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1. When one straight line meets another straight line in such a manner that two right angles are formed by the lines, the two lines are said to be perpendicular to each other.

2. Two lines side by side extending in the same direction are said to be parallel. 3. Of the lines given above: AC is

to BD BD is

to AC. MN is

to OP. OP is

to MN.

4. A line extending in the direction of the horizon is said to be horizontal. A line on the floor of the room is horizontal; a line on the ceiling is horizontal; a line on the blackboard, every point in which is equally distant from the floor, is horizontal. For convenience, lines drawn upon paper, that are parallel with the top and bottom of the paper, may be regarded as representing horizontal lines.

5. A piece of lead (plumbum), or other heavy material suspended by a cord, is called a plumb-line. A line in the direction of a plumb-line is said to be vertical. A vertical line is perpendicular to a horizontal line. Lines on the blackboard may or may not be vertical or horizontal. For convenience, lines drawn upon paper, that are parallel with the sides of the paper may be regarded as vertical lines.

93. MISCELLANEOUS REVIEWS.

1. The angle formed by a vertical line meet A ing a horizontal line, is an angle of de

B grees. 2. An angle that is equal to one half of a

D

С right angle, is an angle of — degrees.

3. The angle ADB is an angle of degrees. 4. The angle BDC is an angle of de

F grees. 5. If from a right angle, an angle of 30 H

G degrees be taken, the remaining angle is an angle of degrees.

6. The angle FHG is an angle of degrees.

7. During the month of November, 1897, there were consumed at the Illinois Institution for the Education of the Blind, 64 loads of coal. The weight of each load in pounds is given below. Find the total weight. 6100 8020 5490

5190
8380
6860
6800

7130
4850
6230
6560

7090
8010
6780
6690

7790
7080
6980
5780

6810
6620
6240
6980

8600
6450
6420
5990

9100
6570
6310
4740

6740
7950
6300
5520

5380
4750
6530
3630

7640
8840
6950
4930

5650
7290
6980
5150

5900
4960
4920
6420

6200
8330
5880
67.70

6620
6300
7030
6220

7170
7080
5160
6020

9210

MULTIPLICATION.

94. Multiplication is the process of taking a number (of things) a number of times.

NOTE 1.—The word number as first used in the above statement, stands for measured magnitude. The second word number does not stand for measured magnitude, but rather for pure number, representing simply the times the number (of things) is to be repeated.

95. The multiplicand is the number (of things) taken or repeated. *

96. The multiplier is the number that shows how many times the multiplicand is to be repeated.

97. The product is the number (of things) obtained by multiplying.

98. The sign, x, which is read multiplied by, indicates that the number preceding the sign is a multiplicand, and the number following it, a multiplier.

NOTE:-For other uses of this sign, see Werner Arithmetic, Book II., page 274.

99. PRINCIPLES. 1. The multiplier is always an abstract number.

2. The denomination of the product is always the same as that of the multiplicand.

100. PRIMARY FACTS OF MULTIPLICATION. There are sixty-four primary facts of multiplication. See Werner Arithmetic, Book II., p. 275.

* “The multiplicand, however written, must always be understood to express measured quantity; it is always concrete.Psychology of Number, McClellan & Dewey, page 76.

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7. Tell which is the multiplicand, which the multiplier, and which the product, in each of the above examples.

102. Observe that in each of the above examples the multiplier is a pure number.

103. Observe that in each of the above examples the denomination of the product is the same as the denomination of the multiplicand.

104. MULTIPLICATION AND ADDITION COMPARED. Find the sum of each of the following groups of numbers and compare the result with the product in the corresponding problem in article 101.

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MULTIPLICATION-SIMPLE NUMBERS.

105. Find the product of 563 and 7. Operation.

Explanation. 563

Seven times 3 units of the first order are 21 units of the 7

first order; they are equal to 1 unit of the first order and 2 3941 units of the second order. Write the 1 unit of the first

order and add the 2 units of the second order to the next partial product

Seven times 6 units of the second order are 42 units of the second order; 42+ 2 = 44; 44 units of the second order equal 4 units of the second order and 4 units of the third order. Write the 4 units of the second order and add the 4 units of the third order to the next partial product.

Seven times 5 units of the third order are 35 units of the third order; 35 + 4

39; 39 units of the third order equal 9 units of the third order and three units of the fourth order. Write the 9 units of the third order and the 3 units of the fourth order.

The product of 563 and 7 is 3941.

106. Find the product of 3426 and 57. Operation.

Explanation. 3426

Seven times 3426 equals 23982. Fifty times 3246 equals 57

171300.* Fifty times the number plus 7 times the num23982

ber equals 57 times the number. Therefore adding

23982 and 171300 gives 57 times 3426. 195282

The product of 3426 and 57 is 195282.

17130

107. PROBLEMS.

1. Multiply 3241 by 27. 6. 6521 x 54
2. Multiply 6759 by 27. 7. 3572 x 74
3. Multiply 4328 by 36. 8. 6428 x 71
4. Multiply 5672 by 36. 9. 3521 x 29

5. Multiply 3479 by 54. 10. 6479 x 29 (a) Find the sum of the ten products.

* Ten times five times a number equals fifty times the number. Arithmetic, Book II., p. 61.

See Werner

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