26. If one triangle be the polar triangle of another, the latter will be the polar triangle of the former.

Let ABC be any triangle, A'B'C' the polar triangle; then ABC will be the polar triangle of A'B'C'.

For since B' is a pole of AC, the arc AB' is a quadrant, and since C' is a pole of BA, the arc AC is a quadrant (Art. 10); therefore A is a pole of B'C' (Art. 11). Also A and A' are on the same side of B'C'; for A and A' are by hypothesis on the same side of BC, therefore A'A is less than a quadrant; and since A is a pole of B'C' and AA' is less than a quadrant, A and A′ are on the same side of B'C'.

Similarly it may be shewn that B is a pole of C'A', and that B and B are on the same side of C'A'; also that C is a pole of A'B', and that C and C' are on the same side of A'B'. Thus ABC is the polar triangle of A'B'C'.

27. The sides and angles of the polar triangle are respectively the supplements of the angles and sides of the primitive triangle.

For let the arc B'C', produced if necessary, meet the arcs AB, AC, produced if necessary, at the points D and E respectively; then since A is a pole of B'C', the spherical angle A is measured by the arc DE (Art. 12). But B'E and C'D are each quadrants; therefore DE and B'C' are together equal to a semicircle; that is, the angle subtended by B'C' at the centre of the sphere is the

supplement of the angle A. This we may express for shortness thus; B'C' is the supplement of A. Similarly it may be shewn that C'A' is the supplement of B, and A'B' the supplement of C.

And since ABC is the polar triangle of A'B'C', it follows that BC, CA, AB are respectively the supplements of A', B', C′; that is, A', B, C are respectively the supplements of BC, CA, AB.

From these properties a primitive triangle and its polar triangle are sometimes called supplemental triangles.

Thus, if A, B, C, a, b, c denote respectively the angles and sides of a spherical triangle all expressed in circular measure, and A', B', C', a', b', c' those of the polar triangle, we have

A'= ·α, В=π-b, С'=π-c,

α =π- A, b = π – B,
П

28. The preceding result is of great importance; for if any general theorem be demonstrated with respect to the sides and angles of any spherical triangle it holds of course for the polar triangle also. Thus any such theorem will remain true when the angles are changed into the supplements of the corresponding sides and the sides into the supplements of the corresponding angles. We shall see several examples of this principle in the next chapter.

29. Any two sides of a spherical triangle are together greater than the third side. (See the figure of Art. 18.)

For any two of the three plane angles which form the solid angle at are together greater than the third (Euclid, x1. 20). Therefore any two of the arcs AB, BC, CA, are together greater than the third.

From this proposition it is obvious that any side of a spherical triangle is greater than the difference of the other two.

30. The sum of the three sides of a spherical triangle is less than the circumference of a great circle. (See the figure of Art. 18.)

For the sum of the three plane angles which form the solid angle at O is less than four right angles (Euclid, XI. 21); therefore

therefore,

AB + BC + CA is less than 2π × ОA;

that is, the sum of the arcs is less than the circumference of a great circle.

31. The propositions contained in the preceding two Articles may be extended. Thus, if there be any polygon which has each of its angles less than two right angles, any one side is less than the sum of all the others. This may be proved by repeated use of Art. 29. Suppose, for example, the figure has four sides, and let the angular points be denoted by A, B, C, D. Then

therefore,

AB+ BC is greater than AC;

AB+ BC + CD is greater than AC + CD,

and à fortiori greater than AD.

Again, if there be any polygon which has each of its angles less than two right angles, the sum of its sides will be less than the circumference of a great circle. This follows from Euclid, XI. 21, in the manner shewn in Art. 30.

32. The three angles of a spherical triangle are together greater than two right angles and less than six right angles.

Let A, B, C be the angles of a spherical triangle; let a', b', c' be the sides of the polar triangle. Then by Art. 30,

a+b+c is less than 2,

And since each of the angles A, B, C is less than π, the sum A+B+C is less than 3π.

33. The angles at the base of an isosceles spherical triangle are

equal.

B

T

Let ABC be a spherical triangle having AC = BC; let O be the centre of the sphere. Draw tangents at the points A and B to the arcs AC and BC respectively; these will meet OC produced at the same point S, and AS will be equal to BS.

Draw tangents AT, BT at the points A, B to the arc AB; then ATTB; join TS. In the two triangles SAT, SBT the sides SA, AT, TS are equal to SB, BT, TS respectively; therefore the angle SAT is equal to the angle SBT; and these are 'the angles at the base of the spherical triangle.

The figure supposes AC and BC to be less than quadrants; if they are greater than quadrants the tangents to AC and BC will meet on CO produced through O instead of through C, and the demonstration may be completed as before. If AC and BC are quadrants, the angles at the base are right angles by Arts. 11 and 8.

34. If two angles of a spherical triangle are equal, the opposite sides are equal.

Since the primitive triangle has two equal angles, the polar triangle has two equal sides; therefore in the polar triangle the angles opposite the equal sides are equal by Art. 33. Hence in the primitive triangle the sides opposite the equal angles are equal.

35. If one angle of a spherical triangle be greater than another, the side opposite the greater angle is greater than the side opposite the other.

Let ABC be a spherical triangle, and let the angle ABC be greater than the angle BAC; then the side AC will be greater than the side BC. At B make the angle ABD equal to the angle BAD; then BD is equal to AD (Art. 34), and BD + DC is greater than BC (Art. 29); therefore AD + DC is greater than BC; that is, AC is greater than BC.

36. If one side of a spherical triangle be greater than another, the angle opposite the greater side is greater than the angle opposite the other.

This follows from the preceding Article by means of the polar triangle.

Or thus; suppose the side AC greater than the side BC, then the angle ABC will be greater than the angle BAC. For the angle ABC cannot be less than the angle BAC by Art. 35, and the angle ABC cannot be equal to the angle BAC by Art. 34; therefore the angle ABC must be greater than the angle BAC.