653. SCH. I. Let it be required to find the area of a spherical triangle whose angles are 105°, 80°, and 95°, on a sphere the area of whose surface is 144 sq. in.

The spherical excess of the spherical triangle is 100°, or 10 referred to a right angle as the unit.

And the area of a tri-rectangular triangle is of 144, or 18 sq. in.

Hence, the required area is 2 of 18, or 20 sq. in.

654. SCH. II. It may be proved, as in § 652, that

If the unit of measure for angles is the right angle, the volume of a triangular spherical pyramid is equal to the spherical excess of its base, multiplied by the volume of a tri-rectangular pyramid.

EXERCISES.

13. Find the area of a spherical triangle whose angles are 103°, 1120, and 127°, on a sphere the area of whose surface is 160.

14. Find the volume of a triangular spherical pyramid the angles of whose base are 92°, 119°, and 134°; the volume of the sphere being 192.

15. What is the volume of a spherical wedge the angle of whose base is 127° 30', if the volume of the sphere is 112 ?

16. The area of a lune is 28 sq. in. If the area of the surface of the sphere is 120 sq. in., what is the angle of the lune?

17. What is the ratio of the areas of two spherical triangles on the same sphere, whose angles are 94°, 135°, and 146°, and 87°, 105°, and 118°, respectively?

18. The area of a spherical triangle two of whose angles are 78° and 99°, is 34. If the area of the surface of the sphere is 234, what is the other angle?

19. The volume of a triangular spherical pyramid the angles of whose base are 105°, 126°, and 147°, is 601; what is the volume of the sphere ?

20. If two straight lines are tangent to a sphere at the same point, their plane is tangent to the sphere.

21. The sum of the arcs of great circles drawn from any point within a spherical triangle to the extremities of any side, is less than the sum of the other two sides of the triangle.

655. If the unit of measure for angles is the right angle, the area of any spherical polygon is equal to the sum of its angles, diminished by as many times two right angles as the figure has sides less two, multiplied by the area of a tri-rectangular triangle.

Let K denote the area of any spherical polygon; n the number of its sides; s the sum of its angles referred to a right angle as the unit; and T the area of a tri-rectangular triangle.

To prove

K = [s-2 (n − 2)] × T.

The spherical polygon may be divided into spherical triangles by drawing diagonals from any vertex; the number of such spherical triangles being equal to the number of sides of the spherical polygon, less two.

Now the area of each spherical triangle is equal to the sum of its angles, less two right angles, multiplied by T. (§ 652.)

Hence, the sum of the areas of the spherical triangles is equal to the sum of their angles, diminished by as many times two right angles as there are triangles, multiplied by T.

But the number of triangles is n 2.

Therefore, the area of the spherical polygon is equal to the sum of its angles, diminished by n 2 times two right angles, multiplied by T.

That is,

K = [s — 2 (n − 2)] × T.

656. SCH. It may be proved, as in § 655, that

If the unit of measure for angles is the right angle, the volume of any spherical pyramid is equal to the sum of the angles of its base, diminished by as many times two right angles as the base has sides less two, multiplied by the volume of a tri-rectangular pyramid.

657. COR. Let P denote the volume of a spherical pyramid; and let K denote the area of the base, n the number of its sides, and s the sum of its angles referred to a right angle as the unit.

Let V represent the volume of the sphere; S the area of its surface; T the area of a tri-rectangular triangle; and T the volume of a tri-rectangular pyramid.

That is, the volume of a spherical pyramid is to its base as the volume of the sphere is to its surface.

EXERCISES.

22. Find the area of a spherical hexagon whose angles are 120°, 139°, 148°, 155°, 162°, and 167°, on a sphere the area of whose surface is 280.

23. Find the volume of a pentagonal spherical pyramid the angles of whose base are 109°, 128°, 137°, 153°, and 158°; the volume of the sphere being 180.

24. The arcs of great circles bisecting the angles of a spherical triangle meet in a point equally distant from the sides of the triangle. (Exs. 3, 4, p. 337.)

25. A circle may be inscribed in any spherical triangle.

26. State and prove the theorem for spherical triangles analogous to Prop. IX., I., Book I.

27. State and prove the theorem for spherical triangles analogous to Prop. V., Book I.

28. State and prove the theorem for spherical triangles analogous to Prop. LI., Book I.

29. The volume of a quadrangular spherical pyramid, the angles of whose base are 110°, 122°, 135°, and 146°, is 12 cu. ft. What is the volume of the sphere?

30. The area of a spherical pentagon, four of whose angles are 112°, 131°, 138°, and 168°, is 27. If the area of the surface of the sphere is 120, what is the other angle?

31. If the side AB of a spherical triangle ABC is equal to a quadrant, and the side BC is less than a quadrant, prove that ▲ A is less than 90°.

32. If PA, PB, and PC are three equal arcs of great circles drawn from a point P to the circumference of a great circle ABC, prove that P is the pole of ABC.

33. The spherical polygons corresponding to a pair of vertical polyedrals are symmetrical.

34. Either angle of a spherical triangle is greater than the difference between 180° and the sum of the other two angles.

35. If a polyedron be circumscribed about each of two equal spheres, the volumes of the polyedrons are to each other as the areas of their surfaces.

36. If ABC and A'B'C' are a pair of polar triangles on a sphere whose centre is O, prove that the radius OA' is perpendicular to the plane OBC.

37. The intersection of two spheres is a circle, whose centre lies in the line joining the centres of the spheres, and whose plane is perpendicular to this line.

38. The distance between the centres of two spheres, whose radii are 25 in. and 17 in., respectively, is 28 in. Find the diameter of their circle of intersection, and the distance of its plane from the centre of each sphere.

BOOK IX.

MEASUREMENT OF THE CYLINDER, CONE,

AND SPHERE.

THE CYLINDER.

DEFINITIONS.

658. A prism is said to be inscribed in a cylinder when its bases are inscribed in the bases of the cylinder.

A prism is said to be circumscribed about a cylinder when its bases are circumscribed about the bases of the cylinder. The lateral area of a cylinder is the area of its lateral surface.

A right section of a cylinder is a section perpendicular to its elements.

659. If a regular polygon be inscribed in, or circumscribed about, a circle, and the number of its sides be indefinitely increased, its perimeter and area approach the circumference and area of the circle respectively as limits (§ 363).

Hence, if a prism whose base is a regular polygon be inscribed in, or circumscribed about a circular cylinder (§ 553), and the number of its faces be indefinitely increased,

1. The lateral area of the prism

approaches the lateral area of the cylinder as a limit.

2. The volume of the prism approaches the volume of the cylinder as a limit.