MEASUREMENT OF SURFACES 66 PROBLEMS RELATING TO SURFACES 71 THE CIRCLE AND ITS LINES 73 ARCS AND DEGREES 78 PARTS OF THE CIRCLE 82 PART II. AXIOMS AND THEOREMS. AXIOMS. ILLUSTRATED 85 THEOREMS. ILLUSTRATED 88 FIRST LESSONS IN GEOMETRY. PART FIRST. LESSON FIRST. LINES. NOTE TO THE TEACHER.—In all the development-lessons, the pupils are to be occupied with the diagrams, and not with the printed matter. See Note A, Appendix. Refer to DIAGRAM 1, and show that What are here drawn are intended to represent _length_ only. They have a little width, that they may be seen. They are called _lines_. _A line is that which has length only._ POINTS Show that Position is denoted by a point. It occupies no space. It has _some_ size, that it may be seen. The ends of a line are points. A line may be regarded as a succession of points. The intersection of two lines is a point. A point is named by placing a letter near it. [Illustration: Diagram 1.] A point may be represented by a dot. The point is in the center of the dot. _A point is that which denotes position only._ A line is named by naming the points at its ends. Read all the lines in Diagram 1. CROOKED LINES. See Note B, Appendix. Does the line _m_ _n_ change direction at the point 1? At what other points does it change direction? It is called a crooked line. _A crooked line is one that changes direction at_ some _of its points_. CURVED LINES. The line _o p_ changes direction at every point. It is called a curved line. _A curved line is one that changes direction at_ every _point_. STRAIGHT LINES. Does the line _i j_ change direction at any point? It is called a straight line. _A straight line is one that does_ not _change direction at any point_. OTHER LINES. The line _q r_ winds about a line. It is called a _spiral line_. The line _w x_ winds about a point. It also is called a spiral line. _A spiral line is one that winds about a line or point._ The line 7 8[1] looks like waves. Footnote 1: To be read seven, eight, not seventy-eight. It is called a wave line. * * * * * What kind of a line is _a b_? Why? What is a straight line? What kind of a line is 11 16? Why? What is a crooked line? What kind of a line is _o p_? Why? What is a curved line?