A line is curved when its direction constantly changes.

A bent line consists of two or more straight lines, the direction being changed abruptly at one or more points.

A straight line is usually designated by two letters placed at its extremities; as the line A B, for example

(Fig. 4). The length of the straight line, A B, is also termed the distance between the points A and B.

A

-B

FIG 4

Through one given point any number of straight lines may be drawn having different directions, but through two given points only one straight line can be drawn.

The Vertical. Special names are given to certain directions from any given point on the earth's surface, but these are of much greater importance in connection with the application than with the theory of Geometry. If a thread be fixed to a point at one extremity and a weight be suspended from the other, the thread will hang along a straight line, the direction of which is termed the vertical. A cord used in this way is termed a plumb-line (Fig. 5). Hence the vertical at any point is the direction of a plumb-line suspended from that point.

A Moving Point.-When two points are given we may always imagine them to be joined by a straight line. If the points are on the surface of a sheet of paper, we may, with a very fine-pointed pencil, mark the position of the straight line joining them (Fig. 3). The line, in this case, is the path over which the point of the pencil passes,

FIG. 5.

and it is frequently convenient to consider a line as the path described by a moving point. This path is a straight line when the point moves always in the same direction.

Plane Surfaces.-Other properties of the straight line will be better understood by reference to what is termed a plane surface.

Take a smooth board, such as a drawing-board, and place its surface in different positions against a straight thread. If the thread throughout its length touches the surface of the board we say that the board is even, or that its surface is a plane. Hence a plane surface is such that a straight line applied to it in any direction coincides with the surface in every part.

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.

The Horizontal.-Planes may slope in different directions, and one of these plane-directions has a particular name.

When a vessel of water is at rest at a place on the surface of the earth, the surface of the water assumes the curvature of the general surface of the earth at the place. If a plane touches or coincides with the surface of the water at any point, the plane is termed the horizontal plane at that point; and any straight line lying in such a plane is termed a horizontal line.

The terms vertical and horizontal are applied only to two particular directions with regard to the earth.

Intersection of Surfaces.-When two surfaces meet together or cut one another, they form a line. For example :

C

If we take a drawing-board, and partly immerse it in water; the line A B (Fig. 6), which separates the

FIG. 6.

part immersed from the part above the water, is called the line of intersection of the plane of the board and the surface of the water.

Intersection of Planes. -The line of intersection of two planes is a straight line.

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

Surfaces and Lines have no Thickness.-We can now further illustrate the fact that surfaces and lines do not take up solid room; that is to say, that they have no thickness. If they had thickness, then, by adding new surfaces or new lines to a body without altering the old ones, the size of the body would be increased. But this is not the case. For example, take a block of wood having six plane surfaces which form by their intersection twelve straight lines. The block is a solid, and has length (A B), breadth (A C), and thickness (AD). If it be cut into two parts, as in the figure, a surface will clearly be added to each part; whereas the total thickness of the block remains unchanged.

A

FIG. 7.

Hence a surface has not thickness, but length and breadth only. Again, if one of the surfaces of a body be cut into two parts, two new lines are obtained without increasing the total breadth of the surface. Hence a line has neither breadth nor thickness, but length only.

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

Another

FIG. 8.

easy method of obtaining a straight line is to fold a sheet of writing-paper as in the figure.

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

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. 9). If the ruler touches the line in every part, it is straight; if not, it is curved.

FIG. 9.

This test is an application of the fact that if two straight lines have two points in common, they coincide entirely; or, as it is sometimes stated, two straight lines cannot enclose a space.

Equality of Lines. We have said that straight lines have only one kind of extension, namely, length. When two straight lines have the same length, they are said to be equal. Let A B and C D be two

straight lines, and let it be required to prove that they are equal. Place the straight line C D on the straight line A B, so that the point C falls on A, and the line C D falls on A B; then, if the point D falls on B, it will be evident that A B is equal to C D.

In placing one line upon another, only two things require attention : 1, One extremity of the first must be placed on an extremity of the second; 2, The direction of the one must be made to coincide with the direction of the other.

This being done, the position of the superposed line is completely determined. For example, let us suppose that the lines A B and C D are known to be two equal straight lines, and that it is required, for some purpose or other, to apply one to the other. The straight line C D may be placed on the straight line A B, so that the point C is on A, and so that C D falls along A B. The position of the straight line C D will then be completely determined, and it will follow, as a matter of course, that because C D is equal to A B, the point D will fall on B. We must not say, "Place the line C D on A B, so that C is on A and D on B; for if the point C be fixed on A, and the line CD be directed along the line A B, the length of the line C D will determine where the point D shall fall. If C D be shorter than A B, D will fall within A B; if C D be longer than A B, D will fall beyond A B; and if C D be equal to A B, then D will fall on the point B. A comparison of the lines may be easily effected by placing at C and D the points of a pair of compasses (Fig. 10), and then, without altering the distance between the points, moving the instrument to the line A B. If the two straight lines are not equal, it will be easy, by means of the compasses, to mark off from the greater a part equal to the less.

The Measurement of Lines.-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.

FIG. 10.

SCALE OF THREE INCHES.

FIG II

7

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.

exact copies of this bar are deposited in secure places throughout the kingdom. Drapers and others use rods equal in length to the standard yard. Carpenters and mechanics, for the measurement of shorter lines, use a measure called a foot-rule, which is equal in length to onethird of the yard. The foot is divided into twelve inches, and the inch is subdivided into eighths or twelfths (Fig. 11).

FIG. 12.

Several

Surveyors use a chain, twenty-two standard yards long, consisting of 100 equal links (Fig. 12). Since 22 yards 7.92 inches.

Builders use a tape (Fig. 13) of the same length as the surveyor's chain, but subdivided into feet and inches. So that the length of lines is estimated in England by comparing them with the standard yard at Westminster, by means of measures which are either copies of the yard or multiples or parts of the

same.

=

792 inches, a link

FIG. 13.

To measure lines on paper, an ivory scale, divided into inches or parts of an inch, is used. A line shorter than the scale may be conveniently measured by placing the points of a pair of compasses (Fig. 10) at the two extremities, and carrying the compasses to the scale. If one point be placed at the beginning of the scale, the numbers opposite the other will give the length of the line. A line longer than the scale may be measured by marking off with the compasses the length of the scale from the line as many times as it is contained therein, and then taking the measure of the remainder as recommended above.

How to Measure a Straight Line on Land.-If the line be not very long, we may stretch a cord between the extremities, and measure the cord. If the line be long, it will be necessary to find intermediate