B and containing equal angles, as A, B, C, D, E, F. If the length or breadth of one of its sides be twice multiplied by itself, the last product will give the number of cubic or solid feet, &c. that the fig ure contains. Thus, if the length C of one of the sides of the adjacent cubic figure be 3 inches, the contents, 3X3X3, will be 27 cubic inches. Geometry teaches us that all similar solid bodies are to each other as the cubes of their like dimensions. Thus, if one ball, or globe, have a diameter of 2 inches, and another have a diameter of 4 inches, the latter will be 8 times the size of the former, since 4x4X4=64, while 2x2x2=8. For the same reason, a cube whose sides are 3 feet long will be 27 times as large as one of 1 foot long, since the cube of 1 (1x1x1) is 1, while the cube of 3 (3X3X3) is 27. 4. If the product of a number multiplied by itself be again multiplied by the same number, the last product is called the cube, or third power, of that number. Thus, 4X4X4=64; 64 is the cube, or third power of 4. 5. The term power designates the product arising from multiplying a number a certain number of times, and the number 80 multiplied is called the root. The powers are distinguished from each other by the number of times that the root is used as factor. Thus, 16 is the second power of 4, because it contains that factor twice ; 64 is the third power of 4, because it contains the factor 4 three times; and 256 is the fourth power of 4, because it contains 4 as a factor four times ; and so on. 6. A power is sometimes denoted by a number placed at the right of the root, and a little above it, which is called the index, or exponent of the power. Thus, 4signifies the third power, or cube of 4; and 45 the fifth power of 4. 7. The root of a number is designated by a small figure placed in what is called the radical sign. Thus, 24 signifies the square root of 4; and J16 the cube root of 16. The sign and number are called the exponent of the root. The figure is generally omitted in the radical sign for the square root. Thus, 9 and ✓9 both signify the same number, the square root of 9, namely 3. Roots and powers are also frequently denoted by numbers placed in a fractional form, the numerator expressing 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 8=N8X8=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, N2, N5, N10, 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 ? Of 42X22 ? Suggestive Questions. — How much is 42 ? How much 34 ? How much, then, is 42 X 34 ? 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 28 ? Between 32 and 33 ? Between 42 and 43 ? Ans. 4; 18; 48. 7. Find the square and the cube of each of the digits, arrange them in tabular form as follows, and commit them to memory. Roots, 1 2 3 4 5 6 7 8 9 Square, or 2d power, Cube, or 3d power, 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 SECT. III.] separate. 3. Involve the number in the usual manner. the multiplication, but keep the products of the different ranks indicate, without performing the multiplication. 2. Perform MULTIPLICATION. No. 1. 20+4 24= 1st Product by the units, 20.4+4 SQUARE of 24, 20'+twice 20.4+42 20 + 4 2d Product by units, 202.4+twice 20.4+4 2d Product by tens, 203 + twice 202 • 4+ 20.42 CUBE of 24, 20+thrice 202.4+thrice 20.4+48 8000+-4800+960+64 13824 Suggestive Questions. — What is the product (Position.] What is the sum of the products by 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 20% 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 multiplied by the and three times the square of the multiplied by 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 němber? 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, 1; Two, three; five, two. 18 17 26 30 25 17 23 37 29 27 66 74 27 5 3 8 4 8 9 5 8 4 8 9 8 35 12 13 17 18 18 22 22 22 19 19 19 23 8 3 4 4 5 4 8 3 4 5 6 34 34 34 34 68 57 38 37 52 13 13 13 15 5 9 4 7 8 6 9 4 6 2 5 3 4 17 37 65 76 15 22 22 23 16 19 17 34 27 9 7 7 9 7 5 7 4 9 9 7 9 6 3. Name the quotients, and remainders where they occur, of the following numbers : |