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OBS. The accents used to distinguish the different denominations below feet, are called Indices.

333. Duodecimals may be added and subtracted in the same manner as other compound numbers. (Arts. 168, 169.)

MULTIPLICATION OF DUODECIMALS.

334. Duodecimals are principally applied to the measurement of surfaces and solids. (Arts. 153, 154.) Ex. 1. How many square feet are there in a board 8 ft. 9 in. long and 2 ft. 6 in. wide?

Operation.

8 ft. 9' length, 2 ft. 6′ width, 17 ft. 6'

4 ft. 4' 6"

9

We first multiply each denomination of the multiplicand by the feet in the multiplier, beginning at the right hand. Thus, 2 times 9' are 18', equal to 1 ft. and 6'. Set the 6' under inches, and carry the 1 ft. to the next product. 2 times 8 ft. are 16 ft. Again, since 6' of a 9' is of a ft. 54" or 144 place to the right of inchproduct. Then 6' or of a ft. or 4 ft. Now sum is 21 ft. 10' 6",

21 ft. 10' 6" Ans.
and 1 to carry makes 17 ft.
ft. and 9' of a ft., 6' into
4' and 6". Write the 6" one
es, and carry the 4' to the next
of a foot multiplied into 8 ft.
adding the partial products, the
which is the answer required.

OBS. It will be seen from this operation, that feet multiplied into feet, produce feet; feet into inches, produce inches; inches into inches, produce seconds, &c. Hence,

335. To find the denomination of the product of any two factors in duodecimals.

Add the indices of the two factors together, and the sum will be the index of their product.

Thus feet into feet, produce feet; feet into inches, produce inches; feet into seconds, produce seconds; feet into thirds, produce thirds, &c.

QUEST.-333. How are duodecimals added and subtracted? 334. To what are duodecimals chiefly applied? 335. How find the denomination of the product in duodecimals? What do feet into feet produce? Feet into inches? Feet into seconds?

Inches into inches, produce seconds; inches into seconds, produce thirds; inches into fourths, produce fifths,&c. Seconds into seconds, produce fourths; seconds into thirds, produce fifths; seconds into sixths, produce eighths, &c.

Thirds into thirds, produce sixths; thirds into fifths, produce eighths; thirds into sevenths, produce tenths, &c. Fourths into fourths, produce eighths; fourths into eighths, produce twelfths, &c.

Note.-The foot is considered as the unit, and has no index.

336. From these illustrations we have the following RULE FOR MULTIPLICATION OF DUODECIMALS.

I. Place the several terms of the multiplier under the corresponding terms of the multiplicand.

II. Multiply each term of the multiplicand by each term of the multiplier separately, beginning with the lowest denomination in the multiplicand, and the highest in the multiplier, and write the first figure of each partial product one or more places to the right, under its corresponding denomination. (Art. 335.)

III. Finally, add the several partial products together, carrying 1 for every 12 both in multiplying and adding, and the sum will be the answer required.

OBS. It is sometimes asked whether the inches in duodecimals are linear, square, or cubic. The answer is, they are neither. An inch is 1 twelfth of a foot. Hence in measuring surfaces an inch is 12 of a square foot; that is, a surface 1 foot long and 1 inch wide. In measuring solids, an inch denotes of a cubic foot. In common language, inches in duodecimals are called carpenter's inches.

2. How many square feet are there in a board 18 feet 9 inches long, and 2 feet 6 inches wide?

3. How many square feet are there in a board 14 feet 10 inches long, and 11 inches wide?

QUEST.-What do inches into inches produce? Inches into thirds? Inches into fourths? Seconds into seconds? Seconds into thirds? Seconds into eighths? Thirds into thirds? Thirds into sixths? 336. What is the rule for multiplication of duodecimals? Obs. In duodecimals, what kind of inches are those spoken of in measuring surfaces? In measuring solids? What are they called?

