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where p is put for the semi-sum of the angles of the triangle.

But, if one of the angles, as A, be a right angle, then,

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and, by applying these two spherical forms to the parts of the complemental triangle, there result the remaining four theorems, which are used in solving right-angled spherical triangles.

Again, the solution of oblique-angled triangles, it is well known, may be reduced to that of right-angled triangles.

Let D and D' denote the segments of the base, V and 'the segments of the vertical angle A", of a plane or spherical triangle, made by a perpendicular let fall from A", on S"; so that D is adjacent to the angle A, and opposite to the segment ; then,

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Whence, by the properties of the polar triangle,

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and these two spherical forms, together with those which solve right-angled spherical triangles, are sufficient for the solution of all the cases of oblique-angled spherical triangles.

A recollection, therefore, of the fundamental principles of Plane Trigonometry, and of the very striking analogy which has been pointed out, together with the well-known properties of the polar, and the complemental triangles, may serve to re-produce all that is wanted, for the solution of the most important problems of Spherical Trigonometry.

(279.) The expression for the surface of a spherical triangle in terms of its three angles and the radius of the sphere, is too simple and too remarkable, to need, at all, the aid of an artificial memory and, when the three angles are not given, they may be found, if the data be sufficient, by a solution of the triangle.

(280.) Upon the whole, the last of the three methods of relieving the memory, ought, perhaps, chiefly to be recommended to those who are new to this subject, and who are not likely to have very frequent need to employ the principal theorems of Spherical Trigonometry: they will thus easily recover the requisite forms, whenever an occasion calls for them. And, if these forms be often wanted, they will, undoubtedly, be fixed in the memory, by repeated use; so that the aid of any indirect method of arriving at them, will become altogether unnecessary.



Spherical Trigonometry.




(281.) Problem. THE numerical values of any three parts of a proposed spherical triangle being given, to find the remaining three parts, by geometrical constructions, made on the surface of a given sphere.


Let the values of the three sides of the triangle be given, to find the three angles.

On the given sphere, describe (Art. 68.) three arches of great circles; and, by means of a compass and a scale


of chords, cut off from them parts (Introd. 3.) equal in value to the three given sides of the triangle, each to each then, (Art. 94.) make on the sphere's surface, a triangle, the sides of which shall be equal to the three arches so described, each to each: describe also (Art. 76.) the polar triangle; and the measures of its sides, taken by the compass and the scale of chords, will be (Art. 78.) the supplements of the required angles: whence the angles themselves will become known.


Let the values of the three angles be given, to find the three sides of the triangle.

Describe, as in the first case, a triangle on the sphere's surface, having its three sides, each the measure of the supplement of one of the given angles; then the measures of the sides of its polar triangle, taken by a compass and a scale of chords, will, manifestly, give the values of the three sides, which are sought.


Let the values of two sides, and of the included angle be given, to find the other parts of the triangle.

On the given sphere, describe, as in the first case, an arch, AD of a great circle; from either of its extremities, A, as a pole, at the distance of a quadrant, describe the great circle DE; and make, by means of

the compass and the scale of chords, the arch DE equal to the measure of the given angle; join A, E; and from





AD and AE, produced if necessary, cut off, (Introd. 3.) AB and AC equal in value to the two given sides of the triangle, each to each; join (Art. 66,) B, C: the sides BC may then be measured, by means of the same instruments; as may, likewise, the angles B and C, if from each of those angular points, as a pole, a great circle be described, cutting the two sides, which contain the angle to be measured.


Let there be given the values of any one of the sides, and of the two angles adjacent to it, to find the other parts of the triangle.

Describe an arch of a great circle of the sphere, and make it equal, in value, to the given side, as in the former cases: make also, as in the third case, at each extremity

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