Analogies: the last two may be demonstrated without recurring to the polar triangle by starting with the formulæ in Art. 39.

53. In equation (4) of the preceding article, cos (a - b) and

C

cot are necessarily positive quantities; hence the equation.

2

shews that tan (A + B) and cos (a + b) are of the same sign; thus (A + B) and 4 (a + b) are either both less than a right angle or both greater than a right angle. This is expressed by saying that (4+ B) and (a+b) are of the same affection.

54. To demonstrate Gauss's Theorems.

We have cos c = cos a cos b + sin a sin b cos C; therefore,

1 + cos c =

1 + cos a cos b + sin a sin b (cos2 C sin2 C)

= {1 + cos (a - b)} cos3 1⁄2 C + {1 + cos (a + b)} sin2 1⁄2 C ; therefore cos c = cos (a - b) cos2 C + cos2 (a + b) sin2 C. 1⁄2 1⁄2 ¦

Similarly, sin2 c = sin2 1⁄2 (a - b) cos31⁄2 C + sin2 (a + b) sin2 1 C.

Now add unity to the square of each member of Napier's first two analogies; hence by the formulæ just proved

Extract the square roots; thus, since (A+B) and 1⁄2 (a + b) are of the same affection, we obtain

3

cos (A + B) cos c = cos (a + b) sin & C
sin (a + b) sin C ....................... (2).

.........

(1),

cos (4 – B) sin & c =

Multiply the first two of Napier's analogies respectively by these results; thus

sin (A + B) cos c = cos (a - b) cos C....................... (3), sin (AB) sin c = sin (a - b) cos & C......... (4).

The last four formulæ are called Gauss's Theorems, although they are really due to Delambre.

55. The properties of supplemental triangles were proved geometrically in Art. 27, and by means of these properties the formulæ in Art. 47 were obtained; but these formulæ may be deduced analytically from those in Art. 39, and thus the whole subject may be made to depend upon the formulæ of Art. 39.

For from Art. 39 we obtain expressions for cos A, cos B, cos C'; and from these we find

cos A+ cos B cos C

cos b cos c) sina + (cos b - cos a cos c) (cos c

sin2 a sin b sin c

In the numerator of this fraction write 1 - cos2 a for sin2 a; thus the numerator will be found to reduce to

cos a (1 - cos3 a - cos3b - cos2 c + 2 cos a cos b cos c), and this is equal to cos a sin B sin C sin2 a sin b sin c, (Art. 41); therefore cos A+ cos B cos C = cos a sin B sin C. Similarly the other two corresponding formula may be proved.

Thus the formulæ in Art. 47 are established; and therefore, without assuming the existence and properties of the Polar Triangle, we deduce the following theorem: If the sides and angles of a spherical triangle be changed respectively into the supplements of the corresponding angles and sides, the fundamental formulæ of Art. 39 hold good, and therefore also all results deducible from them.

56. The formulæ in the present chapter may be applied to establish analytically various propositions respecting spherical triangles which either have been proved geometrically in the preceding chapter, or may be so proved. Thus, for example, the second of Napier's analogies is

sin (a - b) tan (A-B) = sin (a + b)

this shews that (A – B) is positive, negative, or zero, according as (a - b) is positive, negative, or zero; thus we obtain all the results included in Arts. 33-36.

57. If two triangles have two sides of the one equal to two sides of the other, each to each, and likewise the included angles equal, then their other angles will be equal, each to each, and likewise their bases will be equal.

We may shew that the bases are equal by applying the first formula in Art. 39 to each triangle, supposing b, c, and A the same in the two triangles; then the remaining two formulæ of Art. 39 will shew that B and C are the same in the two triangles.

It should be observed that the two triangles in this case are not necessarily such that one may be made to coincide with the other by superposition. The sides of one may be equal to those of the other, each to each, but in a reverse order, as in the following figures.

Two triangles which are equal in this manner are said to be symmetrically equal; when they are equal so as to admit of superposition they are said to be absolutely equal.

58. If two spherical triangles have two sides of the one equal to two sides of the other, each to each, but the angle which is contained by the two sides of the one greater than the angle which is contained by the two sides which are equal to them of the other, the base of that

which has the greater angle will be greater than the base of the other; and conversely.

Let b and c denote the sides which are equal in the two triangles; let a be the base and A the opposite angle of one triangle, and a' and A' similar quantities for the other. Then

sin (a + a) sin

this shews that

= cos b cos c + sin b sin c cos A';

cos a = sin b sin c (cos A – cos A');

(a − a) = sin b sin c sin (A + A ́) sin § (A – A^); (a − a) and (A – A') are of the same sign.

59. If on a sphere any point be taken within a circle which is not its pole, of all the arcs which can be drawn from that point to the circumference, the greatest is that in which the pole is, and the other part of that produced is the least; and of any others, that which is nearer to the greatest is always greater than one more remote; and from the same point to the circumference there can be drawn only two arcs which are equal to each other, and these make equal angles with the shortest arc on opposite sides of it.

This follows readily from the preceding three articles.

60. We will give another proof of the fundamental formulæ in Art. 39, which is very simple, requiring only a knowledge of the elements of Co-ordinate Geometry.

Suppose ABC any spherical triangle, O the centre of the sphere, take O as the origin of co-ordinates, and let the axis of pass through C. Let x,, y, z, be the co-ordinates of 4, and x, Y2, those of B; let r be the radius of the sphere. Then the square of the straight line AB is equal to

2

2

2

2

2

and x2+ y2+z ̧3 = r3,
' = r2, x2 + y2 + z ̧2 = r2, thus

2

2

2

Now make the usual substitutions in passing from rectangular to polar co-ordinates, namely,

cos 0, cos 0, + sin 0, sin 0, cos (4, - 4,) = cos AOB,

2

1

2

that is, in the ordinary notation of Spherical Trigonometry,

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

This method has the advantage of giving a perfectly general proof, as all the equations used are universally true.

EXAMPLES.

1. If A = a, shew that B and b are equal or supplemental, as also C and c.

2. If one angle of a triangle be equal to the sum of the other two, the greatest side is double of the distance of its middle point from the opposite angle.

3. When does the polar triangle coincide with the primitive triangle?

4. If D be the middle point of AB, shew that

cos AC + cos BC= 2 cos AB cos CD.

5. If two angles of a spherical triangle be respectively equal to the sides opposite to them, shew that the remaining side is the supplement of the remaining angle; or else that the triangle has two quadrants and two right angles, and then the remaining side is equal to the remaining angle.