equal parts, still calling the parts grains, and these are the basis of the table of Troy weight, by which are weighed gold, silver and jewelry. Apothecaries' weight is the same as Troy weight, only having different divisions between grains and ounces. Apothecaries make use of this weight in compounding their medicines, but they buy and sell their drugs by Avoirdupois weight. In buying and selling coarse and drossy articles, it became customary to allow a greater weight than that used for small and precious articles, and this custom at length established Avoirdupois, or common weight, by which all articles are weighed with the following exceptions. Avoirdupois weight is about one sixth part more than Troy weight, the former being 7000 grains, and the latter 5750 grains. In buying and selling by the hundred weight, 28 pounds have been called a quarter, 112 pounds a cut. ; but this practice of grossing, as it is called, is now pretty generally laid aside, and 25 pounds are considered a quarter, and 4 quarters (100 pounds) a hundred weight." Avoirdupois, or Common Weight. *"The original standard of English Long Measure, was a barleycorn taken from the middle of the ear, and well dried. Three of these in length were called an inch, and then the others as in the table. Long Measure is employed for denoting the distance of places, and for measuring where length only is concerned. When measure is applied to surface where both length and breadth are concerned, it is called Square Measure. A square inch is a square measuring an inch on every side. The table of Square Measure is made from that of Long Measure, by multiplying the several numbers of the latter into themselves. Thus 12 inches are a foot in length; a square foot, then, is a square which measures 1 foot, or 12 inches on every side, and contains 12×12=144 square inches. Three feet in length make a yard. A square yard is a square measuring 3 feet on each side; but such a square contains (see figure) nine (3×3—9) square feet; and when we say that a surface contains so many square feet, or square yards, we mean that the surface is equal to such a number of squares-meaning a foot, or a yard, on each side.” VIII. Cloth Measure. 24 inches make 1 nail, IX. Square Measure. na. 144 inches make 1 squ. foot, ft. 2724 feet 4 nails 1 quarter, gr. 9 feet 1 squ. yard, yd. 4 quarters 1 yard, yd. 304 yards 1 squ. rod, rd. 3 quarters 1 ell Flemish, E. Fl. 1 squ. rod, rd. 5 quarters 1 rood, 6 quarters 37.2 in. 1 ell English, E. E. 40 rods X. Solid, or Cubic Measure. [1728 inches make 1 foot, 1 squ. mile, mi. NOTE. The wine gallon contains 231 cubic inches. *"Four pounds of Troy Weight of wheat, gathered from the middle of the ear, and well dried, were called one gallon; and this was the original standard of all English measures, both liquid and dry; and this was the same as the present wine gallon. But, in time it became customary to use a larger measure in selling cheap liquors; and this custom at length established the Beer Measure, which bears about the same proportion to Wine Measure that Avoirdupois does to Troy Weight. The Dry Measure was also made larger than the Wine Measure, and was at length established at about a mean be tween Wine and Beer Measure. By Wine Measure, are measured wine, all kinds of spirits, cider, vinegar, oil, &c. By Beer Measure are measured ale and beer, and by Dry Measure all kinds of dry goods-corn, grain, salt, roots, fruit, &c. A standard bushel is 18 inches diameter, and 8 inches deep. The statute bushel for measur ing coal, ashes and lime, in Vermont, contains 38 quarts, or 2553.6 cubic inches." When measure is applied to solids which have length, breadth and thickness, it is called solid, or cubic measure. A solid inch is a body or block, having six sides, each of which is an inch square, and the number of inches in a solid foot is equal to the number of such blocks that would be required to make a pile a foot square and a foot high. Now it would require 144 blocks to cover a square foot one inch high. Hence to raise the pile 12 inches high would require 12 times 144-1728 blocks or inches. In like