3. If a point move from an assumed position, it may start in any one of an infinite number of directions. If it continue in the direction in which it starts, so as to pass through the position of every point lying in the same direction (§ 2), it will generate what is called a straight line.

Since a straight line is determined by direction, and two points determine direction, two points determine the position of a straight line.

If separate straight lines have any point in common there can be but one, for if they have two coincident points the two lines will form one line.

The straight line is infinite in extent.

The portion of a straight line between any two points of the line is called a segment.

Unless otherwise specified, the indefinite straight line is to be understood when two points are named which locate the line, as: the line AB (the segment is represented by AB).

An infinite number of straight lines may pass through a point.

If a point move with ever-changing direction, it will generate a curved line; the law of change determining the character of the curve.

As any fixed point has a fixed direction from another fixed point, only one straight line can join two points.

4. Our idea of distance involves the ideas of motion and of time. Time is, and advances with uniformity. Motion may or may not be uniform. If motion be uniform, the greater the time the greater the distance. In equal times the more rapid of two motions will cover the greater distance.

If a point move from position A to position B without change of direction, it will reach position B after having moved over a less distance, than if on the way it had made any changes of direction.

We therefore say that the shortest distance between two points is that portion of the straight line determined by them (§ 3) which lies between them.

Although the straight lines with which we deal are in general infinite in length, our representations of them are limited by the extent of the surface upon which our representations are made.

Henceforth when the word line is used without any qualification a straight line is understood.

5. ROTATION AXIOM. If two straight lines intersect, either or both may be moved without changing the point of intersection. They may be brought to coincide; in which case all the points of the two lines are in common. The lines are then said to be congruent.

PLANE GEOMETRY.

6. A plane is a surface such that if any two of its points are joined by a straight line it will contain the entire line.

An infinite number of straight lines may lie in a plane. In drawing, a dot upon a plane is used as the representative of a point, and a mark by chalk, pencil, or pen is used to represent a line.

AN ANGLE.

7. Let the surface of this page represent a plane, and AB a line of the plane.

If the direction of B from A be positive (+), the direction of C from A will be negative (−).

If the straight line rotate about the point A as a pivot, and remain in the plane of the page, it will make a change in direction. If the motion be considered as having stopped when the line has arrived at the position AD, the change in direction is called an angle; and A is called its vertex.

The angle may be described as: the angle BAD (≤ BAD), or by a single symbol as 0.

The distances AB or AD have nothing whatever to do with the magnitude of the angle; it is purely a matter of direction.

FIGURES.

8. Volumes, surfaces, lines, angles, and points, or any combination of them that we may make, constitute geometric figures.

The relations of parts of the same figure to each other, and the relations of different figures to each other, constitute the subject matter of geometry.

The Elementary Geometry is ordinarily separated into two parts, viz. :

(a) That which relates to figures in a plane, and is called Plane Geometry.

(b) That which relates to figures that do not lie in one plane only, and is called Solid Geometry.

From now on until we arrive at § 105, we shall be concerned with Plane Geometry.

TRANSLATION AXIOM. A geometric figure may be moved at pleasure without changing the relations existing between the parts which compose it.

MORE ABOUT PLANES.

9. A plane may be made to pass through a given point for a plane may be moved so as to cause any one of its points to coincide with the given point.

Any infinite number of planes may be made to pass through a given point: because the manner of moving any plane so as to cause it to contain a given point has not been limited.

If a plane pass through a point, any liné which contains the given point, and lies in the plane, may, by motion of the plane, be brought to pass through a second point, located anywhere.

Thus we see that a plane may be passed through any two points. It will contain the straight line joining those points. Every possible position of a plane which contains two given points, may be reached by causing a selected plane which contains them to rotate about the line joining them as an axis, until it returns to the position it had at starting. Thus we see that an infinite number of planes may be passed through two points. These planes will each contain the line determined by the two points.

A plane passed through a line and making a complete rotation on the line as an axis, will encounter every point in space.

10. If a line in each of two planes be made to coincide, and then one of the planes be rotated about this line until at least one point, not in the common line, shall be common to the two planes, any line which may be drawn through this point and any point of the common line will lie in both planes. Lines may therefore be drawn through this point so as to reach every point of both planes; in which case the two planes would have every point in common. Hence the statement or