Putting for AC', BC', BAC', and ABC' their values, we

and we obtain the formula (74) to (78) as before.

144. The formulæ of Arts. 139 to 142 are collected below for the convenience of the student:

By comparing the formulæ for the sines, cosines, and tangents of A and B with the corresponding forms for plane triangles as given in Arts. 10 and 14, no difficulty will be found in retaining them in the memory.

NAPIER'S RULES OF CIRCULAR PARTS.

145. These are two artificial rules which include all the formulæ of the preceding article.

In any spherical right triangle, the elements a and b, and the complements of the elements A, B, and c (written in abbreviated form, co. A, co. B, and co. c), are called the circular parts.

If we suppose them arranged in the order in which the letters occur in the triangle, any one of the five may be taken and called the middle part; the two immediately adjacent are called the adjacent parts, and the remaining two the opposite parts.

Then Napier's rules are:

I. The sine of the middle part is equal to the product of the tangents of the adjacent parts.

II. The sine of the middle part is equal to the product of the cosines of the opposite parts.

146. Napier's rules may be proved by taking each of the circular parts in succession as the middle part, and showing that the results agree with the formulæ of Art. 144.

1. If a is the middle part, b and co. B are the adjacent parts, and co.c and co. A the opposite parts.

Then the

2. If b is the middle part, a and co. A are the adjacent parts, and co. c and co. B the opposite parts. Then,

sin b = tan a tan (co. A)

= tan a cot A,

and

sin b =

= cos (co. c) cos (co. B)

= sin c sin B;

which agree with (79) and (77).

3. If co.c is the middle part, co. A and co. B are the adjacent parts, and a and b the opposite parts. Then,

or

and

or

cos ccot A cot B;

sin (co. c) = cos a cosb,

cos c = cos a cosb;

which agree with (83) and (74).

4. If co. A is the middle part, b and co. c are the adjacent parts, and a and co. B the opposite parts. Then,

sin (co. A) = tan b tan (co. c), or cos Atanb cotc,

and sin (co. A) = cos a cos (co. B),

which agree with (76) and (82).

or cos A

= cos a sin B ;

5. If co. B is the middle part, a and co. c are the adjacent parts, and b and co. A the opposite parts. Then,

sin (co. B) = tana tan (co.c), or cos B = tana cotc, and sin (co. B) = cosb cos (co. A), or cos B: = cos b sin A; which agree with (78) and (81).

147. Writers on Trigonometry differ as to the practical value of Napier's rules; but in the opinion of the highest authorities, it seems to be regarded as preferable to attempt to remember the formulæ by comparing them with the analogous forms for plane triangles, as stated in Art. 144.

SOLUTION OF SPHERICAL RIGHT TRIANGLES.

148. To solve a spherical right triangle, two elements must be given in addition to the right angle.

There may be six cases:

1. Given the hypotenuse and an adjacent angle.

2. Given an angle and its opposite side.
3. Given an angle and its adjacent side.
4. Given the hypotenuse and another side.
5. Given the two sides a and b.

6. Given the two angles A and B.

149. Either of the above may be solved by aid of Art. 144. The formula for computing either of the remaining elements when any two are given may be found by the following rule :

Take that formula which involves the given parts and the required part.

If all the remaining elements are required, the following rule may be found convenient in selecting the formulæ :

Take the three formulæ which involve the given parts.

150. It is convenient in the solution to have a check on the logarithmic work, which may be done in every case without the necessity of looking out any new logarithms.

Examples of this will be found in Art. 153.

The check formula for any particular case may be selected from the set in Art. 144 by the following rule :

Take that formula which involves the three required parts.

Note. If Napier's rules are used, the following rule will indicate which of the circular parts corresponding to the given elements and any required element is to be regarded as the middle part:

If these three circular parts are adjacent, take the middle one as the middle part, and the others are then adjacent parts.

If they are not adjacent, take the part which is not adjacent to either of the others as the middle part, and the others are then the opposite parts.

For the check formula, proceed as above with the circular parts correspond ing to the three required elements.

Thus if c and A are the given elements,

1. To find a, consider the circular parts a, co. c, and co.A; of these, a is the middle part, and co.c and co. A are opposite parts. Then, by Napier's rules,

sin a cos (co. c) cos (co. A) = sin c sin A.

2. To find b, the circular parts are b, co. c, and co. A; in this case co. A is the middle part, and b and co. c are adjacent parts. Then,

sin (co. A) = tan b tan (co. c), or cos A= tan b cot c.

co. A; co.c is

3. To find B, the circular parts are co. B, co. c, and the middle part, and co. A and co. B are adjacent parts. Then,

sin (co. c) = tan (co. A) tan (co. B), or cos c = cot A cot B.

4. For the check formula, the circular parts are a, b, and co. B; a is the middle part, and b and co. B are adjacent parts. Then,

sin a = tanb tan (co.B)= tan b cot B.

151. In solving spherical triangles, careful attention, must be paid to the algebraic signs of the functions; the cosines, tangents, and cotangents of angles greater than 90° being taken negative.

It is convenient to place the sign of each function just above or below it, as illustrated in the examples of Art. 153; the sign of the function in the first member being then determined in accordance with the principle that like signs produce +, and unlike signs produce —.

Note. In the examples after the first of Art. 153, the signs are omitted in every case where both factors of the second member are +.

152. In finding the angles corresponding, if the function is a cosine, tangent, or cotangent, its sign determines whether the angle is less or greater than 90°; that is, if it is +, the angle is 90°; and if it is, the angle is > 90°, and the supplement of the acute angle obtained from the tables must be taken (Art. 47).

If the function is a sine, since the sine of an angle is equal to the sine of its supplement, both the acute angle obtained from the tables and its supplement must be retained as solutions, unless the ambiguity can be removed by the principles of Art. 138.