give fourths; seconds multiplied by thirds, give fifths, &c.; that is, the product of any two denominations will always be of the denomination expressed by the sum of their indices.

RULE.-Begin with the lowest denomination of the multiplicand, and multiply it by the highest denomination of the multiplier, and place each term of the product according to its respective value. Multiply in the same manner by each remaining denomination of the multiplier, and place the product of each succeeding multiplication one or more places farther to the right, according to the denomination. The several products thus obtained, when added together, will give the required product.

Note. It will be remembered to carry by 12 in all cases. Ex. 1. Multiply 4 feet 4 inches, by 4 feet 4 inches.

OPERATION.

ft. in.
4 4'

4 " 4'

17

4'

1

5' 4"

Beginning with the 4 feet in the multiplier, we say 4 times 4' are 16'1 ft. 4'. The 4' is set down and the 1 ft. carried to the next product, making it 17 feet. Then multiplying by 4', we say 4'x4' 16"-1' and 4"; set down the 4" and carry the 1', and say 4' times 4 are 16' and 1' is 17'1 ft. and 5', which being set down, and the two products added, we obtain 18 ft. 9' 4" as the required product.

18 ft. 9' 4"

11. How many square feet does a board 28 ft. 10' 6" long, and 3 ft. 2' 4" wide contain? Ans. 92 ft. 2′ 10′′ 6''.

12. In a board 16 ft. 9' long, and 2 feet 3 broad, how many square feet? Ans. 37 ft. 8' 3".

13. There is a wall 82 ft. 6 in. high, and 13 ft. 3 in. wide. How many square feet does it contain? Ans. 1093 ft. 1' 6".

14. There is a room 20 feet square and 7 ft. 6 in. high, to be plastered at 10 d. New York currency, per square yard. How many dollars will it cost? Ans. $6.94.+

15. There is a yard 58 ft. 6 in. in length, and 54 ft. 9 in. in breadth. How many dollars will it cost to pave it, at 5 d. New York currency, per square yard? Ans. $18.53.

16. If a floor be 59 feet 9 inches long, and 24 feet 6 inches broad, how many square yards does it contain? Ans. 162 yards, 5 ft. 10'6".

Note 2d. If three dimensions, viz. length, breadth, and depth be given, the solid content is found by multiplying them successively into each other.

17. There is a pile of wood 12 feet 6 inches long, 4 feet high, and 8 feet 6 inches wide. How many cords does it Ans. 3 cords, 41 feet.

contain ?

To reduce solid feet to cords, divide by 128, that being the number of solid feet in one cord. The required dimensions of the cord, are 8 feet long, 4 feet wide, and 4 feet high; since 8×4 x4=128.

18. How many solid feet are there in a block 6 feet 8 inches in length, 4 feet 6 inches in height, and 3 feet 4 inches in width? Ans. 100 feet.

19. There is a certain pile of wood measuring 24 feet in length, 16 feet 9 inches in depth, and 12 feet 6 inches in width. How many cords are there; and how many solid feet may be daily consumed to have it last one year? Ans. 39 cords, 33 feet; daily allowance, 133 feet, nearly.

20. How many square feet are there in a board, which measures 16 feet, 9 inches in length, and 2 feet, 3 inches in breadth? Ans. 37 feet, 8'3".

QUESTIONS.-What are Duodecimals? How is the foot divided? What is each part called? How is the inch divided? What is each part called, &c.? When is the foot divided duodecimally? Repeat the table of denominations. How may duodecimals be added? What difficulty will be encountered in Multiplication of Duodecimals? Give an illustration of the difficulty. Of what denomination will the product of any two denominations be? What is the rule? What is Note 1st? What is Note 2d?

INVOLUTION.

A number is involved by being multiplied into itself. The number thus multiplied by itself is called the root. A power of any number is the product obtained by multiplying that number into itself. The particular power produced depends on the number of successive multiplications; the given number always being the first power, and also the root of the succeeding higher powers. The first multiplication then produces the second power; the second multiplication, the third power; the third multiplication, the fourth power, &c., the power obtained being always one in advance of the number of multiplications.

