of the faces of the poor," by the already overgrown capitalist, would be meritorious. But, as wealth will neither save its possessors from private, nor from public calamities; as it will secure to a people neither power nor freedom; nor health nor happiness to an individual; but, on the contrary, as it invites the conqueror, and avenges the wrongs of those from whom it is unduly drawn, by afflicting the greedy with numerous ailments, both of body and of mind; as it is manifestly the will of the Most High, that such shall be the consequences of excessive accumulation; so has usury, that is, "the lending for gain," which is the great means of accumulation, been forbidden, and held in wellmerited abhorrence, by the wise and the good, in all ages and nations. (32) What is the present payment on £6075 10s. payable, that is, due in twelve months, discount at the rate of 5 per cent, per annum? Now, mark, this discount is very like interest; so like, indeed, that they are very commonly confounded, and, as interest is a little more than discount, so, those who pay money, generally avail themselves of the near resemblance, and take that which is a little more advantageous to them. There is, however, this difference; INTEREST is a charge for the use of a sum of money, and is to be paid AFTER the use of it has been had, at the stipulated time; whilst DISCOUNT is an allowance made for it, on its being paid BEFORE it becomes due: that is to say, it is a payment of interest beforehand; which payment, therefore, according to the rules of usury itself, ought to be something less than when it is deferred. The difference, for instance, between interest and discount, on £100, payable in one year, at 5 per cent, per annum, is this; the interest to be paid at the end of the year, being £5, the discount paid on the commencement is £4 15s. 24d.: that is 4s. 94d. less: being exactly the interest on the discount, for the time that it is paid in advance; and this difference, as you see, amongst those who love money, is a thing not to be overlooked.-To find this discount on £100 payable in a year, at 5 per cent. per ann, according to our rule, you state it thus, If £105, due after the lapse of a year, may now be paid by £100, what is the present worth of £5? And the foregoing question, that is, No. 32, is to be stated thus, If £105 : £100 :: 6375 10s Od. (33) What is the present payment for £5760, payable in half a year, discount at 7 per cent, per annum ? Do not fail to observe, that the question here is, not as to the amount of money to be paid a year before it becomes ude, but only half a year before; we must, therefore, take only half of the rate per cent per ann. that is £3 15s, and adding it to £100 for the first term, we state the question thus, As £103 15s due in six months, may now be paid by £100, what is the present payment for £5760 ? OF DOUBLE RULE OF THREE, Or Rule of Five; AND. OF COMPOUND PROPORTION, Or the CHAIn-Rule. 301. The proportional numbers of which we have yet treated, exist in pairs, or, as we have called them, couplets; and the business of the Rule of Three, as has been stated, is this; that having one pair of terms, and a single, or odd term, we thereby find a fourth, which, bearing a certain proportion to the others, completes the second couplet. And this, which is almost the simplest office of the rule of proportion, is called, when it is to be distinguished from more complex operations of a similar kind, and of which I am now about to treat; this more simple form of the rule of proportion is called, Single Rule of Three. 302. As there is Single, so you will justly infer that there is DOUBLE RULE OF THREE. And, as the office of the single consists in finding a fourth term, and thus completing a single pair of proportional terms, so that of the double rule, is to find a sixth term, which shall bear a required proportion to five terms previously ascertained. For example, the questions which this rule determines are of this nature;-If 12 men can build a wall 200 yards long, and 16 feet high in 20 days, how many men can build a similar wall, but 300 yards long, in 30 days? Or; if a family of 14 persons expend £42 per month, of thirty days; how much, at the same rate, will a family of 21 persons spend in 45 days?— Questions of this sort, and a great variety of others, having, as these have, five terms, which five terms form an essential part of the question; with questions of this sort, it is, that the Double Rule of Three; or, as it is sometimes called, the Rule of Five, has to deal. And it is by a sort of blending of several terms together, into two distinct sums, and a division of one of those sums by the other, that the question is resolved: much in the same manner as in the Rule of Three, in which, by multiplying, we blend certain of the terms together, and divide the product thereof by the other term, and so obtain the term sought. 303. The questions I have stated above are of that very simple form, and I have purposely chosen them so, in order that you may see the process more clearly; they are of so simple a form, that you can answer them without working them with a pen: and this simplicity will enable you to see, that the process I am about to describe, leads you to the right answer. Let us, then, work the second of them, and trace the working in such a manner, as to lead clearly into the principle on which the rule is founded, by which rule such questions as these are stated and worked. The question is; If a family of 14 persons expend £42 in 30 days, how much, at the same rate, will a family of 21 persons spend in 45 days 304. Now, in the first place, let us see how much £42 allows to each of the 14 persons. We find that it is £3 for each. Next, this money is to serve each person 30 days; and this, we find, is 2s. per day for each person; and this we have found, you must bear in mind, by dividing the money, which, |