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 hori. zontal; 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 A meeting a horizontal line, is an angle of B degrees. 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 F degrees. 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 7130 7090 7790 6810 8600 9100 6740 5380 7640 5650 5900 6200 6620 7170 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. 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. 1. 2. 3. $375.12 $375.42 $375.42 4. 5. 6. 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 units of the second order. Write the 1 unit of the first 3911 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 number 23982 equals 57 times the number. Therefore adding 23982 and 17130 171300 gives 57 times 3426. The product of 3426 and 57 is 195282. 195282 107. PROBLEMS. 1. Multiply 3241 by 27. 6. 6521 x 54= 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. See Werner Arithmetic, Book II., p. 61. |