a arn The limit of the sum of a series is a number, to which the sum of a finite number of terms becomes more and more nearly equal, as the number of terms is increased; but to which the sum is never actually equal, however many terms be taken. The algebraical expression for the sum of a geometric series, whose first a - apni term is a, and common ratio r, is which = 17 1- 1Now if r be a proper fraction, ru may be made as small as we please, and arp therefore may be made as small as we please, by increasing n. Hence 1 the sum of an infinite decreasing geometric series continually approximates, though no finite number of terms can ever equal, the quantity Therefore the limit of the sum of such a series is the ratio of the first term to the difference between 1 and the common difference. 1一个 Prop. 84.- To prove and explain the Rule for finding the time at which several sums due at different times may be paid together. The principle on which the ordinary rule is founded is the following. It is said that the debtor, by holding the debts in his hands, gains the advantage of the interest on them for the several times, for which he is allowed to hold them; and that therefore, if he pay them all at one time, this time should be determined by the condition that he should still gain the same aggregate advantage as before. In other words, it is said that the sum of the interest on the several debts for the times, at which they are due, ought to be equal to the interest on the sum of the debts for the equated time. This implies that the sum of the interest on the debts paid after they are due should be equal to the sum of the interest on the debts paid before they are due, for the difference of the times at which they are, and ought to be, paid; and also that the sum of the amounts of the several debts for the difference between the longest time and their own times, should be equal to the amount of the sum of the debts for the difference between the same time and the equated time. On these principles, if £ S1, £S2, £ S3, &c. be several sums due at times T1, T2, T3, &c. terms respectively, and be the interest on £1 for one term, and t the number of terms in the equated time, we have s. T S, Ti + S, T, + S, T, + &c. whence t = Si +52 +5, + &c. which forinula expresses the ordinary rule. t But with reference to these principles, it may be said, that the debtor by paying at the equated time gains the full advantage of leaving the debts unpaid their own times, sooner than he would in this case do. And again, by paying a debt before it is due, the debtor loses not the interest but the discount, and therefore the interest on the debts paid after they are due, should be equal to the discount on those paid before they are due, which would give the equated time smaller than the above formula. But the ordinary Rule is sufficiently exact for practical use. See note. Prop. 85.—To prove the Rule for finding the amount of an annuity. If A be the number of pounds in the annuity, n the number of terms for which it is to be paid, r the interest on £1 for 1 term, then the amount of the 1st payment, which is forborne for n - 1 years is A (1+r) - 1; that of the second payment is A (1+r)n—?; and so on; that of the last is A. Hence the sum of the amounts. =A{(1+mn-i +(1+x)= – 4 + &c. +1} (1+r)n - 1 (1 + r)n - 1 (Prop. 81.) (1+r) — 1 which expresses the Rule, (1 + r)n being the amount of £l at Compound interest. Prop. 86.- To prove the Rule for finding the present value of an annuity. Let P be the number of pounils in the sum, which, put out to interest, will be sufficient to pay the annụity A for n years. Then Principal for 1st year = P 2nd year = P(1+r) — A or nth year = P(1+r)n-1-A(1+r)n – 2-A(1+r)n-1-&c.-A the amount of this last should be just sufficient to pay the nth annuity; P(1+r)" — A (1 + r)n-1 - A (1+r)n – 2 – &c. – A=0 (1+r)n - 1 .: P(1+r)n – A X =0 (1+r) — 1 (1 + r)n 1 P(1+r)n = AX (1 + rn-1 P= AX (1 + r)n Xr { r which expresses the Rule, and shows that P is such a sum, as put out at Compound Interest would amouut to the amount of the annuity. 1 Cor. 1. PEA which as n increases approximates r(1+r)" A A to —; so that is the limit of the value of the annuity, as the number of years for which it is paid is indefinitely increased. Hence if the annuity be A perpetual P =-. Cor. 2. This Prop. enables us to find the value of a leasehold, or the fine payable for a lease of any number of years, this being the value of an annuity equal to the difference between the rack-rent and the lease-rent. Also Cor. 1 gives the value of a freehold, which is a perpetual annuity. Prop. 87.--To prove the Rule for finding the value of a deferred annuity. The value of an annuity to commence after m years, and to continue n years, is evidently the difference between the values of the annuity for m+n years, and for m years. Hence if P be the present value, (1 + r)m +1 -1 (1 + r)m P=AX (1+r)mto xn A 1 Х (1 + ron (1+r)m+n) which expresses the Rule. Cor. This Prop. enables us to find the value of Reversionary Annuities. 1 {at r : : Prop. 88.