src="@public@vhost@g@gutenberg@html@files@40240@40240-h@images@i019.jpg" alt="Fig. 1." title="" tag="{http://www.w3.org/1999/xhtml}img"/> Fig. 1.

The complement of an arc, or angle, is what it wants of ninety degrees. Thus, since A D is an arc of ninety degrees, B D is the complement of A B, and A B is the complement of B D. If A B denotes a certain number of degrees of latitude, B D will be the complement of the latitude, or the colatitude, as it is commonly written.

The supplement of an arc, or angle, is what it wants of one hundred and eighty degrees. Thus, B A is the supplement of G D B, and G D B is the supplement of B A. If B A were twenty degrees of longitude, G D B, its supplement, would be one hundred and sixty degrees. An angle is said to be subtended by the side which is opposite to it. Thus, in the triangle A C K, the angle at C is subtended by the side A K, the angle at A by C K, and the angle at K by C A. In like manner, a side is said to be subtended by an angle, as A K by the angle at C.

Let us now proceed with the doctrine of the sphere.

A section of a sphere, by a plane cutting it in any manner, is a circle. Great circles are those which pass through the centre of the sphere, and divide it into two equal hemispheres. Small circles are such as do not pass through the centre, but divide the sphere into two unequal parts. The axis of a circle is a straight line passing through its centre at right angles to its plane. The pole of a great circle is the point on the sphere where its axis cuts through the sphere. Every great circle has two poles, each of which is every where ninety degrees from the great circle. All great circles of the sphere cut each other in two points diametrically opposite, and consequently their points of section are one hundred and eighty degrees apart. A great circle, which passes through the pole of another great circle, cuts the latter at right angles. The great circle which passes through the pole of another great circle, and is at right angles to it, is called a secondary to that circle. The angle made by two great circles on the surface of the sphere is measured by an arc of another great circle, of which the angular point is the pole, being the arc of that great circle intercepted between those two circles.

In order to fix the position of any place, either on the surface of the earth or in the heavens, both the earth and the heavens are conceived to be divided into separate portions, by circles, which are imagined to cut through them, in various ways. The earth thus intersected is called the terrestrial, and the heavens the celestial, sphere. We must bear in mind, that these circles have no existence in Nature, but are mere landmarks, artificially contrived for convenience of reference. On account of the immense distances of the heavenly bodies, they appear to us, wherever we are placed, to be fixed in the same concave surface, or celestial vault. The great circles of the globe, extended every way to meet the concave sphere of the heavens, become circles of the celestial sphere.

The horizon is the great circle which divides the earth into upper and lower hemispheres, and separates the visible heavens from the invisible. This is the rational horizon. The sensible horizon is a circle touching the earth at the place of the spectator, and is bounded by the line in which the earth and skies seem to meet. The sensible horizon is parallel to the rational, but is distant from it by the semidiameter of the earth, or nearly four thousand miles. Still, so vast is the distance of the starry sphere, that both these planes appear to cut the sphere in the same line; so that we see the same hemisphere of stars that we should see, if the upper half of the earth were removed, and we stood on the rational horizon.

The poles of the horizon are the zenith and nadir. The zenith is the point directly over our heads; and the nadir, that directly under our feet. The plumb-line (such as is formed by suspending a bullet by a string) is in the axis of the horizon, and consequently directed towards its poles. Every place on the surface of the earth has its own horizon; and the traveller has a new horizon at every step, always extending ninety degrees from him, in all directions.

Vertical circles are those which pass through the poles of the horizon, (the zenith and nadir,) perpendicular to it.

The meridian is that vertical circle which passes through the north and south points.

The prime vertical is that vertical circle which passes through the east and west points.

The altitude of a body is its elevation above the horizon, measured on a vertical circle.

The azimuth of a body is its distance, measured on the horizon, from the meridian to a vertical circle passing through that body.

The amplitude of a body is its distance, on the horizon, from the prime vertical to a vertical circle passing through the body.

Azimuth is reckoned ninety degrees from either the north or south point; and amplitude ninety degrees from either the east or west point. Azimuth and amplitude are mutually complements of each other, for one makes up what the other wants of ninety degrees. When a point is on the horizon, it is only necessary to count the number of degrees of the horizon between that point and the meridian, in order to find its azimuth; but if the point is above the horizon, then its azimuth is estimated by passing a vertical circle through it, and reckoning the azimuth from the point where this circle cuts the horizon.

The zenith distance of a body is measured on a vertical circle passing through that body. It is the complement of the altitude.

The axis of the earth is the diameter on which the earth is conceived to turn in its diurnal revolution. The same line, continued until it meets the starry concave, constitutes the axis of the celestial sphere.

The poles of the earth are the extremities of the earth's axis: the poles of the heavens, the extremities of the celestial axis.

The equator is a great circle cutting the axis of the earth at right angles. Hence, the axis of the earth is the axis of the equator, and its poles are the poles of the equator. The intersection of the plane of the equator with the surface of the earth constitutes the terrestrial, and its intersection with the concave sphere of the heavens, the celestial, equator. The latter, by way of distinction, is sometimes denominated the equinoctial.

The secondaries to the equator,—that is, the great circles passing through the poles of the equator,—are called meridians, because that secondary which passes through the zenith of any place is the meridian of that place, and is at right angles both to the equator and the horizon, passing, as it does, through the poles of both. These secondaries are also called hour circles because the arcs of the equator intercepted between them are used as measures of time.

The latitude of a place on the earth is its distance from the equator north or south. The polar distance, or angular distance from the nearest pole, is the complement of the latitude.

The longitude of a place is its distance from some standard meridian, either east or west, measured on the equator. The meridian, usually taken as the standard, is that of the Observatory of Greenwich, in London. If a place is directly on the equator, we have only to inquire, how many degrees of the equator there are between that place and the point where the meridian of Greenwich cuts the equator. If the place is north or