4. Through p, o draw the diameter ps, and another Aa, at right angles to it.

5. Set off om equal to the semitangent of the complement of the A (42°); then through the points A, m, a, describe a circle, cutting DC in в; and ABC will be the triangle required.

To measure the required parts.

Through the points p, в, draw the line pвr, cutting the circle E Dec in r; then ar, taken on the chords, gives A B 64° 40', AC on the same scale is 54° 43', and OB, taken on the line of semitangents, and then subtracted from 90°, gives BC 42° 12′.

LB.

Which side is less than 90°, being like its opp. B.

S

9.6313060

Which side is less than 90°, because 4s A and B are like.

64° 40'

INSTRUMENTALLY.

1. Extend the compasses from 48° (ZA) to 90° on the sines, and that extent will reach, on the same line, from 25° 25′ (complement of ▲ B) to 35° 17′, the complement of a C.

2. Extend from 64° 35′ ( ≤B) to 90° on the sines, and that extent will reach, on the same line, from 42° (compt. of A) to 47° 48', the complement of в C.

3. Extend from 48° (ZA) to 25° 25′ (complement of B) on the tangents, and this extent will reach, on the sines, from 90° to 25° 20′ the comp'. of AB (e).

2. In the right-angled spherical triangle ABC,

4. In the right-angled spherical triangle A B C,

(e) The analogy for obtaining the value of AB by the instru

ment, is got by substituting for cot A, in the numeral solu

tan A

tion, which then becomes tan A: Cot B:: rad : cos AB,

OF QUADRANTAL SPHERICAL TRIANGLES,

The different cases, or varieties, that may happen in the volation ci quainmal spherical triangles, in which two things, together with the quadrantal side, are always given, to find a third, are the same as in rightangled spherical triangles.

And since the sides and angles of any quadrantal spherical triangle are the supplements of the opposite angles and sides of a right-angled spherical triangle, described from its angular points as poles, the three general formulæ which have been given for the latter, may be readily converted into the following ones, which are equally applicable to all the cases of quadrantal spherical triangles:

1. r X sin eith. <=

2. rx cos cith.side

3. rx cos hyp'. <=

cot one side X cot other side.

Where it is to be remarked, that the angle opposite the quadrantal side is called the hypothenusal angle, and the other parts simply the sides and angles. And in applying these forms to practice, it is only necessary to attend to the observations that were made for rightangled spherical triangles.

AFFECTIONS OF

QUADRANTAL SPHERICAL TRIANGLES.

1. The sides are of the same kind as their opposite angles; and conversely.

2. The hypothenusal angle is greater or less than 90°, according as a side and its adjacent angle, or the two sides, or the other two angles, are like or unlike.

3. An angle at the quadrant is obtuse or acute according as its adjacent side and the hypothenusal angle, or the other angle and hypothenusal angle, are like or unlike.

4. A side is greater or less than 90°, according as its adjacent angle and the hypothenusal angle, or the other side and the hypothenusal angle, are like or unlike.

OTHER PROPERTIES OF

QUADRANTAL SPHERICAL TRIANGLES.

1. If the hypothenusal angle be 90°, one of the other angles and its opposite side will be each 90°, and the other side and angle will be measured by the same number of degrees.

2. If an angle, or a side, be 90°, the opposite side, or angle, and the hypothenuse will be each 90°; and the other angle and side will be measured by the same number of degrees.

3. If an angle at the quadrant be less than the hypothenusal angle, their sum will be less than 180°; and if it be greater than the hypothenusal angle, their sum will be greater than 180°.

4. If a side be less than its opposite angle, their sum will be less than 180°; and if it be greater than its opposite angle, their sum will be greater than 180°.

5. The difference of the two sides is less than 90°; and their sum is greater than 90°, and less than 270°. 6. The three angles are either all equal to, or less than, 90°, or two of them are greater than 90°, and the other less.

The six cases of quadrantal spherical triangles already mentioned, may be ranged as follows:

When a side and its opposite angle are given, to find the rest.

1. To find the other 4.

As rad; tan giv. 4: cot opp. or giv. side: sin other 4. Which may be either an acute or its supt. 2. To find the other side.

As cos giv.: cos opp. or giv.side :: rad: sin other side. Which side may be an arc less than 90°, or its supt.

3. To find the hypothenusal L.

As sin giv. side sin opp. or giv. :: rad; sin hyp'. 4.

:

Which hypothenusal

or its supplement.

may be either an acute 4,