Place the tracing-paper circle (Fig. 208) on Fig. 209 so that point B falls on point B1. Then ✪ B coincides

with OB1. Why?

Make ZB coincide

with ZB1.

Point A must then

coincide with A1, and

C with C1; for radii of equal circles are equal.

A

FIG. 208

FIG. 209

Therefore arcs AC and A, C1 coincide, and are equal.

An angle having the vertex at the center of a circle is a central angle.

The discussion above shows that in equal circles equal central angles cut off (intercept) equal arcs. The same is true for two equal central angles in the same circle.

EXERCISES

1. Review carefully the preceding proof until you can repeat it correctly.

2. Show, as in $104, that equal arcs, in equal circles, have equal central angles.

105. Theorem. A statement to be proved, such as the statement proved in §104, is called a theorem.

106. How to measure angles by means of circle arcs. Draw a circle, and at the center B (Fig. 210) draw an angle, as ABC.

Measure angle ABC with the protractor and note the number of degrees it

B

FIG. 210

contains. Each degree in angle ABC cuts off (intercepts) a small arc on the circle, called arc-degree.

State the number of arc-degrees for a central right angle; a straight angle a perigon; a 60-degree angle; a 45-degree angle.

What part of a circle is an arc-degree?

Arc-degrees are used to measure arcs. Since there are as many angle-degrees in a central angle as there are arc-degrees in the arc, we say that a central angle and the arc it cuts off have the same measure, or that the central angle is measured by the intercepted arc.

EXERCISES

1. If an arc contains 90 arc-degrees, what is the measure of the central angle of the arc?

2. State the measure of the central angles if the arcs contain 20°; 15°; 180°; 270°.

107. Quadrant. Semicircle. One-fourth of a circle is a quadrant. One-half of a circle is a semicircle.

108. Using the circle in locating places. Knowledge of the division of the circle into arc-degrees enables us to locate places exactly. This may be seen from the following. A man asked to be directed to a certain building in Chicago. He was told to walk three blocks west and then two blocks north. This illustrates a method by which places in a city are commonly located. A similar method is used in geography to locate places on the surface of the earth. However, in place of streets or roads, imaginary circular lines are laid out running from north to south and from east to

west, and covering the surface of the earth. The northsouth lines are called meridians, the east-west lines parallels. Figs. 211 and 212 represent

Equator

A

B

the surface of the earth and show some of these circular lines. Thus circle ABCDEF (Fig. 212) passes around the earth midway between the poles P and P1. This line is the equator. It divides the surface of the earth into the Northern and Southern Hemispheres. It is divided into 360 arc degrees (Fig. 213).

180°

225° 270° 315°

135

90°

FIG. 213

P

FIG. 212

The equal circles passing through the poles are the meridians (Fig. 214). The meridian which passes through the Royal Observatory at Greenwich, England, is the prime meridian. This meridian divides the surface of the earth into the Eastern and Western Hemispheres.

FIG. 214

90°

109. Longitude and latitude. The number of degrees east or west of the prime meridian is called longitude. All places east of the prime meridian have east longitude, all places west have west longitude.

The number of degrees measured along a meridian, northward or southward from the equator, is called latitude.

Thus, the longitudes of places may vary from 0° to 180°, the latitudes from 0° to 90°.

For example, the location of the custom house at Portland, Maine, is: Lat. 43° 39′ 28′′ N., Long. 70° 15′ 18" W. To represent the location of Portland by a drawing, make a sketch of the Western Hemisphere

180°

N

90° A 70°15'18

(Fig. 215). From the 0° mark on the equator, pass along the equator to the left to point A, which is 70° 15′ 18′′ W., and from A toward the north to point P, which is 43° 39′ 28′′ north of the equator. Then the location of P is: Lat. 43° 39' 28" N., Long. 70° 15′ 18" W. Find the latitude and longitude of your city and make a drawing like Fig. 215 to show the location.

FIG. 215

EXERCISES

1. Practice making good drawings like Figs. 213 to 215.

Using the method given on page 160, make sketches locating the places in the table below:

10. On an outline map locate the places in the above table from the directions given in Exercises 2 to 9.

11. From a map find the approximate positions of Washington, D. C.; San Francisco; Chicago; Liverpool; Paris; Berlin; Rome.

12. A ship in Long. 10° W. sails west 16 degrees. What is the new longitude?

13. A ship starts in Long. 47° E. and sails 83° west. What is the new longitude?

Find the differences between the longitudes of important buildings in the following cities: