PROPOSITION XVII. THEOREM.

730. In two polar triangles each angle of either is the supplement of the side lying opposite to it in the other.

B

Let A BC and A'B'C' be two polar triangles.

We are to prove

of the sides B'C', A' C'

A, B and C respectively the supplements and A' B'.

Let the sides A B and A C, produced if necessary, meet the side B'C' in the points b and c.

Since the vertex A is a pole of the arc B'C',

A is measured by bc,

§ 721

8714

(a spherical is measured by the arc of a great circle described about its vertex as a pole and intercepted by its sides).

Now, since B' is the pole of the arc A c, B' c =

Since C is the pole of the arc A b, C' b

=

90°.

.. B'c+C'b⇒ B'C' + b c = 180°.

90°.

..A (bc) is the supplement of the side B'C'.

In like manner it may be shown that each of either ▲ is the supplement of the side lying opposite to it in the other.

Q. E. D.

731. SCHOLIUM. In two polar triangles each side of either is the supplement of the angle lying opposite to it in the other. If A, B and C denote the angles, and a, b and c the sides of a triangle, the angles of the polar triangle will be 180° — a, 180° b and 180° c; and the sides of the polar triangle will be 180° A, 180°. B and 180° – C.

By reason of these relations polar triangles are often called supplemental triangles.

PROPOSITION XVIII. THEOREM.

732. The sum of the angles of a spherical triangle is greater than two, and less than six, right angles.

B

a'

Let ABC be a spherical triangle.

We are to prove Z A + Z B + C greater than 2, and less than 6, right angles.

Denote the sides of the polar A opposite the A, B, C respectively, by a', b', c.

- b' and C = § 730

=

(in two polar ▲ each of either is the supplement of the side lying opposite to it in the other.)

--

By adding, A+ZB+ZC= 540° — (a' + b' + c').

But a' + b'+d is less than 360°,

$724

(the sum of the sides of a spherical polygon is less than the circumference of a great circle).

Also,

:. ZA+ZB+ZC> 180°.

since each is less than 2 rt. 4,

their sum is less than 6 rt. .

Q. E. D.

733. COROLLARY. A spherical triangle may have two, or even three right angles; or two, or even three obtuse angles.

734. DEF. A spherical triangle having one right angle is called rectangular; having two right angles, bi-rectangular; having three right angles, tri-rectangular.

Each of the sides of a tri-rectangular triangle is a quadrant, and the triangle is called, when reference is had to its sides, triquadrantal.

PROPOSITION XIX. THEOREM.

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

B

Let A, B and C be the angles of the spherical tri

angle ABC.

We are to prove A greater than the difference between 180° and (B+

I.

II.

C).

Suppose (B+ C) < 180°.

Now ZA+ZB+ZC> 180°.

By transposing, <A> 180° (LB+ZC).

Suppose (B+ <C) > 180°.

§ 732

Now of the three sides (180° - A), (180° — ≤ B), (180° - C), of the polar A, each is less than the sum of the other two,

§ 722

(either side of a spherical ▲ is less than the sum of the other two sides). ... (180°

or,

B) + (180°C) > 180° - ZA;

360° (B+ C) > 180° - A.

By transposing, Z A > (Z B + Z C) — 180°.

Q. E. D.

Ex. 1. The volume of a cone is 1728 cubic inches; what is the volume of a similar cone whose surface is 4 times as great? 2. The volume of a cone is V; what is the volume of a similar cone whose surface is n times as great?

736. DEF. Equal spherical triangles are triangles which have their corresponding sides and angles equal each to each and arranged in the same order, so that when applied to each other they will coincide. Thus in Fig. 1, A B C and A' B'C' are equal

737. DEF. Symmetrical spherical triangles are triangles which have their corresponding sides and angles equal each to each, but arranged in reverse order.

Thus, in Fig. 2, ABC and A'B'C' are symmetrical spherical triangles. For, since the face angles of the two trihedrals are equal respectively, but are arranged in reverse order, the sides of the spherical triangles, which measure these face angles, are equal, each to each, and are arranged in reverse order; and since the dihedral angles of the two trihedrals are equal respectively, but are arranged in reverse order, the angles of the spherical triangles, which are equal to these trihedrals, are equal, each to each, and are arranged in reverse order.

In like manner we may have symmetrical spherical polygons of any number of sides, and corresponding symmetrical spherical pyramids.

Two symmetrical spherical triangles cannot be made to coincide. For, if their convexities lie in opposite directions, they evidently will not coincide; and if their convexities lie in the same direction, and we apply A B to A'B', the vertices C and C will lie on opposite sides of A' B'.

738. There is, however, one exception. Two symmetrical isosceles spherical triangles can be made to coincide.

B

Thus, if ABC be an isosceles spherical triangle, A B = A C and in its symmetrical triangle A' B'A' C'. Hence A B = A'C' and A CA' B'. And, since A and A' are equal, if A B be placed on A' C', A C will fall on its equal A' B'.

In consequence of the relations established between polyhedral angles and spherical polygons, from any property of polyhedral angles, we may infer a corresponding property of spherical polygons. Reciprocally, from any property of spherical polygons, we may infer a corresponding property of polyhedral angles.

Ex. 1. The altitude of a cone of revolution is 12 inches; at what distances from the vertex must three planes be passed parallel to the base of the cone, in order to divide the lateral surface into four equal parts?

2. The altitude of a given solid is 2 inches, its surface 24 square inches, and its volume 8 cubic inches; find the altitude and surface of a similar solid whose volume is 512 cubic inches.

3. The volumes of two similar cones of revolution are 6 cubic inches and 48 cubic inches respectively, and the slant height of the first is 5 inches; find the slant height of the second.