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Note. Every great circle of the sphere is the horizon of a certain place. And when the sphere is projected on an oblique circle, it is easy to determine the latitude of the place, where the primitive would be horizontal, and also its difference in longitude from the place, for which the projection is made.

Thus, in this projection, VP=le= the latitude ; and Akh the difference, in longitude from Cambridge, of the place, where this oblique circle would be the horizon.

END OF SPHERICAL GEOMETRY.

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1. SPHERICAL TRIGONOMETRY teaches the relations and calculation of the sides and angles of spherical triangles.

2. A spherical triangle is a figure on the surface of a sphere, bounded by three arcs of great circles.

3. A spherical triangle, like a plane one, is equilateral, isoscelar, or scalenous, according as the three sides, two of them only, or no two of them, are equal.

4. A spherical triangle is right-angled, or rectangular, if it have one right angle, or more ; quadrantal, or rectilateral, if one of the sides be a quadrant ; and oblique, if it have neither a right angle, nor quadrantal side.

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5. In a right-angled spherical triangle, as in a plane one, the side, opposite to the right angle, is called the bypotenuse ; and the other two the legs, or sides.

6. Two sides of a spherical triangle are said to be alike, or of the same affection or kind, when they are each greater or less than a quadrant, and unlike, or of different affec

; tion or kind, when one is greater and the other less than a quadrant. Also two angles are alike, or of the same affection or kind, when they are both acute or obtuse ; and unlike, or of different affection or kind, when one is acụte and the other abtuse.

7. The circular parts of a right-angled spherical triangle are five, namely, the two legs, and the complements of the hypotenuse and the two oblique angles,

8. The right angle is not considered as separating the legs ; and if no other part be. situated between two of three circular parts, and thus separate them, that, which is in the middle, is called the middle part; and the other two conjunct extremes. But if one of three circular parts be

. separated from the other two, it is called the middle part, and the other two disjunct extremes,

Nore. To illustrate the meaning and application of these trigonometrical terms, first introduced into this branch of science by Lord NAPIER, a spherical triangle is annexed, together with the conjunct and disjunct extremes, corresponding to any given mean or middle part,

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The following are among the properties, common to plane and spherical triangles, and the demonstrations in both cases are similar.

1. The greater, equal, or less side subtends the greater, equal, or less angle ; and the contrary.

2. Two triangles are equal, 1. When the three sides of one are respectively equal to the three sides of the other. 2. When two sides and the included angle of one are respectively equal to two sides and the included angle of the other. 3. When two angles and the included side of one are respectively equal to two angles and the included side of the other. 4. When they have equal angles above equal bases.

Among the differences of plane and spherical triangles the following are very remarkable.

7. In a plane triangle, if only the three angles be given, only the relative lengths of the sides can be found. But in a spherical triangle, the sides being circular arcs, whose

values are expressed in degrečs as well as those of the angles, if the three angles be given, the sides may be thence determined.

2. Two angles always determine the third in a plane triangle ; but never in a spherical triangle.

THEOREM I.

The sum of any two sides of a 'spherical triangle is greater than the third. *

THEOREM II.

The sum of the three sides of any spherical triangle is less than 360°.t

* DEMONSTRATION. The shortest distance between any two points on the surface of a sphere is the arc of a great

circle

passing through them. The sides of a spherical triangle then are the shortest distances betwen the angular points on the sphere, as the sides of a plane triangle are the shortest distances between the angular points on a plane. Therefore the sum of any two sides of a spherical triangle is greater than the third. Q. E. D.

+ DEMONSTRATION.

Let the sides AB, AC, containing any angle A, be produced till they meet again in D ; and the arcs ABD, ACD, will be each 180°, since all great circles bisect each other ; therefore ABD+ACD=360°. But by last Theo.

. DB+DCBC ; consequently, the sum of the three sides is less than 360°. Q. E. D.

[See the figure on next page.]

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