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PART II.

SPHERICAL TRIGONOMETRY.

CHAPTER I.

FUNDAMENTAL PRINCIPLES.

93. Def. Spherical trigonometry treats of the relations among the six parts of a trihedral angle.

Def. The six parts of a trihedral angle are its three face-angles and its three edge-angles.

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94. Representation of a trihedral angle by a spherical triangle. If O be the vertex of a trihedral angle, and OM, OP, and OQ its three edges, we may construct a sphere having its centre at O, and having an arbitrary radius OA. The spherical surface will then cut the edges at the three points A, B, and C equally distant from 0.

The three faces OMP, OPQ, OQM will intersect the spherical

M

surface in three arcs of great circles, AB, BC, CA, which arcs form a spherical triangle.

It is shown in geometry that the three angles A, B, and C of the spherical triangle are equal to the respective edge-angles OM, OP, and OQ of the trihedral angle. It is also shown that the arcs AB, BC, and CA, which form the sides of the triangle, measure the respective face-angles MOP, POQ, QOM of the trihedral angle.

Therefore the six parts of the trihedral angle are represented by the corresponding parts of the spherical triangle, and the relations among the parts of the one are the same as the relations among the parts of the other.

The term spherical trigonometry is applied because the investigations are generally made by means of the spherical triangle.

A trihedral angle, with its corresponding spherical triangle, may be readily constructed as follows: Cut a circular disk of pasteboard or stiff paper, from four to six inches or more in diameter. From this disk cut out a sector of any magnitude. It will be well to have several disks with sectors ranging from 45° to 200° cut out. Divide the remainder of the disk by two radii into three sectors, such that the greatest shall be less than the sum of the other two. Bend the disk along each of the two dividing radii, cutting the latter part of the way through if necessary, and bring the extreme radii together. We shall then have a figure like O-ABC of the preceding diagram, the three plane sides forming the trihedral angle, and the three arcs bounding the edge of the disk forming the spherical triangle.

95. General remarks upon spherical triangles. A spherical triangle may be defined as that figure which is formed by joining any three points on the surface of a sphere by arcs of great circles. The three points will then be the vertices of the triangle.

But between any two points we may draw two arcs of a great circle, which together make up a complete great circle through the points. One of these arcs will be less, the other greater, than 180°. To avoid ambiguity, the arc less than 180° is supposed to be taken, unless otherwise expressed. We therefore adopt the rule:

Each side of a spherical triangle is supposed less than 180°, unless otherwise expressed.

This rule is a mere convention, which may be set aside whenever we desire to give greater generality to our conclusions. Nothing prevents us from supposing ourselves to pass from one vertex to another by passing several times around the sphere. The corresponding side of the triangle will then consist of several coincident great circles plus either of the arcs joining the vertices. If we suppose a to be the shorter arc joining two vertices, the general arc measure of the side through those vertices will be

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96. Every spherical triangle encloses a portion of the spherical surface, forming the area of the triangle. We then have the theorem:

Three great circles divide the surface of the sphere into eight triangular portions.

This is shown as follows: One great circle divides the surface into two equal parts. A second great circle intersects the first in two points, and divides each of those parts into two lunes, so that the whole surface is then divided into four lunes. A third great circle cuts through all four of these lunes, and forms eight spherical triangles.

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The eight spherical triangles formed by three great circles.

In the same way, any two planes divide the space around their line of intersection into four parts. A third plane intersecting them divides the space around their point of intersection into eight parts, forming eight trihedral angles.

Remark. The student should guard himself against considering a figure of which either side is a small circle of the sphere as a spherical triangle. For example, the figure formed by two arcs of meridians and a parallel of latitude is not a spherical triangle. Such figures do not represent the parts of a trihedral angle, and so do not correspond to the definition of a spherical triangle. All the important problems connected with them may be reduced to problems of spherical trigonometry, so that there is no need of giving them special consideration.

EXERCISES.

The following exercises are introduced to test the student's fundamental conceptions of spherical geometry, and especially of the relations of great circles of the sphere. Their successful performance will show that he is prepared to take up the subject of spherical trigonometry with advantage. A globe, on which figures may be drawn at pleasure, will be of great service in assisting his conceptions, and should be made use of whenever practicable.

1. A and A' are two opposite points on a sphere. If any third be taken on the sphere, to what constant arc will the sum XA' be equal, and what will be the angle AXA'?

point

XA

NOTE. Opposite points are those at the ends of a diameter.

2. If one side of a spherical triangle be equal to a semicircle, what relations will then subsist between the other two sides? What will be the magnitude of the opposite angle?

3. Let A, B, and C be the three vertices of a spherical triangle ; a, b, and c, the sides opposite these vertices respectively; A', B', and C', the points opposite the vertices. It is then required:

(a) To express by the letters at their vertices the eight triangles which will be formed when each side of the original triangle ABC is produced into a great circle.

(b) To express the sides of each of these eight triangles in terms of a, b, and c, making use of the theorem that any two great circles intersect each other in two opposite points.

(c) To express the angles in terms of the angles of the original triangle, which we may represent by the letters A, B, and C marking their vertices.

(d) It being found that the eight triangles are divisible into four pairs, such that the sides and angles of each pair are equal, it is required to show the relations of each pair.

4. If one angle of a spherical triangle is A, show that the sum of the other two angles is contained between the limits 180° — A and 180°

A.

NOTE. If the student finds any difficulty in this question he may begin by supposing the triangle to be isosceles, and the two equal sides to increase from 0° to 180°.

5. Hence show that the spherical excess cannot exceed twice the smallest angle.

6. If the three sides of an equilateral spherical triangle be continually and equally increased, what is the limit of their sum? What is the limit of the angles as the sum approaches its limit?

97. Fundamental equations. Let us put

a, b, c, the three face-angles of the trihedral angle—that is, the angles subtended by the three sides of the spherical triangle; A, B, C, the opposite edge-angles of

the trihedral angle, or the angles of the spherical triangle.

Then if O-ABC be any trihedral

angle, we shall have

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Through any point A of OA pass a plane perpendicular to OA, and let B and C be the points in which it meets the other two edges. We shall then have

A = angle BAC,

while OAB and OAC will both be right angles.

In the triangle BOC we have

=

BC OB + OC2 - 20B. OC cos a.

In the triangle BAC we have

BC AB+ AC2

=

(§ 58)

2AB. AC cos A.

Equating these two values of BC3, and transposing, we find 20B. OC cos a = OB2 – AB2+ OC2 − AC2+2AB. AC cos A. But, because OAB and OAC are right angles,

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Substituting these values, and dividing by 20B. OC, we have

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By treating the other edges in order in the same way, we obtain two more equations, which may be written by simply permuting a, b, and c and A, B, and C circularly; that is, by substituting for each letter the one next in order, a following c.

Thus

we have the system of three equations:

cos a = cos b cos c + sin b sin c cos A;
cos b = cos c cos a + sin c sin a cos B;

(1)

cos c = cos a cos b+ sin a sin b cos C.

These three equations are the fundamental equations of spherical trigonometry, because by means of them, when three parts are

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