PART I.

THE ELEMENTS OF

Spherical Geometry.

SECTION IV.

ON THE RELATIVE SPECIES OF THE SIDES AND ANGLES OF A SPHERICAL TRIANGLE.

DEFINITIONS.

(120.) 1. Ir a spherical triangle have one, at least, of its sides a quadrant, it is called a Quadrantal Triangle.

2. If a spherical triangle have one, at least, of its angles a right angle, it is called a Right-angled Spherical Triangle.

3. If a spherical triangle have none of its sides a quadrant, nor any of its angles a right angle, it is called an Oblique-angled Spherical Triangle:

4. And, if each of its angles be less than a right angle, it is called an Acute-angled Spherical Triangle.

PROP. I.

(121.) Theorem. If two angles of a spherical triangle be right angles, the sides opposite to them shall be quadrants: and, conversely, if two sides of a spherical triangle be quadrants, the angles opposite to them shall be right angles.

Let FEG be a spherical triangle; and first, let the angles E and G be right angles: then are FE and FG quadrants.

E

f

For, (Art. 51.) F is the pole of EG, and consequently, (Art. 72. and 36.) FE and FG are quadrants.

Secondly, let FE and FG be quadrants: then, the angles E and G (Art. 37. 50.) are right angles.

(122.) COR. 1. If all the angles of a spherical tri

angle be right angles, all the sides are quadrants: and, if all the sides be quadrants, all the angles are right angles.

(123.) COR. 2. Hence, it is manifest, that, on the semi-surface of a sphere, there may be as many such quadrantal, and right-angled, triangles, as there are quadrants in the great circle, which bounds that surface, and no more: wherefore, four such triangles are exactly equal to half of the surface, and eight to the whole surface, of the sphere.

PROP. II.

(124.) Theorem. In a right-angled spherical triangle, if either of the sides containing a right angle be a quadrant, the hypotenuse of that right angle shall also be a quadrant.

For, (Art. 50. and 36.) the extremity of the side which is a quadrant is a pole of the great circle, the arch of which constitutes the other side of the triangle: wherefore, (Art. 36.) the third side, namely, the hypotenuse, is a quadrant.

PROP. III.

(125.) Theorem. In a right-angled spherical triangle, if the hypotenuse of a right angle be a quadrant, one of the two sides, containing that right angle, shall also be a quadrant; and one other angle a right angle.

Let BAC be a right-angled spherical triangle, and

let the side BC, opposite to the right angle 4, be a quadrant. Then, either AB, or AC, is a quadrant; and either C or B a right angle.

B

D

For, from Cas a pole, at the distance CB, describe the circle BD, which (Art. 36.) is a great circle; and let it cut CA in D: then, if BD pass through A, it is evident, that CA (Art. 36.) is a quadrant; but if not, the angle ADB (Art. 50.) is a right angle: and the angle BAD is, by the hypothesis, a right angle; wherefore, (Art. 51.) B is the pole of AC: BA is, therefore, (Art. 36.) a quadrant, and (Art. 121.) the angle BCA is a right angle.

(126.) DEF. If two sides of a spherical triangle be each of them quadrants; or if each of them be greater, or each less than a quadrant; or, if two angles of a spherical triangle be each of them a right angle; or each greater, or each less than a right angle; they are said to be of the same species in all other cases they are said to be of different species.

A side, also, which is a quadrant, or greater, or less,

than a quadrant, is said to be of the same species as an angle, which is a right angle, or greater, or less than a right angle, respectively: and in all other cases, a side and an angle are said to be of different species.

PROP. IV.

(127.) Theorem. In a right-angled spherical triangle, which has only one right angle, the two sides containing that angle are, each, of the same species as the angle opposite to it.

Let ACB be a right-angled spherical triangle, having the angle C, and no other angle, a right angle: then is the side AC of the same species as the angle B, and CB is of the same species as the angle A.

?

For, produce (Art. 65.) CA and CB, until they meet D, having first found (Art. 64.) the pole P of CB; which point, because the angle C is a right angle, will (Art. 50.) be in CAD; and join (Art. 66.) P, B: where