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By solid measure is meant, the measure used to measure bodies that have length, breadth, and thickness. 1728 inches
make 1 foot.
a ton or load. 128 solid feet, i. e. 8 feet long,
a cord. 4 feet high, and 4 feet wide,
make a minute, marked ". ! 60 minutes
1 degree, 30 degrees
1 sign, 12 signs, or 360 degrees, the great circle of the Zodiack.
Measures and Numbers not included in the Tables.
12 particular things make a dozen.
make 1 line.
A hand is used to measure the height of horses.
COMPOUND SUBTRACTION. The subtraction of compound numbers is the same as simple numbers, except borrowing and carrying.
Pup. From what you have taught me concerning Compound Addition, I expect I must borrow and carry, according to the increase of numbers from one denomination to another.
Tut. You must. The following is the rule for Compound Subtraction.
Write the less number under the greater, so that its several denominations may stand under the corresponding denominations of the greater.
Begin with the lowest denomination mentioned, and subtract the lower from the upper number, setting the difference beneath. If the upper number is not so large as the number under it, borrow as many as it takes of this denomination to make one of the next greater, and add to it ; then subtract, carrying one to the lower number of the next higher denomination for the one which you borrowed. In every other respect it is like simple subtraction.
Pup. These rules are very plain, because they are so much like the simple rules. But it must have been much easier, I think, if these weights, measures, &c. had all increased in a decimal ratio, as the simple numbers do.
Tut. It would be much easier, and all the rules of arithmetic, in which these weights, measures, &c. are concerned, would be much simplified, if they increased in a decimal ratio. The inconvenience of the present division of weights and measures has been long felt, and the French, in the latter part of the last century, altered all their weights, measures, time, &c. and reduced them to a decimal ratio. This division of weights and measures still continues in France, and will probably soon extend itself to all civilized countries. There can be no serious objection to the change, but a little confusion which it might cause. This would be but trifling compared with the advantages wbich would result from it. The division of time was abolished a few years after its adoption, and will probably never receive any very essential alteration from its present division.
Although these numbers do not increase in a decimal ratio, yet they can be reduced to the same denomination, and then the operations with them may be performed as upon decimal numbers. If you have any operation to perform upon 2 lb. 6 oz. 16 pwt. 12 gr. the denominations may be reduced to grains, which is done as follows.
As 12 oz. make 1 pound, if you multiply pounds by 12 the product will be in ounces. In the above example there are 2 pounds, which, multiplied by 12, gives 24, the number of ounces in 2 pounds, and as there are 6 oz. beside the pounds, add them to 24, which make 30, the whole number of ounces in 2 lb. 6 oz. Now as 20 pwt. make an ounce, if you multiply the ounces by 20, the product will be in pennyweights, to which add the given pennyweights, and you have the number of pennyweights there are in 2 lb. 6 oz. 16 pwt. Multiply this sum by 24, the number of grains it takes to make a pennyweight, and you have a product in grains, to which add the given grains, and you bave the number of grains there are in 2 lb. 6 oz. 16 pwt. 12 gr.
After you have reduced a number of different denominations to one denomination, you may wish to reduce it back again to its original expression. To do this, you must proceed directly the reverse from what you do to reduce it to one denomination. To reduce 14796 grains to their highest denominations, you will proceed as fol. lows.
2 lb. 6 oz. 16 pwt. 12 gr.
Here you have grains to reduce to higher denominations, and as 24 grains make 1 pennyweight, dividing the grains by 24, gives the number of pennyweights contained in them. In 14796 grains there are 616 pwt. and 12 grş.
Pennyweights divided by 20, give ounces, Because 20 pwt. make 1 oz. In 616 pwt. there are 30 oz. and 16 pwt. over. Ounces divided by 12, give pounds, because 12 ounces make 1 pound. In 30 oz. there are ? pounds, and 6 ounces over, so that you have the original number as first given, viz. 2 lb. 6 oz. 16 pwt. 12 grs.
Reducing numbers from a higher to a lower denomination is called reduction descending, because the value of
the figures, taken separately, descends. When numbers are reduced from a lower to a higher denomination, it is called reduction ascending, because the value of the fig. ures, taken separately, ascends or increases. The first example given is in reduction descending, and the second is in reduction ascending.
The examples and what I have said, will be sufficient to convince you of the reason of the following rules for
To reduce compound numbers from higher to lower denominations :
Multiply the highest denomination by such a number as a unit of that denomination will make of the next lower denomination, and to the product add the number given, belonging to the next lower denomination. Proceed in this manner with all the denominations, and the last sum will be the number required.
Reduce 59 lb. 13 pwt. 5 grs. to grains. Ans. 340157. Reduce 40 cwt. 3 qrs. 24 lb. 12 oz. 14 dr. to drams.
Ans. 1174734. Reduce 365 d. 5 h. 48'48" to seconds. Ans. 31556928.
To reduce numbers from a lower to a higher denomi
Divide the given number by such a number as it takes of this lower denomination to make one of the next higher denomination, and the quotient will be the number of the next higher. Then divide this higher by as many as it takes of it to make one of the next higher denomination, and proceed in this manner with the whole ; when the last quotient, with the several remainders, will be the answer.
Reduce 801213 grains to pounds, &c.
Ans. 1390 lb. 11 oz. 18 pwt. 19 gr. Reduce 9758 pints of brandy to pipes, &c.
Ans. 9 p. 1 bhd. 22 gal. 3 qts. Reduce-89763 square yards to acres.
Ans. 18 a. 2 r. 7 p. 101 ft. 36 in.