altitudes above the earth's surface does not change abruptly, but by insensible degrees, a ray of light in passing through it does not describe a series of right lines, but a curve concave towards the surface of the earth, as represented in the figure. When the ray arrives at the earth's surface, at O, an observer, situated at this point, will receive it according to its last direction OS'; and, as all bodies appear in the directions in which the visual rays reach the observer, the heavenly body S will appear to be at S'. If, in this case, the apparent zenith distance be measured, it will be the angle ZOS' instead of ZOS, which is the true zenith distance. The difference of these two angles, or the angle S'OS, is called the Astronomical Refraction; the effect of which is, therefore, to cause all the heavenly bodies to appear more elevated above the horizon than they really are. When the refraction takes place with respect to terrestrial bodies, it is denominated Terrestrial Refraction; but as this refraction is neither caused by terrestrial nor celestial bodies, but by the refractive power of the atmosphere, this distinction does not appear to be proper; and it would be more consistent with the phenomena to denominate the whole Atmospheric Refraction. Experience seems to prove that this effect is produced by some kind of action which bodies exercise upon light, analogous to that which chemists call affinity. This being the case, at least as to its effects, if we suppose a plane to pass through the centre of all these concentric strata of air, and which is the centre of the earth, and continued to their utmost limits, where the ray first enters the atmosphere, each of these spherical beds will be divided into two equal and symmetrical parts by this plane; and, therefore, there will be an equal attraction on each side, which being exerted in opposite directions, the ray will continue to move in that plane from its first entrance into the atmosphere till it arrive at the eye of the observer. Hence it is concluded, that the effect of refraction is wholly in a vertical direction, so as to augment the apparent altitudes of all the heavenly bodies, and to diminish their zenith dis tances. The effects of refraction, however, are not the same at all altitudes. When rays of light fall perpendicularly on the refracting medium, they do not suffer any refraction; and as the effect of this deflecting power increases with the obliquity of the incident ray, the refraction of the heavenly bodies increases from the zenith, where it is nothing, to the horizon, where it is greatest. Nor are the effects of refraction always the same at equal altitudes; for they vary with both the temperature and pressure of the atmosphere. The limits of this variation are generally between 30 and 36′ of a degree, at the horizon; and these follow nearly the same proportion as far as 8° or 10° of altitude; above which the atmosphere is subject to less changes, and consequently the refraction follows the same rule. The mean horizontal refraction will, therefore, be about 33' or 33′ 15′′. Hence, refractions are calculated for the mean state of the atmosphere, and arranged in tables, according to the apparent altitude of the heavenly bodies, for the use of the practical astronomer; and these are sufficiently accurate for all common purposes; but when great nicety is required they must be corrected for the height of both the barometer and thermometer. Since refraction increases the apparent altitude of all the celestial bodies, it accelerates their rising, and retards their setting; or, in other terms, it causes them to appear before they actually ascend above the horizon, and to remain in sight after they have really descended below it; and thus adds to the length of the day. Refraction has also an effect in changing the apparent shape of these bodies as well as their places; and it is from this cause that the Sun and full Moon appear of an oval shape at the time of their rising and setting. The lower limb, or edge, being more refracted than the upper limb, they are, in appearance, brought nearer to each other, and the vertical diameters shortened; and as the horizontal diameter is not shortened in the same proportion, it gives rise to the oval appearance so often observed at those times. Another effect of refraction is, that of causing two celestial bodies to appear nearer to each other than they really are. Let ZH and ZR (fig. 2, p. 67) be quadrants of two vertical circles, and S, S' any two bodies in these arcs, their true distance SS' will be an arc of a great circle: but as these two bodies are elevated by refraction to s and s', their apparent distance will consequently be ss', which is evidently less than their true distance SS'; because the two verticals approach each other, and meet at the zenith Z. If the altitudes of both bodies be the same, and the refraction known, by subtracting it from the apparent altitudes, the true altitudes SR and S'H will be obtained, and then the true distance of the two bodies can easily be found by proportion. For in the spherical triangles Zss' and ZSS', as the sine of Zs the apparent zenith distance: the sine of ZS the true zenith distance :: ss' the apparent distance between the two bodies: SS' their real distance. When the apparent altitudes of the two bodies are different, then, in the same triangle Zss', the two complements Zs and Zs' of the apparent altitudes, and the apparent distance ss', are known, from which the angle Z can be found by the common principles of spherical trigonometry. Then by adding the refraction to the complements of the altitudes, the two sides ZS and ZS', and the angle Z of the triangle ZSS', will be given to find the true distance SS'. Again, if the two bodies be on the same vertical ZI, as at a and b, the effect of refraction will cause them to appear at a' and b'; then, since refraction is the greatest at the least altitude, the body at a will be more elevated by it than that at b, and consequently the apparent distance a'b' will be less than the true distance ab; and their difference will be equal to the difference of the refractions at the apparent altitudes a' and b'. The following easy and practical methods of ascertaining the refraction are presented to the attention of the young astronomer. Observe the altitude of the Sun or a star, the right ascension and declination of which are known; and, by means of a good watch, or a chronometer, find the exact time between the moment of observing the altitude and the time of the Sun or star's passing the meridian; from which the horary angle is easily obtained by saying, as 1 hour: the observed time :: 15°: the degrees in the required angle. Then, having the complement of the latitude, the complement of the Sun or star's declination, and the horary angle, the complement of the altitude may easily be found by the common principles of spherical trigonometry; and the difference between the observed and the calculated altitude will be the refraction required. This method was put in practice by M. Cassini, when he observed the altitude of the Sun's centre at 20 m. past 5 on the morning of the 1st of May 1738, in latitude 48° 50' 10" N., and found it to be 5° 0' 14"; from which he deduced the refraction at that altitude equal to 10′ 30′′. Another method of finding the refraction is, by taking the greatest and least altitudes of some circumpolar star, which passes the upper part of the meridian near the zenith, and consequently is at that time nearly free from refraction. Then, having the latitude of the place of observation, the apparent distance of the star from the pole, at each observation, will be known; and the less of these distances taken from the greater will give the refraction at the least altitude. M. de la Caille employed this method in ob serving a star to pass the meridian of Paris within 6' of the zenith; and, when it passed the lower part of the meridian, its altitude was 7° 52' 25". The altitude, as deduced from the polar distance, he found to be 7° 46' 20"; and consequently the refraction at the apparent altitude of 7° 52′ 25′′ was 6′ 5′′, according to his determination. Both these methods, however, serve rather for verifying the refraction than for finding it in the first instance, as they require the latitude of the place of observation to be known, which can only be accurately ascertained after the refraction has been correctly determined. When the refraction has been found for a few apparent altitudes, it then becomes desirable to ascertain the law of its variation, in order to adapt it to all other altitudes. The celebrated astronomer, Dr. Bradley, gave the following simple and general rule for finding the refraction r at any altitude a; viz. as radius 1: cotang. (a + 3r) :: 57′′: r; which expresses the refraction in seconds, agreeing very nearly with observations made at a mean state of the barometer and thermometer. This formula has likewise been improved in point of accuracy by the labours of later astronomers; but it has at the same time been rendered more complicated. The following general rule for finding the refraction, answering to any observed altitude, has been deduced from the formule given by Laplace, in his celebrated work the Mécanique Céleste, viz : 1. Add the logarithmic cotangent of the observed altitude to -2.8230506, and the sum will be the log. tangent of an arc. 2. Add the log. tangent of half this arc to -2.5225024, and the sum will be the log. tang. of a second arc, which is to be reduced into seconds. 3. Then to the log. of this number of seconds, |