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pound apparatus as we have described in art. 147. may be advantageously mounted, is that in which the principal axis occupies a vertical position, and the one circle, A B, consequently corresponds to the celestial horizon, and the other, G H, to a vertical circle of the heavens. The angles measured on the former are therefore azimuths, or differences of azimuth, and those on the latter zenith distances, or altitudes, according as the graduation commences from the upper point of its limb, or from one 90° distant from it. It is therefore known by the name of an azimuth and altitude instrument. The vertical position of its principal axis is secured either by a plumb-line suspended from the upper end, which, however it be turned round, should continue always to intersect one and the same fiducial mark near its lower extremity, or by a level fixed directly across it, whose bubble ought not to shift its place, on moving the instrument in azimuth. The north or south point on the horizontal circle is ascertained by bringing the ver. tical circle to coincide with the plane of the meridian, by the same criterion by which the azimuthal adjustment of the transit is performed (art. 137.), and noting, in this position, the reading off of the lower circle, or by the following process.

(152.) Let a bright star be observed at a considerable distance to the east of the meridian, by bringing it on the cross wires of the telescope. In this position let the horizontal circle be read off, and the telescope securely clamped on the vertical one. When the star has passed the meridian, and is in the descend. ing point of its daily course, let it be followed by moving the whole instrument round to the west, without, however, unclamping the telescope, until it comes into the field of view; and until, by continuing the horizontal motion, the star and the cross of the wires come once more to coincide. In this position it is evident the star must have the same precise altitude above the western horizon, that it had at the moment of the first observation above the erstern. At this point let the mo. CHAP. II. ALTITUDE AND AZIMUTH INSTRUMENT. 101

tion be arrested, and the horizontal circle be again read off. The difference of the readings will be the azimuthal arc described in the interval. Now, it is evident that when the altitudes of any star are equal on either side of the meridian, its azimuths, whether reckoned both from the north or both from the south point of the horizon, must also be equal, consequently the north or south point of the horizon must bisect the azimuthal arc thus determined, and will therefore become known.

(153.) This method of determining the north and south points of a horizontal circle (by which, when known, we may draw a meridian line) is called the “ method of equal altitudes,” and is of great and con. stant use in practical astronomy. If we note, at the moments of the two observations, the time, by a clock or chronometer, the instant halfway between them will be the moment of the star's meridian passage, which may thus be determined without a transit; and, vice versa, the error of a clock or chronometer may by this process be discovered. For this last purpose, it is not neces. sary that our instrument should be provided with a hori. zontal circle at all. Any means by which altitudes can be measured will enable us to determine the moments when the same star arrives at equal altitudes in the eastern and western halves of its diurnal course; and, these once known, the instant of meridian passage and the error of the clock become also known.

(154.) One of the chief purposes to which the altitude and azimuth circle is applicable is the investigation of the amount and laws of refraction. For, by following with it a circumpolar star which passes the zenith, and another which grazes the horizon, through their whole diurnal course, the exact apparent form of their diurnal orbits, or the ovals into which their circles are distorted by refraction, can be traced; and their devi. ation from circles, being at every moment given by the nature of the observation in the direction in which the refraction itself takes place (i.e. in altitude), is made a matter of direct observation.

(155.) The zenith sector and the theodolite are peculiar modifications of the altitude and azimuth instrument. The former is adapted for the very exact observation of stars in or near the zenith, by giving a great length to the vertical axis, and suppressing all the circumference of the vertical circle, except a few degrees of its lower part, by which a great length of radius, and a consequent proportional enlargement of the divi. sions of its arc, is obtained. The latter is especially devoted to the measure of horizontal angles between terrestrial objects, in which the telescope never requires to be elevated more than a few degrees, and in which, therefore, the vertical circle is either dispensed with, or executed on a smaller scale, and with less delicacy; while, on the other hand, great care is bestowed on securing the exact perpendicularity of the plane of the telescope's motion, by resting its horizontal axis on two supports like the piers of a transit-instrument, while themselves are firmly bedded on the spokes of the hori. zontal circle, and turn with it.

