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 divisions of its arc, is obtained. The latter is especially devoted to the measures 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, which themselves are firmly bedded on the spokes of the horizontal circle, and turn with it.

(193.) The next instrument we shall describe is one by whose aid the 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 reflection 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.1

(194.) The principle of this instrument is the optical property of reflected rays, thus announced :-"The angle between the first and last directions of a ray which has suffered two reflections 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 CE 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 C E,

1 Newton communicated it to Dr. Halley, who suppressed it. The description 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.

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 reflection at C, have been thrown upon the silvered part of D, and are thence directed by a second reflection into the

Fig. 26.

B

telescope. The two images so formed will both be seen in the field of view at once, and by moving the radius C E 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 CP of one object, and F Q of the other, is twice the angle E C A included between the fixed and moveable radii CA, CE. Now, the graduations of the limb being purposely made only half as distant as would correspond to degrees, the arc, A E, when read off, as if the graduations were whole degrees, will, in fact, read double its real amount, and therefore the numbers so read off will express, not the angle E C A, but its double, the angle subtended by the objects.

(195.) 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, plumb-line, or artificial horizon can be used, the sea-offing affords the only resource; and the image of the star observed, seen by reflection, 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. 23) gives the true altitude of the star. On land, an artifi

cial horizon may be used (art. 173), and the consideration of dip is rendered unnecessary.

(196.) The adjustments of the sextant are simple. They consist in fixing the two reflectors, the one on the revolving radius C E, the other on the fixed one CB, so as to have their planes perpendicular to the plane of the circle, and parallel to each other, when the reading of the instrument is zero. This adjustment in the latter respect is of little moment, as its effect is to produce a constant error, whose amount is readily ascertained by bringing the two images of one and the same star or other distant object to coincidence; when the instrument ought to read zero, and if it does not, the angle which it does read is the zero correction and must be subtracted from all angles measured with the sextant. The former adjustments are essential to be maintained, and are performed by small screws, by whose aid either or both the glasses may be tilted a little one way or another until the direct and reflected images of a vertical line (a plumb-line) can be brought to coincidence over their whole extent, so as to form a single unbroken straight line, whatever be the position of the moveable arm, in the middle of the field of view of the telescope, whose axis is carefully adjusted by the optician to parallelism with the plane of the limb. In practice it is usual to leave only the reflector D on the fixed radius adjustable, that on the moveable being set to great nicety by the maker. In this case the best way of making the adjustment is to view a pair of lines crossing each other at right angles (one being horizontal, the other vertical) through the telescope of the instrument, holding the plane of its limb vertical, then having brought the horizontal line and its reflected image to coincidence by the motion of the radius, the two images of the vertical arm must be brought to coincidence by tilting one way or other the fixed reflector D by means of an adjusting screw, with which every sextant is provided for that purpose. When both lines coincide in the centre of the field, the adjustment is correct.

(197.) 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.

(198.) We must not conclude this part of our subject 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, annihilated. Let PQ be two objects which we may suppose fixed, for purposes of mere explanation, and let K L be a telescope move

able on O, the common axis of two circles, A M L and a be, of which the former, A M L, 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 O a 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 O 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 are ABCD 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.

(199.) Micrometers are instruments (as the name imports') for measuring, with great precision, small angles, not exceeding a few minutes, or at most a whole degree. They are very various in construction and principle, nearly all, however, depending on the exceeding delicacy with which space can be subdivided by the turns and parts of a turn of fine screws. -in the parallel wire micrometer, two parallel threads (spider's lines are generally used) stretched on sliding frames, one or both moveable by

Fig. 28.

Thus

screws in a direction perpendicular to that of the tnreads, are placed in the common focus of the object and eye-glasses of a telescope, and brought by the motion of the screws exactly to cover the two extremities of the image of any small object seen in the telescope, as the diameter of a planet, &c., the angular distance between which it is required to measure. This done, the threads are closed up by turning one of the screws till they exactly cover each other, and the number of turns and parts of a turn required gives the interval of the threads, which must be converted into angular measure, either by actual calculation from the linear measure of the threads of the screw and the focal length of the object-glass, or experimentally, by measuring the image of a known object placed at a known distance (as a foot-rule at a hundred yards, &c.) and therefore subtending a known angle.

(200.) The duplication of the image of an object by optical means furnishes a valuable and fertile resource in micrometry. Suppose by any optical contrivance the single image A of any object can be converted into two, exactly equal and similar, A B, at a distance from one another,