17. The value of 5 horses and 7 males is $2436; now if the value of each mule is $208, what is the value of each horse? Ans. $196.

18. A man left $2535 each to his four children, but one of them dying, the three remaining children divided the money; how much did each receive? Ans. $3380.

19. Mr. Smith left $6264 to each of three sons and $7240 to each of two daughters, but one daughter dying, her share was equally divided among the remaining children; what did each receive? Ans. Son, $8074; daughter, $9050.

20. The income of a man who "struck oil" was $480 a day; how many teachers would this employ at a salary of $438 a year?

Ans. 400.

21. A stock dealer bought 325 cows at $28 each, and sold 124 of them at cost; how much must he receive a head for the remainder to gain $804? Ans. $32.

22. Mr. Galton buys a farm of 110 acres at $75 an acre, $2200 to be ps d down and the remainder in five yearly installments; what must he pay each year? Ans. $1210.

23. A farmer raised 765 bushels of oats, of which he kept 65 bushels for seed, and after retaining enough for the use of his horses till next harvest, allowing 60 bushels to each horse, sold the balance at 85 cents a bushel, and received $442; how many horses had he? Ans. 3 horses.

24. Mr. Milman bequeathed $6500 to each of two sons, to a third son $1000, $5000 to each of 3 daughters, and the balance of his estate, amounting to $25,000, to several benevolent institutions; the will, however, being set aside, the property was divided equally among his children; what was the share of each? Ans. $9000.

25. If a soldier enlisting in the late war for 3 years, received a bounty of $930; then served one year as a private, at $13 a month, 6 months as a corporal, at $14 a month, and 18 months as a sergeant at $17 a month; what was the whole amount of his pay and his average pay per month? Ans. $41 a month

GENERAL PRINCIPLES

OF THE FUNDAMENTAL RULES.

PRINCIPLES OF ADDITION.

1. The sum of all the parts equals the whole.

2. The whole diminished by one or more parts equals the sum of the other parts

PRINCIPLES OF SUBTRACTION.

1. The Remainder equals the Minuend minus the Subtrahend. 2. The Minuend equals the Subtrahend plus the Remainder. 8. The Subtrahend equals the Minuend minus the Remainder.

PRINCIPLES OF MULTIPLICATION.

1. The Product equals the Multiplicand into the Multiplier. 2. The Multiplicand equals the Product divided by the Multiplier. 3. The Multiplier equals the Product divided by the Multiplicand.

PRINCIPLES OF DIVISION.

1. The Quotient equals the Dividend divided by the Divisor. 2. The Dividend equals the Divisor multiplied by the Quotient. 3. The Divisor equals the Dividend divided by the Quotient.

4. The Dividend equals the Divisor multiplied by the Quotient plus the Remainder.

5. The Divisor equals the Dividend minus the Remainder, divided by the Quotient.

OTHER PRINCIPLES OF DIVISION.

1. Multiplying the Dividend or dividing the Divisor by any num. ber, multiplies the Quotient by that number.

2. Dividing the Dividend or multiplying the Divisor by any number, divides the Quotient by that number.

3. Multiplying or dividing both Dividend and Divisor by the same number, does not change the Quotient.

4. A General Law. -A change in the Dividend by multiplication or division produces a similar change in the Quotient; but such a change in the Divisor produces an opposite change in the Quotient.

NOTE TO TEACHER.-Let the pupils be required to show the reason for the above principles, and give illustrations of them. No demonstrations are given, since it is better for the pupil to learn to depend somewhat upon himself, that he may become, not a mere imitator, but an original thinker.

INTRODUCTION TO SECONDARY OPERATIONS.

MENTAL EXERCISES.

1. What numbers multiplied together. will produce 41 6? 8? 10? 18? 14? 16? 18? 20? 24? 26? 28? 80?

2. What numbers can be produced out of the numbers 2 and 8? 8 and 5? 2, 3, and 5? 3, 4, and 5? 2, 3, 4, and 5?

3. Will the product of any two numbers, each greater than a unit produce 1, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 81, and 37 ?

4. What may we call a number which is composed by multiplying several numbers together? Ans. A Composite Number.

5. What shall we call numbers that cannot be produced by multiplying several numbers together? Ans. Prime Numbers.

6. Which are prime and which composite numbers in the follow ing list: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15?

