8.72

12. Determine to three figures the ratio

9.36

13. Write a paper on one of the following topics:

a. Uses of the triangle in ornamental work; in construction; in designing; in surveying.

e.

Various methods of finding unknown distances.

f. Importance of congruence, similarity.

g. The life and work of Thales.

CHAPTER VI

THE CIRCLE

HOW THE CIRCLE IS USED

99. Meaning and uses of the circle. In the previous chapters, we have been studying figures formed by straight lines. We shall now take up the study of a well-known curved line, the circle. The circle is found so frequently in everyday life that everyone should know how to draw and use this curve.

The earliest nations showed great skill in the use of the circle in designs with which they beautified their temples and other important buildings. At a very early date men learned how to make use of the wheel for moving objects, having recognized that it required

OF THE CIRCLE

only a forward movement and no lifting as in carrying.

A similar use of the circle has found its widest application in modern machinery, as in watches, lathes, sewing machines, and automobiles. We find it on tanks, boilers, pillars, doilies, phonograph records, baskets, and numerous other objects. It is used in designs and many other types of ornamental work.

Note the circles and circular parts in the steam engine (Fig. 185) which is used to supply power for farms.

Gas and electric-light companies use the circle in measuring devices, such as gas and electric-light

meters.

In geography we learn how circles determine the position of places on the earth's surface (Fig. 215). In astronomy stars are located and the exact time is determined by means of circles.

In geometry, drawings, or constructions, are based on the properties of the circle.

The circle is also used to represent numerical facts in a graphical way (Figs. 226 to 230).

It is the purpose of this chapter to acquaint you with some of these uses of the circle and to give you a clear understanding of the circle itself.

Drawing various designs and making the fundamental geometric constructions will help you develop skill in the use of the compass.

We shall see that the problem of finding lengths of circular lines leads to valuable experiences in solving equations, working with formulas, and making arithmetical computations.

100. How to draw a circle. Cut a piece of cardboard, as AB (Fig. 186), making it about 2 inches long and inch wide. With a

pin, prick a hole near each

end, at A and B.

Place the strip AB on a sheet of paper and fasten the end at A by means of a pin stuck through the hole.

C

FIG. 186

B

Through the hole at B insert the point of a sharp pencil, and turn the strip about point A. The point of the pencil then traces a curved line, as BC. When a complete turn is made the curve closes. The closed curved line is called a circle.

All points on this line have the same distance from A. Thus, a circle is defined as a closed curved line all points of which are equally distant from a fixed point. The fixed point is the center of the circle.