manner it would require 9 solid blocks, a foot each way, to cover a square yard to the height of one foot, and 3 times 9=27, to raise it 3 feet, or make one solid yard. A cord of wood is sometimes called 8 feet. In this case four feet in length, four in breadth, and one in height=16 solid feet; or 8 feet in length, 4 in breadth, and 6 inches in height, a foot; that is, of a cord is called one foot, &,two feet, &c. In measuring lands, roads, &c., the distances are usually taken in chains and links. In ordinary business, feet and inches are the most common measures. By forty feet of round timber, in the table of solid measure, is meant so much round timber as will make forty feet after it is squared. 54 gallons I hogshead, hhd. 8 bushels 2 hogsheads 1 butt, butt. 4 quarters 1 bushel, pk. bu. gr. 1 quarter, 2 butts NOTE. The Ale Gallon contains 282 contains 2684 cubic inches. cubic or solid inches. 1 tun, tun. NOTE. A gallon dry measure A Winchester bushel is 18 inches in diameter, 8 inches deep, and contains 2150 cubic inches. The coal bushel must be 19 inches in diameter, and 36 bushels heaped up make a London chaldron of coal, the weight of which is 3156 lbs. A voirdupois. *"Every circle, without regard to its size, is supposed to be divided into 360 equal parts, called degrees, and these again to be subdivided into minutes and seconds; so that the absolute quantity expressed by any of these denominations must always depend upon the size of the circle. In this measure are reckoned latitude, longitude, and the planetary motions." NOTE. Points and lines are applied to measuring pendulums of clocks; When folded into 4 leaves! it is called Quarto. When folded into 8 leaves it is called Octavo. When folded into 12 leaves it is called Duodecimo, or 12mo. When folded into 18 leaves it is called 18mo. When folded into 24, it is called 24mo. hands to measuring horses ; fathoms When folded into 48, it is to measuring depths of the sea. called 48mo. A gallon of train oil weighs A stone of butcher's meat A gallon of molasses A stone of iron A firkin of butter A barrel of four 71⁄2 pounds. 11 14 56 191 cwt. 196 pounds. 200 REDUCTION is the changing of numbers from one denomination to another, without altering their value. 1. In £6, 7s. 8d. 2qrs., how many farthings? For Operation, see next page. QUESTIONS. 1. What is Reduction? 2. By what numbers do you multiply in example 1st? and why? Operation. 20 127 12 1532 4 It is plain, that, if in one pound, there are 20s. in six pounds there are 6 times as many, or 120s.; and in £6, 7s. there are 127s. Again, if, in one shilling there are 12d. in 127s. there are 127 times 12, which, with the 8d. added, equals 1532d. Last 6130qrs., Ans. ly, it is evident that, if in 1 penny there are 4 farthings, in 1532d. there are 4 times the number of farthings, or 6130, the 2qrs. in the given question being added. Hence it appears, that in any given number of pounds there are 20 times as many shillings as pounds, 12 times as many pence as shillings, and 4 times as many farthings as pence. This process is called Reduction Descending, because higher denominations are brought into lower. 2. In 6130 farthings, how many pounds? Operation. 4)6130 12)1532 2qrs. 2|0)12/7 8d. It will be seen that this question is the reverse of the former; and as 4 farthings are equal to 1 penny, so the number of pence in 6130qrs. will equal the number of times it contains 4, or 1532d. and 2qrs. over. Again, Ans. £6 7s. 8d. 2qrs. as 12d. are equal to 1 shilling, so in 1532d. the number of shillings will equal the number of times it contains 12, or 127s. and 8d. over. Lastly, since 20s. is equal to 1 pound, 127s. must equal 6 pounds and 7s., because 20 is contained in 127s. 6 times, and 7s. over. It is always to be remembered, that the remainder is of the same denomination as the dividend, whatever may be the divisor. This latter process is called Reduction Ascending, because lower denominations are brought into higher. By these examples, it appears that Reduction Ascending and Descending mutually prove each other. From the preceding operations we derive the following RULES: |