have the three sides terminated in the one extremity equal; and if there be given the three sides terminated in the other extremity, and the differences of the three angles which they severally make with the base; to find the values of functions of the common base, of the equal sides, and of the angles, of which the differences are given.

Let S' be put for the common base, S for each of the three equal sides of the proposed triangles S", S'', S'' for the three given sides ; A for the least of the angles opposite to the side S; and (A +a), (A+b), for the two other angles at the same extremity of the base, a and b being the given differences.

cos s cos S

Then (Art. 230. I.) cos S=cos S'cos S” + sin S' sin S" cos A, cos S=cos S'cos S" + sin S' sin S" cos (A+a), cos S=cos S'cos S" + sin S' sin S"" cos (A+b);

=cos S” + sin S" (cos A. tan S') =cos S'"+sin S"cosa (cos A tan S') - sin S"sin a(sin Atan S)

(Introd. 28.) =cosS"'+sin S"" cos b(cos Atan S)-sin S'' sin b(sin Atan S'). And, since the above three equations involve no more unknown quantities than the functions of the three quantities which are required to be found, it is evident, that the three unknown quantities may be, each of them, eliminated, by the usual processes of algebraical calculation, and by substituting (Introd. 17.) 7(1+tan? S')

for cos S': and the equations will be the more easily solved if



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

(250.) Cor. If 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;

cos S" - cos S" .. (Introd. 17.) tan S':

sin S" cos A - sin S'"' cos (4+a)

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(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.

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

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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.



Spherical Trigonometry.





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

(253.) Der. 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. 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.

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cos A" = tan S'cot S (Art. 232. VIII.)

cos S" =

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Let the hypotenuse S, and either of the two oblique angles A' or A", be given.


sin S' = sin S sin A'. (232. V.)
tan S” = tan Scos A' (VIII.)
cot A" = cos S tan A' (IV.)

sin S" = sin S sin A".
tan s' = tan Scos A".
cot A' = cos Stan A".


Let the two sides, S'and S", be given.

tan si tan A'

sin S" (Art. 232. VII.)

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