each : wherefore (Art. 99.) GH is equal to BA, HL to AC, and LG to CB.

Again, it follows from E. 33. 6. that GH has to the whole circumference, of which it is a part, the same ratio, as DE has to the whole circumference, of which DE is a part : therefore, (E. 16. 5.) GH and DE are to each other, as the circumferences of great circles, in their respective spheres : and the same ratio, it may be, likewise, shewn, has HL to EF, and LG to FD: therefore, (E. 11. 8.) the sides of the two triangles GHL, DEF, about equal angles, are proportionals, and those are homologous sides, which are opposite to the equal angles. But the sides and the angles of the triangle ABC have been proved to be equal to the sides and the angles of GHL: wherefore, the sides of the triangles ABC, DEF, about equal angles are, likewise, proportionals.

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.

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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.

For, (Art. 51.) Fis 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 triangle 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 A, be a quadrant. Then, either AB, or AC, is a quadrant; and either C or B a right angle.

A

D

For, from C as 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,

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