(156.) The last instrument we shall describe is one by whose aid the direct angular distance of any two objects may be measured, or the altitude of a single one determined, either by measuring its distance from the visible horizon (such as the sea-offing, allowing for its dip), or from its own reflexion on the surface of mercury. It is the sextant, or quadrant, commonly called Hadley's, from its reputed inventor, though the priority of invention belongs undoubtedly to Newton, whose claims to the gratitude of the navigator are thus doubled, by his having furnished at once the only theory by which his vessel can be securely guided, and the only instrument which has ever been found to avail, in applying that theory to its nautical uses. *

* Newton communicated it to Dr. Halley, who suppressed it. The de. scription of the instrument was found, after the death of Halley, among his papers, in Newton's own handwriting, by his executor, who communicated the papers to the Royal Society, twenty five years after Newton's death, and eleven after the publication of Hadley's invention, which might be, and probably was, independent of any knowledge of Newton's, though Hutton insinuates the contrary.

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(157.) The principle of this instrument is the optical property of reflected rays, thus announced:—“The

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angle between the first and last directions of a ray which has suffered two reflexions in one plane is equal to twice the inclination of the reflecting surfaces to each other.” Let A B be the limb, or graduated arc, of a portion of a circle 60° in extent, but divided into 120 equal parts. On the radius C B let a silvered plane glass D be fixed, at right angles to the plane of the circle, and on the moveable radius C E let another such silvered glass, C, be fixed. The glass D is permanently fixed parallel to A C, and only one half of it is silvered, the other half allowing objects to be seen through it. The glass C is wholly silvered, and its plane is parallel to the length of the moveable radius CE, at the extremity E, of which a vernier is placed to read off the divisions of the limb. On the radius A C is set a telescope F, through which any object, Q, may be seen by direct rays which pass through the unsilvered portion of the glass D, while another object, P, is seen through the same telescope by rays, which, after reflexion at C, have been thrown upon the silvered part of D, and are thence directed by a second reflexion into the telescope. The two images so formed will both be seen in the field of view at once, and by moving the radius CE will (if the reflectors be truly perpendicular to the plane of the circle) meet and pass over, without obliterating each other. The motion, however, is arrested when they meet, and at this point the angle included between the direction C P of one object, and FQ of the other, is twice the angle ECB included between the fixed and moveable radii CB, CE. Now, the graduations of the limb being purposely made only half as distant as would correspond to degrees, the arc B E, when read off, as if the graduations were whole degrees, will, in fact, read double its real amount, and therefore the numbers to read off will express not the angle ECB, but its double, the angle subtended by the objects.

(158,) To determine the exact distances between the stars by direct observation is comparatively of little service; but in nautical astronomy the measurement of their distances from the moon, and of their altitudes, is of essential importance; and as the sextant requires no fixed support, but can be held in the hand, and used on ship-board, the utility of the instrument becomes at once obvious. For altitudes at sea, as no level, plumbline, or artificial horizon can be used, the sea-offing affords the only resource; and the image of the star observed, seen by reflexion, is brought to coincide with the boundary of the sea seen by direct rays. Thus the altitude above the sea-line is found; and this corrected for the dip of the horizon (art. 24.) gives the true altitude of the star. On land, an artificial horizon may be used (art. 139.), and the consideration of dip is rendered unnecessary.

(159.) The reflecting circle is an instrument destined for the same uses as the sextant, but more complete, the circle being entire, and the divisions carried all round. It is usually furnished with three verniers, so as to admit of three distinct readings off, by the average of which the error of graduation and of reading is reduced. This is altogether a very refined and elegant instrument.

(160.) We must not conclude this chapter without mention of the “ principle of repetition ;” an invention of Borda, by which the error of graduation may be diminished to any degree, and, practically speaking, an

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