PLANE GEOMETRY.

CHAPTER I.

LINES AND PLANES.

1. In observing the forms of the different objects about us, we perceive upon their surfaces lines of various kinds, and we learn, by comparing them, to distinguish a straight line from one which is not straight. If a very fine

and flexible cord could be perfectly stretched between two points, the figure presented by this cord would be a straight line (fig. 1). When, however, a heavy cord is extended, its weight causes it to bend and to form a curve (fig. 2); so that telegraph wires, cables of ships, and towing lines, although stretched, do not form straight lines; and, in

fact, the finest thread,

Fig. 1.

Fig. 2.

when stretched as in fig. 1, is not exactly straight.

B

Fig. 3.

A

If, however, the thread be fixed at one end and a weight be suspended from the other, the figure formed by the cord will be exactly a straight line (fig. 3).

Such a cord is termed a plumb-line, and its direction is said to be vertical.

2. A straight line is usually designated by two letters placed at its extremities, as the line AB, for example (fig. 4.)

B

Fig. 4.

3. Take now a smooth board, such as a drawingboard, and place its surface in every direction against the straight line. If the line throughout its length touches the surface, we say that the board is even, and that its surface is a plane.

4. A plane is a surface such that a straight line applied to it in any direction, coincides with the surface in every part.

5. If our drawing-board had become warped, the line would not be in contact with its surface in every part. A ruler is used to test surfaces in this way by joiners, stone-cutters, and the workmen who prepare plates for the engraver. If, when the ruler is applied to the surface, no light can be seen between them, the surface is true.

6. When water in a pond or open vessel is at rest,

its surface is level; that is, it is not inclined on any side to the vertical line. Such a surface is termed a horizontal plane. A straight line lying in such a plane is termed a horizontal line.

7. Take now the drawing-board and partly immerse it in water, the line A B (fig. 5) which separates the part immersed from

the part above the water, is a straight horizontal line. This line is called the line of intersection of the plane of the board and the plane of the water. Thus the line of intersection of two planes is a straight line.

Fig. 5.

The lines of intersection of the ceiling and walls of a room afford examples of straight lines formed by intersecting planes.

8. Take a piece of wood, and cut upon it two smooth and even faces; the line of intersection, A B (fig. 6), of these planes is a straight line, and if the strip of wood containing this line be cut from the board, it will form a straight-edge or ruler.

Fig. 6.

Another easy method of obtaining a straight line is to fold a sheet of paper.

9. By means of the even drawing-board and straight

ruler we can draw, with pen or pencil, a straight line on any plane whatever, as, for example, on a sheet of paper stretched upon the drawing-board..

How to test a ruler.

10. Rulers for drawing usually have all their edges straight, and are made flat and thin, in order that a small pressure of the hand may be sufficient to apply them to the board. In consequence of heat and dampness, the wood of a ruler frequently warps, and then, the planes being untrue, the line of their intersection is no longer straight; it is therefore necessary to test a ruler before using it.

To do this, draw a line with the edge of the ruler, and apply the same edge to the opposite side of the line (fig. 7). If the ruler touches the line in every part it is straight; if not, it is curved.

Fig. 7.

Rulers of glass or metal are not so liable to warp as wooden ones, but they are more liable to slip when in

A and B is evidently less than the slack one. Similarly, if any two points be joined by a straight line, this line will be shorter than any

curved line (fig. 8) or

Fig. 9.

any bent line (fig. 9) connecting the same points.

The measurement of lines.

12. When two straight lines are given, we may find which is the longer by applying one to the other, and in the same way we may find how many times one is contained in the other. The two lines may also be compared by applying to them a third line, and observing how many times it is contained in each. The line which is applied to the others is termed a unit or measure, and the process is termed measuring.

The standard unit of length used in England is the yard, which is the length of a straight line joining two marks on a bar of metal preserved in the Houses of Parliament, at Westminster. Several exact copies of this bar are deposited in secure places throughout the kingdom. Drapers,

surveyors, masons, and others, use rods, equal in length to the standard yard. Carpenters, smiths, and mechanics, for the measurement of shorter lines, use a measure called a foot-rule, which is equal in length to one third of the

Fig. 10.