arms will form one line; and then the two angles are called Supplemental Adjacent Angles.

66. A Right Angle is half a straight angle.

A

67. A Perpendicular to a line is a line that makes a right angle with it.

68. When the sum of two angles is a right angle, each is said to be the Complement of the other.

69. An Acute Angle is one which is less than a right angle. 70. An Obtuse Angle is one which is greater than a right angle, but less than a straight angle.

71. The whole angle which a sect must turn through, about one of its end points, to take it all around into its first position,

or, in one plane, the whole amount of angle round a point, is called a Perigon.

A

72. Since the angular magnitude about a point is neither increased nor diminished by the number of

lines which radiate from the point, the sum

of all the angles about a point in a plane is a perigon.

73. A Reflex Angle is one which is greater than a straight angle, but less than a perigon.

74. Acute, obtuse, and reflex angles, in distinction from. right angles, straight angles, and perigons, are called Oblique Angles; and intersecting lines which are not perpendicular to each other are called Oblique Lines.

75. When two lines intersect, a pair of angles contained by the same two lines on different sides of the vertex, having no arm in common, are called Vertical Angles.

76. That which divides a magnitude into two equal parts is said to halve or bisect the magnitude, and is called a Bisector. 77. If we imagine a figure moved, we may also suppose it to leave its outline, or Trace, in the first position.

78. A Triangle is a figure formed by three lines, each intersecting the other two.

B

А

79. The three points of intersection are the three Vertices of the triangle.

80. The three sects joining the vertices are the Sides of the triangle. The side opposite A is named a; the side opposite B is b.

81. An Interior Angle of a triangle is one between two of

the sects.

82. An Exterior Angle of a triangle is one between either sect and a line which is a continuation of another side.

83. Magnitudes which are identical in every respect except the place in space where they may be, are called Congruent. 84. Two magnitudes are Equivalent which can be cut into parts congruent in pairs.

II. Properties of Distinct Things.

85. The whole is greater than its part.

86. The whole is equal to the sum of all its parts.

87. Things which are equal to the same thing are equal to

one another.

88. If equals be added to equals, the sums are equal.

89. If equals be taken from equals, the remainders are equal.

90. If to equals unequals are added, the sums are unequal, and the greater sum comes from adding the greater magnitude. 91. If equals are taken from unequals, the remainders are unequal, and the greater remainder is obtained from the greater magnitude.

92. Things that are double of the same thing, or of equal things, are equal to one another.

93. Things which are halves of the same thing, or of equal things, are equal to one another.

III. Some Geometrical Assumptions about Euclid's Space.

94. A solid or a figure may be imagined to move about in space without any other change. Magnitudes which will coincide with one another after any motion in space, are congruent; and congruent magnitudes can, after proper turning, be made to coincide, point for point, by superposition.

95. Two lines cannot meet twice; that is, if two lines have two points in common, the two sects between those points coincide.

96. If two lines have a common sect, they coincide throughout. Therefore through two points, only one distinct line can pass.

97. If two points of a line are in a plane, the line lies wholly in the plane.

98. All straight angles are equal to one another.

99. Two lines which intersect one another cannot both be parallel to the same line.

IV. The Assumed Constructions.

100. Let it be granted that a line may be drawn from any one point to any other point.

101. Let it be granted that a sect or a terminated line may be produced indefinitely in a line.

102. Let it be granted that a circle may be described around any point as a center, with a radius equal to a given sect.

103. REMARK. Here we are allowed the use of a straight edge, not marked with divisions, and the use of a pair of compasses; the edge being used for drawing and producing lines, the compasses for describing circles and for the transferrence of sects.

But it is more important to note the implied restriction, namely, that no construction is allowable in elementary geometry which cannot be effected by combinations of these primary constructions.