Зображення сторінки
PDF
ePub

PRACTICAL GEOMETRY.

CHAPTER I.

THE STRAIGHT LINE.

LINES.

1. DEFINITIONS.-A mathematical point is a position in space of which no part can be taken; a mathematical line is formed by the motion of a point in space; when the motion is always in the same direction, a right or straight line is produced. A physical line is drawn by the motion of the point of a pen or pencil over any surface. A mathematical surface is produced by the motion of a line in any direction; when the moving line is straight and the motion all in one direction, the surface is called a plane. When a portion of a surface is entirely enclosed by a line or lines, the boundary is called the perimeter or circumference. When two right lines meet together at a point they are said to form an angle with each other, which is greater or less according to the amount of turning or inclination which one line makes with the other; for example, the angle AOC (fig. 1) which OC makes with OA is greater than the angle AOB which BO makes with O A. Let a circle be drawn from O as a centre and with any radius, then the angle at O is measured by the arc which it subtends, the arc and the angle increasing and decreasing together; when

B

the angle COA is double the angle BOA, the arc AC is

Fig. 1.

twice as long as the arc A B. If the line A O is produced to any point E, and the perpendicular OD is set up so as to make the angle A OD equal to the angle EOD, then each of these angles is called a right angle.

The angle A OB, which is less than

a right angle is called an acute angle, and the angle A OC, which is greater than a right angle, is called an obtuse angle. An angle is said to be included by the two lines forming it. The point O is called the vertex or angular point of the angles AOC, AOD, AO B.

[graphic]

Fig. 2.

2. To divide a right line of given length into two or more equal parts.-Let AB (fig. 2) be the given line. From the points A and B as centres, and with any radius, B G or A F, greater than half the length of A B, describe circles cutting each other in D and E; then the straight line DCE will bisect A B in C, making CA equal to CB; also DC will be perpendicular to A B, the angles ACD and BCD being equal.

[graphic]

If we bisect each of the equal parts AC and BC in the manner above described, we shall have the line A B divided into four equal parts; and if each of these is again bisected the line will be divided into eight equal parts. In this manner we can divide a line into any number of equal parts, that number being a power of two, as 4, 8, 16, 32, and so on..

When the given line is too long to allow of the arcs being struck from the two ends, its middle point can be found by measuring equal distances A G and BF from each end, and then bisecting, by the foregoing method, the line between the points F, G in the point C which will also be the middle point of the given line A B.

3. From a given point in a given right line to erect a perpendicular.-Let C (fig. 2) be the given point in the given line AB. From C as a centre and with any radius CA describe a circle cutting the line in A and B. From A and B as centres, and with any radius AF greater than A C, describe circles cutting each other in the points D and E; then the right line DCE will be perpendicular or at right angles to A B.

Another method.

Fig. 3.

Let CAB (fig. 3) be the given line, A the given point. Take any length A D and measure AC equal to three times AD, and A B equal to four times AD. From A as a centre, and with AC for radius, describe a circle; from B as a centre and with BD as a radius, draw an arc cutting the circle in the point E. Then

[graphic]

the straight line EA is perpendicular to A B.

This method is used by surveyors in setting out a right angle; let a cord or measuring tape be held tight at three points so as to form a triangle ABE, having the lengths of its sides in the proportion of 3, 4, and 5; then the angle BA E, which is opposite the longest side or hypotenuse BE, will be a right angle.

4. To draw a perpendicular to a given right-line from a

given point outside of it.-Let A B (fig. 4) be the given

Fig. 4.

line, C the given point.

Take

any point X on the opposite side of the line to that on which C lies, and join CX. From C as a centre, and with CX for radius, describe a circle cutting the given line in the points A and B; join AC, BC. From A and B as centres, with AC as radius, describe circles intersecting in C and D; draw CD cutting A B in the point H, then the line CHD is perpendicular to AB.

In applying this process to surveying, it is only necessary to measure two equal lengths from C to A B, namely, CA and CB, and to bisect the line A B in H; then the line HC will be perpendicular to A B.

[graphic]

Fig. 5.

5. From a given point to draw a right line equal in length to a given line.-Let A B (fig. 5) be the given line, C the given point. Join AC. From A and C as centres, and with A C for radius, describe circles cutting each other in the point D; join DC and D A. From A as a centre, and with AB as a radius, describe a circle and produce DA to meet it in E. From D as a centre,

[graphic]

and with DE for radius, describe a circle, and produce DC to meet it in F; then CF is equal to A B, and is drawn from the given point C.

« НазадПродовжити »