124. The Circle. It is learned in geometry that

c = 2 πr, and α = πι,

where c circumference, r radius,

=

=

=

radius, a = area, and T = 3.14159+ 3.1416 about 34. For practical purposes 22 may be taken. Furthermore, if we have a sector with angle n degrees,

From these formulas we can, by the help of the formulas relating to triangles, solve numerous problems relating to the circle.

r

n

Exercise 63. The Circle

1. A sector of a circle of radius 8 in. has an angle of 62.5°. A chord joining the extremities of the radii forming the sector cuts off a segment. What is the area of this segment?

2. A sector of a circle of diameter 9.2 in. has an angle of 29° 42'. A chord joining the extremities of the radii forming the sector cuts off a segment. What is the area of the remainder of the circle?

B

A

3. In a circle of radius 3.5 in., what is the area included between two parallel chords of 6 in. and 5 in. respectively? (Give two answers.)

4. A regular hexagon is inscribed in a circle of radius 4 in. What is the area of that part of the circle not covered by the hexagon? 5. In a circle of radius 10 in. a regular five-pointed

star is inscribed. What is the area of the star? What is the area of that part of the circle not covered by the star?

6. In a circle of diameter 7.2 in. a regular fivepointed star is inscribed. The points are joined,

thus forming a regular pentagon. There is also a regular pentagon formed in the center by the crossing of the lines of the star. The small pentagon is what fractional part of the large one?

7. A circular hole is cut in a regular hexagonal plate of side 8 in. The radius of the circle is 4 in. What is the area of the rest of the plate?

8. A regular hexagon is formed by joining the mid-points of the sides of a regular hexagon. Find the ratio of the smaller hexagon to the larger.

125. Plane Sailing. A simple and interesting application of plane trigonometry is found in that branch of navigation in which the surface of the earth is considered a plane. This can be the case only when the distance is so small that the curvature of the earth may be neglected.

This chapter may be omitted if further applications of a practical nature are not needed.

126. Latitude and Departure. The difference of latitude between two places is the arc of a meridian between the parallels of latitude which pass through those places.

Thus the latitude of Cape Cod is 42° 2′ 21′′ N. and the latitude of Cape Hatteras is 35° 15′ 14′′ N. The difference of latitude is 6° 47′ 7′′.

The departure between two meridians is the length of the arc of a parallel of latitude cut off by those meridians, measured in geographic miles.

1 60

1 360

The geographic mile, or knot, is the length of 1' of the equator. Taking the equator to be 131,385,456 ft., of of this length is 6082.66 ft., and this is generally taken as the standard in the United States. The British Admiralty knot is a little shorter, being 6080 ft. The term "mile" in this chapter refers to the geographic mile, and there are 60 mi. in one degree of a great circle.

Calling the course the angle between the track of the ship and the meridian line, as in the case of N. 20° E., it will be evident by drawing a figure that the difference in latitude, expressed in distance, equals the distance sailed multiplied by the cosine of the course. That is distance x cos C.

diff. of latitude =

In the same way we can find the departure. This is evidently given by the equation

For example, if a ship has sailed N. 30° E. 10 mi., the difference in latitude, expressed in miles, is

10 cos 30° = 10 × 0.8660 = 8.66,

and the departure is 10 sin 30° = 10 × 0.5 = 5.

127. The Compass. The mariner divides the circle into 32 equal parts called points. There are therefore 8 points in a right angle, and a point is 11° 15'. To sail two points east of north means, therefore, to sail 22° 30' east of north, or northnortheast (N.N.E.) as shown on the compass. Northeast (N.E.) is 45° east of north. One point east of north is called north by east (N. by E.) and one point east of south is called south by east (S. by E.). The other terms used, and their significance in angular measure,

will best be understood from the illustration and the following table:

The compass varies in different parts of the earth; hence, in sailing, the compass course is not the same as the true course. The true course is the compass course, with allowances for variation of the needle in different parts of the earth, for deviation caused by the iron in the ship, and for leeway, the angle which the ship makes with her track.

Exercise 64. Plane Sailing

1. A ship sails from latitude 40° N. on a course N.E. 26 mi. Find the difference of latitude and the departure.

2. A ship sails from latitude 35° N. on a course S. W. 53 mi. Find the difference of latitude and the departure.

3. A ship sails from a point on the equator on a course N.E. by N. 62 mi. Find the difference of latitude and the departure.

4. A ship sails from latitude 43° 45′ S. on a course N. by E. 38 mi. Find the difference of latitude and the departure.

5. A ship sails from latitude 1° 45′ N. on a course S.E. by E. 25 mi. Find the difference of latitude and the departure.

6. A ship sails from latitude 13° 17' S. on a course N.E. by E. 3 E., until the departure is 42 mi. Find the difference of latitude and the latitude reached.

7. A ship sails from latitude 40° 20′ N. on a N.N.E. course for 92 mi. Find the departure.

8. If a steamer sails S.W. by W. 20 mi. what is the departure and the difference of latitude?

9. If a sailboat sails N. 25° W. until the departure is 25 mi., what distance does it sail?

10. A ship sails from latitude 37° 40′ N. on a N.E. by E. course for 122 mi. Find the departure.

11. A yacht sails 6 points west of north, the distance being 12 mi. What is the departure?

12. A steamer sails S.W. by W. 28 mi. It then sails N.W. 30 mi. How far is it then to the west of its starting point?

13. A ship sails on a course between S. and E. 24 mi., leaving latitude 2° 52' S. and reaching latitude 2° 58' S. Find the course and the departure.

14. A ship sails from latitude 32° 18′ N., on a course between N. and W., a distance of 34 mi. and a departure of 10 mi. Find the course and the latitude reached.

15. A ship sails on a course between S. and E., making a difference of latitude 13 mi. and a departure of 20 mi. Find the distance and the course.

16. A ship sails on a course between N. and W., making a difference of latitude 17 mi. and a departure of 22 mi. Find the distance and the course.

128. Parallel Sailing. Sailing due east or due west, remaining on the same parallel of latitude, is called parallel sailing.

129. Finding Difference in Longitude. In parallel sailing the distance sailed is, by definition (§ 126), the departure. From the departure the difference in longitude is found as follows: Let AB be the departure. Then in rt. ▲ OAD

The triangles DAB and OEQ are similar, the arcs being (§ 125) considered straight lines.

That is, the number of minutes in the difference in longitude is the product of the number of miles in the departure by the secant of the latitude, the nautical, or geographic, mile being a minute of longitude on the equator.

Exercise 65. Parallel Sailing

1. A ship in latitude 42° 16' N., longitude 72° 16' W., sails due east a distance of 149 mi. What is the position of the point reached?

2. A ship in latitude 44° 49' S., longitude 119° 42' E., sails due west until it reaches longitude 117° 16' E. Find the distance made.

3. A ship in latitude 60° 15' N., longitude 60° 15' W., sails due west a distance of 60 mi. What is the position of the point reached?