the power, and the denominator the root. Thus 4 signifies the second or square root of the first power of 4, which in this case is simply the square root of 4, since the first power of a number is the number itself; and 83 signifies the third, or cube root of the second power of 8. Thus 88×8=√/64=4.

8. Numbers whose roots can be exactly found, are called perfect powers, and their roots rational numbers. Numbers whose roots cannot be exactly expressed in numbers, are called imperfect powers, and the approximation to their roots are called surds, or irrational numbers. Thus, 1, 4, 9, are perfect powers, because they have exact roots, namely, 1, 2, 3. But 2, 5, 10, are imperfect powers, and their roots, No2, No 5, No 10, are surds, because they cannot be exactly expressed in numbers. Exercises for the Black-board or Slate.

1. What is the square, or second power of 16? 2. What is the cube, or third power of 12? 3. What is the cube of 3'6 ? 4. What is the numerical value of 42x34? Suggestive Questions. - How much is 42? How much, then, is 42×34?

Of 42×22? How much 342

5. What is the sum of the squares of the prime numbers (see Oral Arithmetic, p. 84) between 1 and 10 inclusive? Of the cubes of the composite numbers between 1 and 10 inclusive? Ans. 88; 2521.

6. What is the difference between 22 and 23? Between 32 and Between 42 and 43 ? Ans. 4; 18; 48.

33? 7. Find the square and the cube of each of the digits, arrange them in tabular form as follows, and commit them to memory.

Exemplification for the Black-board.

8. Involve 24 to the second and third power; in other words, find the square and cube of 24. Perform this in three ways; 1. Keep the units and tens separate throughout, and merely

the multiplication, but keep the products of the different ranks indicate, without performing the multiplication. 2. Perform 3. Involve the number in the usual

separate.

manner.

CUBE of 24,

13824

203+thrice 202 • 4+thrice 20.42+48 8000+4800+960+64

Suggestive Questions. What is the product | [Position.] What is the sum of the products by by the units in No. 1? What does the dot be- the units and by the tens in No. 1? Read it. tween the 20 and 4 signify? See p. 58, 1. 5. Why is this called the square of 24? Whence What does the character between the two fours comes the twice 204? [Addition.] Read the signify? The small 2 after the last 4? Read 2d product by the units in No. 1, and explain the product by the units in No. 1. [Here direct how it is produced; thus, 4 times 42-43, &c. the attention of the class to this product in Nos. Do the same with the second product by the 2 and 3, and show it to be the same.] What is tens. Read it over. In No. 3, is this product the product by the tens in No. 1? What is the 1152, or 11520? Why so? Read the sum of product by the tens in No. 3; 48 or 480? Why? the second product by the units and by the tens

in No. 1. Why is it called the cube of 24? Examine the cube in the three processes, and see if they agree. What does the square of 24 contain besides the squares of the units and of the tens? Ans. Twice their Would that be so, whatever was the number of the tens and units? To what does 202 of the fifth line of No. 1 correspond in the same line in No. 2? To what does 42 correspond? What does the cube of 24 contain besides the cubes of the tens and of the units? Ans. Three times the square of the multiplied by and three times the square of the multiplied by

the the

Would this be so, whatever was the number of the tens and units? Has not, then, the following been developed as the tenth principle of arithmetic ? namely,

X. 1. The SQUARE of any number of tens and units is equal to the squares of the tens and of the units taken separately, plus twice the product of the tens and units. 2. The CUBE of any number of tens and units is equal to the cubes of the tens and units taken separately, plus three times the square of the tens multiplied by the units, and three times the square of the units multiplied by the tens.

9. Involve 18 to the third power, as in process No. 1 of the last exercise, and repeat the 10th principle from the process that will stand before you on the slate.

10. Involve 65 to the third power, as in the last exercise, and repeat as above.

Questions by the teacher.-What is multiplication? See p. 56, 2. What is the multiplicand? The multiplier? The product? What are the factors? How may multiplication be proved? Should there be ciphers on the right of either factor, or of both factors, will the product be correct, or too small, or too large, if these ciphers be neglected in the multiplication? How, then, can this product be rectified? If decimal fractions occur in either or in both factors, will the product be correct, or too large, or too small, if the separatrix be neglected in the multiplication? How may it be rectified? What is involution? What is the square, or second power of a number? What is a cube, or third power of a number? What is a power of a root? What is the index, or exponent of a root? What is the index, or exponent of a power?

SECTION IV.-Division.

[For an explanation of the terms and characters used in division, See p. 56, 4, and 58, 8.]

Exercises for the Slate and Black-board.

1. Name the quotients of the following numbers [to be repeated as a daily exercise till the quotients can be given correctly at a glance, without naming the divisors or dividends]:

4÷2, 8÷2, 6÷2, 12÷÷2, 18÷÷2, 10÷2, 14÷2, 16÷2, 9÷3, 18÷÷3, 12÷÷3, 6÷÷3, 24÷3, 16÷4, 25÷÷5, 18÷3, 20÷5, 27 ÷3, 24÷6, 12÷6, 6÷6, 15÷÷3, 21 : 3, 21 : 7, 14 : 7, 10 : 5, 30: 5, 24: 8, 8: 8, 18: 9, 32 : 8, 54; 9, 28 : 7, 64 : 8, 49: 7, 36: 6, 48: 6, 63: 7, 72: 8, 56: 7, 81: 9, 32: 4, 40: 5, 35: 5, 36: 4, 45: 5.

2. Name the quotients and remainders of the following numbers, in this manner, namely, 4, 7; Two, three; five,

3. Name the quotients, and remainders where they occur, of

the following numbers :

18 34 24 25 32 31 24 29 39 21 18 14 25

4 5 6 5 6 7 5 7 9 3 4 2 4

Exemplification for the Black-board.

Short Division; that is, where the divisor does not exceed 12.

4. Divide 63543 by 4.

Divisor, 4)63543 Dividend.

Quotient, 15885" 3 undivided remainder.
Divisor, 4

Proof, 63543

Suggestive Questions.- How many fours in 6 of the fifth rank? The 2 that remain of the fifth rank make how many of the fourth rank? How many fours in 23 of the fourth rank, then? The 3 that remain of the fourth rank make how many of the third rank? How many fours in 35 of the third rank, then? How many of the second rank are the 3 that remain of the third rank? How many fours in 34 of the second rank, then? How many of the first rank are the 2 that remain of the second rank? How many fours in 23, then? Our quotient, then, appears to be 15885 and 3 remainder. Is the remainder carried to the right or to the left in division, as above? Which way are numbers carried in all other operations; that is, in addition, subtraction, and multiplication? Why, then, do we commence at the left in division, and at the right in all other operations?

Proof. In division, what terms are factors of the dividend? See p. 57, 5. What terms, then, multiplied together, will reproduce the dividend, if the work be correctly done? Is the remainder a part of the quotient? Is it also a part of the dividend? Why must it be added in when the dividend is reproduced by multiplying the divisor and quotient?

5. Divide 264852 by each of the digits severally from 2 to 9, also by 11 and 12, proving each problem by multiplication, as above.

6. Divide 65382432 by each digit separately from 2 to 9, also by 11 and 12, and prove by multiplication.

7. Divide 97862432 by each digit separately from 2 to 9, also by 11 and 12, and prove by multiplication.