Pole

Pole. The imaginary ends of the axis about which the earth turns are known in geography as the north and south poles, and this is the most popular use of the word. In the geometry of a sphere the pole of any circle is the point in which a perpendicular to the plane of the circle through its centre cuts the sphere. Every point on the circle is therefore equidistant from it, and, if the circle be a great circle, the distance is 90°. The north and south poles are thus the geometrical poles of the earth's equator. In astronomy they are the points in which the earth's axis, if produced, cuts the celestial sphere; for the sake of distinction, these are often known as the celestial poles. No stars occur actually at these points (where they would be extremely convenient for astronomical measurements); but the Pole Star (q.v.) is slightly distant from the north celestial pole. Two other points of reference in astronomical measurements are the zenith and nadir, these being the poles of the horizon.

Another use of the word in geometry is to denote the point at which two tangents to a conic meet, this being called the pole of the chord joining the two points of contact. This chord is known as the polar of the point, and the theory of pole and polar is of great use in projective geometry, where their property of reciprocity often gives extremely pretty solutions to certain problems.