rallelism of the central mirror. If the first distance exceeds the second, the error is subtractive; otherwise additive, the mirror being in its first position; but the contrary when in its second position. Thus, if the first distance was 119° 59′ 21′′ and the second 120° 0′ 39", the error would be 39", additive when the mirror was in its first position, subtractive for the second. The error for any other angle may be found by means of col. 2d. Table XXXIV. when the inclination of the plane of the horizon glass to the axis of the telescope is 80°, by saying, as the tabular correction corresponding to 120° `(=4′ 5′′) is to the error of the glass 39′′, so is the tabular error for any other angle as 85° (=1′ 15′′) to the corresponding error of the glass 12". In this manner a table of errors may be made for all angles.*

The angle between the plane of the horizon glass and axis of the telescope produced, being constant in all observations and adjustments of the sextant, no error can arise from the want of parallelism of its surfaces.

Verification of the parallelism of the surfaces of the coloured glasses. Turn down the glass at D, which is to be examined, and another at E, to defend the eye from the sun; direct the telescope to the sun and move the index till its direct and reflected images coincide; then turn the dark glass at D, so that the surface which was farthest from the horizon glass may now be nearest to it, and if the contact of the same two limbs be complete, the surfaces of this glass are parallel, but if they lap over, or separate, the index must be moved to bring them again in contact, then half the arch passed over by the index will be the error, arising from the want of parallelism of the glass at D.

DESCRIPTION AND USES

OF THE

CIRCLE OF REFLECTION.

HE Circle of Reflection was invented by the celebrated professor CHEVALIER DE BORDA, Mr. TROUGHTON, and Mr. MENDOZA Y RIOS. In its present improved state it has a decided superiority over the sextant in measuring the distance of the moon from the sun, or a star, on account of its correcting, in a great measure, the errors arising from a faulty division of the limb, want of parallelism in the surfaces of the mirrors and coloured glasses, and entirely avoiding the error which might arise in a sextant from the mirrors not being parallel when the index is on 0.

Fig. 1. Plate VIII. represents the circle of reflection, as given by DE BorDA: in fig 2 is a section of the same instrument, marked with the same letters of reference as in fig. 1. The principal parts of this instrument are, the circular limb LMV; the central index EF; the horizon index MD; the central glass or mirror A; the horizon glass or mirror B: the telescope GH; the coloured glasses fig. 3, 4; the handle fig. 5; the ventelle fig. 6; and the adjusting tool fig. 7.

The limb of the instrument LMV, is a complete circle of metal, and is connected with a perforated central plate by six radii; it is divided into 720° because of the double reflection; each degree is generally divided into three equal parts, and the division is carried to minutes, or lower, by means of the verniers of the two indices.

The two indices are moveable round the same axis, which passes exactly through the centre of the instrument; the central index EF carries the cen

* The method of calculating the above tabular numbers when the angle of inclination of the telescope and horizon glass differs from 800 is given in the explanation of Table XXXIV. prefixed to

tral mirror A; and the horizon index MD carries the telescope GH and the horizon mirror B; both indices are furnished with verniers and tangent screws at O and N.

The central mirror A is placed on the central index immediately above the centre of the instrument; the plane of this mirror makes an angle of about 30° with the middle line of the index, and is adjusted perpendicular to the plane of the instrument, by means of the screws placed on the back part of the frame of the mirror.

The horizon glass B is placed on the horizon index near the limb, so as to interfere as little as possible with the rays proceeding from objects situated on the opposite side of that index with respect to the central mirror. The horizon glass is adjusted perpendicular to the plane of the instrument in a similar manner to that of a horizon glass of a sextant; and in some circles this mirror is moveable about an axis perpendicular to the plane of the instrument; by which means the situation with respect to the telescope may be adjusted.

The telescope GH, attached to the other end of the horizon index, is an astronomical one inverting the observed objects, and has two parallel wires in the common focus of the glasses, distant from each other between two and three degrees. These wires, at the time of observation, must be placed parallel to the plane of the instrument. To effect this, marks are made on the eye-piece, and on the tube at G, and by making them coincide, the wires may be brought to their proper position. The telescope may be raised or depressed by two screws 1, K, so as to be directed to any part of the horizon glass; and, by means of the graduations on the two standards i, k, the telescope may be rendered parallel to the plane of the instrument.

