3. Glasgow, 22d July, 1819. Messrs. John and Peter Fenton bought of William Smith, Mule twist No 54. 40 lbs. at 7/9; No 56, 15 lbs. at 7/11; No 58, 20 lbs. at 8/2; No 60. 25 lbs. at 8/5; and No. 62. 30 lbs. at 8/8. Discount 35 per cent. For which they granted their bill at 4 months. Write the bill of parcels and bill. 9. Edinburgh, 11th Nov. 1819. Mr. William Anderson bought of Robert Plenderleath, Scotch linen, 10 pieces, 273 yards, at 3/2; do. 6 pieces, 158 yards, at 2/11; Irish linen, 8 pieces, 253 yards, at 2/44; do. 9 pieces, 292 yards, at 1/10; sheet and ropes, 6/6. The account was paid, and 5 per cent. discount allowed. Write the bill of parcels. INVOICES. An Invoice is an account of goods sent off, generally by sea, either in consequence of an order from the person to whom they are sent, and at his risk; or consigned to him for sale at the risk of the proprietor. FORM OF AN INVOICE. Invoice of 6 hogsheads of tobacco, shipped by John Jamieson on board the Peggy, Captain Cleghorn, for Gottenburg, on account and risk of Messrs. Wersean & Rydberg, merchants there. W.&R. No 1. 18 1 10 tare 1 2 19 ACCOUNTS SALES. An Account Sales is an account of goods sold on commission, transmitted to his employer by the agent, to whom they were consigned. FORM OF AN ACCOUNT SALES. ACCOUNT SALES of four bales Osnaburgs, per the Neptune, Drummond, on account of Mr. James Ford, Montrose, 1819. 2 bales FLAX OSNABURGS, at 14 months. May 4. No 1. 14ps. 2007 yds. at 77d. £63 8 3 Wrapper, 26 yds. at 6d. 0 13 0 5. N° 2. 15ps. 2156 yds. at 7d. 63 12 7 6 2 bales Tow OSNABURGS, at 12 months. May 26. No 3. 15 ps. 2148 yds. at 7d. £62 13 0 Wrapper, 28 yds. at 5d. 0 12 10 No 4. 14 ps. 1997 yds. at 611d. 57 11 0 Wrapper, 25 yds. at 5d. 0 10 5 128 7 51 08 8 Landing, wharfage, carting, and housing Insurance against fire, 3/6 per cent. Net proceeds to the credit of Mr. James Ford, London, 2d June, 1819. 16 1 2 £233 13 6 Errors excepted. COULEY & SON. Remark. When the agent does not guaranty the debts, the buyer's name is inserted, and the phrase "Errors and bad debts excepted" is prefixed to the subscription. Before the learner proceeds to the next Section, he should be made acquainted with the manner of using Tables of Logarithms. The following are the properties of logarithms. 1mo. The sum of the logarithms of two numbers is equal to the logarithm of their product. Thus, Log. a b Log. a+Log. b. 2do. The difference of the logarithms of two numbers is equal to the logarithm of their quotient. Thus, Log. (ab)=Log. aLog. b. 3lio. The logarithm of any power of a number is equal to the logarithm of the root multiplied by the index of the power. Thus, Log. rt (Log. r)xt, and Log. (1·04)60 Log. (1.04)×60. 4to. The logarithm of any root of a number is equal to the logarithm of that number divided by the index of the root. Thus, Log.tv/r=(Log. r)÷÷t, and Log. m, "√rt={t×Leg. r) -÷m. SECT. II. COMPOUND INTEREST Compound Interest is an allowance of so much per cent., given not only for the use of the principal, or sum borrowed, but also for the use of the interest after it becomes due: so that at every term when the interest is due, it is added to the principal, and the amount becomes a new principal for the succeeding term. Let p the principal, r = the simple interest of £1 for a year, t=the number of years, a=the amount at the end of t years, and R=(1+r) = the amount of £1 at the end of a year. Since £1 at the end of the first year becomes R, and since the amount is increased each year in the same ratio, therefore 1: So that at the end of t years, the amount is p Rt, that is, a=pR¢; from which equation are deduced the following theorems. | III. | r=t√(ap)—1; or Log. R=(Log. a—Log. p)—t IV. t=(Log a-Log. p)÷Log R. Note. These theorems will answer when the interest is payable, (or convertible into principal, which shall reproduce interest,) any number of times in the year, if t denote the number of payments of interest, or the number of conversions of interest into principal, and r, the interest of £1 in one of those intervals; that is, r must denote half a year's interest of £1, when the interest is payable halfyearly; a quarter of a year's interest of £1, when the interest is payable quarterly; and so on. EXAMPLES. 