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S=4qp=4X3.14 X4 X4
=200.96 sq. ft.

4 4
V προς X3.14 X4 X4X4

3 3
=267.95 cu. ft.

e. Exercises.--(1) Find the volume of a-
(a) Sphere whose radius is 3 inches.
(6) Hemisphere whose radius is 5 inches.

261.67 cubic inches. Answer. (c) Cylinder whose altitude is 10 inches and circular base with radius 372 inches. (d) Prism with base area of 25 square inches and altitude of 2 feet.

50 cubic inches. Answer. (e) Cone whose altitude is 12 inches and whose base is a circle of 9-inch diameter.

() Prism with rectangular base (sides 10 and 7 inches) and altitude 11 inches.

770 cubic inches. Answer. (9) Pyramid with triangular base (b=6 inches, h=474 inches) and altitude of 11 inches.

(2) Find the surface area of a(a) Sphere of 5 units radius. (6) Sphere of 12 units diameter. 452.16 square units. Answer. (c) Sphere with a great circle area of 252 square inches. (3) Find the radius of a

22 (a) Sphere whose surface area is 2,464 square inches use ==

7

22 (6) Sphere whose volume is 113% cubic inches use =

7

3 ins. Answer. 60. Intersection of a sphere and a plane.-a. Circular intersections.-If a plane intersects a sphere, the intersection is a circle.

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(1) When a plane which intersects a sphere passes through the center of the sphere, the circle cut out is called a great circle of the sphere. The center of the great circle is the center of the sphere.

(2) When the intersecting plane does not pass through the center of the sphere, the circle cut out is a small circle of the sphere.

b. Arcs on sphere.-Through any two given points on a sphere, an arc of a great circle may be drawn.

(1) When the two points are not extremities of a diameter, one, and only one, great circle can be drawn.

(2) When the two points are the extremities of a diameter, every circle through them is a great circle.

c. Tangent planes and lines.-A plane which intersects a sphere at just one point is the tangent to the sphere at that point.

(1) A tangent plane is perpendiular to the radius of the sphere drawn to the point of tangency.

(2) Every line in the tangent plane and through the point of tangency is a tangent line to the sphere.

SECTION IX

SPHERICAL GEOMETRY

Paragraph Basic definitions--

61 Physical applications of spherical geometry-

62 61. Basic definitions.—a. Quadrant.-One-fourth of a great circle, for example, arc AB, figure 109, is a quadrant.

b. Distance between two points on a sphere is measured along the great circle which connects them, for example arc BD connecting points B and D, figure 109.

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c. Dihedral angle.--The angle formed by two intersecting planes. A dihedral angle is measured numerically by the plane angle formed by one perpendicular in each plane drawn to a common point on the edge of the dihedral angle. ZBOD measures the dihedral angle between the great circle A through D, and the great circle A through B.

d. Acis of a circle.—The diameter of a sphere, perpendicular to a circle of that sphere, is called the axis of that circle. The axis of the earth is generally taken as the diameter perpendicular to the equator, that is, as the axis of the equator.

e. Pole of a circle. --The poles of a circle are the points at which the circle's axis meets the sphere. Points A and C are poles of circle BDEF.

NOTE.—To draw the great circle which has a given point as pole, measure off a quadrant's distance on the surface of the sphere from the given point.

Example: The distance from either the north pole or the south pole to the equator is a quadrant.

f. Spherical angles.-A spherical angle is formed by two minor arcs of great circles having a common end point. Arcs AB and AD, figure 109, form a spherical angle. It is measured as follows:

(1) It equals numerically the angle formed by the tangents to the arcs at the vertex of the angle.

(2) It equals numerically the dihedral angle formed by the planes of its sides.

NOTE.—The sides of a spherical angle are arcs of great circles and each of these great circles lies in a plane. These planes determine the dihedral angle.

(3) It' equals numerically the arc intercepted on the great circle of which its vertex is a pole.

(4) It is apparent that any angle on the earth's surface is in reality a spherical angle. For small distances these angles are sometimes considered as plane angles, since the great circles determining the sides are almost straight lines if the distance is small enough.

g. Spherical polygon.-A closed figure formed by three or more minor arcs of great circles on a sphere is called a spherical polygon.

h. Spherical triangles.-A spherical polygon of three arcs is a spherical triangle. Each of the arcs is called a side. The figure determined by arcs DE, DC, and CE, figure 109, is a spherical triangle.

