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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 tele. scope 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, plumba line, 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

nihilated. Let PQ be two objects which we may suppose fixed, for purposes of mere explanation, and let KL be a

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KD telescope moveable on 0, the common axis of two circles, AML and abc, of which the former, A ML, is absolutely fixed in the plane of the objects, and carries the graduations, and the latter is freely moveable on the axis. The telescope is attached permanently to the latter circle, and moves with it. An arm Oa A carries the index, or vernier, which reads off the graduated limb of the fixed circle. This arm is provided with two clamps, by which it can be temporarily connected with either circle, and detached at pleasure. Suppose, now, the telescope directed to P. Clamp the index arm 0 A to the inner circle, and unclamp it from the outer, and read off. Then carry the telescope round to the other object Q. In so doing, the inner circle, and the index-arm which is clamped to it, will also be carried round, over an arc A B, on the graduated limb of the outer, equal to the angle POQ. Now clamp the index to the outer circle, and unclamp the inner, and read off: the difference of readings will of course measure the angle POQ; but the result will be liable to two sources of error — that of graduation and that of observation, both which it is our object to get rid of. To this end transfer the telescope back to P, without unclamping the arm from the outer circle ; then, having

made the bisection of P, clamp the arm to b, and unclamp it from B, and again transfer the telescope to Q, by which the arm will now be carried with it to C, over a second arc, BC, equal to the angle POQ. Now again read off; then will the difference between this reading and the original one measure twice the angle POQ, affected with both errors of observation, but only with the same error of graduation as before. Let this process be repeated as often as we please (suppose ten times); then will the final arc ABC D read off on the circle be ten times the required angle, affected by the joint errors of all the ten observations, but only by the same constant error of graduation, which depends on the initial and final readings off alone. Now the errors of observation, when numerous, tend to balance and destroy one another; so that, if sufficiently multiplied, their influence will disappear from the result. There remains, then, only the constant error of graduation, which comes to be divided in the final result by the number of observations, and is therefore diminished in its influence to one tenth of its possible amount, or to less if need be. The abstract beauty and advantage of this principle seem to be counterbalanced in practice by some unknown cause, which, probably, must be sought for in imperfect clamping.

CHAP. III.

OF GEOGRAPHY.

OF THE FIGURE OF THE EARTH. ITS EXACT DIMENSIONS.

ITS FORM THAT OF EQUILIBRIUM MODIFIED BY CENTRIFUGAL FORCE. - VARIATION OF GRAVITY ON ITS SURFACE. -STATICAL AND DYNAMICAL MEASURES OF GRAVITY. -THE PENDULUM. - GRAVITY TO A SPHEROID. - OTHER EFFECTS OF EARTH'S ROTATION. -TRADE WINDS. -DETERMINATION OF GEOGRAPHICAL POSITIONS. - OF LATITUDES. OF LONGITUDES. CONDUCT OF A TRIGONOMETRICAL SURVEY, — OF MAPS. - PROJECTIONS OF THE SPHERE. - MEASUREMENT OF HEIGHTS BY THE BAROMETER.

(161.) GEOGRAPHY is not only the most important of the practical branches of knowledge to which astronomy is applied, but is also, theoretically speaking, an essential part of the latter science. The earth being the general station from which we view the heavens, a knowledge of the local situation of particular stations on its surface is of great consequence, when we come to enquire the distances of the nearer heavenly bodies from us, as concluded from observations of their paralax as well as on all other occasions, where a difference of locality can be supposed to influence astronomi. cal results. We propose, therefore, in this chapter, to explain the principles, by which astronomical observation is applied to geographical determinations, and to give at the same time an outline of geography so far as it is to be considered a part of astronoiny.

(162.) Geography, as the word imports, is a deline. ation or description of the earth. In its widest sense, this comprehends not only the delineation of the form of its continents and seas, its rivers and mountains, but their physical condition, climates, and products, and their appropriation by communities of men. With physical and political geography, however, we have no

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