113. What is a composite factor of a number? Has 25 a composite factor? Has 36? 40? 49? 12? 81? Is oue or unity considered a factor? Are all even numbers except 2, prime or composite? By what prime number can they be divided? Is any number prime, when the sum of digits is divisible by 3? By what prime number can such a number be divided? Can we divide any number ending with a cipher or 5? By what prime number can we divide it? Can any prime number except 5 end with a five or a cipher? Is 3003 a prime number? By what is it divisible? 114. Give the rule for finding the prime factors of a composite number. When the same prinie factor is found in a number more than once; how is it expressed? What does the small figure denote ? Write the third power of five on the blackboard.

115. What does a parenthesis denote? Show the use of the parenthesis on the blackboard Show the use of a vinculum.

116. What is cancellation? Upon what principle does it depend? Is a number changed by being both multiplied and divided by the same number? How is a multiplier known in arithmetic? AN3.-By the sign of multiplication placed before it. How is a divisor known? ANS.-By the sign of division placed before it. When a sign of division is placed before a parenthesis, is a part or all the numbers within it used as the divisor? ANS.-All of them. What is to be done with the signs within the parenthesis, when the sign of division is placed before it? When the sign of multiplication is placed before it? 17 + (18 + 4 × 3) =what? 17 X (18 X 4 X 3) what?

117. What is a common divisor of two or more numbers? Is the common divisor of two numbers a factor of each? Why? Must a common divisor of two or more numbers be a prime or composite number?

118. What is the greatest common divisor of two or more numbers? What is said of numbers which have no common divisor? When two numbers have no common divisor, have they a common factor? Are composite numbers ever prime to each other? Give the first rule for finding the prime factors of a composite number. Give the second.

.

119. What is the multiple of a number? What is a multiple of 7? Of 9? Of 5? Of 4? Eight is a multiple of what number? Six is a multiple of what number? 120. What is a common multiple of two or more numbers? What is a common multiple of 4 and 6? Of 5 and 8? Of 3, 4, and 7? Twelve is a common multiple of what numbers? Twenty is a common multiple of what numbers? 121. What is the least common multiple of two or more numbers? What is the least number that can be divided by 4, 3, and 2? What is the least number that can be divided by 5, 3, and 10? What is this number called? Must a least common multiple be a prime or a composite number? Give the second rule for finding the least common multiple of two or more numbers. When finding the least common multiple of two or more numbers, why can factors that divide any of the other numbers be rejected?

122. What is a fraction? What is meant by? By? By of 7? How many kinds of fractions are there?

123. What is a common fraction? What is meant by a power of ten? .124. What is a decimal fraction? How are decimal fractions usually expressed? Can decimal fractions be treated in the same way as common fractions? How are common fractions expressed? How are decimal fractions usually expressed?

125. What does the denominator of a fraction show? From what does the fraction take its name? How may it be written ?

126. What does the numerator of a fraction show? How may it be written ? Show on the blackboard in how many ways can be written. In writing what is the number above the line called? What does it show? What is the number below the line called? What does it show? What kind of a fraction is? Is? Give an example on the blackboard. Is it

127. What is a proper fraction? greater or less than a unit ?

128. What is an improper fraction? Give an example of a fraction equal to a unit. Give an example of a fraction greater than a unit. Give an example of a fraction less than a unit. How many kinds of fractions are there? ANS.-Two, common and decimal. How are fractions divided in respect to value? ANS.-Into proper

and improper fractions.

129. What is a mixed number? What sign is understood between the integer and the fraction? Write a mixed number on the blackboard. Write an integer. Write a fraction.

130. What is a simple fraction? Give an example of a simple fraction.

131. What is a complex fraction? How does a simple fraction differ from a com. plex fraction? Give an example of each.

132. What is a compound fraction? Give an example of a compound fraction. ANS.

What is meant by 3 of 7? By of ? What does of signify? When is of used?

When the multiplier is a fraction. When is times used? ANS. When the mul tiplier is an integer.