(1) Properties of spherical triangles:

(a) Any side of a spherical triangle is less than the sum of the other two sides. (The sides are arcs of great circles and are measured in degrees. The number of degrees is the same as the number of degrees of the central angle subtended at the center of the sphere in the plane of the circle.)

(6) The sum of the angles of a spherical triangle is greater than 180° and less than 540°. Any angle of a spherical triangle must be less than 180°.

(c) The sum of the sides of a spherical triangle is less than 360°. A very important application of the spherical triangle is met in celestial navigation.

(2) All triangles whose sides are distances between points on the earth's surface are spherical triangles. Again, for very short distances the triangle formed may be considered a plane figure.

(3) Examples: (a) If two angles of a spherical triangle are 65° and 105° what can be said about the third side?

Solution: 65° plus 1050 equals 170° and there must be more than 180° Therefore the third angle must be more than 10°. Also, since every angle of a spherical triangle is less than 180°, the third angle must be less than 180°.

(6) If two sides of a spherical triangle are 35° and 76°, what can be said about the third side?

Solution: The 35° side plus the third side must be greater than the 76° side, since the sum of two sides is greater than the third side. Therefore the third side must be greater than 41o. Also the third side must be less than the sum of 76° and 35° or less than 111°.

(4) Exercises.-(a) If two sides of a spherical triangle are 70° and 90°, what can be said about the third side?

(6) If two angles of a spherical triangle are 100° and 70°, what can be said about the size of the third angle?

Greater than 10° but less than 180°. Answer. (c) Is it possible to construct a spherical triangle with angles 75°, 65°, 35°? Why?

(d) May a side or an angle of a spherical triangle equal 180°?

(e) If two sides of a spherical triangle are 125° and 160°, what can be said about the third side?

Greater than 35° and less than 75°. Answer. 62. Physical applications of spherical geometry.a. Arbitrary landmarks.-For practical purposes, the earth may be considered to have a spherical surface. Upon this surface, for reasons of determining position, the following real and imaginary landmarks are set up:

(1) Equator.-A great circle on the earth's surface perpendicular to the usual "earth's axis.” This is of course an imaginary landmark.

(2) Latitude.—The distance in degrees of arc on a great circle north or south of the equator. These degrees may be converted to nautical miles by remembering that 1 minute of arc (of a great circle) is equal to one nautical mile.

(3) Meridian.— Any great circle passing through the North and South Poles.

(4) Prime Meridian.The meridian passing through Greenwich, England. It is chosen as the reference or zero point from which all other meridians are measured.

(5) Longitude.--The angular distance from any meridian to the Prime Meridian. The angles run both east and west of the Prime Meridian and never exceed 180° in either direction. It should be noted that the longitudinal distance between two meridians in nautical miles at the equator is not the same as at some point above or below the equator. One minute of arc of longitude varies from 1 nautical mile at the equator to 0 nautical miles at the poles.

NOTE.—The number of degrees of arc between two meridians is equal to the number of degrees of the dihedral angle formed by the two meridians, and is also equal to the number of degrees in the spherical angle formed at the pole by the meridians.

b. Determination of distances.-(1) Example: Find the distance in nautical miles between the following places, which have the same longitude: (a) 12°15' N and 30°11' N.

Solution: The difference in latitude between the two places is obtained by subtraction.

30°11' N
12°15' N
17°56'=difference in latitude

To change degrees latitude to minutes latitude, multiply by 60.

17 X 60=1,020

56
1,076'=difference

in minutes. Using the fact that 1 minute of latitude equals 1 nautical mile, the distance is 1,076 nautical miles.

NOTE.—In the foregoing subtraction it was necessary to borrow a degree in order to subtract the 15'.

(6) 15°16' N and 27°23' S.

Solution: Since these are on opposite sides of the equator, it is
necessary to add the latitudes.

15°16' N
27°23' S
42°39'difference in latitude

(42X60)+39'=2559 minutes of latitude=2,559 nautical miles.

(2) Exercises.—Find the distance in nautical miles between each of the following places that have the same longitude, if the latitudes are:

(a) 14°14' N and 76°30' N.

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