BILL OF PARCELS. Mr. Timothy Huckster 25lb Bohea tea, at 3s. 6d. per lb. 481b. Cheese, at 9d. per lb.. Newburyport, January 1st, 1808. Bought of Samuel Merchant, 15 Pair worsted hose, at 5s. 8d. per pair. 4 Dozen women's gloves, at 36s. 6d. per dozen. 19 Dozen knives and forks, at 5s. 9d. per dozen. 9 Grindstones at 15s 9d. per stone. Cwt. Brown sugar, at 51s. per cwt. 31 lb. Loaf Sugar, at 1s. 02d. per lb. Received payment in full, Samuel Merchant. .34 3 8 8. TARE AND TRET TARE and Tret are practical rules for deducting certain allowances, which are made by merchants and tradesmen in selling their goods by weight. Tare is an allowance, made to the buyer, for the weight of the box, barrel or bag, &c. which contains the goods bought, and is either at so much per box, &c. at so much per cwt. or at so much in the gross weight. Tret is an allowance of 4lb. in every 104lb. for waste, dust, &c. Cloff is an allowance of 2lb. upon every 3 Cwt. Gross weight is the whole weight of any sort of goods, together with the box, barrel, or bag, &c. which contains them. Suttle is, when part of the allowance is deducted from the gross. Neat weight is what remains after all allowances are made. CASE I.* When the tare is at so much per box, barrel or bag, &c: Multiply the number of boxes, barrels, &c. by the tare, and subtract the product from the gross, and the remainder will be the neat weight required. EXAMPLES. 1. In 6 hogsheads of sugar, each weighing 9cwt. 2qrs, 10lb. gross, tare 25lb. per hogshead; how mnch neat? Cwt. qr. lb. 25×6=1 1 10 This, as well as every other cafe in this rule, is only an application of the rules of Proportion and Practice. 2. In 5 bags of cotton, marked with the gross weight as follows, tare 23lb. per bag; what neat weight? 3. What is the neat weight of 15 hogsheads of tobacco, each 7cwt. 13lb. tare 100lb. per hogshead? Ans. 97cwt. Cqr. 11lb. 1qr. CASE II. When the tare is at so much per cwt.: Divide the gross weight by the aliquot parts of a cwt. subtract the quotient from the gross, and the remainder will be the neat weight. EXAMPLES. 1. In 129cwt. 3qrs. 16lb. gross, tare 14lb. per cwt. what neat weight? Cwt. qr. lb. | 14lb. | | 8 129 3 16 gross. 16 0 26 tare. Ans. 79 3 21 neat. 3. What is the neat weight of 9 barrels of potash, each weighing 305lb. gross, tare 12lb. per cwt.? Ans. 2450lb. 14oz. 44dr. 4. What is the value of the neat weight of 7hhds. of tobacco, at 51. 7s. 6d. per cwt. each weighing 8cwt. 3qrs. 10lb. gross, tare 211b. per ewt.? Ans. £.270 4 44 reckoning the odd ounces. CASE III. When tret is allowed with tare: Divide the suttle weight by 26, and the quotient will be the tret, which subtract from the suttle, and the remainder will be the neat. EXAMPLES. 1. In 247cwt. 2qrs. 15lb. gross, tare 28lb. per cwt. and tret4lb. for every 104 lb. what neat weight? |28|| 247C.2qr.15lb.gross. 61 3 17 12 tare, subtract. 2. What is the neat weight of 4 hhds. of tobacco, weighing as follow The 1st. 5cwt. 1qr. 12lb. gross, tare 65lb. per hhd.; the 2d. 3cwt. 0qr. 19lb. gross, tare 75lb.; the 3d. 6cwt. 3qrs. gross, tare 491b.; and the 4th 4cwt. 2qrs. 9lb. gross, tare 351b. and allowing tret to each as usual? Ans. 17cwt. Oqr. 19.b.+ CASE IV. When tare, tret and cloff are allowed: Deduct the tare and tret as before, and divide the suttle by 168, and the quotient will be the cloff, which subtract from the suttle, and the remainder will be the neat. EXAMPLES. 1. What is the neat weight of 1hhd. of tobacco, weighing 16cwt. 2qrs. 20lb. gross, tare 14lb. per cwt. tret 4lb. per 104, and cloff 2lb. per 3cwt. ? 