that will allow each owner to lay out all his land. How many rods must there be in a lot? RULE. PROBLEM X. Two or more numbers being given, to find their least common multiple; that is, the least number that will contain each of the given numbers a whole number of times. Divide two or more of the given numbers by any prime number that will measure them, repeat the operation upon the quotients and undivided numbers, and thus continue, till they become prime to each other. Multiply the several divisors, the last quotients, and undivided numbers together; the product will be the least common multiple. Is, among the numbers to be divided, any number is a measure of another, the measuring number may be rejected; that is, dropped from the operation. It is obvious, that one number is the multiple of another, when the former contains all the factors of the latter. The factors of 6 are 3 and 2, and the factors of 9 are 3 and 3. Now 54 contains all these factors, (3X2X3X 3=54), and 54 is a common multiple of 6 and 9, but it is not their least common multiple—it is 3 times as great as the least, owing to the existence of the factor, 3, in both 6 and 9. Hence we observe, that a common factor of two or more numbers must enter but once into the multiplication, to give the least common multiple. The above rule effects the necessary exclusion. 23. What is the least common multiple of 12, 25, 30, and 45. 3) 12 25 30 45 We find, after dividing twice, that 4 and 2 ap5) 4 25 10 15 pear; and, by dropping 4 5 2 3 the 2 because it measures the 4, we avoid another 3 X5 X 4 X 5 X 3 -900 division. Ans. 900. 24. What is the least common multiple of 6, 10, 16, and 20 ? 25. What is the least common multiple of 25, 35, 60, and 72! 26. What is the least common multiple of 105, 140, and 245 ? 27. What is the least common multiple of 19, 82, 94, 788, and 356 ? 28. Allowing 63 gallons to fill a hogshead, 42 a tierce, and 32 a barrel, what is the smallest quantity of molasses, that can be first shipped in some number of full hogsheads, then discharged and reshipped in some number of full tierces, and again discharged and reshipped in some number of full barrels ? 29. A certain four dealer, who purchased his flour from a mill on the opposite side of a river, owned four boats, one of which would carry 8 barrels of flour, another 9, another 15, and another 16. What is the smallest number of barrels he could purchase, that would make some number of full freights for either of the boats ? IX. COMPOUND NUMBERS. COMPOUND NUMBERS are those which are employed to express quantities that consist of several denominations; each denomination being denoted separately. Under this head are classed, all the subdivisions of measures; of length, surface, solidity, weights, money, time, &c. The following tables of denominations of compound numbers, show how many units of each lower denomination are equal to a unit of the next higher, and, exhibit each lower denomination as a fraction of the next higher. MONEY, WEIGHTS, AND MEASURES. ENGLISH MONEY. The denominations of English Money are, the pound, £, the shilling, 8., the penny, d., and the farthing, qr. 4 farthings - 1d. of 1 d. 1 s. id. 20 shillings = 1 £. 1s. of 1 £. 12 pence 12 of l s. 24 of idwt zo of 1 oz. = TROY WEIGHT. The denominations of Troy Weight are, the pound, lb., the ounce, oz., the pennyweight, dwt., and the grain, gr. 24 grains 1 dwt. 1 gr. 20 pennyweights 1 oz. . 12 ounces 1 lb. 12 1 of 1 lb. 1 oz. io of 1 oz. 16 ounces 1 lb. 17 28 pounds 1 qr. 1 lb. of 1 qr. 28 4 quarters 1 cwt.|| lqr. of 1 cwt. 20 hundred-weight .