Any number of great circles can pass through any one point, but only one great circle can pass through two points, unless they are diametrically opposite.

The great circle which is parallel to the small circle is often called the parent circle.

A spherical triangle is a triangle each of whose sides is a great circle. It must be appreciated that a small circle cannot form the side of a spherical triangle.

The length of the arc of a circle can be measured by the angle which the arc subtends at the centre of the circle. The sides of a spherical triangle are measured in degrees, minutes, and seconds.

Any two great circles intersect each other in two places diametrically opposite, that is in two places 180° apart. No side of a spherical triangle can therefore exceed 150°.

The three angles of a spherical triangle must together be more than 180° and less than 540°, that is, more than two right angles and less than six right angles. The amount by which the total of the three angles in the triangle exceeds 180° is called the spherical excess.

As in plane trigonometry the greater side is opposite the greater angle, if two sides are equal their opposite angles are equal, as in an isosceles triangle.

If one angle of the triangle is 90° it is called a right-angled triangle, and if one side of the triangle is 90° it is called a quadrantal triangle.

Great Circles.—The diameter of the sphere which is perpendicular to the plane of a great circle cuts the surface of the sphere in two points called the poles of the great circle. The pole of a great circle is 90° away from all points on the great circle. Any great circle drawn through the poles must cut the first great circle at right angles, e.g., a meridian is a great circle passing through the poles of another great circle called the equator. Notice that the angle between two meridians is equal to the arc of the equator between the same two meridians.

As mentioned earlier any two great circles cut each other in two places diametrically opposite, i.e., 180° apart.

Right-angled spherical triangles. 

The formulae for solving right-angled spherical triangles are summarised in Napier’s mnemonic rules. To use these rules we must consider the five parts of the triangle in order. The five parts are the three sides and the other two angles—excluding the right angle. It will be found convenient to represent these parts on a small diagram. Draw a circle and divide it into five parts as shown (fig. 139). Insert the name of the right angle at the top of the figure and working clockwise around the triangle and around the figure insert the names of the other five parts (fig. 184). Now if we consider any sector of the figure, the sectors immediately alongside are called “adjacent,” the other two are called “opposite.” It will be found that if we select any three of the five parts one part will be in the middle (in the sense that it is equidistant from the other two) and the other two will either both be “adjacent” or both be “opposite.” Napier’s rules state

Sine middle part = product cosines of opposite parts. Sine middle part = product of tangents of adjacent parts. Notice “i” in sine and middle, “o” in cosine and opposite, and “a” in tangent and adjacent.

But we must use the complements of the side opposite to the 90° angle and the complements of the other two angles. That is why our diagram has a part marked comp. When using these parts we use the complementary ratio. Using these rules if we are given any two parts in a right-angled spherical triangle we can find any third part.