cos (b + c) = 2 sin (a + b + c) sin § (b + c ·

a),

equation (2) becomes, after dividing both members by 2,

and extract the square root of both members, we have

That is, the cosine of one half of any angle of a spherical triangle is equal to the square root of the sine of one half of the sum of the three sides, into the sine of one half this sum minus the side opposite the angle, divided by the rectangle of the sines of the adjacent sides.

If we subtract equation (1), of this article, member by member, from the number 1, and recollect that

Dividing equation (4) by equation (3), member by mem

82. From the foregoing values of the functions of one half of any angle, may be deduced values of the functions of one half of any side of a spherical triangle.

Representing the angles and sides of the supplemental polar triangle of ABC as in Art. 80, we have

Substituting these values in (3), Art. 81, and reducing by the aid of the formulas in Table III., Art. 63, we find

In a similar way, we may deduce from (4), Art. 81,

83. To deduce Napier's Analogies. From equation (1), Art. 80, we have

cos A+ cos B cos C = sin B sin C cos a

since, from proportion (1), Art. 78, we have

Also, from equation (2), Art. 80, we have

cos B+cos A cos C = sin A sin C cos b

= sin C

sin A
sin a

sin a cos b.

(2.)

Adding (1) and (2), and dividing by sin C, we obtain

taken first by composition, and then by division, gives

Dividing (4) and (5), in succession, by (3), we obtain

But, by formulas (2) and (4), Art. 67, and formula (E"), Art. 66, equation (6) becomes

and, by the similar formulas (3) and (5), of Art. 67, equation (7) becomes

These last two formulas give the proportions known as the first set of Napier's Analogies; viz.,

cos (a+b): cos (a-b) :: cot C:

tan (A+B). (10.)

sin (a+b) : sin(a-b) :: cct: tan † (A—B). (11.) If in these we substitute the values of a, b, C, A, and B, in terms of the corresponding parts of the supplemental polar triangle, as expressed in Art. 80, we obtain

cos (A+B): cos (A−B) :: sin (A+B): sin (A-B) ::

tan c: tan (a+b), (12.)

tanic:

: tan (a-b), (13.)

the second set of Napier's Analogies.

In applying logarithms to any of the preceding formu las, they must be made homogeneous in terms of R, as explained in Art. 30.

In all the formulas, the letters may be interchanged at pleasure, provided that, when one large letter is substituted for another, the like substitution is made in the corresponding small letters, and the reverse: for example, C may be substituted for A, provided that at the same time c is substituted for a, &c.

NOTE. It may be noted that, in formulas (10) and (12), whenever the sign of the first term of the proportion is minus, the sign of the last term must, also, be minus, i. e., whenever (a+b) is greater than 90°, (A+B) must, also, be greater than 90°, and the reverse; and similarly, whenever (a + b) is less than 90°, (A + B) must, also, be less than 90°, and the reverse.

SOLUTION OF OBLIQUE-ANGLED SPHERICAL TRIANGLES.

84. In the solution of oblique-angled triangles six different cases may arise: viz., there may be given,

I. Two sides and an angle opposite one of them.

II. Two angles and a side opposite one of them.

III. Two sides and their included angle.

IV. Two angles and their included side.

V. The three sides.

VI. The three angles.