14lb. is )16 2 20 20 0 gross. 9 8 tare, subtract. Ans. 13 3 22 6 neat. 2. If 9hhds. of tobacco, contain 85cwt. 0qr. 2lb. tare 30lb. per hhd. tret and cloff as usual, what will the neat weight come to at 64d. per per lb. after deducting for duties and other charges, 511. 11s. 8d.? Ans. £.187 18s. 5d. INVOLUTION, OR TO RAISE POWERS. A POWER is the product arising from multiplying any given number into itself continually a certain number of times, thus: 3x3-9 is the 2d. power, or square of 3. 3×3×3-27 is the 3d. power, or cube of 3. 3×3×3×3=81 is the 4th. power, or the biquadrate of 3, &c. =32 =33 =34 The number denoting the power is called the index, or the exponent of that power. Thus, the fourth power of 3 is 81, or 34; the second power of 5 is 25, or 52, &c. 2x2=4, the square of 2; 4x4-16=4th. power of 2; 16×16=256= 8th. power of 2, &c. RULE. Multiply the given number, root, or first power continually by itself, till the number of multiplications be I less than the index of the power to be found, and the last product will be the power required. Note. Whence, because fractions are multiplied by taking the prod ucts of their numerators, and of their denominators, they will be in volved by raising each of their terms to the power required, and if a mixed number be proposed, either reduce it to an improper fraction, or reduce the vulgar fraction to a decimal, and proceed by the rule. EXAMPLES. Ans. 000004100625. 3. What is the fourth power of ⚫045 ? Here we see, that in raising a fraction to a higher power, we de orease its value. EVOLUTION, OR THE EXTRACTION OF ROOTS. THE Root is a number whose continual multiplication into itsel. produces the power, and is denominated the square, cube, biquadratef or 2d. 3d. 4th. root, &c. accordingly as it is, when raised to the 2d, 3d. &c. power, equal to that power. Thus, 4 is the square root of 16, because 4x4=16, and 3 is the cube root of 27, because 3×3×3=27, and so on. Although there is no number of which we cannot find any power exactly, yet there are many numbers, of which precise roots can never be determined. But, by the help of decimals, we can approximate towards the root to any assigned degree of exactness. The roots, which approximate, are called surd roots, and those which are perfectly accurate, are called rational roots. Roots are sometimes denoted by writing the character before the power, with the index of the power over it; thus the 3d. root of 3 36 is expressed✓ 36, and the 2d. root of 36 is ✔ 36, the index 2 being omitted when the square root is designed. 3 If the power be expressed by several numbers, with the sign + or between them, a line is drawn from the top of the sign over all the parts of it. Thus the 3d. root of 47+22 is ✔47+22, and the 2d. root 17 is 59—17, &c. of 59 Sometimes roots are designed like powers, with fractional indices. Thus, the square root of 15 is 152, the cube root of 21 is 213, 4th. root of 37 — 20 is 37—204, &c. Y and A Third Surfolids, or 11th. Pow 1 2048 531441 Fourth Surfolids, or 13th. Pow1 8192 1594323 16777216 31381059609 244140625 2176782336 13841287201 68719476736 282429536481 67108864 1220703125 13060694016| 96889010407 549755813888 2541865828329 2d. Surfolids Sqd. or 14th. Pow! 16384 4782969 268435456 6103515625 78364164096 678223072849 4398046511104 22876792454961 Surfolids Cubed, or 15th. Pow. 1 32768 14348907 1073741824 30517578125 470184984576 4747561509943 35184372088832 205891132094649 1048576 177147 4194304 48828125 362797056 1977326743] 8589934592 60466176 282475249 1073741824) 3486784401 |