Εικόνες σελίδας
PDF
Ηλεκτρ. έκδοση

will be known, when those of all arches from 0° to 45°

have been found.

But it is manifest, that a and b in the two forms,

[blocks in formation]

denote whole numbers of seconds; and therefore, the arch (30° + 2b) of which the tangent is thus found, must always consist of an even number of seconds.

The considerations, however, which have been suggested, in this last article, indicate a very great abridgement of labour in the actual computation of the trigonometrical functions of circular arches.

Α

TREATISE

ON

Spherics.

PART II.

SPHERICAL TRIGONOMETRY.

:

PART II.

THE ELEMENTS OF

Spherical Trigonometry.

SECTION I.

ON THE INVESTIGATION OF SUCH GENERAL PROPOSITIONS AS ARE APPLICABLE TO THE PURPOSES OF SPHERICAL

TRIGONOMETRY.

DEFINITION.

(227.) SPHERICAL Trigonometry is that part of Spherics, in which are investigated the relations existing between the sides, the angles, and the surface of a spherical triangle, with a view to the resolution of the following general problem: "Of the six first quantities, namely, those made up of the three sides, and the three angles, of a proposed spherical triangle, any three being given, to determine the rest *."

* That this problem involves no impossibility, is very evident, from what has been already proved, in treating of Spherical Geometry.

N

(228.) DEF. If a given right-angled spherical triangle have two angles, that are not right angles, and from the summit of either of them, as a pole, a great circle be described, cutting the opposite side and the hypotenuse produced, if necessary, the triangle contained by the segments of the circumference so described, and of the two sides which it cuts, is called the Complemental Triangle of the given triangle.

Thus, let the angle C, and no other angle, of the spherical triangle ABC, be a right angle; and from either

[ocr errors][merged small][merged small]

of the oblique angles, B, as a pole, let there be described the great circle FG, cutting the opposite side AC, and the hypotenuse BA, produced, if necessary *, in Fand G: the triangle FAG is one of the complemental triangles of

*In the figure, the two sides of the given spherical triangle are supposed to be of the same species, and to be, each of them, less than a quadrant: there are, therefore, two other cases, which might be illustrated by separate figures: but, as the reasoning is general and very easy to be understood, it seems unnecessary to exhibit more than one of the cases.

« ΠροηγούμενηΣυνέχεια »