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256. MULTIPLICATION OF CIRCULATING DECIMALS.

Rule I. Turn both terms into their equivalent vulgar fractions, and multiply those fractions together. II. Reduce the product to its equivalent decimal, and let the work be continued until the decimal figures repeat; then mark the first and last repeating figures, and it will be the product required". ExAMPLEs. 1. Multiply .i.2 by 34. OPERATION. • - 19 4 • 31 * =;=55. 34 =; 31 124 62

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Earplanation.

I first turn the decimals into their equivalent vulgar fractions; I then multiply the fractions together, and reduce the product to a decimal, continuing the quotient until the figures recur.

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. Multiply 4, .4, and .0004 together. Prod. .0000; 90123456.

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* The reason of this operation will be plain; for if two quantities be respectively equal to other two, the product of the first two will equal the product of the last two; wherefore, in the present case, the product of the given decimals is evidently found when the product of their equivalent vulgar fractions is found.

257. DIVISION OF CIRCULATING DECIMALS.

RULE I. Change the divisor and dividend into their equivalent vulgar fractions, (Art. 249, 250, or 25l.) and divide the latter by the former, by Art. 204.

II. Reduce the quotient to its equivalent decimal, and it will be the answer".

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gether, and the result # is next reduced to a decimal, which is the quotient.

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Then # X # -: ; = .3967213 quotient.
5 61

. Divide 34 by 7. Quot. 311688
. Divide 234.6 by 13. Quot. 176.
. Divide 9.928 by 26. Quot. 37:23.
. Divide 13.5ió9533 by 4.297. Quot. 3.j45.
Divide 123456 by oosi. Quot. 1508.61.
Divide 36 by 35. Quot. 14229240011857.70780988.

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• What has been said of the product (in the foregoing note) is equally true of the quotient, as is sufficiently evident from Art. 204.

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DUODECIMALS.

258. If an unit be divided into 12 equal parts, each of these parts into 12, each of these latter into 12, and so on without end; such fractions are called Duodecimal Fractions.

259. The parts into which the unit is divided are called primes, and marked thus ‘; those into which the prime is divided, are called seconds, and are marked thus "; those into which the second is divided are called thirds, and marked thus"; and so on for the succeeding divisions, viz. fourths"; fifths", sirths"; &c.

260. Duodecimals", or Cross Multiplieation, is a method of finding the content of any rectangular surface, the length and breadth being given in feet, inches, and duodecimal parts, and is employed by artificers in computing their work.

Rule I. Under the multiplicand write the multiplier, so that feet may stand under feet, primes under primes, &c. II. Multiply each term in the multiplicand (beginning at the lowest) by the feet in the multiplier, and write the result of each under its respective term; that is, carry one for every 12 that arises, and set down the remainder. III. Multiply in the same manner by the primes, and let the result of each term stand one place to the right of that term in the multiplicand. IV. Multiply in like manner by the seconds, and set each re

P The name is derived from the Latin duodecim, twelve, and is expressive of the nature of the division and subdivision of unity, which take place in these operations. The term Cross Multiplication arises from the cross method of operating, or multiplying cross-ways. This rule is much in use among Artificers, as it supplies them with a ready method of determining the dimensions of their work and materials.

Brick-layers, masons, glaziers, and others, measure their work by the square foot; painters, paviors, plasterers, &c. by the square yard; tiling, slating, and flooring, are usually measured by the square of 100 feet, and brick-work is frequently measured by the rod of 16+ feet.

sult two places to the right: by the thirds, and set each result three places to the right; and so on to the end.

V. Add all the products together, (observing continually to carry one for every 12, and to set down the remainder,) the sum will be the answer".

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or 13f. 8' 4", as in the example; and the same may be shewn in all

4 12. 12 cases.

It will be useful to remember the following particulars; namely, that

Feet multiplied into feet produce feet. Feet multiplied into inches produce inches. Feet multiplied into seconds produce seconds. &c. &c. Inches multiplied into inches produce seconds. Inches multiplied into seconds produce thirds. &c. " &c. Seconds multiplied into seconds produce fourths. Seconds multiplied into thirds produce fifths. &c. - &c. And in general, the product of any two terms will be of that denomination, which is denoted by the sum of the numbers which express the denominations of those terms. Thus seconds multiplied by thirds produce fifths, for 2 + 3 = 5; and the same universally.

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ExAMPLEs. 1. "Multiply of 3' 1" by of 7' 9". Explanation. OPERATIon. I first multiply the multiplicand by 5,

carrying 1 for every 12, and putting

f : ". - - - -- down the remainders under their like for 3 3 1 Multiplicand. the first line. I next multiply by the 7 5 7 9 Multiplier. primes, putting the results one place to the 1 l 3 5 right of the last, and carrying as before for 1 3 9 7” the second line. I then multiply by the 9 4. seconds, putting the results two places to l 8 3 91 v the right, and carrying as before for the

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* These examples may be proved by Practice, and by Decimals; thus, Ex. I.

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It may be remarked, that the opera- 8.904.07224

tion by practice gives the exact answer, 12 while that by decimals is 7 fourths (or Togossssss 12

7- of a square foot) short of the ‘f. 20736 10.18640.256 12

truth, in consequence of both factors being infinite, and too few figures taken Toggs'ssors for the operation: if the factors had been —- i. continued, they would have been respectively 2.2569416, and 5.64593; whence (by proceeding according to articles 256 and 251.) the exact answer would have been obtained.

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