7. What may we call the numbers whose product makes a composite number? Ans. Makers of the numbers.

8. If the word Factor means the same as maker, what may we call the makers of a composite number? Ans. Factors.

9. Form composite numbers out of the factors 3 and 4; 8, 4, and 6; 4, 5, and 6. What are the factors of 12? 15? 18? 20? 21? 24?

10. Form a composite number by using 2 twice as a factor; 8 twice as a factor; 2 three times as a factor; 3 four times as a factor.

11. Required one of the two equal factors of 9; of 16; of 25; of 86 one of the three equal factors of 8; of 27; of 64; of 125.

12. A number composed of two equal factors is called the second power of that factor; of three equal factors, the third power, etc.

13. Required the second power of 3; of 4; of 6; of 7; of 8: the third power of 2; of 3; of 4; of 6.

14. One of the two equal factors of a number is the second root of a number; one of the three equal factors is the third root, etc.

15. What is the second root of 16? of 25? of 86? of 49? What is the third root of 8? of 27? of 64? of 125 ?

16. What would it seem natural to call the process of making composite numbers? Ans. Composition.

17. What would it seem natural to call the process of finding the factors of a number? Ans. Factoring.

18. What are the first four operations of arithmetic called? Ane. The Fundamental or Primary Operations of arithmetic.

19. What would it be natural to call these operations which are derived from the fundamental operations? Ans The Derivative or Secondary Operations.

SECTION III.

SECONDARY OPERATIONS.

100. The Primary Operations of arithmetic are those of synthesis and analysis, including the four fundamental rules.

101. The Secondary, or Derivative Operations, are those which arise from or grow out of the primary opera tions of synthesis and analysis.

102. The Secondary Operations are Composition, Factoring, Greatest Common Divisor, Least Common Multiple, Involution, and Evolution.

COMPOSITION.

103. Composition is the process of forming composite numbers when their factors are given.

104. A Composite Number is a number which can be produced by multiplying together two or more numbers, each greater than a unit; as 8, 12, 15, etc.

105. The Factors of a composite number are the numbers, which, when multiplied together, will produce it; thus 4 and 2 are the factors of 8.

106. A Prime Number is one that cannot be produced by multiplying together two or more numbers, each greater than a unit; as, 2, 5, 7, 11, etc.

107. A composite number consisting of two equal factors is said to be the 2d power of that factor; of three equal factors, the 3d power, etc.; thus, 9 is the 2d power of 3, and 64 is the 3d power of 4.

NOTE-An even number is one that is exactly divisible by 2; an odd number is one that is not exactly divisible by 2.

MENTAL AND WRITTEN EXERCISES.

1. Tell which of the following numbers are prime or somposite: 2, 8, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16.

2. Name the prime numbers from 1 to 53. Name the prime numbers from 53 to 101.

3. Write the numbers from 1 to 100, and cut out all the composite numbers, leaving the primes.

4. What is the second power of 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20?

5. What is the 3d power of each of the above? The 4th power? The 5th power? The 6th power?

PRINCIPLES.

1. Every composite number is equal to the product of its factors.

2. A factor of a number is a factor of any number of times that number.

108. To form composite numbers out of any factors 1. Form a composite number out of 4, 9, and 5. SOLUTION.-A composite number formed out of the factors, 4, 9, and 5, is equal to their product, which is 180.

OPERATION.

4x9x5=180

WRITTEN EXERCISES.

Form composite numbers out

2. Of 5, 6, 7, and 8.

3. Of two 2's, 3, and 7.

Ans. 1680.
Ans. 84.

4. Of three 3's, four 2's, and two 5's.
5. Find a number consisting of four 5's.
6. Find the fifth power of 3, of 4, of 7.

Ans. 243;

Ans. 10800.
Ans. 625

1024; 16807.

7. Form a composite number out of the first four prime Dumbers after unity.

Ans. 210.

8. Form a composite number out of all the prime numbers between 11 and 29. Ans. 96577. 9. Form all the composite numbers you can out of 2, 3, Ans. 6; 10; 14; 15; 21; etc.

5, and 7.

10. Form all the composite numbers you can out of 2, 3, 5, 7, and 11. Ans. 6; 10; 14; 22; 15; 21; etc. 11. Find a composite number consisting of three factors. the first being 2, the second being twice as great, and the third three times as great.

Ans 48.