133. By what is the denomination of a fraction determined? Which term of a fraction shows the denomination?

134. On what does the value of a fraction of any denomination depend?

135. What is reduction of fractions? Upon what does the value of the denominator depend? ANS-Upon the number of the equal parts into which the unit is divided. Which denomination has the greater value, 7ths or 9ths?

136. What is reduction ascending? When is a simple fraction said to be ex. pressed in its lowest terms? What is meant by the numerator and denominator being prime to each other? Which denomination does the lowest term express?

137. What is reduction descending? In reduction descending is the size of the equal parts increased or diminished? Recite the formula for finding how many quarters in a half. Is the operation reduction ascending, or descending? Which is the greater denomination? Is the value of the fraction changed?

138. To what does the numerator of a fraction correspond in division? To what does the denominator correspond? Repeat Principle I. in the reduction of fractions. Illustrate by, lines at the blackboard. Illustrate the same by a problem. Recite Principle II. Give the reason. Illustrate by lines. Illustrate by an example. Recite Principle III. Give the reason, and illustrate in the same way. Recite Principle V. Illustrate at the blackboard. Illustrate Principle VI. Give the law. Give the formula for finding the number of thirds in 8-12ths. Illustrate the same by lines at the blackboard.

139. What is meant by the lowest terms of a fraction? When is a fraction said to be expressed in its lowest terms? Give the formula for reducing a fraction to lower terms. Recite the rule. Upon what principle does the rule depend Is reduction ascending performed by multiplication or division? By what rule is the reduction ascending of compound numbers performed?

140. What is it to reduce an improper fraction to a mixed number? What is a mixed number? What is an improper fraction? Give the formula for reducing an improper fraction to a mixed number. Write a rule at the blackboard. In how many ways may a fraction be reduced in reduction ascending?

141. Give the formula for reducing an integer to an improper fraction. For reducing a mixed number to an improper fraction. Write a rule at the blackboard. Is the reduction performed by multiplication or division ?

142. What is meant by reducing fractions to a common denominator? 143. When are fractions said to have a common denominator? What is usually the first step in changing fractions to a common denominator? ANs.-Find the least common multiple of the denominators. Is it necessary to do this? ANS.-It is never ne cessary, but usually more convenient. What does the least common multiple of the denominator show? Recite the rule.

144. What are the analytical steps for adding fractions? Write a rule at the blackboard. What must be done before fractions of unlike denominations can be added?

145. Write at the blackboard a rule for subtracting fractions. Give the formulas for subtracting fractions.

146. What is the rule for adding two fractions which have a unit for their numerator? Give the reason for the rule. Give the rule and the reason for finding the dif ference between two fractions having a unit for their numerators.

147. What is the rule for adding two fractions having common numerators? What is the rule for finding the difference of two fractions having common numerators?

148. What is the rule for multiplying a fraction by an integer? Why does dividing the denominator multiply the fraction? Why does multiplying the nume rator multiply the fraction? What is the rule for multiplying a mixed number by an integer?

149. What is the rule for dividing a fraction by an integer? Why does dividing the numerator of a fraction divide the value of the fraction? Why docs multiplying the denominator divide the value of the fraction? What is the rule for dividing a mixed number by an integer? Illustrate at the blackboard the multiplication of a fraction by an integer. The multiplication of a mixed number by an integer.

150. What is the rule for multiplying by a fraction? What is meant by of 8? ANS--} of 3 times 8. By which number is 8 multiplied? Is it the numerator or denominator of a fraction? The product of 3 and 8 is divided by what? ANS.—By 5. Is it the numerator or denominator of the fraction? What is the rule for multiplying a fraction by a fraction? By cancelling? What when the multiplier is a mixed number? What should be done when the multiplicand is a mixed number and less than the multiplier?

