5. The surface of the zone intercepted between any two parallels is the rectangle contained under the circumference of the primary and the perpendicular distance between the parallels.
6. Tho surface of a lime contained between two great circles is such a proportion of the whole surface of the sphere as the angle contained between the two great circles is of four right angles.
7. The part of a lune contained within any zone made by two of its parallels (as Er u A) is such a proportion of the whole zone as the angle of the circles forming the lune is of four right angles.
We are now to show the method of Co-owes:Ares by which points in the sphere are ascertained, and their relative positions described. Take any great circle oval, and choose any point v as an origin, and either direction to be that iu which arcs are measured. Say for in stance that en, in preference to Ise, shall be the direction in which arcs are measured. Tho position of any point in this great circle is then ascertained simply by determining its distance from u, since there is a tacit understanding as to the direction in which that distance shall be measured. If we give a name to that distance, be it longitude,
right ascension, or any other, the point whose right ascension (if it be right ascension) is SO" means the point which is at SO° distance from u in the direction u A. Again, if we wish to describe any other point, not in the great circle chosen, as r : through F draw a secondary to the great circle (e FAQ), then the point r will be known as soon as A is described, in the manner just laid down, and also as soon as the arc A is ie given, and the polo towards which it is measured. These two co ordinates, VA and A r, when described in magnitude and direction, form a complete description of the position of the point F on the sphere ; and the angles subtended by U A and A r are generally used instead of u• and • r.
For the first steps of the application of spherical geometry to astro nomy, see the next article.