If we value the sum of the approximates, it will fall a little short of the sum of the articles, because the decimals are not complete.

It is proper to add 1 to the last decimal place of the approximate, when the following figure would have been 5 or upwards. Thus the full decimal of 3 qrs. 22 lb. is 946-428571, and therefore 94643 is nearer to it than 946-42. Approximates thus regulated will give exacter answers, sometimes above the true one and sometimes below it.

The mark + signifies that the approximate is less than the exact decimal, or requires something to be added. The mark-signifies that it is greater, or requires something to be subtracted.

SECTION III.

DUODECIMALS, OR CROSS MULTIPLICATION. DUODECIMALS (proceeding by twelves) is a term given to a rule or operation of arithmetic, by which the contents of any surface or solid are found by multiplying together its lineal dimensions, expressed in feet, inches, and lines, and is consequently much used by artificers in finding the contents of their work. The rule is also called cross multiplication, from the manner in which the operation is usually performed, and which is as follows: suppose it were required to find the superficial content of a plank 12 ft. 9 in. long, and 3 ft. 7 in. broad. Set down the two dimensions under each other, placing feet under feet, inches under inches, &c., and for the half inch put down its equivalent 6 lines, as in the following example:

... 12 ft. 9 in. 6 lin. 12.7916.

Also, 7 in.583

and 3 ft. 7 in. =3.583

..12-7916 x 3.583-45.8522 ft.

Now since the feet are conceived to be units of measure, the inches are so many 12ths of unity, and the lines so many 12ths of a 12th, or 144th parts of unity. The units consequently form the first column, the 12ths the second, and the 144ths the third. Multiplying, therefore, the first line by 3 ft. or 3 units, we get 38 ft. 4-12ths of a foot, and 6-144ths of a foot. Next multiplying the upper line by 7-12ths, we get first the 6 lines or 6-144ths, multiplied by 7-12ths, equal to 42-1728ths, which is equal to 31-144ths. Then the 9-12ths multiplied by the 7-12ths, give 63-144ths, which, added to the 31, make 661-144ths, or 5-12ths and 64-144ths; therefore 64 is placed in the third column, and the 5-12ths carried on. Lastly, the 12 units multiplied by the 7-12ths, give 84-12ths, which, added to the 5-12ths, make 89-12ths, and this is equal to 7 units or feet and 5-12ths; consequently, 7 is placed in the first column and 5 in the second. Adding the two products together, we get 45 ft. 10-12ths of a foot, and -144ths of a foot. But in square or superficial measure the 144th part of a foot is an inch; and 10-12ths=120144ths; consequently, the result of the operation is 45 sq. feet and 120 sq. inches.

The operation is itself much simpler than the description or explanation, which is found embarrassing to beginners; it would therefore, perhaps, be better to reject the rule altogether from elementary books of arithmetic; and, regarding the inches and lines as part of a foot, to perform the operation by the ordinary rules of practice, or, which is better, by reducing both factors to decimals, and proceeding as in simple multiplication, as already shown, where 12.7916 x 3.583-45.8522 feet 45 feet 10 in., &c.

SECTION IV.

DATA FOR ARITHMETICAL CALCULATIONS. Subsect. I.-TABLES OF WEIGHTS AND MEASURES. THE following is a tabular view of the weights and measures, according to the present state (1847) of the law, throughout

the British empire. In some of the colonies, however, and particularly in India, a variety of other weights and measures besides is still in use. It is only necessary to observe, that all the quantities in the same horizontal line of the same table are equal to each other.

Three inches make a palm, 4 inches a hand, 5 feet a pace, and 6 feet a fathom. In cloth measure, 24 inches=1 nail, 4 nails=1 quarter, and 4 quarters=1 yard.

In the superficial measurement of stone, brick, or slate work, 36 square yards are termed a rood, and 100 square feet of flooring a square. There are 1728 cubic inches in the cubic foot, and 27 cubic feet in the cubic yard; 40 cubic feet of rough, or 50 of hewn timber, make a load or ton; and 42 cubic feet make 1 ton of shipping. A cubic yard of earth is called a load.

The peck, bushel, coomb, and quarter, are dry measures only.

=

In beer measure, the barrel consists of 36 gallons=4 firkins, and the hogshead contains 1 barrel or 54 gallons. The anker, tierce, hogshead, puncheon, pipe, butt, and tun, used for wine and spirits, are so vague and variable in their contents, that they are to be considered rather as the names of the casks, than as expressing any fixed or definite measures. The Scottish ell was 37.0598 imperial inches; and 1920 ells=32 falls=8 furlongs=1 mile=1∙123024 imperial mile. The Scottish chain for land measure, like the imperial, consisted of 100 links, and though often reckoned to have been just 74 imperial feet, was more correctly=24 ells=74.1196 feet; and 5760 square ells=160 falls=10 square chains=4 roods=1 acre 1.261183 imperial acre. Scottish acres will therefore be reduced to imperial, by multiplying them by 1-261183. Twenty-three Scottish acres made about 29 imperial, or, more nearly, 134 Scottish=169 imperial.

In the old Scottish liquid measure, 128 gills=32 mutchkins 16 chopins = 8 pints=1 gallon = 3.00651 imperial gallons.

=

Although several of the old Scottish dry measures were very different in different districts, most of them were similarly subdivided, having 64 lippies or forpets 16 pecks=4 firlots=1 boll. In some of them, however, the bolls, especially where they were large, had very different sorts of subdivisions.

same.

In these two weights the grain, ounce and pound are the The troy is used for the precious metals and for jewels, as also in trying the strength of spirituous liquors, and for comparing different weights with each other.

Four grains troy make a carat. But this term, when applied to gold, denotes its degree of fineness. Thus, the weight of any quantity or compound of that metal being supposed to be divided into 24 equal parts, if the mass be pure gold, it is said to be 24 carats fine. If it consist of 23 parts of pure gold and 1 of alloy, it is said to be 23 carats fine, and so on. Diamonds and pearls are also weighed by carats of 4 grains, but 5 diamond grains are only equal to 4 troy grains. This sort of weight is not very different all over the globe. There are 150 diamond carats in the troy ounce. Apothecaries' weight is chiefly used for medical prescriptions; but drugs are mostly bought and sold, especially in wholesale, by avoirdupois weight.

The above pound of 7000 troy grains was formerly subdivided into 7680 avoirdupois grains, 10 of which made a scruple, 30 a drachm, and 430 an ounce. The troy pound is less than the avoirdupois in the proportion of 144 to 175, or