=1T. 1 cwt. of 1 T. APOTHECARIES' weight. The denominations of Apothecaries' Weight are, the pound, lb, the ounce, 3, the dram, 3, the scruple, , and the grain, gr. 20 grains 13. 1 gr. of 12 3 scruples 13. 12 } of 13. 8 drams 13. 13 of 13. 12 ounces - 1 it. of 1 iba. 1 qr. . of 1 yd 3 quarters 1 Fl. e. 1 of 1 Fl. e. of 1 E. e. 5 quarters 1 Fr.e.l 1 qr. of 1 Fr. e. 1 qt. || 1 pt. of 1 gal. of 1 pk. 4 pecks 1 bu. | 1 pk. of 1 bu. 1 qr 1 na. of 1 qr. lyd. of 1 qt: 2 pints 4 quarts 1 pt. 19t: 1 of 1 pt. of 1qt; 2 pints 4 quarts 1 pt. 19t: 1 of 1 pun. 84 WINE MEASURE. The denominations of Wine Measure are, the tun, T., the pipe, p., the puncheon, pun., the hogshead, hhd., the tierce, tier., the barrel, bl., the galon, gal., the quart, qt. the pint, pt. and the gill, gi. 4 gills 1 gi. 1 gal. of 1 gal. 31 gallons 1 bl. 1 gal. of 1 bl. 42 gallons 1 tier. 1 of 1 tier. 63 gallons 1 hhd. 1 gal. 1 of 1 hhd. 63 84 gallons 1 pun. 1 gal. 126 gallons 1 p. of 1 p. of IT. BEER MEASURE. The denominations of Beer Measure are, the butt, bt', the hogshead, hhd., the barrel, bl., the kilderkin, kil., the firkin, fir., the gallon, gal., the quart, qt., and the pint, pt. 2 pints 1 qt. 4 quarts 1 gal. of 1 gal. 9 gallons 1 fir. of 1 fir. 2 firkins 1 kil. 1 fir. of 1 kil. 2 kilderkins i bl. 1 k:1. of 1 ul. 3 kilderkins 1 hhd. 1 kil. of 1 hhd. 2 hogsheads 1 bt. 1 hhd. of 1 bt. 26 1 pt: of Iqt. . NOTE. In the United States, the Dry gallon contains 268 cubic inches, the Wine gallon 231 cubic inches, and the Beer gallon 282 cubic inches. By an Act of the British government, however, the distinction between the Dry, Wine, and Beer gallon was abolished in Great Britain, in 1826, and an Imperial Gallon was established, as well for liquids as for dry substances. The Imperial gallon must contain “ 10 pounds, Avoirdupois weight, of distilled water, weighed in air, at the temperature of 62° of Fahrenheit's thermometer, the barometer standing at 31 inches.” This quantity of water will be found to measure 277270 cubic inches. The same Act estabJishes the pound Troy at 5760 grains, and the pound Aroirdupois at 7000 grains = 1 ft. of 1yd. =lm. LONG MEASURE. The denominations of Long Measure are, the nile, m., the furlong, fur., the rod or pole, r., the yard, yd., the foot, ft., and the inch, in. 12 inches 1 in. Ia of 1 ft. 3 feet = lyd. 1 st. 5) yards lr. 1 yd. of 1r. 40 rods 1 fur. 1r. of 1 fur. 8 furlongs 1 fur. of im. SQUARE MEASURE. The superficial contents of any figure having four sides and four equal angles, is found in squares, by multiplying together the length and breadth of the figure. The denominations of Square Measure are, the mile, m., the acre, A., the rood, R., the rod, r., the yard, yd., the foot, ft., and the inch, in. 144 inches 1 ft. 1 in. = 1 of 1 ft. 9 feet... = lyd. i 1 ft. of 1 yd. 304 yards. 1r. lyd. I of 1r. 40 rods IR. 4 roods 1 A. IR. À of 1 A. 640 acres = lm. 1 A. CUBIC MEASURE. The cubical contents of any thing which has 6 sides, its opposite sides being equal - is found in cubes, by multiplying together, the length, breadth and depth. The denominations of Cubic Measure are, the yard, yd., the foot, ft., and the inch, in. 1728 inches 1 ft. 1 in. 1728 of 1 ft. 27 feet lyd. | 1 ft. 27 of lyd 40 feet of round timber, or 50 feet of hewn timber make a ton. 16 cubic feet make a foot of wood, and 8 feet of wood make a cord. C 1r. . of 1 m. TIME. 1 s. 1 m. The denominations of Time are, the year, Y., the day, d., the hour, h., the minute, m., and the second, s. 60 seconds to of 1 m. 60 minutes 1h. . 24 hours id. 365 days id. m. bo of 1h. 24 of 1 d. of 1 Y, |