« ForrigeFortsett »
Table for determining when the things found in the pre
ceding are less than a Quadrant (Sp. Ge. 14 and 15).
The angle or arc found is less than 90°.'
When C is less than 90°. When B is less than 90°. | When B and Care of the same affection.
Schol. The rules for the cases of right-angled spherical trigonometry may be reduced to two, called Napier's Rules of the Circular Parts.
In a right-angled spherical triangle, the right angle is neglected, and the hypotenuse, the two angles, and the complements of the two sides, are called circular parts ; and any of these parts being called the middle part; the two adjacent to it, adjacent parts; and the two remaining parts, opposite parts; then, the rectangle under the radius and the cosine of the middle part, is equal to that under the cotangents of the adjacent parts, or the sines of the opposite parts. Or if the middle part be called M; the two adjacent parts, A and a; and the opposite parts 0 and 0,
Rocos M = cot Ā.cot a,
or Rocos M = sin 0 • sin 0. Either of these rules may be converted into a proportion by Pl. Ge. VI. 16.
The cases marked ambiguous are those in which the thing sought has two values, and may either be equal to a certain angle, or to the supplement of that angle. Of these there are three, in all of which the things given are a side, and the angle opposite to it; and accordingly, it is easy to show that two right-angled spherical triangles may always be found, that have a side and the angle opposite to it the same in both, but of which the remaining sides, and the remaining angle of the one, are the supplements of the remaining sides and the remaining angle of the other, each of each.
Though the affection of the arc or angle found may in all the other cases be determined by the rules in the second of the preceding tables, it may be useful to remark, that all these rules, except two, may be reduced to one, namely, that when the thing found by the rules in the first table is either a tangent or a cosine; and when, of the tangents of cosines employed in the computation of it, one only belongs to an obtuse angle, the angle required is also obtuse.
Thus, in the 15th case, when cos AB is found, if C be an obtuse angle, because of cos C, AB must be obtuse; and in case 16, if either B or C be obtuse, BC is greater than 90°, but if B and C are either both acute, or both obtuse, BC is less than 90°.
It is evident that this rule does not apply when that which is found is the sine of an arc; and this, besides the three ambiguous cases, happens also in other two, namely, the 1st and 11th.
Solution of the Cases of Oblique-Angled Spherical
PROBLEM. In any oblique-angled spherical triangle, of the three sides and three angles, any three being given, the other three may be found.
In this table, the references (c. 4), (c. 5), &c. are to the cases in the preceding tables.
Two sides AB, AC, and the included angle
Let fall the perpendicular CD 1.
from the unknown angle
R:cos A=tan AC:tan AD
B and A are of the same
less than AD (Sp. Ge. 16).
Let fall the perpendicular CD 2. from one of the unknown
angles on the side AB.
R:cos A=tan AC :tan AD The third (c. 2); therefore AD is
side known, and cos AD:cos BC. BD::cos AC:cos BC (9);
according as the segments AD and DB are of the same or different affection, AC and CB will be of the same or different affection.
Two angles A and ACB,
and AC, the side between them.
Let fall the perpendicular CD
from one of the given
ACD (c. 3); therefore the