till the arc described by the reflected image appears just to touch the sea-line, the resulting angle will be the true altitude above the visible horizon.

The altitude of either limb of the sun or moon may be found in the same way by making it appear just to sweep the horizon as the instrument is moved backwards and forwards. In order to take off the glare of the sun, the sextant is furnished with a set of glasses of different degrees of darkness, which may be applied at pleasure between the index-glass and the horizonglass. There is also generally a shaded glass beyond the horizonglass, to diminish the light reflected from the sea.

50. The object of all the instruments above described is to give the place of a body at any time, or to determine the time at which a body occupies a place before determined on, in the sphere of observation.

The true place of the body on the sphere of observation is that point in which the sphere is intersected by a straight line passing through the eye and the body. The observed place is that point in which the sphere of observation is intersected by a straight line drawn from the eye in the apparent direction of the body. The observed place and the true place will coincide if light proceeds from the body to the eye in straight lines, for then the apparent and the true direction will be the same. But if the light proceed from a body in a curved instead of a straight line, the apparent direction will be that of the tangent to the curve just as it enters the eye, for the eye is only affected by the light just at the end of its course, and in that case the apparent direction will not necessarily coincide with the true direction.

This is in fact the real state of the case. The Earth's atmosphere deflects the light from the heavenly bodies out of the straight line in which it would proceed if it continued in vacuo, and therefore their apparent directions are not the same as their true directions.

The atmosphere is a refracting medium which surrounds the Earth on all sides, its outer surface being nearly spherical, and

its density decreasing with its distance from the Earth. A ray of light falling on such a surface undergoes continual refraction throughout its passage, and describes a curvilinear path. This may be shewn by taking a supposed case of a series of uniform media of finite breadth, each of less density than that which it envelopes, and then passing to the limit when the number of such media is indefinitely increased and their breadth indefinitely diminished.

Let a ray of light from a body S fall upon a series of such media, they being concentric spherical shells of small thickness

whose common centre is C.

It will be refracted at each surface and proceed in a straight line through each medium, and therefore it will pursue the polygonal path PQRTV. If an eye be situated at V, the direction in which the light will strike it will be TV, giving the impression of a body S' in the direction VT produced. Now if we pass to the limit, the case will be exactly that of light passing through a medium whose density decreases with the distance from the centre, and the path will be the limit of the polygon in the hypothetical case, that is to say, a curve. The direction of the light just as it enters the eye at V will be the tangent to the curve, and in that direction S will appear to be.

The only case in which the apparent and the true directions coincide will be when the incident light is perpendicular to the

surface, in which case it will suffer no deflection.

And, cæteris

paribus, the more oblique the direction of incidence, the greater will be the deflection.

The hypothetical polygon PQRTV lies entirely in one plane; for any two consecutive sides of it, as QR, RT, lie in the same plane with the normal at R, by the law of refraction. Therefore the normals at Q and T, which meet the normal at R in C, both lie in that plane. And thus it may be shewn that the normals at all the points P, Q, R, T, V, lie in the same plane. Hence the whole polygon lies in the same plane with CP, CV, the two extreme radii. In the limit, therefore, the path of the light will be a plane curve, and its tangent at V will lie in the same plane with the radius at V, and the original direction of the light, PS. If we draw VS parallel to PS, it will be the direction in which S, which is very distant, would appear to be if there were no refraction. Hence the apparent direction lies between the true direction and the vertical in the same plane with these lines.

From this we gather that the effect of refraction is to bring a heavenly body nearer to the zenith, or to diminish its zenith distance. And the displacement being directly towards the zenith, the azimuth of the body will not be affected.

Hence, all altitudes have to be corrected for refraction, while the observed and the true azimuth are the same. The amount of the correction of the altitude increases with the zenith distance, being zero at the zenith and about 33′ in the horizon. It does not however increase uniformly, but according to a law which we cannot now enter into. Its value is about 57" at an altitude of 45°.

As the azimuth of a body is not affected by refraction, the observed time of transit over the meridian requires no correction on this account.

It may be here observed that the length of the day is increased by refraction, the Sun's centre when in the horizon being elevated through a space rather greater than its diameter. In fact, when the Sun appears to us to be just on the horizon, it is really entirely below. Where the diurnal path of the Sun

estine ring een trough the silvered part of the object

The flowing a te nai mode finding the altitude of 2 tar The inserver wing he extant n his right hand, is plane vertical. fires he telescape the star, the inder being in such a posinon hat he mirrors are parallel. The redestei mage herefore incides with hat seen by direct Tsich. The inserver ten noves he index aimly forward wia is eit land, using the dected mage u leave the direct image, md, by the practief he insiment, to descend áruigh ice de meie fescrbed he index. The observer follow the image in is fescent by atually lowering the telewape mtl at last the sea-ine years in the feid of view. If the plane of the instrument vers accurately vertical, it would be eficient to bring the star into coincidence with the middle point of the sea-line in the horizon-glass, and then the observed angle wold be the tide of the star above the visible horizon : bus if the plane of the heriment be act vertical the observed angle will be too great, the distance of the star from the point in the offing vertically below is being less than its distance from any other point.

Thus, if 8 be the star, AB the sea-line, it would not do to observe the distance of S from any point P in the offing, the altitude required being SD, where the arc SD is perpen- í dicular to AB.

B

There is, however, a very easy practical method of determining the right altitude.

If the plane of the instrument be turned through any angle, without altering the inclination of the index-glass to the direction of the star, or moving the index, the reflected image of the star will still remain visible, and will appear to describe a circular arc, since it is always at the same angular distance from the star itself. This kind of motion is easily communicated to the instrument with the hand; and if the index be moved

till the arc described by the reflected image appears just to touch the sea-line, the resulting angle will be the true altitude above the visible horizon.

The altitude of either limb of the sun or moon may be found in the same way by making it appear just to sweep the horizon as the instrument is moved backwards and forwards. In order to take off the glare of the sun, the sextant is furnished with a set of glasses of different degrees of darkness, which may be applied at pleasure between the index-glass and the horizonglass. There is also generally a shaded glass beyond the horizonglass, to diminish the light reflected from the sea.

50. The object of all the instruments above described is to give the place of a body at any time, or to determine the time at which a body occupies a place before determined on, in the sphere of observation.

The true place of the body on the sphere of observation is that point in which the sphere is intersected by a straight line passing through the eye and the body. The observed place is that point in which the sphere of observation is intersected by a straight line drawn from the eye in the apparent direction of the body. The observed place and the true place will coincide if light proceeds from the body to the eye in straight lines, for then the apparent and the true direction will be the same. But if the light proceed from a body in a curved instead of a straight line, the apparent direction will be that of the tangent to the curve just as it enters the eye, for the eye is only affected by the light just at the end of its course, and in that case the apparent direction will not necessarily coincide with the true direction.

This is in fact the real state of the case. The Earth's atmosphere deflects the light from the heavenly bodies out of the straight line in which it would proceed if it continued in vacuo, and therefore their apparent directions are not the same as their true directions.

The atmosphere is a refracting medium which surrounds the Earth on all sides, its outer surface being nearly spherical, and