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It may here be observed, that all the analogies and formulae, of spherical trigonometry, in which cosines or cotangents are not concerned, may be applied to plane trigonometry; taking care to use only a side instead of the sine or the tangent of a side; or the sum or difference of the sides instead of the sine or tangent of such sum or difference. The reason of this is obvious: for analogies or theorems raised, not only from the consideration of a triangular figure, but the curvature of the sides also, are of consequence more general; and therefore, though the curvature should be deemed evanescent, by reason of a diminution of the surface, yet what depends on the triangle alone will remain, notwithstanding.

We have now deduced all the rules that are essential in the operations of spherical trigonometry; and explained under what limitations ambiguities may exist. That the student, however, may want nothing further to direct his practice in this branch of science, we shall add three tables, in which the several formulae, already given, are respectively applied to the solution of all the cases of right and obliqueangled spherical triangles, that can possibly occur.

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*Given. Required. Walues of the terms required. quired are less than 90° — sin given le If the given leg be less g g g * I. than 90°. Hypothenuse, Angle adjacent to the Its cos = ‘an given leg If the things given be of and one leg. given leg. tan hypoth." the same affection. Other leg. Its cos = co-hypoth. Idem. cos given leg." in – sin given leg. II. Its sin = singivenang Ambiguous.

One leg, and


:... tangiven leg its opposite Other leg. Its sin = tangiven ang" Idem. o ** angle. Other angle. Its sin = co-given ang. Idem. - o - cos given leg If the thi b - – - — tan given leg the things given be III. Hypothenuse. Itstan=#. of like affection.

One leg, and the adjacent angle.

Other angle. Other leg.

Angle opposite to the given leg.

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Wi. Hypothenuse. Its cos = rect. cot giv. angles. affection. The two angles. • _ cos opposite angle- If the opposite angle be g Either of the legs. Its cos = .djacent angle o e ang - In working by the logarithms, the student must observe that when the resulting logarithm is the log of a

* quotient, 10 must be added to the index; when it is the log. of a product, 10 must be subtracted from the index. Thus when the two angles are given,

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An angle oppos. (Let fall a perpen: ) Tan I seg. of div. side=cos. giv. ang. Xtan side opp. ang, sought.
III. to one of the from the third tan giv. angxsin 1 seg. - -

Two given sides. angle. Tan ang, sought = sin2 seg. of div. side sides and the included. Third side on one of the Cos side sought = conside not div. x co-org angle. giv. sides. cos 1 seg of side divided

Let fall a perpen. } Tan 1, seg. of div. side = cos giv. ang. X tan other given side.
A side opposite (Let fall a perpen. : Cot 1 seg. of div. ang. =cos giv. sidextan ang, opp. side sought.

IV. to one of the dicular on the - – tan giv. side x cos 1 seg. div. ang.
A side given angles. third side. Tan side sought = **:: seg. of divided angle ".
and the -
two ad- Third angl o fall a o: Cot 1 seg, div. ang. = cos giv. side x tan other giv. angle. * *
Jacent hird angle. rom One of the __cosang. not div. x sin 2 seg.
angles. giv. angles. Cos angle sought= sin 1 seg. div. angle -

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