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plane, or the hour lines of the dial will be found, by computing the distances of the hour circles from the meridian on the prime vertical. A dial so constructed is called a vertical dial.

It is evident that the plane of the dial may make any given angle with the prime vertical, and the hour lines be readily computed by a spherical triangle. When the plane of the dial faces the east or west, the stile is placed at a distance from, and parallel to its plane, because the plane of the dial is itself in the plane of the meridian.

CHAPTER XVI.

APPLICATION OF ASTRONOMY TO NAVIGATION-HADLEY'S SEXTANT

-LATITUDE AT SEA-APPARENT

COMPASS-LONGITUDE AT SEA.

TIME VARIATION OF THE

284. THE Uses of astronomy in navigation are very great. It enables the seaman to determine by celestial observations his latitude and longitude, and thence discover his situation with an accuracy sufficient to direct him the course he ought to steer for his intended port, and to guard him against dangers from shoals and rocks. It also enables him to find the variation of his compass, and so affords him the means of sailing his proper

course.

Almost all the astronomical observations made at sea, consist in measuring angles, and the difficulty of taking an angle at sea, on account of the unsteady motion of the ship, is sufficiently obvious. In taking an altitude, the plumb-line and spirit-level are entirely useless. In observing the angular distance of two objects, the unsteadiness of the ship makes it impossible to measure it by two telescopes, or by one telescope successively adjusted to each object.

285. These difficulties were soon seen when nautical astronomy began to be improved. Many attempts were made to invent a proper instrument. The ingenious Dr. Hooke proposed several methods. Many years afterwards Mr. Hadley proposed the instrument called Hadley's quadrant, now however usually called Hadley's sextant, for a reason that will be mentioned. A few years after Mr. Hadley's invention was communicated to

the world, a paper of Sir Isaac Newton's was found, describing an instrument nearly of the same construction. The principle of this invaluable instrument is, that in taking the angular distance of two objects, the image of one of them seen after two reflections, coincides with the other object seen directly; and this coincidence is in no wise affected by the unsteadiness of the ship. The operation by which the coincidence is made, measures the angular distance of the objects.

see the object

286. Let A and B (Fig. 44) be two celestial or very distant objects; HO, IN the sections of two plane mirrors, in the plane passing through the objects and eye. The mirrors are supposed to be perpendicular to this plane. Let a ray of light, AC, from the object A, incident on the mirror IN, be reflected in the direction CR, and so be incident on the mirror HO, from whence it is again reflected in the direction RE, coinciding with the direction of a ray, BR, from the other object, B. Then an eye any where in the direction of the line RE, will A, coincident with the object B, if a portion of the mirror, immediately above the section HO be transparent. Thus we may make two distant objects appear to coincide by a proper position of the mirrors, viz., by inclining the mirrors at an angle equal to half the angular distance of the objects. For produce the sections of the mirrors to meet in M, and produce AC to meet BRE in E. Then E = BRC-RCE (by the principles of reflection) 2 HRC-2 RCM = 2 M, or the angular distance of the objects equals twice the inclination of the reflectors. Hence if we move the reflector IN, so that both objects may appear to coincide, and can then measure the inclination of the reflectors, we shall obtain the angular distance of the objects. This principle is used in Hadley's sextant as follows.

287. ACB (Fig. 45) may represent the sextant. The angle ACB is 60°, but the arch AB extends a few degrees beyond each radius. A moveable radius, CV, called the index, re

volves about the centre C, carrying a plane mirror, IN, perpendicular to the plane of the sextant, which mirror faces another mirror, H, also perpendicular to the plane of the sextant. This latter mirror is fixed with its plane parallel to CA, the position of the mirror IN, when the radius CV passes through zero or (0) of the arch. The upper part of the mirror H is transparent, through which, by help of a telescope fixed at T, parallel to the plane of the sextant, the object S may be seen directly, while the image of M, seen by reflection, appears to touch it. The angular distance of the objects M and S, is then, as has been shewn, twice the inclination of the mirrors H and IN= (because H is parallel to CA) 2 VCA. Hence the degrees, minutes and seconds in VA, shewn by a vernier, attached to the extremity of the index, would give half the angular distance of the objects; but as the arch VA is only half the angular distance of the objects, for convenience each degree, &c. is reckoned double; thus if VA be actually 42°, it is marked 84o, &c.

The mirror IC is called the index glass, and H the horizon glass, because in taking the altitude of the sun at sea, the horizon is seen, directly, through this glass.

In most sextants there is a provision for adjusting the plane of the horizon glass, parallel to the radius passing through zero of the arch, or rather parallel to the plane of the index glass, when the index is at zero of the arch. This is done by making an image coincide with its object seen directly, when the index passes through zero. Or the quantity of the error may be determined by measuring a small angle, for instance, the sun's diameter, on each side of zero of the arch. Half the difference is the error of the index, and it is most convenient to allow for this, as it cannot be corrected so exactly as its quantity can be ascertained.

For a more particular account of this instrument and its adjustments, see Professor Vince's Practical Astronomy.

288. The best instruments, intended for taking the angular distance of the moon from the sun and stars, are made with great exactness. The radius of a sextant varies in length from five to fourteen inches. The usual length is about ten or twelve inches, and these admit of measuring an angle to 10" or less, by help of the vernier. Ordinary instruments are also made, merely for taking altitudes. Plain sights are only used with these, and they are seldom adapted to take altitudes nearer than two or three minutes.

As an altitude is never greater than 90°, it is evident, for an altitude, a greater arch than 45° is not required. The instruments, therefore, made only for taking altitudes, should properly be called octants, instead of quadrants, as they are sometimes named. The angular distance of the moon from a star is sometimes measured when 120°, for such distances an arch of 60° is necessary, and therefore the instruments intended for the longitude at sea are called sextants.

In the octants, particularly, there is often a provision for measuring angles greater than 90°, by measuring the supplement to 180°, by what is called the back observation; this is not often used.

289. The celebrated Mayer, whose lunar tables have been mentioned, recommended a complete circle for measuring the angular distance of the moon from the sun or stars by reflection, as in Hadley's instrument. Some of the advantages proposed, were similar to those of the astronomical circle over the astronomical quadrant; also by making the horizon glass moveable, the same angle could be repeated on different parts of the limb, and by repeating the angle many times, and taking a mean, the errors of division were almost entirely done away.

a Professor Vince's Practical Astronomy.

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