There are two sets of coloured glasses (fig. 3, 4) each set usually containing four glasses of different shades; the glasses of the larger set (fig. 4) which are placed before the central mirror at a, a, should have each about half the degree of shade with which the corresponding glasses (fig. 3) of the other set, placed at C, are tinged-because the rays from the luminous object pass twice through the coloured glass placed before the central mirror, and only once through the other. The glasses placed at a, a, are kept tight in their places by small pressing screws at their ends, or by slides passing in front, through perforations in the stems of their frames: when fixed for observation they make an angle of about 85° with the plane of the instrument, by which means the image from the coloured glass is not reflected to the telescope. When the angle to be measured is between 5° and 35°, one of the largest set is to be fixed at a, a; in other cases, one of the smaller set is to be placed in the socket C. The reason of using the large glass is this-when the small glass is placed at C, it intercepts the direct light of the luminous object in its passage towards the central mirror, if the object happens to be situated within the angular space, included by the lines from the centre A, by the sides of the frame of the glass placed at C. This is avoided by using the larger glasses.

The handle (fig. 5) is of wood, and is fixed to the back of the instrument immediately under the centre. By this it is held during the time of observation.

The ventelle (fig. 6) is used in terrestrial observations to diminish the light of the object seen directly, to render it equal in brightness to that of the objects seen by reflection: this is performed by putting the ventelle in the socket D, and raising or depressing it till the objects appear of equal brightness.

There are two adjusting tools of the form represented in fig. 7; they are exactly of the same size, and their height is nearly equal to that of the central mirror; they may be used in adjusting the central mirror perpendicular to the plane of the instrument, and in making the axis of the telescope parallel to that plane.

The instrument, as we have now described it, is the same as it was left by De Borda; Mr. Troughton has since suggested the improvement of fixing to the horizon index the arch WSPR, and providing it with two sliding pieces U, X, in order to facilitate the fixing the indices at their proper

angles with each other in taking successive observations. When the central and horizon glasses are parallel, the central index covers the space SP of the arch, and the spaces SW, PR, are each divided into degrees from S to W and from P to R, and numbered 0 at S and P, and continued to 130° towards Wand R. The use of this arch and sliding pieces will be explained hereafter.*

That ingenious mathematician and navigator, M. Mendoza y Rios, has farther improved the circular instrument by the substitution of a circular ring (moving round the centre of the instrument over or adjacent to the limb TMV) for a vernier instead of those attached to the indices by De Borda; and by fixing this circular vernier alternately to each of the indices it serves as a vernier for both, and after any number of observations, gives the compound motion of both indices, and thus double the number of distances are obtained by this instrument that can be obtained by De Borda's circle with the same number of observations. Mr. Rios has also improved the form of the handle for holding the instrument. In theory the instrument, as improved by Mr. Rios, appears to be superior to that of De Borda, but not being used one of the former kind, I cannot, from my own experience, deCe whether it is so much superior in practice; but Mr. Rios says that he found it answered his expectations. As the method of taking the obserVation is nearly the same with both instruments, shall confine myself to the explanation of the uses of De Borda's, from which the method of using the other will be easily discovered.

Adjustments of the Circle of Reflection.

Before entering upon an explanation of the adjustments of this instrument, it will be proper to premise that there are three different methods of observing the angular distance of two objects with this instrument, viz. (1) by what is called an observation to the right, (2) by an observation to the left, and (3) by a cross observation.

An observation to the right is that, where the object, whose image is to be reflected and the central mirror are on the same side of the telescope. An observation to the left, when the object to be reflected and the central mirror are on opposite sides of the telescope, which in both cases is supposed to be directed to the other object; and a cross observation is a combination of the forementioned observations; the first being generally taken to the left, and the second to the right.

