Surveying and Levelling Instruments, Theoretically and Practically Described. / For construction, qualities, selection, preservation, adjustments, and uses; with other apparatus and appliances used by civil engineers and surveyors in the field.

by William Ford Stanley

619.—Optical Arrangements of the Sextant.—Newton in the description of his instrument placed the mirrors parallel to each other, that is, to zero of the arc, in his illustration for the demonstration of the principle. In this position he showed that the direction of the reflected ray is coincident with the direct ray entering the eye from the same object or star. This scheme the author has generally found the clearest for illustrating the principle to persons not well acquainted with optics, there being some difficulty in explaining the law just given, art. 615, from a more complicated scheme.

Fig. 283.—Reflection in direct line from two plain mirrors.

Larger image

620.—If two mirrors be placed with their faces parallel to each other in such a manner that a ray of light may continue after two reflections from them, the ray will continue its path parallel in its direction to its incidence upon the first mirror.

Let MM′, Fig. 283, be two mirrors placed with their faces opposite and parallel to each other. Let the incident ray IM fall on the mirror M whose normal is a. Then, as the angles of incidence and reflection are equal, art. 54, it will be reflected at equal and opposite angle to the normal to M′. Let the normal of M′ be a′. Then again, the incident line MM′ will be reflected at equal angles to the normal to D′, that is, as shown by the diagram, it will continue parallel with the incident ray and in such a position that an object at P would appear to the eye, placed at D′, as though it were at P′ in the direct line of sight.

621.—Parallax.—It will be seen by the figure that the point P does not appear to the eye at D′ in its true position but at P′ therefore with the mirrors MM′ quite parallel, the points P and P′ appear coincident, and would read as one point with the index of the sextant set at zero, that is, at the position when the mirrors are parallel to each other; whereas the points P and P′ really subtend a small angle if direct lines be drawn from them to D′. It is therefore clear that the angle read by coincident reflection and direct or, as it is sometimes called, visual image is less than the true angle at about the position shown. This difference is called the error of parallax. When the object is distant this error is immeasurably small. The parallax error varies proportionately to the distance of the mirrors apart and with their angular position. If the mirrors are in such an angular position that the rays proceeding from an object impinging upon the centre of the first mirror would, if continued, reach the eye, there would be no error of parallax. This occurs in the nautical sextant at about 60°, and the parallax error increases on either side of this point.

622.—In the practice of surveying this small error is neglected. When the box sextant is used the mirrors are placed at a very small distance apart, and the parallax error therefore is extremely small even for near objects. Where two objects are to be triangulated, the one near and the other distant, the parallax error is much decreased or eliminated by taking the near object by direct vision, and the distant object by reflection. In this case, if the near object be towards the right hand, the sextant must be used in an inverted position. If the two objects be both near, a distant object may be sighted in the direction of one of them for the reflected image.

623.—It is readily seen that if the parallelism of the glasses shown in the figure be disturbed, say by a change in the relative angular position of M′ so that the planes M and M′ continued to subtend an angle to each other, then the normal of M′ must also be changed in direction equal to this; but the ray MM′ remaining constant, as there is no movement of M, this ray will therefore be displaced in its reflection from M′ an amount equal to the angle of incidence on M′ from its normal, plus the angle of reflection from the opposite side of the normal, that is, to double the amount of angular change of the position of the mirror or of its normal, which is the same thing. The sextant therefore reads, by change of position of one of its mirrors, half the angle of reflection upon its arc; and to make it read to the angular value of its reflection the divisions on the arc are made twice as close, that is, half degrees are made to read as degrees. This will be better explained by the following scheme.—