Sum of the Sides of a Spherical Polygon.

Proof. From the centre O (fig. 180) of the sphere draw the radii OA, OB, OC, &c. to the vertices A, B, C, &c. of the spherical polygon ABC &c. The plane angles AOB, BOC, &c. form a solid angle at O; and the sum of these angles is, by § 339, less than four right angles. The sum of the arcs AB, BC, CD, &c. is, consequently, less than a circumference of a great circle.

440. Corollary. If then, we denote the sides of a spherical triangle by a, b, c, we have

a+b+c<360°.

441. Theorem. The angle formed by two arcs of great circles is measured by the arc described from its vertex as a pole, and included between its sides.

Proof. The arc AM (fig. 177) measures the angle ACM, which, by § 315, measures the angle of the planes DCA and DCM; and therefore, by § 430, it measures the angle ADM.

442. Corollary. The value of the arc AM expressed in degrees, minutes, &c., is the same as that of ADM.

443. Theorem. If from the vertices of a given spherical triangle as poles, arcs of great circles are described, another triangle is formed, the vertices of which are the poles of the sides of the given triangle.

Proof. Bet ABC (fig. 181) be the given triangle; let EF, DF, and DE be described, respectively, with A, B, C as poles.

Then, since E is in the arc EF, the distance from E to A is, by 425, a quadrant; and since E is in the arc DE, the distance from E to C is also a quadrant; and, therefore, by 427, E is a pole of AC.

In the same way it may be shown, that D is a pole of BC, and F a pole of AB.

Sides and Angles of polar Triangle.

444. Definition. The triangle DEF is called the polar triangle of ABC, and in the same way ABC is the polar triangle of DEF.

As several different triangles might be formed by producing the sides DE, EF, and DF, we shall limit ourselves to the one DEF, such that the pole D of BC is on the same side of BC with the vertex A; E is on the same side of AC with the vertex B ; and F is on the same side of AB with the vertex C.

445. Theorem. If the sides and angles of a spherical triangle and of its polar triangle are expressed in degrees, minutes, &c., the sides of either triangle thus expressed are respectively supplements of the angles of the other triangle.

Proof. Produce the sides AB, AC (fig. 181), if neces-. sary, to G and H.

Since is the pole of AB, and E the pole of AC, we have, by § 425,

EF the angle BAC

that is, the side EF and the angle BAC are supplements of each other.

In the same way it may be shown, that DF and the an

Sum of the Angles of a Spherical Triangle.

gle ABC, DE and the angle ACB, AB and the angle F, BC and the angle D, AC and the angle E, are respectively supplements of each other.

446. Corollary. If therefore we denote the angles of a spherical triangle by A, B, C ; and the sides respectively opposite by a, b, c ; the angles of the polar triangle must a, 180°—b, 1800 c; and the sides of the polar triangle 180°-A, 180° — B, 180°

be 1800

C.

447. Theorem. The sum of the angles of a spherical triangle is greater than two right angles.

Proof. Let A, B, C be the angles of the spherical triangle. The sides of its polar triangle are 180°-A, 180° — B, and 180°C. Now the sum of these sides,

is, by 440, less than 360°, that is,

or,

360°> (180°-A) + (180°-B) + (180°C)

360° 540°-A-B-C,

or, by transposition,

or,

A + B + C > 540° — 360°,

A+B+C > 180°;

that is, the sum of the angles A, B, C is greater than 180°.

448. Theorem. Each angle of a spherical triangle is greater than the difference between two right angles and the sum of the other two angles.

Proof. Let A, B, C be the angles of a spherical triangle; we are to prove that either of these angles, as A, is greater than the difference between 180° and B + C. a. That is, if B + C is less than 180°, we are to prove A > 180° — (B + C).

Equilateral Spherical Triangles are equiangular.

We have, from the preceding proposition,

A+B+C> 180°,

whence, by transposition,

A 180°(B + C).

b. But if BC is greater than 180°, we are to prove A> (B+C) - 180°.

Now, of the three sides 180° - A, 180° — B, 180° — C of the polar triangle, each is, by § 438, less than the sum of the other two; that is,

or

(180° B) + (180° — C) > 180° — A

449. Theorem. If two spherical triangles on the same sphere, or on equal spheres, are equilateral with respect to each other, they are also equiangular with respect to each other.

Proof. Let ABC, DEF (fig. 182) be the spherical triangles, of which the sides AB= DE, ACDF, and BC EF.

=

Draw the radii OA, OB, OC, O'D, O'E OF. The angles AOB and DO'E are equal, because they are measured by the equal arcs AB and DE; in the same way, AOC DOF, BOC=EO'F, and therefore, by § 340, the angle of the planes AOB, AOC is equal to that of the planes DOE, DO'F, that is, BAC EDF.

=

In like manner, ABCDEF, and ACB=

DFE.

Equal Spherical Triangles.

450. Definition. Two spherical triangles are symmetrical, when they are equilateral and equiangular with respect to each other, but cannot be applied to each other, as ABC, ABC (fig. 183).

451. Theorem. If two triangles on the same sphere, or on equal spheres, have a side, and the two adjacent angles of the one respectively equal to a side and the two adjacent angles of the other, they are equal, or else they are symmetrical.

Proof. If the two triangles ABC, DEF (fig. 183) have the side AB=DE, the angle BAC— EDF, and the angle ABCDEF; the side DE can be placed upon AB, and the sides DF, FE will fall upon AC, BC, or upon the sides AC', BC' of the triangle ABC', symmetrical to ᎯᏴ C.

452. Theorem. If two triangles on the same sphere, or on equal spheres, have two sides, and the included angle of the one respectively equal to the two sides and the included angle of the other, they are equal, or else they are symmetrical.

Proof. For one of the triangles may be applied to the other, or to its symmetrical triangle.

453. Theorem. In every isosceles spherical triangle the angles opposite the equal sides are equal.

Proof. Let AB (fig. 184) be equal to AC. From A draw AD to the middle of BC.

In the triangles ABD, ACD, the side AD is common, the side BD=DC, and the side AB = AC; hence, by § 449, the angle ABC- the angle ACB.

454. Corollary. Also the angle ADB ADC, and,