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The foot being viewed as the unit, duodecimals present the following relations.

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Addition and subtraction of duodecimals are performed as addition and subtraction of other compound numbers 12 of a lower denomination making one of a higher Multiplication, however, when both the factors are duodecimals, is peculiar, and will now be considered.

When feet are multiplied by feet, the product is in feet. For instance, if required to ascertain the superficial feet in a board 6 feet long and 2 feet wide, we multiply the length by the breadth, and thus find its superficial, or square feet to be 12. But when feet are multiplied by any number of inches, [primes], the effect is the same as that of multiplying by so many twelfths of a foot, and therefore the product is in twelfths of a foot, or inches. Thus a board, 6 feet long and 6 inches wide, contains 36 inches, because the length being multiplied by the breadth, that is, 6 feet by of a foot, the product is of a foot, or 36'3 feet. When feet are multiplied by seconds, the product is in seconds. Thus 6 feet multiplied by 6 seconds, that is, of a foot by 1 of 12 of a foot, the product is of a foot, or 36" 3 inches.

36 144

12

Feet multiplied by feet, produce feet.
Feet multiplied by primes, produce primes.
Feet multiplied by seconds, produce seconds.
Feet multiplied by thirds, produce thirds.

&c.

Primes multiplied by primes, produce seconds.
Primes multiplied by seconds, produce thirds.
Primes multiplied by thirds, produce fourths.

&c.

Seconds multiplied by seconds, produce fourths.
Seconds multiplied by thirds, produce fifths.
Seconds multiplied by fourths, produce sixths.

&c.

If we would find the length, and 4 f. 6′ 5′′ in

6 f. 4' 8"

4

2/ 7"

6' 5"

3 2' 4" 0""

25 6' 8"

square feet in a floor 6 f. 4′ 3′′ in breadth, we proceed as follows.

We begin on the right hand, and multiply the whole multi

11 4 plicand, first by the seconds in the multiplier, then by the inches, and lastly by the feet. We then add the results together, and thus obtain the answer.

28f. 11' 7" 11" 4"""

We are now led to a general rule for the multiplication of duodecimal numbers.

RULE. Place the several terms of the multiplier under the corresponding ones of the multiplicand. Beginning on the right hand, multiply the several terms of the multiplicand by the several terms of the multiplier successively, placing the right hand term of each of the partial products under its multiplier. Then add the partial products together; observing to carry one for every twelve, both in multiplying and adding. The sum of the partial products will be the answer.

Questions in duodecimals are very commonly performed by commencing the multiplication with the highest denomination of the multiplier, and placing the partial products as in the first of the two following operations. The result is the same, whichever method is adopted. The second operation, however, is according to the rule we have given, and is more conformable to the multiplication of numbers accompanied by decimals.

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When there are not feet in both the factors, there may not be any feet in the product; but, after what has been said, there will be no difficulty in determining the places of the product.

1. Multiply 14f. 9' by 4f. 6'.

2. What are the contents of a marble slab, whose length is 5f. 7, and breadth 1f. 10'?

3. How many square feet are there in the floor of a hall, 48f. 6' long, and 24 f. 3' wide?

4. Multiply 4f. 7' 8" by 9f. 6'.

5 How many square feet are there in a house lot, 43 f. 3 in length, and 25 f. 6' in breadth?

6. What is the product of 10f. 4' 5" by 7f. 8′ 6′′? 7. Calculate the square feet in an alley 44 f. 2′ 9′′ long, and 2f. 10′ 3′′ 2′′ 4′′ wide.

8. How many square feet are there in a garden, 39 f. 10' 7" long, and 18 f. 8′ 4′′ wide?

9. What is the product of 24f. 10' 8" 7" 5" by 9f. 4' 6"?

10. Compute the solid feet in a wall, 53 f. 6′ long, 12f. 3' high, and 2f. thick.

11. The length of a room is 20 feet, its breadth 14 feet 6', and its height 10f. 4. How many yards of painting are there in its walls, deducting a fire place of 4f. by 4f. 4'; and two windows, each 6f. by 3f. 2?

12. How many yards of carpeting, yard wide, will be required for a room 21 f. 6' long, and 18f. wide?

13. What will the plastering of a ceiling come to, at 10 cents a square yard, supposing the length 21 feet 8 inches, and the breadth 14 feet 10 inches?

14. How many yards of papering on the four walls of a hall, 58 f. 8' long, 21 f. 4' wide, and 13 f. 9' high; deducting 2 doors, each 7 f. 6' high and 4f. wide; 7 windows, each 6 f. 2′ high and 3f. 10' wide; and a mop-board, 9 mches wide around the hall?

END OF PART SECOND.

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