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 to be 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:

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

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 quadant 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 40; and the tangent AG, drawn in the plane of the arc AC, is perpendicular to the same radius. 40: 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

H

the arcs AD and AE are both quad

A

G

F

B

C

E

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

the angle DOE is equal to the angle of the planes ABDH, ACEH: hence, the arc DE is the measure of the angle contained by these planes, or of the angle CAB; which was to be proved.

Cor. 1. The angles of spherical triangles may be com pared by means of the arcs of great circles described from their vertices as poles, and included between their sides.

A spherical angle can always be constructed equal to a given spherical angle.

Cor. 2. Vertical angles, such as ACO and BCN are equal; for either of them is the angle formed by the two planes ACB, OCN When two arcs ACB, OCN, intersect, the sum of two adjacent angles, as ACO, OCB, is equal to two right angles.

N

PROPOSITION V. THEOREM.

If from the vertices of the angles of a spherical triangle, as poles, arcs be described forming a spherical triangle, the vertices of the angles of this second triangle will be respectively poles of the sides of the first:

From the vertices A, B, C, as poles, let the arcs EF, FD, ED, be described, forming the triangle DFE: then will the vertices D, E, and F, be respectively poles of the sides BC, AC, AB.

For, the point A being

E

B

A

the pole of the arc EF, the distance AE, is a quadrant; the point being the pole of the arc DE, the distance C CE, is likewise a quadrant: hence, the point E is at a quadrant's distance from the points A and C: hence, it is the pole of the arc AC (P. III., C. 4). It may be shown, in like manner, that D is the pole of the arc BC, and F that of the arc AB; which was to be proved.

Scholium. The triangle ABC, may be described by means of DEF, as DEF is described by means of ABC. Triangles thus related are called polar triangles, or supplemental triangles.

PROPOSITION VI. THEOREM.

Any angle, in one of two polar triangles, is measured by a semi-circumference, minus the side lying opposite to it in the other triangle.

Let ABC, and EFD, be any two polar triangles: then will any angle in either triangle be measured by a semi-circumference, minus the side lying opposite to it in the other triangle.

For, produce the sides AB, A C, if necessary, till they meet EF, in G and H. The point A being the pole of the arc GH, the angle A is measured by that arc (P. IV.). But, since E is the pole of AH, the arc EH is a quadand since F is the

rant ;

M

K

E

B

H

pole of AG, FG is a quadrant: hence, the sum of the arcs EH and GF, is equal to a semi-circumference.

But,

the sum of the arcs EH and GF, is equal to the sum of the arcs EF and GH hence, the arc GH, which A, is equal to a semi-circumference,

measures the angle
minus the arc EF
any other angle, in
circumference, minus
other triangle; which was to be proved.

In like manner, it may be shown, that either triangle, is measured by a semithe side lying opposite to it in the

Scholium. Besides the triangle DEF, three others may be formed by the intersection of the arcs DE, EF, DF. But the proposition is applicable only to the central triangle, which is distinguished from the other three by the circumstance, that the two vertices, A

D

E

and D, lie on the same side of BC; the two vertices, B and E, on the same side of AC; and the two vertices, C and F, on the same side of AB.

PROPOSITION VII. THEOREM.

If from the vertices of any two angles of a spherical triangle, as poles, arcs of circles be described passing through the vertex of the third angle; and if from the second point in which these arcs intersect, arcs of great circles be drawn to the vertices, used as poles, the parts of the triangle thus formed will be equal to those of the given triangle, each to each.

Let ABC be a spherical triangle situated on a sphere whose centre is 0, CED and CFD arcs of circles described about B and A as poles, and let DA and DB be arcs of great circles: then will the parts of the