ON SPHERICAL POLYGONS AND PYRAMIDS.

715. DEF. A spherical Polygon is a portion of a surface of a sphere bounded by three or more arcs of great circles.

The sides of a spherical polygon are the bounding arcs; the angles are the angles included by consecutive sides; the vertices are the intersections of the sides.

716. DEF. The Diagonal of a spherical polygon is an arc of a great circle dividing the polygon, and terminating in two vertices not adjacent.

The planes of the sides of a spherical polygon form by their intersections a polyhedral angle whose vertex is the centre of the sphere, and whose face angles are measured by the sides of the polygon.

717. DEF. A spherical Pyramid is a portion of a sphere bounded by a spherical polygon and the planes of the sides of the polygon.

The spherical polygon is the base of the pyramid, and the centre of the sphere is its vertex.

718. DEF. A spherical Triangle is a spherical polygon of three sides.

A spherical triangle, like a plane triangle, is right, or oblique ; scalene, isosceles or equilateral.

719. DEF. Two spherical triangles are equal if their successive sides and angles, taken in the same order, be equal each to each.

720. DEF. Two spherical triangles are symmetrical if their successive sides and angles, taken in reverse order, be equal each to each.

721. DEF. The Polar of a spherical triangle is a spherical triangle, the poles of whose sides are respectively the vertices of the given triangle.

Since the sides of a spherical triangle are arcs, they may be expressed in degrees and minutes.

PROPOSITION XIII. THEOREM.

722. Any side of a spherical triangle is less than the sum of the other two sides.

Let ABC be any spherical triangle.

We are to prove BC <BA + A C. Join the vertices A, B and C with the centre of the sphere.

Then, in the trihedral ▲ O-A BC thus formed, the face AOC, AOB and BOC are measured, respectively, by the

sides A C, A B and B C.

Now, BOC<BOA+AOC,

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(the sum of any two of a trihedral is greater than the third 4).

.. BC <BA+A C.

Q. E. D.

723. COROLLARY. Any side of a spherical polygon is less than the sum of the other sides.

Ex. 1. Given a cone of revolution whose side is 24 feet, and the diameter of its base 6 feet; find its entire surface, and its volume.

2. Given the frustum of a cone whose altitude is 24 feet, the circumference of its lower base 20 feet, and that of its upper base 16 feet; find its volume.

3. The volume of the frustum of a cone of revolution is 8025 cubic inches; its altitude 14 inches; the circumference of the lower base twice that of the upper base. What are the circumferences of the bases?

4. The frustum of a cone of revolution whose altitude is 20 feet, and the diameters of its bases 12 feet and 8 feet respectively, is divided into two equal parts by a plane parallel to its bases. What is the altitude of each part?

PROPOSITION XIV. THEOREM.

724. The sum of the sides of a spherical polygon is less than the circumference of a great circle.

D

Let ABCDE be a spherical polygon.

We are to prove A B+ BC etc. less than the circumference of a great circle.

Join the vertices A, B, C etc., with O the centre of the sphere.

The sum of the face AO B, BOC etc., which form a polyhedral at O, is less than four rt. .

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.. the sum of the arcs A B, BC etc., which measure these face, is less than the circumference of a great circle.

Q. E. D.

725. COROLLARY. If we denote the sides of a spherical triangle by a, b and c, then a + b + c < 360°.

Ex. 1. The surface of a cone is 540 square inches; what is the surface of a similar cone whose volume is 8 times as great?

2. The lateral surface of a cone is S; what is the lateral surface of a similar cone whose volume is n times as great?

PROPOSITION XV. THEOREM.

726. A point upon the surface of a sphere, which is at the distance of a quadrant from each of two other points, is one of the poles of the great circle which passes through these points.

Let P be a point at the distance of a quadrant from each of the two points A and B.

We are to prove P a pole of the great circle which passes through A and B.

Since PA and PB are quadrants,

POA and PO B are rt. s.

.. PO is to the plane of the

ABC,

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(a straight line to two straight lines drawn through its foot in a plane is I to the plane).

.. P is a pole of the O ABC.

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Q. E. D.

Ex. 1. Show that two symmetrical polyhedrons may be decomposed into the same number of tetrahedrons symmetrical eacli to each.

2. Show that two symmetrical polyhedrons are equivalent. 3. Show that the intersection of two planes of symmetry of a solid is an axis of symmetry.

4. Show that the intersections of three planes of symmetry of a solid are three axes of symmetry; and that the common intersection of these axes is the centre of symmetry.

PROPOSITION XVI. THEOREM.

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

Let A B C be the given triangle; and, from its vertices A, B and C as poles, let the arcs B'C', A'C' and A'B' respectively be described.

We are to prove vertices A', B' and C' poles respectively of arcs BC, A C and A B.

Since B is the pole of the arc A'C', and C the pole of the arc A' B',

A' is at a quadrant's distance from each of the points B and C. .. A' is a pole of the arc BC,

$726

(a point upon the surface of a sphere which is at a quadrant's distance from each of two other points is one of the poles of the great circle which passes through those points).

In like manner, it may be shown that B' is a pole of the arc A C, and C a pole of the arc A B.

Q. E. D.

728. SCHOLIUM 1. A A'B'C' is the polar of ▲ A B C, and, reciprocally, ▲ A B C is the polar of ▲ A'B' C'.

729. SCH. 2. The arcs of great circles described about A, B and C as poles will, if produced, form three triangles exterior to the polar. The polar triangles are distinguished by having their homologous vertices A and A' on the same side of BC and B'C', B and B' on the same side of AC and A'C', and C and C' on the same side of A B and A' B'.