Algebraic Addition.

54. Regarding the following positive numbers as representing gains and the negative numbers as representing losses, find the total gain (or loss) in each case; that is, find the algebraic sum of the numbers in each group:

NOTE.-The positive sums of Nos. 1, 2, 4, and 5 indicate actual gain; the negative sums of Nos. 3 and 6 indicate actual loss.

55. Regarding each of the following positive numbers as representing a rise and each of the negative numbers as representing a fall of the mercury in a thermometer, find the total rise (or fall) in each case; that is, find the algebraic sum of the numbers in each group:

NOTE. The positive sums of Nos. 1, 2, 4, and 5 indicate actual rise; the negative sums of Nos. 3 and 6 indicate actual fall.

57. A circle is a plane figure bounded by a curved line, every point in which is equally distant from a point within, called the center.

58. The line that bounds a circle is a circumference.

59. A straight line passing through the center of a circle and ending in the circumference is a diameter.

60. A straight line from the center of a circle to the circumference is a radius.

61. Any part of a circumference is an arc. 62. For the purpose of measurement, every circumference is considered as divided into 360 equal parts, called degrees.

63. Two lines meeting at a point form an angle. The point in which the two lines meet is the vertex of the angle.

64. Every angle may be regarded as having its vertex at the center of a circle, and the angle is measured by the part of the arc intercepted; thus, the angle hlk is measured by the arc mn.

65. The angle efg is an angle of 90 degrees, called also a right angle. The angle abc is an angle of 90 degrees. The angle hlk is an angle of 45 degrees.

e

f

g

h

k

66. All the angles about a point together equal four right angles.

67. MISCELLANEOUS REVIEW.

1. The angle abd is an angle of about degrees.

The angle dbc is an angle of

degrees.

The angle mls is an angle of about — degrees.

3. Earnings of ten persons for six days.

C

S

Copy the following figures and add by column and by line. Prove by comparing the sum of the sums of the columns with the sum of the sums of the lines. That pupil who can solve this problem without an error, on first trial, has taken an important step toward making himself valuable as an accountant.

Mon. Tues. Wed. Thurs. Frid. Sat.

Total.

SUBTRACTION.

68. Subtraction (in arithmetic) is the process of taking one number from (out of) another.

NOTE 1.-The word number, as here used, stands for measured magnitude, or number of things.

NOTE 2.-Subtraction (in general) is the process of finding the difference of two magnitudes.

69. The minuend is the number from which another number is taken.

70. The subtrahend is the number taken from another number.

71. The difference is the number obtained by subtracting.

72. The sign, which is read minus, indicates that the number that follows the sign is to be taken from (out of) the number that precedes it; thus, 8-3, means, that 3 is to be taken from (out of) 8.

73. PRINCIPLES.

1. Only like numbers can be subtracted.

2. The denomination of the difference is the same as that of the minuend and the subtrahend.

74. PRIMARY FACTS OF SUBTRACTION.

There are eighty-one primary facts of subtraction which should be learned while learning the facts of addition. See Werner Arithmetic, Book II., p. 274, note.

7. Tell which is the minuend, which the subtrahend, and which the difference, in each of the above examples.

76. Observe that in written problems in subtraction the subtrahend is usually written under the minuend and the difference under the subtrahend; and that, as in addition, the units of the same order are written in the same column.

1. In example 2, what figures represent units of the third decimal order? of the second integral order? of the first decimal order? of the first integral order?

2. In example 5, what figures represent units of the first integral order? of the second integral order?

77. Observe that in subtraction of denominate numbers the figures that stand for units of the same denomination and order are usually written in the same column.

1. In example 4, what figures represent tens of pounds? hundreds of pounds?

2. In example 5, what figures represent bushels and units of the first order?

78. Observe that in both addition and subtraction the decimal points, if there are any, usually appear in a column.