Illustration: 2=the first power, and is also the root of the succeeding higher powers.

2x2 4, the 2d power, or square of 2. 2×2×28, the 3d power, or cube of 2. 2×2×2×2=16, the 4th power, or biquadrate of 2. 2×2×2×2×2=32, the 5th power of 2, &c.

The power to which a number is to be raised, is frequently expressed by a small figure, called the index of the required power, placed on the right of that number, thus: 42 denotes the second power of 4-16; and 43 denotes the third power of 4-64; and 95 denotes the fifth power of 9=59049, &c.

i

A fraction is involved by multiplying the numerator and the denominator each into itself, the required number of times; thus the square of is 3x=5. The square of is 36; and the cube of the same is 216 &c. 3432

If the given quantity be a mixed number, it should first be reduced to an improper fraction, before being involved; thus the second power of 2 is 24, and x=25-6. Or, proper and improper fractions may both be reduced to decimals, and then involved.

If any number be raised to two different powers, the power which is obtained by multiplying these two powers together, is expressed by adding their indices, thus: 22×23=25=32; for 22-4, and 23-8, and 8×4=32; and 2×2×2×2×2= 32. Or, 33x35=3°=6561, for 33=27, and 35=243; and 243 X27=6561.

Or, again, any power of a given number is divided by another power of the same number by subtracting the index of the divisor from the index of the dividend, thus: 25÷23=23, for 25 =32, and 23=8, and 32÷8=4, the second power of 2. Or 34÷32=32, for 3=81, and 32=9, and 81-9-9, the second power of 3.

Ex. 1. What are the square, cube, and biquadrate of 3? Ans. 3×39, the square; 3×3×3=27, the cube; and 3×3×3 X3 81, the biquadrate.

2. What are the square, cube, and biquadrate of 5? Ans. 25, 125, and 625.

3. What are the cube and biquadrate of 12? Ans. 1728 and 20736.

4. Multiply the second and third powers of 4 together. What is the product, and what power of 4 is it? Ans. The product, 1024, or fifth power.

5. What power of 3 is obtained by multiplying its third power and its fourth power together; and what is the number? Ans. 7th power, or 2187.

Note. When the number to be raised to some given power consists of whole numbers and decimals, the number of decimals to be cut off in the required power is ascertained by multiplying the number of decimals in the given number by the index of the required power.

6. What is the square of 26.13? 26.13 x 26.13=6827769. Now to determine how many decimals are to be cut off, we

first notice that the number of decimals in the given number is two, and also, that the index of the required power is 2, therefore, 2x2=4, the number of decimals to be cut off. Therefore, 682.7769 is the required power.

7. What is the cube of 25.4? Ans. 16387.064.

8. Divide 2o by 23, and what power of 2 will be obtained? Ans. 8, the cube of 2.

EVOLUTION.

Evolution is the reverse of Involution. In Involution we have the root given to find some required power; but in Evolution a power is given, and a root required.

The relation between roots and powers requires to be clearly understood.

A root of a number is obtained whenever that number is resolved into several equal factors; and a power of a number is obtained whenever that number taken as a root, is multiplied into itself once or more. Thus, 2 is the cube root of 8, because it may be resolved into three 2's; or because 2 raised to its third power equals 8, for 2×2×2=8. Also, 8 is the square root of 64, because the second power of 8 is 64, or 8x 8=64. Again, 9 is the square of 3, and 3 is the square root of 9; 27 is the cube of 3, and 3 is the cube root of 27. Roots and powers are therefore correlative terms.

Such

The exact root of some numbers cannot be obtained. numbers are called irrational powers, and their roots are called surds. Thus, no root can be obtained, which, when multiplied into itself, will produce 2; 2 is therefore an irrational power, and its root is a surd. But a number whose root can be exactly extracted is a perfect or complete power, and its root is called a rational number. Thus, 16 is a complete power, for 4 is its exact root; 4 therefore is a rational number. There are two methods of expressing roots. The first and more common method is, by using the character called the radical sign; written thus, V. This sign, without any accom