The adjustments of a circle consist in placing the mirrors perpendicular to the plane of the instrument, and in making the axis of the telescope parallel to that plane. These are all the adjustments necessary in measuring an angular distance by cross observations; but if one observation only be taken to the right, or to the left, it will be necessary to find the division, on which the horizon index must be placed, to make the horizon glass parallel to the central glass, when the central index stands on 0. These adjustments are similar to those of a sextant, but a particular explanation of each will here be given.

To set the central glass perpendicular to the plane of the instrument.

This adjustment may be made by placing the eye in front of the central glass at L, a little above the plane of the instrument, and observing if the reflected image of that part of the limb nearest the eye appears to make one continued circular line with the parts of the limb towards T, seen to the

Mr. Troughton suggested another alteration in the circle, but (as Mr. Rios justly observes) the instrument thus altered may be considered as a sextant, the limb of which is completed to the whole circumference. A circle of this description is usually furnished with three indices and verniers, by each of which every observation must be read off. This is very troublesome, particularly in the night. It is true that this method corrects in a very great degree the error of not having the index 6xed exactly on the centre, or that of not having an instrument perfectly circular; but errors of this kind in Borda's circle may be reduced in any ratio by taking a number of observations, and the error will in general be extremely small in taking a suficient number to bring the index nearly to the point set out from; so that in those important points I should, on the whole, prefer an instru

right and left of the central glass; for in this case the glass is perpendicular to the plane of the instrument; otherwise, it must be adjusted by means of the screws till the two images coincide.*

By examining this adjustment in different parts of the limb, it will be known if the limb be in the same plane. If any difference should be found, the central glass must be so fixed that the reflected image of the limb may appear as much above the direct image in some places as below it in others.

To set the horizon glass perpendicular to the plane of the instrument. The central glass being previously adjusted, and the telescope directed to the line separating the silvered from the transparent part of the horizon glass, hold the instrument nearly vertical, and move either index till the direct and reflected image of the horizon, seen through the telescope, coincide; then incline the instrument till it is nearly horizontal, and if the images do not separate, the horizon glass is perpendicular to the plane of the instrument; but if they do separate, the position of the glass must be rectified by means of the screws in its pedestal."

This adjustment may be also made by directing the sight through the telescope to any well defined object; then, if by moving the central index, the reflected image passes exactly over the object seen directly, the glass is perpendicular; otherwise its position must be adjusted by means of the screws attached to the pedestal of the glass.

A planet, or star of the first magnitude, will be a good object for this purpose. If the sun is used, one of the coloured glasses must be placed at C and another at D.

To make the axis of the telescope parallel to the plane of the instrument.

The telescope may be raised or depressed by means of two screws attached to the standards i, k, (fig. 2) and passing through two pieces of brass connected with the tube of the telescope. On each of these pieces is a mark or index by which the telescope is to be adjusted. For, by bringing the indices to the same mark on each standard, the telescope will be parallel to the plane of the instrument.†

To find that division to which the horizon index must be placed to render the mirrors parallel when the central index is on 0.

Place the central index on 0; direct the telescope to the horizon glass, se that the line joining the silvered and transparent parts of that glass may appear in the middle of the telescope; hold the instrument vertically, and move the horizon index, till the direct and reflected horizons agree-and the division shown by the horizon index will be that required.

This adjustment may also be made by measuring the diameter of the sun in contrary directions; thus, the central index being fixed on 0, place a dark glass at C and another at D; direct the telescope (through the transparent part of the horizon glass) to the sun, and move the horizon inde till his reflected image appear in the telescope; bring the upper edge of the direct image to coincide with the lower of the other, and note the angle shown by the index; then, by moving the horizon index, bring the lower edge of the

* When the instrument is furnished with adjusting tools, this adjustment may be made in the following manner. Set the two tools on opposite parts of the limb at T and L; place the eye at e, at nearly the same height as the upper edge of the tools, so that part of the tool at T may be hid by the central glass; move the central index till the reflected image of the tool nearest the eye appears in the central glass at the side of the other tool seen directly; then if the upper edges of the tools are apparently in the same straight line, the central glass is perpendicular to the plane of the instrument; otherwise its position must be adjusted by the screws at the back of the frame.

