69. If the right hand figure of the denominator of the fraction to be reduced is either 1, 3, 7, or 9, it is manifest there will always be a remainder after division, and consequently the decimal can never be brought exactly equal to the vulgar fraction in such cases continue the division to as many places. of figures as may be thought necessary.

OF DUODECIMALS.

70. DUODECIMALS are vulgar fractions whose denominators are 12, or 144, or 1728, &c. The division and sub-division of the integer being in 12ths instead of 10ths as in the decimal scale. The method of operating with these fractions (sometimes called cross multiplication) seems to have been introduced for the purpose of computing the contents of artificer's work when the dimensions are in feet and inches, or in feet, inches, and lines or 12ths of an inch.

71. Rule. Set down the dimensions to be multiplied together so that feet may stand under feet, inches under inches, &c.

Begin with the feet in the multiplier and the lowest dimen. sion in the multiplicand. Carry 1 to the left for every 12 in each product, setting down the remainders directly under the corresponding terms of the multiplicand. Next, multiply by the inches, and carry 1 for every 12 as before, remembering to set down the result one place to the right of the product by the feet. And in the same manner proceed with a third dimension, &c. Then add the several products together, carrying 1 for every 12 as before, and the sum will be the answer.

The reason for setting down the products in the manner directed by the rule will be evident if we retain the denominators, but multiply by the parts separately as before (see examp. 8. in the multiplication of vulgar fractions):

or

to be carried and

Secondly, to set down (because 12 of any denomination make 1 of the next superior or that on the left):

And 7x which with (that were carried) make 4, or 311⁄2 to set down.

The answer however, in this example, is sooner obtained by the usual method of vulgar fractions, thus:

And 4 × 12 = 24 = 423 feet, as before.

Ex. 2. Required the number of square feet in a rectangular board whose

F. in.

F. in.

length is 17 5, and breadth 1 83?

N. B. The two dimensions multiplied together give the answer.

Ex. 3. Required the number of cubic feet in a rectangular ditch whose breadth is 8, depth 43, and length 21 feet?

Here the three dimensions must be continually multiplied together.

By this weight are weighed Gold, Silver, Jewels, and some Liquids. Jewellers sometimes express the weight of a diamond in carats of 4 grains (troy-weight) each. A carat however, signifies the of any maзs of gold, or of gold with alloy, and is generally used to denote its degree of fineness.

The Pound is the same as the Pound Troy, only differently divided

Apothecaries use this weight in compounding their medicines, but buy and sell their Drugs by Avoirdupois Weight.

4 Quarters, or 112lb. 1 Hundred Weight cwt.

And slb. is a Stone in the London Markets.

14lb. a Stone, Horseman's Weight.

28lb. a Tod.

ton.

By this Weight are weighed all Groceries, Chandler's Wares, some Liquids, and all Metals except Gold and Silver.