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20. The line of defence in a certain polygon being 236 yards, and that part of it which is terminated by the curtain and shoulder being 146 yards 1 foot 4 inches; what then was the length of the face of the bastion?
Ans. 89 yds Ift 8 inches.
COMPOUND MULTIPLICATION shows how to find the amount of any given number of different denominations repeated a certain proposed number of times; which is performed by the following rule.
SET the multiplier under the lowest denomination of the multiplicand, and draw a line below it. Multiply the number in the lowest denomination by the multiplier, and find how many units of the next higher denomination are contained in the product, setting down what remains. In like manner, multiply the number in the next denomination, and to the product carry or add the units, before found, and find how many units of the next higher denomination are in this amount, which carry in like manner to the next product, setting down the overplus. Proceed thus to the highest denomination proposed: so shall the last product, with the several remainders, taken as one compound number, be the whole amount required.
I. If the multiplier exceed 12, multiply successively by its component parts, instead of the whole number at once.
II. If the multiplier cannot be exactly produced by the multiplication of simple numbers, take the nearest number to it, either greater or less, which can be so produced, and multiply by its parts, as before. Then multiply the given multiplicand by the difference between this assumed number and the multiplier, and add the product to that before found, when the assumed number is less than the multiplier, but subtract the same when it is greater.
COMPOUND DIVISION teaches how to divide a number of several denominations by any given number, or into any number of equal parts. It is performed as follows:X
PLACE the divisor on the left of the dividend, as in simple division. Begin at the left hand, and divide the number of the highest denomination by the divisor, setting down the quotient in its proper place. If there be any remainder after this division, reduce it to the next lower denomination, which add to the number, if any, belonging to that denomination, and divide the sum by the divisor. Set down again this quotient, reduce its remainder to the next lower denomination again, and so on through all the denominations to the last.
I. If the divisor exceed 12, find what simple numbers, multiplied together, will produce it, and divide by them separately, as in simple division, as below.
2. If 20 cwt of tobacco come to 150/ 6s 8d, what is that per cwt ?
II. If the divisor cannot be produced by the multiplication of small numbers, divide by the whole divisor at once, after the manner of long division, as follows:
EXAMPLES OF WEIGHTS AND MEASURES.
1. Divide 17 lb 9 oz 0 dwts 2 gr by 7.
6. Divide 77 ac 1 ro 33 po by 51.
7. Divide 206 mo 4 da by 26.
Ans. 2 lb 6 oz 8 dwts 14 gr. Ans 1 lb 5 oz 3 dr 1 sc 12 gr.
Ans. 3 cwt 1 qr 14 lb. Ans. 3 mi 5 fur 26 po 2 ft 8 in. Ans. 11 yds 1 qr 2 na.
Ans. 1 ac 2 ro 3 po. Ans. 7 mo 3 we 5 ds.
THE GOLDEN RULE, OR RULE OF THREE.
THE Rule of Three enables us to find a fourth proportional to three numbers given for which reason it is sometimes called the Rule of Proportion. It is called the Rule of Three, because three terms or numbers are given, to find a fourth. And because of its great and extensive usefulness, it was often called, by early writers on Arithmetic, the Golden Rule. This Rule is usually by practical men considered as of two kinds, namely, Direct and Inverse. The distinction, however, as well as the manner of stating, though retained here for practical purposes, does not well accord with the principles of proportion; as will be shown farther on.
The Rule of Three Direct is that in which more requires more, or less requires less. As in this: if 3 men dig 21 yards of trench in a certain time, how much will 6 men dig in the same time? Here more requires more, that is, 6 men, which are more than 3 men, will also perform more work in the same time. Or when it is thus: if 6 men dig 42 yards, how much will 3 men dig in the same time? Here, then, less requires less, or 3 men will perform proportionably less work than 6 men in the same time. In both these cases, then, the Rule, or the Proportion, is Direct; and the stating must be
thus, as 3: 21 6 42, or as 3: 6:21: 42.
But the Rule of Three Inverse, is when more requires less, or less requires more. As in this: if 3 men dig a certain quantity of trench in 14 hours, in how many hours will 6 men dig the like quantity? Here it is evident that 6 men, being more than 3, will perform an equal quantity of work in less time, or fewer hours. Or thus: if 6 men perform a certain quantity of work in 7 hours, in how many hours will 3 men perform the same? Here less requires more, for 3 men will take more hours than 6 to perform the same work. In both these cases, then, the Rule, or the Proportion, is Inverse; and the stating must be 14:3: 7, or as 6: 3 14: 7. 7:6: 14, or as 3: 6: 7: 14.
thus, as 6 And, as 3 And in all these statings, the fourth term is found, by multiplying the 2d and 3d terms together, and dividing the product by the 1st term.
Of the three given numbers: two of them contain the supposition, and the third a demand. And for stating and working questions of these kinds, observe the following general Rule:
STATE the question, by setting down in a straight line the three given numbers, in the following manner, viz. so that the 2d term be that number of supposition which is of the same kind that the answer or 4th term is to be; making the other number of supposition the 1st term, and the demanding number the 3d term,