A Treatise on Spherics

cos A tan S', and sin A tan S', be first considered as the
unknown quantities.

(250.) COR. If S", S", and the angles A and
(A+a) be given, the tangent of S' may be found in terms
of the functions of the given quantities by subtracting
the second of the first three equations, in Art. 249, from
the first: for then,

os S'(cos S" - cos S") + sin S' (sin S" cos A- sin S" cos (A+a)) = 0;

.. (Introd. 17.) tan S' =

cos S""-cos S"
sin S” cos A – sin S" cos (4+a) *

(251.) SCHOLIUM. The two preceding articles contain so much, as properly belongs to Spherics, of the solutions of two astronomical problems, which may be thus enunciated.

1. Three known Stars having been observed in the same almacantar, to find the latitude of the place of observation, the hour, and the common zenith-distance of the three Stars.

2. Two known Stars having been observed, at a given hour, in the same almacantar, to find the latitude of the place of observation.

The process, by which the expressions, investigated in Art. 249. 250, are adapted to logarithmical calculation, may be seen in the first Volume of Delambre's Astronomy.

PART II.

THE ELEMENTS OF

Spherical Trigonometry.

SECTION II.

ON THE SOLUTION OF RIGHT-ANGLED AND QUADRANTAL SPHERICAL TRIANGLES.

DEFINITION.

(252.) THE three sides, and the three angles, are called the Parts of a Spherical Triangle.

(253.) DEF. Any three of the six parts of a proposed spherical triangle being given, the finding of the remaining three, or of their trigonometrical functions, is called the Solution of that triangle: and when, by the help of Spherical Trigonometry, the three parts, or their functions, which were, at the first, unknown, have been found, the triangle is said to be solved.

(254.) COR. 1. A spherical triangle, having all its angles right angles, and consequently, (Art. 122.) all its

sides quadrants, is solved without the help of trigonometry.

(255.) COR, 2. If two of the angles of a spherical triangle be right angles, or two of its sides quadrants, then, of the remaining side and angle, if the one be given, the other (Art. 121. and 54.) is known; and the triangle is, in that case, also, solved, without the help of trigonometry.

(256.) Problem.

PROP. I.

Two parts, besides the right angle, being given, of a spherical triangle, which has only one of its angles a right angle, to solve the triangle.

Let A denote the right angle, and S the hypotenuse; A', A" the two oblique angles; and S', S" the two sides respectively opposite to them.

CASE 1.

Let the hypotenuse, S, and either of the other sides, S' or S", be given.