151. What is the rule for dividing by a fraction? Give the analysis of the following at the blackboard ÷ What is the second rule for the division of fractions? What is to be done with the divisor when it is a mixed number? What is the rule for finding the contents of piles of wood, in cords? In how many ways can fractions be reduced? In how many ways can fractions be multiplied? In how many can they be divided? Explain the process of finding the prime factors of 84. of finding the greatest common divisor of 36 and 72. Of finding the least multiple of 7, 2, and 8. Give the formulas for reducing to its lowest terms. For reducing 7 to an improper fraction. For reducing to a mixed number. For adding 3 to §. For finding the difference between 31 and 23. For multiplying by 13. For dividing 8 by 11.

152. Give the rule for simplifying complex fractions; simplify of 9.

153. Give the analytical steps for changing £ to farthings. Give the formulas for each step.

154. Give the steps and formula for each of reducing far. to the fraction of a pound.

155. What is the comparison of numbers? What is the standard? What is the axiom used in the comparison of numbers? 6 is what part of 7? Which is the standard? Which is the number measured? How may it be expressed fractionally! Which is the numerator? Which is the denominator? Is the standard the numerator or denominator?

156. Recite the rule for adding or subtracting denominate fractions. Give the formulas for finding the contents of a pile of wood 124 ft. long, 13 ft. wide, and 14 ft. 6 in. high. Recite the rule.

SECTION X.

LESSON I.

DECIMAL FRACTIONS.

157. The Decimal Scale consists of an indefinite number of places increasing in value from right to left in a tenfold ratio.

158. For convenience, one of these places is called Units' Place. All numbers on the left of units are multiples of the unit. All numbers on the right of units are fractional parts of the unit.

159. For the purpose of distinguishing units' place, a period (.) is placed at the right of it; thus, 3.4; 4.37; 3 units and 4 tenths; 4 units and 37 hundredths.

(a.) NUMERATION OF THE DECIMAL SCALE.

160. A Mixed Number is an integer and a decimal written together.

REVIEW.-What is a unit? (1.) What is the difference between abstract and concrete numbers? (4.) (3.) Of what does arithmetic treat? (5.) What is notation? (6.) What is the difference between the Roman and Arabic notations? (7.) (8.) Recite the table of Roman notation. (7., a.) What is the effect of repeating a letter? (7., c.)

QUESTIONS.-What is meant by a "tenfold ratio"? How is units' place known? What is meant by multiples? Units are how many times as great as tenths? Tenths are how many times as great as hundredths? Tens are how many times as great as tenths?

LESSON II.

EXERCISES IN READING MIXED NUMBERS.

(a.) For convenience in reading, numbers are divided into periods of three figures each, commencing with the right hand figure in decimals, and the units' figure in integers; thus

47,864,134.4,178,642

is to be read, forty-seven millions eight hundred and sixtyfour thousand one hundred and thirty-four, and four million one hundred and seventy-eight thousand six hundred and forty-two ten-millionths.

NOTES.-The name, or denomination, of the decimal is determined by the name of the first significant figure at the right hand of the number; thus, in the above example, the figure 2 is the right hand figure, and as it stands in the ten-millionths' place, the denomination of the decimal is ten-millionths.

N. B.-The pupil will observe that the denominator is always one order higher than the number; thus, in the above example, since 4 is in millions' place, the denomination is ten-millionths. Two periods give the denomination of millionths, one period and two places give hundred-thousandths, &c.

*The following exercises should be read aloud by the pupils at recitation. They should be still further exercised, if necessary, by reading similar exercises from the blackboard, until they become quick and accurate.

The teacher should then require that the books be closed, and proceed to dictate the same exercise for the class to copy. When they have finished let them exchange slates with each other, and let the teacher read the lesson from the book, requiring each pupil to note his neighbor's errors, by drawing a line under the numbers incorrectly written.

REVIEW.-What is the effect of placing a letter of less value before one of greater value? (7., d.) After one of greater value? (7., e.) What is the effect of placing a dash over a letter or combination of letters? (7., f.) In what is the Arabic notation chiefly used? (8.) What is the difference between the simple and the local value of a figure? (9.) (10.) What is the use of the cipher? (11.)