6. A SPHERICAL PYRAMID is a portion of a sphere bounded by a spherical polygon and sectors of circles whose common centre is the centre of the sphere.

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

7. A POLE OF A CIRCLE is a point on the surface of the sphere, equally distant from all the points of the cir cumference of the circle.

8. A DIAGONAL of a spherical polygon is an arc of a great circle joining the vertices of any two angles which are not consecutive.

PROPOSITION I. THEOREM.

Any side of a spherical triangle is less than the sum of

Let ABC be a

whose centre is 0

the other two.

spherical triangle situated on a sphere

then will any side, as AB, be less

than the sum of the sides AC and BC.

For, draw the radii OA, OB, and OC: these radii form the edges of a triedral angle whose vertex is 0, and the plane angles included between them are measured by the arcs AB, AC, and BC (B. III., P. XVII., Sch.). But any plane angle, as AOB, is less than the sum of the plane angles AOC and BOC (B. VI., P. XIX.): hence,

B

the arc AB is less than the sum of the arcs AC and BC; which was to be proved.

Cor. 1. Any side AB, of a spherical polygon ABCDE, is less than the sum of all the other sides.

D

For, draw the diagonals AC and AD, dividing the polygon into triangles. The arc AB is less than the sum of AC and BC, the arc AC is less than the sum of AD and DC, and the arc AD is less than the sum of DE and EA; hence, AB is less than the sum of BC, CD, DE and EA.

B

A

Cor. 2. The arc of a great circle joining any two points on the surface of a sphere, is less than the arc of a small circle joining the same points.

For, divide the arc of the small circle into equal parts, and through the extremities of each part pass the arc of a great circle. The arc of the great circle joining the given points will be less than the sum of these arcs (C. 1), whatever may be their number. But when this number is infinite, the arcs of the great circle coincide with the corresponding arcs of the small circle, and their sum is equal to the entire arc of the small circle.

Cor. 3. The shortest distance between two points on the surface of a sphere, is measured on the arc of a great circle joining them.

PROPOSITION II. THEOREM.

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

Let ABCDE be a spherical polygon a spherical polygon situated on a sphere whose centre is 0: then will the sum of its sides be less than the circumference of a great circle.

For, draw the radii OA, OB, OC, OD, and OE:

these radii form the edges of a polyedral angle whose vertex is at 0, and the angles included between

them are measured by the arcs AB, BC,

CD, DE, and EA. But the sum of these angles is less than four right angles (B. VI., P. XX.): hence, the sum of the arcs which measure them is less than the circumference of a great circle; which was to be proved.

E

Af

B

If a diameter of a sphere be drawn perpendicular to the plane of any circle of the sphere, its extremities will be poles of that circle.

Let C be the centre of a sphere, FNG any circle of the sphere, and DE a diameter of the sphere perpendicular to the plane of FNG: then will the extremities D and E, be poles of the circle FNG.

The diameter DE, being perpendicular to the plane of FNG, must pass through the centre 0 (B. VIII., P. VII., C. 3). If arcs of great circles DN, DF, DG, &c., be drawn from D to different points of the circumference FNG, and chords of these arcs be drawn, these chords will be equal (B. VI.,

F

M

M

P. V.), consequently, the arcs themselves will be equal. But these arcs are the shortest lines that can be drawn from the

point D, to the different points of the circumference (P. I., C. 2) hence, the point D, is equally distant from all the points of the circumference, and consequently is a pole of the circle (D. 7). In like manner, it may be shown that the point E is also a pole of the circle: hence, both D, and E, are poles of the circle FNG; which was proved.

Cor. 1. Let AMB be a great circle perpendicular to DE: then will the angles DCM, ECM, &c., be right angles; and consequently, the arcs DM, EM, &c., will each be equal to a quadrant (B. III., P. XVII., S.): hence, the two poles of a great circle are at equal distances from the circumference.

Cor. 2. The two poles of a small circle are at unequal distances from the circumference, the sum of the distances being equal to a semi-circumference.

Cor. 3. The line DC being perpendicular to the plane AMB, any plane, as DMC, passed through it, will also be perpendicular to the plane AMB: hence, the spherical angle DMA, is a right-angle; that is, if any point, in the circumference of a great circle, be joined with either pole by the arc of a great circle, such arc will be perpendicular to the circumference of the given circle.

Cor. 4. If the distance of a point D, from each of the points A and M, in the circumference of a great circle, is equal to a quadrant, the point D, is the pole of the arc AM.

For, let C be the radii CD, CA, CM. right angles, the line straight lines CA, CM: it is, therefore, perpendicular to their

centre of the sphere, and draw the Since the angles ACD, MCD, are CD is perpendicular to the two

plane (B. VI., P. IV.): hence, the point D, is the pole of the arc AM.

Scholium. The properties of these poles enable us to describe arcs of a circle on the surface of a sphere, with the same facility as on a plane surface. For, by turning the arc DF about the point D, the extremity F will describe the small circle FNG; and by turning the quadrant DFA round the point D, its extremity A will describe an arc of a great circle.

PROPOSITION IV. THEOREM.

The angle formed by two arcs of great circles, is equal to that formed by the tangents to these arcs at their point of intersection, and is measured by the arc of a great circle described from the vertex as a pole, and limited by the sides, produced if necessary.

Let the angle BAC be formed by the two arcs AB, AC: then is it equal to the angle FAG formed by the tangents AF, AG, and is measured by the arc DE of a great circle, described about A as a pole.

For, the tangent AF, drawn in the plane of the arc AB, is perpendicular to the radius A0; and the tangent AG, drawn in the plane of the arc AC, is perpendicular to the same radius AO: hence, the angle FAG is equal to the angle contained by the planes ABDH, ACEH (B. VI., D. 4); which is that of the arcs AB, AC. Now, if the arcs AD and AE are both quad

G

F

B

C

rants, the lines OD, OE, are perpendicular to OA, and