1. What will £500 amount to in 6 years, at the rate of 3 per cent per annum compound interest? Here p=500, R=1·03, and t=6; and by theorem 1st. a=pRt= 500×1.036=500×1·1940523=£597·0261=£597 :0:61. Ans. 2. What will £500 amount to in 75 years 184 days, allowing compound interest at the rate of 5 per cent. per annum ? Here p=500, R=1·05, and t=75181; and by theorem 1st. Log. a=t (Log. R)+Log. p = (Log, 1·05) × 75184 + Log. 500= 021189299×7518+2-69897000-4-29884915, which is the log. of 19899-819, or £19899: 16:41. Ans. 3. What principal lent at compound interest, at the rate of 3 per cent. per annum, will at the end of 6 years amount to £597: 0:61 ? Here a=597-0261, R— 1·03, and t-6; and by theorem 2d. p=aRtax 597.0261 × 83748426 — £500. Ans. 1 = R 4. Suppose a person returns £600 for the loan of £300, at the rate of 5 per cents compound interest; how long was it lent ? Here a―600, p=300, and R=1·05; and by theorem 4th. t= (Log. 600-Log. 300)—Log. 1·05—(2·778151 — 2·477121) → -021189-14-206 years 14 years 75 days. 5. Suppose a person returns £926: 2 for the loan of £800 for 3 years, what rate of interest is allowed ? Here a=926·1, p= 800, and t = 3; and by theorem 3d. r = 3√(926·1-800)-1=3/1·157625—1—·05, or 5 per cent. Remark. Although the laws of this country do not allow money to be lent at compound interest, yet the interest may be demanded when it is due, and lent again, and thus, in a great measure, the advantages of compound interest may be realized; and in the purchase of annuities, reversions, leases, &c. it is usual to allow compound interest to the buyer for his ready money: this subjeet, therefore, deserves consideration. EXERCISES. 1. If £800 were lent for 10 years at compound interest, at the rate of 4 per cent., what would be the amount ? 2. What sum lent at compound interest, at the rate of 4 per cent., would amount to £200 in 15 years? The learner should calculate all the exercises in compound interest and annuities by logarithms. 3. At what rate of compound interest would £1000 amount to £1989: 15:9, in 20 years? 4. In what time would £300 amount to £500, at 5 per cent. compound interest? 5. In what time would any sum double itself, at 4 per cent. compound interest? 6. Find the amount of £1000, lent at compound interest, at the rate of 4 per cent. for 25 years, one half of the interest for a year being converted into principal every half year. 7. Find the amount of £1000, lent at compound interest, at the rate of 4 per cent. for 25 years, one-fourth of the interest for a year being converted into principal every quarter of a year. 8. Find the present value of £3000 due 20 years hence, reckoning compound interest at the rate of 5 per cent. payable yearly; find also the value, when the interest is payable half-yearly. ANNUITIES. An Annuity is a term denoting any periodical income, payable either annually, or at any other equal intervals. I. On the Amount of Annuities. The amount (m) of any annuity (A), unpaid for the time (t), reckoning compound interest on each payment, from the time it becomes due until the end of the term, may be found as follows: Let r denote the simple interest of £1 for a year, and R, the amount of £1 at the end of a year, that is (1+r). Since the amount of £1 in t years is Rt, its increase in that time is R-1; and since this increase arises entirely from r, the simple interest of £1, being laid up at the end of each year, and improved at compound interest during the remainder of the term, it is plain, that Rt-1 is the amount of an annuity of £r in that time, but r': A :: Rt .1 Rt-1: m = Ax the following theorems : From this equation, are deduced R—1, or, Log. m=Log. (A÷r)+Log. (R'—1) r II. Amr(Rt—1), or, Log. A=Log. (m r)-Lóg. (Rt—1) III. t=Log. (1+m r÷A)÷Log. R. |