—To prove the Rule for finding the annuity, which can be purchased for a given sum. By Prop. 86 it appears, that the present value varies as the annuity, and hence the purchasable annuity varies as the sum invested. Therefore if £A be the annuity, which may be purchased with a sum, £ P, and £pi be the present value of £1 annuity, £P £A £1 P .. £A = £ Pi Prop. 89.— To explain the methods of working questions in Exchanges. 1st. Let it be required to determine how much coin of one country is equivalent to a given amount of another, the course of exchange between the two being given. Evidently the amount of coin varies directly as the sum which is to be exchanged for it; therefore the solution of these questions is obtained by a Simple Proportion, which, when the first term is unity, may be solved by ordinary Multiplication, or by Practice. 2nd. Let it be required to determine the course of exchange between two places, the courses between each of the two and a third being known. Thus let it be given that a coins of the 1st place are equal to b coins of the 3rd, and c coins of the 3rd equal to d coins of the 2nd, then it is required to find how many coins of the 2nd are equivalent to a coins of the 1st. This is of course the same number as are equivalent to b coins of the 2nd; so that the question is:-If d coins of a given place are equivalent to c coins of another, how many of the former are equivalent to b of the latter ? Which is a question precisely similar to the 1st case, and the solution therefore is obtained in the same way. 3rd. Let it be required to find the course of exchange between two places, which are connected with several other places by having given the courses between the 1st and 3rd, the 3rd and 4th, the 4th and 5th, &c. the last and the 2nd. Thus let it be given that a, coins of the 1st are equivalent to bị coins of the 3rd place; a, of the 3rd equivalent to be of the 4th ; ag of the 4th to b3 of the 5th ; ( 4 of the 5th to 64 of the 2nd; then it is required to find how many coins of the 2nd are equivalent to aq coins of the Ist. The solution of this question might be effected by a series of Simple Arbitrations by last case: thus the course between the 1st and 4th places might be found from the known courses between the 1st and 3rd, and the 3rd and 4th : then the course between the 1st and the 5th might be found in the same way; and then that between the 1st and 2nd. Thus: Let the course between the 1st and 4th be x, that between the 1st and 5th be y; that between the 1st and 2nd be z: then the following proportions will give x, y, %. b1 62 b3 y 04 64 Therefore, compounding these proportions, we have :а, Хаз Ха4 bi L, Xbs X14 b. XbX 63 X B4 : : : y : : а, Хаз Ха4 The same result would be obtained by writing a series of equations expressiug the given courses of exchange, (thus a, coins of Ist kind = b, coins of 3rd kind a 2 3rd kind = b2 4th kind ყვ 4th kind = bg 5th kind 04 5th kind 64 2nd kind) and multiplying the numbers on the right hand side, and dividing the product by the product of all but the 1st on the left hand side. Whence has arisen the Rule called the Chain Rule. Prop. 90.-To explain the Rules for questions in Barter. 1st. The value of goods given in exchange is to be equal to that of the goods received ; therefore the quantity of the goods exchanged must be the quantity which might be purchased for the value of the goods received. Hence thc first Rule. But evidently questions in Barter are questions in Exchanges, in which, instead of different coins, we have different articles concerned. Hence the solution may be effected by the Chain Rule. 2nd. If it be given that a certain quantity of one article is exchanged for a certain quantity of another, whose rate of cost is known, and it be required to find the quality, or the cost of an unit, of the former, evidently the value of the number of units in the former quantity is equal to the value of the latter, which being determined, the value of one unit is obtained by division by the number of units. 3rd. If the cash price of one article is altered in a bartering transaction, in strict justice that of the article exchanged ought to be altered in the same ratio : or the ratio of the nett to the bartering prices ought to be the same for both articles. For instance, if in bartering the rate of cost of an article be doubled, only one half as much would be received in exchange for a quantity of another article, as would have been received, if the price had been unchanged. But if the rate of cost of the latter be doubled also, then the same quantity will have to be received as if neither price had been altered. Hence the nett, or bartering price of an article may be determined when either is given, the nett and bartering prices of another being given, by a Simple Proportion. Prop. 91.-To explain the Rule of Alligation. The value of the mixture is the sum of the values of the ingredients ; hence if these values be added, and divided by the number of units of quantity in the mixture, the result will be the value of one unit of the mixture. Hence the Rule. Prop. 92.—To explain the method of passing from one scale of Notation to another. To express a number in any given scale, as the duodenary, we have to find what number of collections of single units, of twelves of units, of twelves of twelves of units, &c. (the number of each kind being less than |