68. Using the protractor. The protractor is an instrument used in measuring and in drawing angles. The semicircle is divided into 180 equal parts, each of which is one degree. If these points of division are connected by straight lines with the center of the semicircle, angles of one degree are formed.

To measure AOB with the protractor, the center is placed at the vertex of the angle, and the radius OX is made to lie along the side OA of the angle, Figure 31. The number of angle degrees in ZAOB is the same as the number of arc degrees in arc XY. To find the number of degrees in ZAOB we read the number of degrees on the protractor from the

point X to the point Y. In Figure 31 ≤ AOB is seen to contain 50°.

The protractor may be used to construct an angle of a given size, for example an angle of 65°. To do this draw a line OA, Figure 31. Place the center of the protractor at 0, and lay the side of the protractor, OX, along OA. Place point C opposite 65°. Draw a line from 0 to C. Then ZAOC is the required angle.

The ruler, compasses, and protractor are drawing instruments which the pupil should learn to use with accuracy and skill. Geometric drawings should be accurate, neat, and legible.

Exercise 71

1. Measure the angles in Figure 18 with the protractor. 2. With the protractor make an angle of 90°; of 30°; of 57°; of 120°; of 180°.

3. Use the protractor to draw two lines that are perpendicular.

4. Make a figure such as Figure 24 on page 135, showing the directions. Make the angles as accurate as you can. 5. Construct a square with a side of 3 in. What kind of angles must be made at the corners ?

6. In playing the game of "fox and geese" in the snow a figure like this one is made. Using ruler and compasses, make such a figure. Suppose the radius of the outer circle to be 40 ft. and that of the inner circle 30 ft. Use the scale of 20 ft. to 1 in.

FIG. 32.

7. A wheel 36 in. in diameter has a hub 4 in. in diameter and has 16 spokes. Make a diagram of this wheel on the scale of 9 in. to 1 in.

8. Make a figure like Figure 33. Make the sides of the square 2 in. Use the protractor in making the right angles. Use the side of the square as radius in making the arcs.

9. Make a copy of Figure 34. First find the point of intersection of the diagonals. Use the vertices of the square as centers and half of the diagonal as a radius in drawing the arcs.

69. Circular and angular measure. The following table is used in measuring both angles and arcs.

1 degree (°)=60 minutes ().

1'=60 seconds (').

Exercise 72

1. Reduce 15° 24′ 8′′ to seconds.

2. Reduce 60° 48′ 35′′ to seconds.
3. Reduce 430' to degrees and minutes.
4. Reduce 1235" to minutes and seconds.

5. Reduce 125" to minutes; to degrees.`

6. Over how many degrees does the end of the minute hand of a clock pass in 1 hr.?

7. Over what arc does the end of the hour hand pass in 1 hr.? In 1 min.? In 1 sec.?

8. In Italy clock faces are marked from 1 to 24. Over what arc does the end of the hour hand pass in 1 hr.? In 1 min.? In 1 sec.?

9. The latitude of New York is 40° 42′ N., of London, 51° 30' N. Find the difference in their latitudes.

10. The latitude of New Orleans is 29° 51′ 45′′ N., of Buenos Aires, 34° 26' 21" S. Find the difference in their latitudes.

11. Find the difference in latitude between Chicago, latitude 41° 53' 6" N., and Constantinople, latitude 41° 0' 16" N.

12. A ship sails south at the average rate of 1° 42′ 37′′ a day for 8 days. Find the total distance sailed south measured in degrees, minutes, and seconds.

LONGITUDE AND TIME

70. How longitude is measured. It has been learned in geography that longitude is measured east and west from a principal meridian. The longitude of places in the United States is measured from the principal meridian running through Greenwich, England.

Below are given the longitudes measured from Greenwich, of eleven cities, correct to 1'.

1. The captain of a ship finds that he is in longitude 56° W. How many degrees east of New York is he?

2. A ship is in longitude 8° W. when it gets a wireless message from a ship in longitude 12° E. What is the difference in longitude of the two ships?

3. Find the difference in longitude of two places A and B (a) if A is in longitude 30° W. and B 73° W. (b) If A is in longitude 40° E. and B 147° E. (c) If A is in longitude. 56° W. and B 98° E. (d) If A is in longitude 145° E. and B 32° W.

4. A ship in longitude 8° W. sails west 24°. longitude then?

What is its

5. A ship in longitude 15° E. sails west 84°. In what longitude is it then?

6. If a ship sails east 47° from longitude 150° E. what is its new longitude? Show this on a globe.

7. What is the difference in longitude between 178° W. and 178° E.?

8. A captain finds that he is in longitude 150° W. He knows the longitude of Melbourne. How can he find out how many degrees of longi

tude he must pass over to reach Melbourne?

Hint. In Figure 35, suppose that the ship is at S, that Melbourne is at M, and that PGP' is the meridian of Greenwich. How many degrees in the arc OC? How many degrees in the arc OB? How many in the arc COB? How many in the shorter arc BDC? What is the difference in

longitude of Melbourne and the ship?

FIG. 35.

9. State how to find the difference in longitude of two places on the same side of the principal meridian.

10. State how to find the difference in longitude of two places on opposite sides of the principal meridian, when the sum of their longitudes is less than 180°.

11. State how to find the difference in longitude of two places on opposite sides of the principal meridian, when the sum of their longitudes is greater than 180°.

12. Find the difference in longitude between Chicago and New York.

13. Find the difference in longitude between San Francisco and Constantinople.

14. Find the difference in longitude between Boston and Manila.

15. Find the difference in longitude between Washington and Tokio.