257. DIVISION OF CIRCULATING DECIMALS.

LE I. Change the divisor and dividend into their equivavulgar fractions, (Art. 249, 250, or 251.) and divide the by the former, by Art. 204.

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

Divide 9.928 by .26. Quot. 37.23,

Divide 13.5169533 by 4.297. Quot. 3.145.

Divide 12.3456 by .0081. Quot. 1508.91.

Divide 36 by 25. Quot. 1.4229249011857707509881.

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

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'; sixths vi; &c.

iv

260. Duodecimals", or Cross Multiplication, 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 Artifcers, 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 ".

8

12'

12'

and 2f. 5′ 2 + wherefore 5+ ×2+

This rule may be proved by vulgar fractions; thus, ex. 3. 5f. 8′ = 5 +

5

8

12

12

12

4

12.12

cases.

or 13ƒ. 8′ 4′′, as in the example; and the same may be shewn in all

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

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 same universally.

5; and the

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

6'

By Practice.

By Decimals. 2.2569 &c. = 2f. 8′ 1′′

5.6458 &c. = 5 7 9

180552

3

112845

3 1

90276

1

6

112845

10.18640256

12

2.23683072

It may be remarked, that the operation by practice gives the exact answer, while that by decimals is 7 fourths (or

7 of a square foot) short of the

20736

truth, in consequence of both factors being infinite, and too few figures taken for the operation: if the factors had been continued, they would have been respec

tively 2.2569416, and 5.64593; whence (by proceeding according to articles 256 and 251.) the exact answer would have been obtained.

6. Multiply 10f. 4' 5" and 7f. 8' 6" together. Prod. 79f. 11' 0" 6" 6iv.

7. Multiply 5f. 4' 8" by 9f. 8' 6". Prod. 52f. 3′ 9′′ 8"".

8. Multiply 311f. 7′ 5′′ by 36f. 4′ 11′′. Prod. 11345ƒ. 11′ 1′′ 5" 7IV.

9. What will a mahogany side-board cost, which is 5 feet 11 inches long, and 3 feet 7 inches wide throughout, at 4s. 6d. per square foot ? Ans. 41. 15s. 4d..

10. What sum will pay for new glazing a hall window containing 60 squares, each 1f. 2′ 3′′ long, and 11′ 5′′ wide, at 3s. Sd. per square foot? Ans. 121. 8s. 6d..

11. What is the price of a marble slab, the length of which is 5 feet 7 inches, and breadth 3 feet 8 inches, at 9s. per square foot? Ans. 91. 4s. 3d.

12. A room which is 66 feet 10 inches about, was wainscotted 3 feet 11 inches upwards from the floor; what did it come to at ̧ 2s. 7d.. per square foot? Ans. 341. 7s. 1d..

13. A drawing-room which is 32 feet 8 inches long, and 25 feet 9 inches wide, is surrounded with a cornice 34 inches wide, the gilding of which cost 31. 5s. 6d. required what sum was charged per square foot? Ans. 1s. 11d. 7507.

14. The paving of a brew-house, 24 feet 11 inches long, and 34 feet 6 inches broad, cost 7s. 9d. per square yard; what did the whole amount to? Ans. 371. Os. 2d.

15. The expense of digging, planting, and manuring a kitchen-garden amounted to 14l. 1s. 8d. how much is that per square yard, supposing the length to be 109 feet 6 inches, and the breadth 58 feet 6 inches? Ans. 4d..

16. What sum must I pay for painting a room 48 feet 10 inches about, and 9 feet 10 inches high, at 2s. 8d. per square yard? Ans. 71. 5s. 7d.4.