If you suspect that the marks on the standards are inaccurate, you may examine them in the following manner. Lay the circle horizontally on a table; place the two adjusting tools on opposite parts of the limb at T and L; and at about 12 or 15 feet distance let a well defined mark be placed, so as to be in the same straight line with the tops of the tools; then raise or lower the telescope till the mark is apparently in the middle between the two wires: then the axis of the telescope will be parallel to the plane of the instrument, and the difference (if any) between the divisions pointed out by the indices on the graduation of the standards i, k, (fig. 2) will be the error of the indices, by knowing which, it will be easy in future adjustments to make allowance for the error.

direct image to coincide with the upper edge of the reflected one, and note als the angle pointed out by the index; half the sum of these two angles will be the point of the limb where the horizon index must be placed to render the mirrors parallel. Thus, if the index in the first observation stood on 473 30. and in the second on 474° 34', the half sum of the two 474° 2′ would be the point where the horizon index must be placed to make the mirrors parallel.

These are all the adjustments necessary to be made preparatory to measır, ing any angular distance. When the angle is measured by cross observations, the error arising from the want of parallelism of the surfaces of the mirrors, and screens, will in general be very small; however, the method of verifying those glasses and making allowance for any error in them will be given hereafter.

To observe the meridian altitude of any celestial object, either by an observation to the right or to the left.

The method of observing the meridian altitude of an object with a circle is exactly similar to that with a quadrant or sextant. The central index must be fixed on 0, and the horizon index on the point which renders the two mirrors parallel; then the altitude may be taken either by an observation to the right or to the left; but the former method, in which the large coloured glasses are not necessary, is in general to be preferred; because those large glasses are more liable to cause an error in the observation than the small ones.

If an observation to the right is to be taken, a small dark glass must be placed at C, if the object be bright, then hold the instrument in the right hand in a vertical position; move the central index, according to the order of the divisions of the limb, till the reflected image of the object, seen in the telescope nearly touches the direct image of the horizon; tighten the index by the screw at the back of the instrument; make the contact complete in the middle between the parallel wires of the telescope, by the tangent screw, and by sweeping, exactly in the same manner, as when observing with a quadrant, and the central index will point out the altitude of the object.

If an observation to the left is taken, and the object be bright, a large dark glass must be placed at aa, if the altitude be between 5 and 35°; otherwise a small glass at C; hold the instrument in the left hand, in a vertical position, move the central index contrary to the order of the divisions: and bring the reflected image in contact with the horizon as above: the angle shown by the central index being subtracted from 720°, will be the sought altitude.

In both these methods of observing the meridian altitude of an object, the circle, the radius of which is only five inches, will hardly be so accurate as a good sextant of a larger radius; but, by the help of a well regulated watch, the meridian altitude may be obtained, by the circle, to a much greater degree of accuracy than by a sextant, by observing in the following manner. A few minutes before the object passes the meridian, begin to observe the altitude by cross observations (in the manner to be described in the next article) and note the time of each observation by the watch: continue to observe till a few minutes after the object has passed the meridian: then the angles shown by the central index being divided by the whole number of observations, will give the approximate meridian altitude; the correction to be applied to it, to obtain the true meridian altitude, may be found by means of Tables XXXII. and XXXIII. by a method which will be explained

* In some instruments there is an adjustment of the horizon glass, to place it at its proper angle with the axis of the telescope: if an adjustment of this kind is necessary, it ought to be mate before the other adjustments, in such manner, that if a coloured glass be fixed at C, none of the rays from the central glass can be reflected to the telescope from the horizon glass without passing the coloured glass To effect this, the ventelle must be placed at D, and lowered so as to intercept the direct Telt entirely then place the coloured glass at C, and direct the tele-cope to the silvered part of the horizon glass; move the central index, and if no uncoloured images appear (reflected from the central glass) but all have the same tinge as that of the coloured glass used, the bo rizon glass is in its proper position; otherwise it must be turned on its axis till the uncoloured