When the angle observed is very near 180 degrees, the correction may be omitted; for then it will be easy to keep the plane of the instrument so near that of the before mentioned great circle, as not to want any, if the situation of that circle be known; if it be not, the observer, when he sees the two objects together, may turn the instrument on the axis of the telescope till he finds that position of it by which he obtains the least angle; and this (if the specula are set truly perpendi cular to the plane of the instrument) will always hap pen when the objects appear to coincide in the line g It, as expressed in the figure.
The instrument consists of an octant ABC, Figure 5, having on its limb BC, an arch of 45 degrees, di vided into 90 parts or halt' degrees, each of which an swers to a whole degree in the observation. It has an index ML moveable round the centre to mark the divisions; and upon this, near the centre, is fixed a plane speculum EF perpendicular to the plane of the instrument, and making such an angle with a line drawn along the middle of the index, as will be most convenient. for the particular uses the instrument is designed for; (for an instrument made according to Figure 5, the angle LFM may be of about 65 degrees.) IKGH is another smaller plane speculum, fixed on such part of the octant as will likewise be determined by its particular use, and having its surface in such direction, that when the index is brought to mark the beginning of the divisions (i. e. 0°) may be exactly parallel to that of the other; this speculum being turned towards the observer and the other from him. PR is a telescope, fixed on one side of the octant, hav ing its axis parallel to that side, and passing near the middle of one of the edges Ili or 11-1 of the speculum IKGH; so that half its object glass may receive the rays reflected from that speculum, and the other half remain clear to receive them from a distant object. The two specula must also be disposed is such a man ner that a ray of light coming from a point near the middle of the first speculum may fall on the middle of the second in an angle of 70 degrees or thereabouts, and be thence reflected into a line parallel to the axis of the telescope, and that a clear passage be left for the rays coming from the object to the speculum EF by the side HG. ST is a dark glass fixed in a frame, which turns on the pin V; by which means it may be placed before the speculum EF, when the light of one of the objects. is too strong: of these there may he several. In the distinct base of the telescope repre sented by the circle abcde f, are placed three hairs, two of which, a c and b d, are at equal distances from and parallel to the line g it, which passes through the axis, and is parallel to the plane of the octant: the third f c is perpendicular to g It through the axis.
The instrument, as thus described, will serve to take any angle not greater than 90 degrees: but if it he designed for angles from 90 to 180 degrees, the polished surface of the speculum EF, Figure 5, must he turned towards the observer; the second IKGH must be brought forward to the position NO so as to receive on its middle the rays of light from the mid dle of the first, in an angle about 25 degrees, their surfaces being perpendicular to one another when the index is brought to the end of the divided arch next C; and this second must stand five or six inches wide of the first, that the head of the observer may not in tercept the rays in their passage towards it, when the angle to be observed is near 180°. The smaller spe culum is fixed perpendicularly on a round brass plate, toothed on the edge, and may be adjusted by an end less screw.
In order to make an observation, the axis of the telescope is to be directed towards one of the objects, the plane of the instrument passing as near as may be through the other, which must lie to that hand of the observer as the particular form of the instrument may require; viz. the same way that the speculum IT
does from IKGH, if it be composed according to this figure and description. The general rule is, that when the index is brought to the beginning of the scale, (i. e. to 0° when the instrument is designed for angles under 90°, or to 90° when it is designed for angles from 90° to 180°); if then a line be imagined to be drawn on it, parallel to the axis of the telescope, or line of direction of the sight, so as to point towards the object seen directly; whichever way this line is carried by the motion of the index along the arch from 0° toward 90° in the first case, or from 90° to wards 180° in the second, the same way the object seen by refiexion ought to lie from that which is seen directly. The observer's eye being applied to the telescope, so as to keep sight of the first object; the index must be moved backward and forward till the second object is likewise brought to appear through the telescope, about the same distance from the hair c f (Figure 6) as the first: if then the objects appear wide of one another, as at i and /r, the instru ment must be turned a little on the axis of the teles cope, till they come even or very nearly so, and the index must be removed till they unite in one, or ap pear close to one another in a line parallel to cf, both of them being kept as near the line g h as they can. If the instrument be then turned a little on any axis perpendicular to its plane, the two images will move along a line parallel to g h, but keep the same posi tion in respect of one another; so that in whatever part of that line they may be observed, the accuracy of the observation will be no otherwise affected than by the indistinctness of the objects. if the two ob jects be not in the plane of the instrument, but equal ly elevated on, or depressed below it, they will appear together at a distance from the line g h, when the in dex makes an angle something greater than their nearest distance in a great circle; and the error of the observation will increase nearly in proportion to the square of their distance from that line, but may be corrected by help of the fifth corollary. Suppose the hairs a c and b (I, each at a distance from the line g h, equal to of the focal length of the object glass, so as to comprehend between them the image of an object whose breadth to the naked eye is a little more than 2°i; and let the images of the objects ap pear united at either of those hairs; then as the co sine of half the degrees and minutes marked by the index is to the doubled sine of the same, so is one minute to the error which is always to be subtracted from the observation. Other hairs may also be plac ed in the area abcdef, parallel to g h, and at dis tances from it proportional to the square roots of the numbers 1, 2, 3, 4, &c. and then the errors to be sub tracted from the same observation made at each of those hairs respectively will be in proportion to the numbers 1, 2, 3, 4, &c. This correction will always be exact enough if the observer takes care (especially when the angle comes near 180°) to keep the plane of the instrument from varying too much from the great circle passing through the objects. When the angle is very near 180° the correction may be omitted, for then it will be easy to keep the plane of the instru ment so near that of the before mentioned great cir cle, as not to want any, if the situation of that circle be known; if it be not, the observer, when he sees the two objects together, may turn the instrument on the axis of the telescope till he finds that position of it by which he obtains the least angle; which (if the specu la are set truly perpendicular to the plane of the instru ment) will always happen where the objects appear to coincide in the line ,sr h, as shown in Fig. 6.