SPHERICAL GEOMETRY.

DEFINITIONS.

1. A sphere is a solid such that all points in its surface are equidistant from a certain point within called the center.

2. A radius of a sphere is any straight line drawn from the center to the surface.

All radii of a sphere are equal.

3. A sphere may be described by the revolution of a semicircle about its diameter, the middle of the diameter being the center, and half the diameter a radius of the sphere.

4. A diameter of a sphere is any straight line passing through the center and terminating each way in the surface.

All diameters of a sphere are equal, each of them consisting of two radii.

5. The axis of a sphere is a diameter about which the sphere is supposed to have been described by the revolution of a semicircle.

6. Every intersection of a plane with a sphere is a circle, as will be seen from the demonstration of Prop. VI.

7. The intersection of a sphere with a plane passing through the center is called a great circle, and its intersection with any other plane, a small circle.

8. The axis of a circle of a sphere, is that diameter of the sphere which is perpendicular to the circle.

The extremities of the axis are called the poles of the circle.

9. The angle made by the arcs of two great circles is called a spherical angle, and is to be regarded as the same with the angle between the planes of the circles.

Thus, BAD is a spherical angle, having for its substitute the angle between the c

planes ACB and ACD, supposing C to be the center of the sphere.

10. A spherical lune is a part of the surface of a sphere included between two great semicircles having a common diameter, as ADBE.

C

E

A spherical ungula or wedge is a part of a sphere, bounded by a lune and the two B great semicircles which include the lune, as CADBE. 11. A spherical triangle is a part of the surface of a sphere, included between the arcs of three great circles.

The arcs are called sides of the triangle.

12. Spherical triangles are distinguished as right-an gled, isosceles, equilateral, &c., in the same way as plane triangles.

A quadrantal triangle is that of which one side is a quadrant.

13. A spherical polygon is a portion of the surface of a sphere, bounded by several arcs of great circles ; which arcs are called sides of the polygon.

14. Each side of a triangle or a polygon must be understood to be less than a semicircumference of a great circle, unless the contrary is stated.

15. A spherical pyramid is a part of the sphere, contained by the planes of a solid angle whose vertex is the center, and the spherical polygon included by these planes; as ABCDE. The polygon is called the base of

the pyramid.

When the base is a spherical triangle, the pyramid is called triangular.

B

C

16. A line or plane is said to touch or be tangent to a sphere, when it meets the surface of the sphere in one point only.

And two spheres are said to touch each other, when they meet and do not intersect.

STRAIGHT LINE AND SPHERE.

PROP. I.

If a perpendicular drawn from the center of a sphere to any straight line be equal to the radius of the sphere, this line touches the sphere at the foot of the perpendicular.

For since the perpendicular is equal to the radius, the foot of the perpendicular is in the surface of the sphere; the line therefore meets the surface at the foot of the perpendicular: and every other point in the line is without the surface, being further from the center of

Σ

the sphere, because the perpendicular is the shortest distance from a point to a straight line.

PROP. II.

If a perpendicular from the center of a sphere to 'any straight line be longer than the radius, the line is wholly without the sphere.

For no point of the line can be at less than the perpendicular distance from the center: but this is greater than the radius; wherefore the line must at every point be without the surface of the sphere.

PROP. III.

If a perpendicular from the center of a sphere to any straight line be less than the radius, the line will meet the surface in two points on opposite sides of the perpendicular, and at equal distances from it; the part of the line between these points being within the surface of the sphere, and the rest of the line without it.

Let AB be the straight line, C the center of the sphere, and CD the perpendicular, which is less than the radius. In the plane

passing through AB and C, describe an arc of a circle from the center C, with a radius equal to that of the sphere, and let it meet AB in E and F.

Then as the distances CE and CF are each equal to the radius of the sphere, the points E and F are in the surface; and (Euc. 3, 3) ED is equal to DF.

Also any point in AB between E and F is within the

surface of the sphere, since its distance from C is less than CE or CF; and any point in EF produced either way, is further from C than F or E, and therefore is without the surface of the sphere.*

PLANE AND SPHERE.

PROP. IV.

If the perpendicular from the center of a sphere to any plane be equal to the radius of the sphere, the plane is a tangent to the sphere at the foot of the perpendicular.

For as the perpendicular is equal to the radius, the foot of it is a point in the surface of the sphere, and the plane meets the surface in this point. Moreover it can meet it in no other point: for the distance from any other point in the plane to the center of the sphere is greater than the perpendicular, and therefore greater than the radius of the sphere; so that every such point is without the surface.

PROP. V.

If a perpendicular from the center of a sphere to a plane be longer than the radius of the sphere, the plane is wholly without the sphere.

For this perpendicular is the shortest distance from the center to the plane; and as the distance from the

* The limits of this treatise are such as to require the omission here of a number of obvious propositions concerning the straight line and sphere, similar to several that are given for the circle in the third book of Euclid; as for example, the propositions 7, 8, 14, 15, 35, 36, and 37 · of that book.