4. How many square feet in a gate 12 feet 5 inches wide, and 6 feet 8 inches high?

5. How many square feet in a floor 16 feet 6 inches long, and 12 feet 9 inches wide?

6. How many square feet in a ceiling 53 feet 6 inches long, and 25 feet 6 inches wide ?

7. How many square feet are there in a stock of 6 boards 17 feet 7 inches long, and 1 foot 5 inches wide? 8. How many feet in a stock of 10 boards 12 feet 8 inches long, and 1 foot 1 inch wide?

9. How many cubic feet in a stick of timber 12 feet 10 inches long, 1 foot 7 inches wide, and 1 foot 9 inches thick?

10. How many cubic feet in a block of marble 8 feet 4 inches long, 2 feet 6 inches wide, and 1 foot 10 inches thick?

11. How many cubic feet in a load of wood 6 feet 7 inches long, 3 feet 5 inches high, and 3 feet 8 inches wide?

12. How many feet in a load of wood 7 feet 2 inches long, 4 feet high, and 3 feet wide?

13. How many feet in a load of wood 9 feet long, 4 feet 3 inches wide, and 5 feet 6 inches high?

14. How many feet in a pile of wood 100 feet long, 5 feet high, and 4 feet wide?

15. How many feet in a pile of wood 150 feet long, 8 feet high, and 5 feet wide?

16. How many cubic feet in a wall 40 feet 6 inches long, 5 feet 10 inches high, and 2 feet thick?

17. How many solid feet in a vat 10 feet 8 inches long, 7 feet 2 inches wide, and 6 feet 4 inches deep?

18. How many bricks 8 inches long, 4 inches wide, and 2 inches thick, are there in a wall 20 feet long, 10 feet high, and 11⁄2 feet thick?

19. How much will the flooring of a room which is 20 feet long and 18 feet wide come to, at 64 cents per square foot?

20. How much will the plastering of a room 16 feet square come to, at 12 cents per square yard?

plied by 5?

SECTION XIV.

POWERS AND ROOTS.

INVOLUTION.

ART. 337. Ex. 1. What is the product of 5 multiAns. 5 x5=25. 2. What is the product of 3 multiplied into 3 twice? Ans. 3x3x3=27. 3. What is the product of 2 into 2 three times? Ans. 2x2x2x2=16.

338. When any number or quantity is multiplied into itself, the product is called a power. Thus in the examples above the products 25, 27, and 16 are powers.

339. Powers are divided into different orders; as the first, second, third, fourth, fifth power, &c. They take their name from the number of times the given quantity or number is used as a factor, in producing the given power. The original number, that is the number which being multiplied into itself, produces a power, is called the first power or root of all the other powers of that number; because they are derived from it.

Note.-1. The second power of a number is also called the square; (Art. 153. Obs. 1 ;) for if the side of a square is 3 yards, then the product of 3X3=9 yards, will be the area of the given square. (Art. 163.) But 3X3 9 is also the second power of 3; hence it is called the square.

yards.

3 yards.

3X3=9 yards.

QUEST.-338. What is a power? 339. How are powers divided? From what do they take their name? What is the first power? Note. What is the second power called? The third? The fourth?

[blocks in formation]

8. What is the cube of 3? Of 4? Of 5? Of 6?

340. The number of times a quantity or number is employed as a factor to produce the given power, is usually denoted by a small figure placed above it at the right hand. This figure is called the index or exponent. The index of the first power is 1; but this is omitted; for (2)12. The square or second power of 4, or 4×4, is written (4); the cube or third power of 12, or 12 × 12 × 12, is written 12)3. Hence,

21-2, the first power of 2;
22=2×2, the square, or 2d

power

of 2;

23-2×2×2, the cube, or 3d power of 2;
24=2×2×2×2, the biquadrate or 4th power of 2;
25=2×2×2×2×2, the 5th power of 2.

QUEST.-340. How are powers denoted? What is this figure called What does it show? What is the index of the first power?

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