(73.) COR. The three sides, therefore, of a spherical triangle are together less than six quadrants: and, (Art. 42), each of its angles being less than two right angles, its three angles are together less than six right angles.

PROP. I.

(74.) Theorem. The three sides of a spherical triangle are together less than the circumference of a great circle of the sphere: but any two of them are greater than the third.

Let PBD be a spherical triangle: the three sides

P

B

PD, BD, and DP are, together, less than the circumference of a great circle, but any two of them are greater than the third.

For let C be the sphere's center; and let C, B, C, D and C, P be supposed to be joined: Then, it is evident (Art. 7.) that CB, CD and CP will be the intersections of the planes of the three great circles, the arches of which bound the spherical triangle PBD: it is evident, also,

D

Find (Art. 63.), the direct distance between a great circle of the sphere and either of its poles; which is done,

independently of any great circle having been actually described: then, from P as a pole, at the distance so found, describe (Art. 59.) the great circle ABD; and (Art. 65.) produce the two arches PE and PF, until they meet the circumference ABD, in the points A and B.

The arch AB (Art. 54.) measures the spherical angle EPF.

PROP. XIII.

(70.) Problem. In the surface of a given sphere, to draw an arch of a great circle, which shall pass through a given point, in that surface, and be at right angles to the circumference of a given circle, in the sphere.

Find (Art. 64.) either pole of the given circle

describe (Art. 66.) a great circle, of the sphere, passing through the pole thus found, and through the given point: and (Art. 50.) its circumference shall be at right angles to the circumference of the given circle.

PROP. XIV.

(71.) Problem. A spherical angle, in the surface of a given sphere, being given, to make a plane rectilineal angle, which shall be equal to it.

From the angular point of the given spherical angle, as a pole, describe (Art. 59.) any circle, in the sphere; describe, also, (Art. 61.) a circle, in a plane, that shall be equal to the circle first described, in the sphere: and in this latter circle, place (E. 1. 4.) a straight line equal to the direct distance between the two points, in which the circle in the sphere, first described, cuts the arches containing the given angle; which distance (Art. 59.) may be considered as given: then shall the plane rectilineal angle subtended by this straight line, at the center of the circle in which it is placed, be equal (E. 28. and 27. 3. and Art. 54.) to the given spherical angle.

PART I.

THE ELEMENTS OF

Spherical Geometry.

SECTION III.

ON THE GENERAL RELATIONS OF THE SIDES AND ANGLES

OF SPHERICAL FIGURES.

DEFINITION.

(72.) A Spherical Triangle is a figure, on the surface of a sphere, contained by three arches of great circles, in the sphere, each of which arches is less than the semi-circumference of a great circle *.

* A portion of the sphere's surface may be bounded by three arches of great circles, of which arches, one may be greater than the half of the circumference; and the angle opposite to it may be greater than two right angles. There might, indeed, if the restriction laid down in the above definition were removed, be no fewer than eight spherical triangles formed, by joining three given points on a sphere's surface. But, as the main object of Spherical Geometry is to elucidate Spherical Trigonometry, and as the determination of the unknown parts, of such a trilateral figure, is always reducible to the solution of a spherical triangle, such as we have defined it to be, the properties of the former kind of figure are not investigated in this Treatise.

-

(73.) COR. The three sides, therefore, of a spherical triangle are together less than six quadrants: and, (Art. 42), each of its angles being less than two right angles,

its three angles are together less than six right angles.

PROP. I.

(74.) Theorem. The three sides of a spherical triangle are together less than the circumference of a great circle of the sphere: but any two of them are greater than the third.

Let PBD be a spherical triangle: the three sides

P

B

PD, BD, and DP are, together, less than the circumference of a great circle, but any two of them are greater than the third.

For let C be the sphere's center; and let C, B, C, D and C, P be supposed to be joined: Then, it is evident (Art. 7.) that CB, CD and CP will be the intersections of the planes of the three great circles, the arches of which bound the spherical triangle PBD: it is evident, also,

D